G19rb Reservoir Engineering B(1)

G19RB

Reservoir Engineering B

Heriot-Watt University Edinburgh EH14 4AS, United Kingdom

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Produced by Heriot-Watt University, 2014 Copyright © 2014 Heriot-Watt University All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means without express permission from the publisher. This material is prepared to support the degree programmes in Chemical and Petroleum Engineering.

Distributed by Heriot-Watt University Reservoir Engineering B

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Acknowledgements Thanks are due to the members of Heriot-Watt, School of Energy Geoscience Infrastructure and Society who planned and generated this material. We would like to acknowledge the assistance and contributions from colleagues across the University and students in preparing this and support material.

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TOPIC 1 FLUID FLOW IN POROUS MEDIA 1

INTRODUCTION

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CHARACTERISATION AND MODELLING OF FLOW PATTERNS 2.1 Idealised Flow Patterns 2.2 General Case 2.2.1 Linear Horizontal Model of a Single Phase Fluid 2.2.1.1 Linearisation Of Partial Differential Flow Equation For Linear Flow 2.2.1.2 Conditions of Solution 2.2.2 The Radial Model 2.3 Characterisation of the Flow Regimes by their Dependence on Time

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BASIC SOLUTIONS OF THE CONSTANT TERMINAL RATE CASE FOR RADIAL MODELS 3.1 The Steady State Solution 3.2 Unsteady State Solution 3.2.1 General Considerations 3.2.2 The Line Source Solution 3.2.2.1 Range of Application and Limitations to Use 3.2.3 The Skin Factor 3.3 Semi-Steady-State Solution 3.4 The Application of the CTR Solution in Well Testing

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SUPERPOSITION 4.1 Effects of Multiple Wells 4.2 Principle of Superposition and Approximation of Variable – Rate Pressure Histories 4.3 Effects of Rate Changes 4.4 Simulating Boundary Effects (Image Wells)

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SUMMARY

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LEARNING OBJECTIVES: Having worked through this chapter the student will be able to:

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Understand the nature of fluid flow in a porous medium and the relation between time, position and saturation

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Understand the assumptions used in the derivation of the diffusivity equation

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Describe the characterisation of the reservoir flow regime on the basis of time

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Apply the solutions of the diffusivity equation to steady state flow, semi-steady state flow and transient flow

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Calculate the pressure in a reservoir at a specific radius and a specific time under transient flow conditions

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Calculate the effect of multiple wells and multiple flow rates on reservoir pressure

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From measured bottom hole pressures use the line source solution to determine the reservoir permeability and skin factor Describe the application of semi-steady state solutions to determine reservoir boundaries and their influence on flow rates.

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1

INTRODUCTION

The ability to determine the productivity of a reservoir and the optimum strategy to maximise the recovery relies on an understanding of the flow characteristics of the reservoir and the fluid it contains. The physical means by which fluid diffuses through a rock (or any other porous medium) depends on the interaction between the fluid (and its properties) and the rock (and its properties). In terms of energy, the process may at first sight appear to be similar in concept to the application of the general energy equation to flow through pipes, although in this case the container through which the fluid flows is made of very small tubes. It is precisely because of the geometry and dimensions of the tubes that the application of the general energy equation would be impossible: the description of a real pore network in a whole reservoir would be too complex. Coupled with this is the interaction between the material of the tubes (or pores) and the fluids. Surface chemistry effects start to dominate the flow when very small tubes are considered and when multiphase flow occurs in them. Thus, complex force fields are produced from not only the viscous pressure drop but also the effects of surface tension and capillary pressure. The combination of these factors dictates the nature of the fluid flow and one of the initially unusual aspects is the time taken for pressure to change in the reservoir or for fluid to migrate from one location to another. For instance, if a large body of water, such as a swimming pool were drained, for all intents and purposes, the level of water in the swimming pool would be the same as the water drained out. It would take an appreciable amount of time for the water to drain (i.e. it would not be instantaneous), but the pressure or level of the water in the pool would be the same at all locations of the pool. The pressure in the pool would equilibrate almost immediately. Contrast this with, for example, a water saturated reservoir rock in which the water could flow, but where the permeability of the reservoir and the compressibility and viscosity of the water dictated that the transfer of the water through the reservoir was not instantaneous (as in a swimming pool), but took an appreciable time. In this case pressure changes in one part of the reservoir may take days, even years to manifest themselves in other parts of the reservoir. In this case, the flow regime would not be steady state while the pressure was finding its equilibrium and a major problem, therefore, would be that Darcy’s Law could not be applied until the flow regime became steady state. In some way, the diffusion through the reservoir needs to be examined: Darcy’s Law is one expression of that diffusion process, but time dependent scenarios must also be examined. To illustrate this, consider the following model of a radial reservoir with a well at the left side (Figure 1). The model represents a radial slice of reservoir from wellbore to external boundary.

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Figure 1 Model of a radial reservoir and the pressure response measured after different times

Each tube contains water, the height of which represents the pressure at that part of the reservoir. The tubes are connected to each other at the base by a small diameter tube which restricts the flow. Under initial conditions, the height of the fluid is identical in each of the tubes (assuming the model is level). The outlet at one end is at a lower level than the model and when it is opened the fluid immediately drains from the model and the level of the water in the tubes decreases. The energy to drive this system is the potential energy stored in the height of the water columns: there is no high pressure inlet to the model. As is shown in Figure 1, to reduce the pressure in the model, the fluid needs to be expelled, but because of the permeability of the rock (the restrictions in the bottoms of the tubes) it takes time for the fluid in the tubes nearest the outlet to move (or expand in the case of pressurised fluid in a reservoir) and

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therefore it takes time for the pressure to change. When the flow is started from the outlet, there is an immediate reduction in the pressure in tube 1 and this pressure perturbation moves through the rest of the fluid at a rate dictated by the rock permeability and fluid properties. This produces a variation in the pressure along the model. The pressure profile takes time to develop from the outlet (at tube 1) to the tube farthest from the outlet (tube 10) and at time, t=1, the pressure in tube 10 is still equal to the pressure at the initial time, t=0. This is termed a transient flow condition as the fluid is trying to reach pressure equilibrium. When the fluid in tube 10 starts to expand and flow, all of the fluid in the whole model is now expanding and flowing to the outlet. Tube 10 represents the limit of the fluid volume: there are no more tubes behind to supply fluid at the initial pressure. Therefore, as the pressure perturbation moves through the model from tube 1 to tube 10, the rate of pressure change in the fluid is not limited by the volume of the fluid: it is as if the volume of fluid was infinite in extent. During the transient period, the reservoir is often referred to as infinite acting. On inspection, a profile has been developing across the tubes during the transient period. At the end of the transient period, the fluid in all of tubes is expanding producing a decline in the pressure in all of the tubes. The shape of the pressure profile across all of the tubes remains essentially constant and as time continues, the profile sinks through the model until the water in the tube nearest the outlet empties. During this time, the water in the model has not been replaced so steady state conditions have not been achieved, however, since the gradient between the pressures in each adjacent tube is not changing, the system can be considered to be in pseudo-steady state or semi-steady state: the pressure gradient is constant but the absolute pressure is declining. This mimics the situation in a real reservoir where the pressure is perturbed around a well and the pressure disturbance moves out into the rest of the reservoir until it reaches the outer boundary. If this is sealing and no flow occurs across the boundary, then the reservoir pressure will decline (neglecting any injection into the reservoir) in a pseudo-steady state manner. If the boundary is nonsealing (i.e. it is the water oil contact and the aquifer water is mobile) then the aquifer water will flow into the reservoir and a steady state will be achieved if the flowrates match. The flow described in this model is trivial, but it illustrates the problem of applying Darcy’s Law to real reservoirs: the effect of time on flow may be considerable and if only steady state flow relationships were available then either permeability of the reservoir would remain unknown or unrealistic flow periods would be required to measure an essentially simple rock property.

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CHARACTERISATION AND MODELLING OF FLOW PATTERNS

The actual flow patterns in producing reservoirs are usually complex due mainly to the following factors: (i) The shapes of oil bearing formations and aquifers are quite irregular (ii) Most oil-bearing and water bearing formations are highly hetereogenous with respect to permeability, porosity and connate water saturation. The saturations of the hydrocarbon phases can vary throughout the reservoir leading to different relative permeabilities and therefore flow patterns (iii) The wellbore usually deviates resulting in an irregular well pattern through the pay zone (iv) The production rates usually differ from well to well. In general, a high rate well drains a larger radius than a lower rate well (v) Many wells do not fully penetrate the pay zone or are not fully perforated There are essentially two possibilities available to cope with complexities of actual flow properties. (i) The drainage area of the well, reservoir or aquifer is modelled fairly closely by subdividing the formation into small blocks. This results in a complex series of equations describing the fluid flow which are solved by numerical or seminumerical methods.

(ii) The drained area is modelled by a single block to preserve the global features and inhomogeneities in the rock and fluid properties are averaged out or substituted by a simple relationship or pattern of features (such as a fracture set, for example). The simplifications allow the equations of flow to be solved analytically. The analytical solutions will be examined in this chapter.

2.1

Idealised Flow Patterns

There are a number of idealised flow patterns representing fluid flow in a reservoir: linear, radial, hemispherical, spherical. The most important cases are the linear and

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radial models since both of them can be used to describe the water encroachment from an aquifer into a reservoir, and the radial model can be used to describe the flow of fluid around the wellbore. In the following sections, dealing mainly with oil, the compressibility of the flowing fluid may depend on the pressure. It will always be assumed that the product of compressibility and pressure, cP, is smaller than one, i.e. cP<<1. If it is not (as in the case of a gas) then the pressure dependence of compressibility must be taken into account.

2.2

General Case

Consider the co-ordinate system shown in Figure 2. The X and Y coordinates form a horizontal plane with the Z coordinate perpendicular to this plane. The flow velocity, U, is a vector with components Ux, Uy, Uz.

Figure 2 The specification of the flow velocity in a Cartesian co-ordinate system

The components of the flow velocity vector, U are: Ux = -(kx/)(P/x) Uy = -(ky/)(P/y) Uz = -(kz/)(P/z+g)

(2.1)

where k = permeability (m2) in the direction of X, Y, Z. The Z direction has an elevation term, g, included to account for the change in head P = pressure (Pa)  = viscosity (Pas)

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 = density (kg/m3) g = acceleration due to gravity (m/s2) U = flow velocity (m/s) = (m3/s/m2) These components are similar to Darcy’s law in each of the three directions.

2.2.1 Linear Horizontal Model of a Single Phase Fluid In this geometry, the flow is considered to be along the axis (in the x direction) of a cuboid of porous rock. The total length of the cuboid is L and fluid flows into the rock at the left end (x=0) and exits at the right end (x=L). There is no flow in the other directions at any time i.e. Uy = Uz =0 for all values of x, y, z and time, t (in a real reservoir, there may be flows in different directions in different parts of the reservoir and there may be cross flows from different layers within the reservoir). The rock is 100% saturated with the fluid. The flow equations are:

(2.2a)

(2.2b)

where kx = permeability (in the X direction), (m2)  = density, (kg/m3) Ux = flow velocity (m/s) t = time (s)  = porosity  = viscosity, Pas P = pressure, Pa x = distance, (m) The latter equation is obtained from a mass balance as follows (Figure 3):

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Figure 3 Flow into and out of a cuboid of porous rock.

In Figure 3, fluid flows into the end of the cuboid at position x=0, through the rock only in the X direction and out of the cuboid at x=L. In the middle of the cuboid, an element from position x to position x+dx is examined. The bulk volume of the element is the product of the area, A and the length, dx, i.e. the bulk volume = A*dx. The pore volume of the element is therefore the product of the bulk volume and the porosity, , i.e. the pore volume = A*dx*. If the flow was steady state then the flowrates into and out of the volume (qin and qout) would be identical and Darcy’s Law would apply. If the flow rates vary from the inlet of the volume to the outlet, i.e. q in ≠ qout then either the fluid is accumulating in the element and qin > qout or the fluid is depleting from the element qout > qin (which is possible in a pressurised system since the pressure of the fluid in the element may reduce causing it to expand and produce a higher flow rate

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out of the element). Therefore, there is a relationship between the change in mass, m, along the cuboid and the change in density, , over time as the mass accumulates or depletes from any element. In terms of mass flowrate, Mass flow rate through the area, A = q ((m3/s)*(kg/m3) = kg/s) Mass flow rate through the area, A at position x = (q)x Mass flow rate through the area, A at position x+dx = (q)x+dx Mass flowrate into a volume element at x minus mass flowrate out of element at x + dx =(q)x - (q)x+ dx The mass flow rate out of the element is also equal to the rate of change of mass flow in the element, i.e.

Therefore the change in mass flow rate i.e. if the change in mass flowrate is positive it means the element is accumulating mass; if the change is negative it is depleting mass. This must equal the rate of change of mass in the element with a volume = A*dx*

The rate of change of mass is equal to

hence

since the flow velocity, U = q/A, this becomes

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or

(2.2b)

Substituting the parameters of equation 2.2a in 2.2b gives

(2.3)

Equation 2.3 shows the areal change of pressure is linked to the change in density over time. Realistically, it is pressure and time that can be measured successfully in a laboratory or a reservoir, therefore a more useful relationship would be between the change in pressure areally with the change in pressure through time. The density can be related to the pressure by the isothermal compressibility, c, defined as:

1 dV c=- ( ) V dP T where V is the volume (m3) and P is the pressure (Pa). The density equals mass per unit volume ,  

m hence: V

(Quotient Rule, constant mass system)

(2.4)

Since

(from above) then 



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(2.5)

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This is the partial differential equation for the linear flow of any single phase fluid in a porous medium which relates the spatial variation in pressure to the temporal variation in pressure. If it were applied to a laboratory core flood, it could describe the pressure variation throughout the core from the initial start of the flood when the flowrate was increased from zero to a steady rate (the transient period) as well as the steady state condition when the flow into the core was balanced by the flow out of the core. Inspection of the equation shows that it is non-linear because of the pressure dependence of the density, compressibility and viscosity appearing in the coefficients and c. The pressure dependence of the coefficients must be removed before simple solutions can be found, i.e. the equation must be linearised. A simple form of linearisation applicable to the flow of liquids such as undersaturated oil is to assume their compressibility is small and constant. More complex solutions are required for more compressible fluids and gasses.

2.2.1.1 Linearisation Of Partial Differential Flow Equation For Linear Flow Assuming that the permeability and viscosity terms do not depend on location (i.e. distance along the cuboid), then

(2.6)

The left hand side can be expanded to:

Using equation 2.4 and since

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the above becomes c(P/x)2 + (2P/x2) Usually (P/x)2 is neglected compared to 2P/x2 since the pressure gradient is small, and substituting gives

(2.7)

This is termed the linear diffusivity equation The assumption is made that the compressibility is small and constant, therefore the coefficients

are constant and the equation is linearised. In equation (2.7)

is

termed the diffusivity constant. For liquid flow, the above assumptions are reasonable and have been applied frequently, but can be applied only when the product of the compressibility and pressure is much less than 1, i.e. cP <<1.0. Thus the requirement for small and constant compressibility. The compressibility in this case is the saturation weighted compressibility, i.e. the effect of the oil, water and formation compressibilities: c = coSo + cwSwc + cf

(2.8)

where c is the saturation weighted compressibility co is the compressibility of oil cw is the compressibility of the connate water cf is the compressibility of the formation (pore volume) So is the oil saturation Swc is the connate water saturation

2.2.1.2 Conditions of Solution

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The solution of the equation requires initial conditions and the boundary conditions. (i) Initial Solution Condition. At time t = 0, the initial pressure, Pi, must be specified for every value of x. (ii) Boundary Conditions. At the end faces x = 0 and x = L, the flow rate or pressure must be specified for every value of time, t. Solutions of the linear diffusivity equation are needed when dealing with linear flow from aquifers. For solutions dealing with well problems a radial model is required.

2.2.2 The Radial Model Figure 4 illustrates the geometry of this model in which the flow occurs in horizontal planes perpendicular to the Z axis (i.e. in planes parallel to the XY plane) within a layer of constant height, h. The flow is radial and is either towards the Z axis or away from it.

Figure 4 Radial horizontal flow geometry geometry

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At a distance r from x-axis, the flow velocity, U is now radius dependent: U = q/2rh

(2.9)

From Darcy’s Law (taking account of the flow direction and the co-ordinate direction):

(2.10)

The mass balance gives:

(2.11)

Eliminating U and q through equations 2.9 to 2.11 gives the non-linear equation:

(2.12)

Making assumptions as for linear flow, linearises the equation to:

(2.13)

This is termed the radial diffusivity equation

2.3

Characterisation of the Flow Regimes by their Dependence on Time

To apply the diffusivity equation to real reservoirs requires careful consideration of the boundary conditions. It will be shown that for most practical purposes, the solutions to the diffusivity equation can be grouped according to the flow regime that they

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represent: steady-state, semi-steady-state (pseudo steady state) or unsteady state (transient). Steady-state refers to the situation in which the pressure and the flow rate distribution in the reservoir remain constant with time. Unsteady state is the situation in which the pressure and/or the flow rate vary with time. Semi-steady state is a special case of unsteady state that resembles steady-state flow. These differences in the flow regimes have ramifications in practical reservoir engineering since working solutions to the diffusivity equation are usually limited to a particular flow regime. For instance, in a pressure build up test in a well, the determination of an accurate average reservoir pressure will depend strongly on the flow regime the well is in and therefore which working solution is used.

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3.1

BASIC SOLUTIONS OF THE CONSTANT TERMINAL RATE CASE FOR RADIAL MODELS The Steady State Solution

If a well is produced at a constant flow rate, q, and if the pressure at the external radius, re is maintained constant, flow will finally stabilise to steady state conditions.

i.e. flowrate, q = constant and the pressure gradient, and time, t

therefore,

for all values of radius, r

and the flow equation becomes

integrating between the limits rw and r gives:

(3.1)

Integrating between the limits rw and re gives:

 qμ   re  Pe  Pw    ln    2πkh   rw 

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(3.2)

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which is identical to the relationship described for a radial system by Darcy’s Law. In this case, the pressure at the external radius of the reservoir is required and the only way to measure it in the reservoir would be to drill a well at the external radius. This is uneconomic, therefore a mean reservoir pressure,P , is used. It is found from routine bottom hole pressure measurements and well tests conducted on the wells in a reservoir, it includes the effect of the area of influence of each well. In simple terms, the volume drained by each well is used to weight the bottom hole pressure measurements made in the well; all of the weighted pressures of all of the wells in the reservoir are then averaged. Figure 5 shows a well in a reservoir and its area of influence. Volumetrically, this volume is drained by the well and the mean reservoir pressure,P , is related to the pressure, P of elements of volume, dV being drained. The total volume is V.

Figure 5 Pressure distribution around a well

(3.4a)

where dV = 2rhdr

(3.5)

The volume of the well’s drainage zone, V = (re2-rw2)h and considering rw <
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(3.6)

from equation 3.1,

r 2w assuming 4 is negligible

(3.7)

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3.2 Unsteady State Solution 3.2.1 General Considerations

Figure 6 Wellbore pressure response to a change in flowrate

Figures 6a and 6b show the response of a reservoir at a wellbore when a flow rate, q, is suddenly applied. The pressure of the flowing fluid in the wellbore, Pwf falls from the initially constant value, Pi (static equilibrium) through time and the constant terminal rate (CTR) solution of the diffusivity equation describes this change as a function of time. The CTR solution is therefore the equation of Pwf versus t for a constant production rate for any value of the flowing time. The pressure decline, Figure 6(b), can normally be divided into three sections depending on the value of the flowing time and the geometry of the reservoir or part of the reservoir being drained by the well. This figure represents the pressure change at the wellbore through time which is equivalent to the pressure change (or change in the height of water) in the cylinder nearest the outlet in the model represented in Figure 1. Initially, the pressure response can be described using a transient solution which assumes that the pressure response at the wellbore during this period is not affected by the drainage boundary of the well and vice versa. This is referred to as the infinite reservoir case, since during the transient flow period, the reservoir appears to be infinite in extent with no limits to the fluid available to expand and drive the system.

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The transient period is followed by the late-transient when the boundaries start to affect the pressure response. This is analogous to the pressure disturbance having moved along the line of tubes in the model in Figure 1. The nature of the boundaries affects the type of solution used to describe the pressure change since a well may drain an irregularly shaped area where the boundaries are not symmetrical or equidistant from the well. The next phase in the pressure decline is termed semi-steady state or pseudo steady state where the shape of the pressure profile in the reservoir is not changing through time and the wellbore pressure is declining at a constant rate. It is analogous to the model depicted in Figure 1 where the level of water in all of the tubes is falling and no additional water is being added to tube 10 to maintain absolute pressure profile. If the pressure profile developed in the reservoir around the well had remained constant, true steady state conditions would have occurred and the steady state solutions as mentioned in the previous section would have applied.

3.2.2 The Line Source Solution This solution assumes that the radius of the wellbore is vanishingly small relative to the mean radius of the reservoir. It allows the calculation of the pressure at any point in an unbounded reservoir using the flowrate at the well. The benefits are clear in that no flow rates other than those measured in the producing well are required and from which the pressure at any location can be calculated. The disadvantage is that the solution works for infinite acting reservoirs only and if barriers are met, then the solution fails to represent the true flow regime. The technique of superposition can be used to combine the effect of more than one well in an infinite acting reservoir and this technique can represent the effect of a barrier. The barrier is equivalent to the pressure disturbance produced by a second, imaginary well producing at the same rate and having the same production history as the real well with both these wells in an infinite acting reservoir. This solution has found a lot of use in well test analysis.

(3.8)

The term Ei(-y) is the exponential integral of y (the Ei function) which is expressed as

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It can be calculated from the series

 yn  Ei(y)   lny   n!n  where  = 0.5772157 (Euler’s Constant). On inspection of the similarities in the Ei function and the ln function, it can be seen that when y <0.01, Ei(-y) = + lny and the power terms can be neglected. Therefore,

Ei(y) ln(1.781y) = ln(y ) (  1.781= e  e0.5772157)

Solutions to the exponential integral can be coded into a spreadsheet and used with the line source solution. Practically, the exponential integral can be replaced by a simpler logarithm function as long as it is representative of the pressure decline. The limitation

that y<0.01 corresponds to time, t, from the start of production The equation can be applied anywhere in the reservoir, but is of significance at the wellbore (i.e. for well test analysis) where typical values of wellbore radius, r w, and reservoir fluid and rock parameters usually means that y<0.01 very shortly after production starts. Therefore the line source solution can be approximated by

q cr 2 P  Pi  (ln ) 4kh 4kt or, since -ln(y) = ln(y-1)

P  Pi 

q 4kt (ln 2) 4kh cr

(3.9)

and if the pressure in the wellbore is of interest,

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Pwf  Pi 

q 4kt (ln 2) 4kh crw

(3.10)

The values of exponential integral have been calculated and presented in Matthews and Russel’s Monograph and are produced in Table 1. The table presents negative values, i.e. -Ei(-y). For values of y<0.01, the ln approximation can be used. For values >10.9, the decline in pressure calculated is negligible.

3.2.2.1 Range of Application and Limitations to Use The line source solution has limitations on its application: it cannot represent the initial flow into a wellbore since the assumption that the wellbore is a line is obviously not the case and some time has to elapse for the relative size of the wellbore to have a negligible effect on the flow and expansion of the fluid in the majority of the reservoir. The reservoir must also be infinite acting. Analysis of real reservoir performance has shown that the line source solution is valid for: (i)

flowing time, t >100 crw2/k

(3.11)

where rw is the wellbore radius. The value of 100 has been derived form the analysis of the responses of real reservoirs; it can be varied according to the nature of a specific well and reservoir. The time involved here is not the same as the dimensionless time, tD calculated for other models of fluid flow in a reservoir. (ii)

t < cre2/4k

(3.12)

where re is the external radius. The reservoir boundaries begin to effect the pressure distribution in the reservoir after this time, the infinite acting period ends and the line source solution does not represent the fluid flow. 3.2.3

The Skin Factor

The analysis of fluid flow encountered thus far has assumed that a constant permeability exists within the reservoir from the wellbore to the external boundary. In reality, the rock around the wellbore can have higher or lower permeability than the rest of the reservoir. This results from formation damage which may occur during drilling and completion (where the wellbore fluids alter the wettability of the near wellbore formation as fluid leaks off into it, or solids suspended in the drilling fluids are deposited in the pore spaces and become trapped thereby physically hindering the

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flow of fluid and reducing the permeability) or during production (where sand or precipitates from the hydrocarbon fluids or from formation brines can alter wettability and plug pore spaces). Alternatively, wellbores intersecting fractures may exhibit enhanced permeabilities as the fractures offer much greater conductive paths to the fluids around the wellbore, thus enhancing the permeability. This situation may also be required as part of the reservoir management: hydraulic fractures or acidising workovers are performed on wells to bypass zones of reduced permeability which have developed during production. In these cases, the line source equation fails to model the pressure drop in these wells properly since it uses the assumption of uniform permeability throughout the drainage area of the well up to the wellbore. Figure 7 shows the effect of a reduction in permeability around a wellbore. The skin zone does not affect the pressures in the rest of the formation remote from the wellbore, i.e. it is a local effect on the pressure drop at the wellbore.

Figure 7 Variation of the permeability around the wellbore changes the local pressure profile

It can be shown that if the skin zone is considered equivalent to an altered zone of uniform permeability, ks, with an outer radius, rs, the additional drop across this zone (Ps) can be modeled by the steady-state radial flow equation. It is assumed that after the pressure perturbation caused by the start of production has moved off into the rest of the formation, the skin zone can be thought of as being in a steady state flow regime. The pressure drop associated with the presence of a skin is therefore the difference in the bottomhole flowing pressures at the well when skin is present and when skin is not present, i.e.

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(3.13)

Equation 3.13 simply states that the pressure drop in the altered zone is inversely proportional to the permeability, ks rather than to the permeability, k of the rest of the reservoir and that a correction to the pressure drop in this region must be made. When this is included in the line source solution it gives the total pressure drop at the wellbore:

Pi  Pwf  

 k  r  q q  Ei(y) Ps   Ei(y) 2 1ln s  (3.14) 4kh 4kh  k s  rw 

If at the wellbore the logarithm approximation can be substituted for the Ei function, then:

(3.15)

A skin factor, s, can then be defined as:

(3.16)

and the drawdown defined as:

(3.17) Equation 3.17 shows that a positive value of skin factor will indicate that the permeability around the well has been reduced (by some form of formation damage). The absolute value reflects the contrast between the skin zone permeability and the

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unaltered zone permeability and the depth to which the damage extends into the formation. Part of the essential information from a well test is the degree of formation damage (skin factor) around a well caused by the drilling and completion activities. Alternatively, a well may have a negative skin factor, i.e. the permeability of the skin zone may be higher than that of the unaltered zone, caused by the creation of highly conductive fractures or channels in the rock. The extent of the damage zone cannot be predicted accurately and there may be variations vertically in the extent of the damage zone therefore this simple model may not characterise the near wellbore permeability exactly. An altered zone near a particular well affects only the pressure near that well, i.e. the pressure in the unaltered formation away from the well is not affected by the existence of the altered zone around the well.

3.3

Semi-Steady-State Solution

Once the initial pressure perturbation produced by bringing a well onto production has moved through the reservoir and met the boundaries, the infinite-acting nature of the fluid changes to become finite acting. As stated previously, this is termed pseudo steady state or semi steady state because the pressure drop with time is the same at all points around the flowing well, i.e.

and where there is no flow across the outer boundary at r = re of the drainage zone, i.e.

In a similar manner to the steady state flow regime, the pressure difference between the wellbore and, say, the external radius, or the pressure difference between the wellbore pressure and the initial pressure, or the pressure difference between the wellbore pressure and the average reservoir pressure can be calculated depending on the physical measurements which have been taken. Usually, an average pressure is known in a reservoir and this is used to determine the pressure drop. Figure 8 shows the pressure profile in the reservoir and the values which may be relevant.

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Figure 8 Pressure profile in a reservoir under semi steady state flow conditions

Under semi steady state conditions, the pressure profile can be averaged over the volume of the reservoir. This gives the average reservoir pressure at a particular time in the stage of depletion of the reservoir. If there are several wells in a reservoir, each well drains a portion of the total volume. For stabilised conditions, the volume drained by each well is stable and in effect the whole reservoir can be subdivided into several portions or cells. The average pressure in each cell can also be calculated from the stabilised pressure profile. The calculation of the average pressure is determined from the material balance of the initial pressure and volume of fluid and its isothermal compressibility.

Pe  Pwf 

q  re 1  ln   s 2kh  rw 2 

(3.18)

Pr  Pwf 

q  re 3   ln  + s  2kh  rw 4 

(3.19)

The pressure differences (Pr - Pwf) pressure at radius, r from the wellbore - pressure in the wellbore when flowing, (Pe- Pwf) pressure at the external boundary, re, from the wellbore - pressure in the wellbore when flowing, ( Pr -Pwf) average reservoir pressure - pressure in the wellbore when flowing do not change with time, whereas Pr, Pe, Pr and Pwf do change.

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3.4

The Application of the CTR Solution in Well Testing

The study of fluid flow so far has related the pressure drop expected as a result of a flow rate from a well in a reservoir. If the appropriate parameters, such as porosity, permeability and fluid viscosity are known, then for a particular flow regime, such as unsteady state, the pressure drop at a certain distance from the well at a certain time after production starts can be calculated. In reality, only flow rates and pressures at wells can be measured directly, and the most important unknown factor in the diffusivity equation is the permeability. Therefore, rather than calculate a pressure drop for a given set of conditions, the pressure drop can be measured continuously and the permeability calculated. This is part of the objectives of well testing and for illustration, the following example calculates the permeability and skin factor for a well in a reservoir. It is important to note that these examples all assume that an initially undisturbed reservoir is brought on production, i.e. that there has been no previous production in the reservoir therefore the pressure is at its initial value. In well test analysis, the previous history of a well must be accounted for. The section on superposition will introduce the concepts of a multi-rate history for a well. To illustrate the use of the relationship, the following example of a well test is considered: A well is tested by producing it at a constant flow rate of 238stm3/day (stock tank) for a period of 100 hours. The reservoir data and flowing bottomhole pressures recorded during the test are as follows: Data porosity,  formation volume factor for oil, Bo net thickness of formation, h viscosity of reservoir oil,  compressibility, c wellbore radius, rw initial reservoir pressure, Pi well flowrate (constant)

18% 1.2rm3/stm3 6.1m 1x10-3 Pas 2.18 x10-9Pa-1 0.1m 241.3bar 238stm3/day

The time and bottomhole pressure have been recorded as shown in the table:

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The objective is to calculate the permeability and the skin factor. Looking at the test description, this is the first time the well has been put on production and the reservoir pressure will decline at a rate dictated by the solutions of the diffusivity equation. The pressure decline has been recorded at the wellbore (as in the table of data) and it is expected that there will be an unsteady state (transient) period initially followed by a semi steady state or steady state flow period. It is thought to be an isolated block therefore there would be a depletion of the reservoir pressure under semi steady state conditions expected. The initial unsteady state or transient flow period can be used to determine the permeability and skin factor of the well, and the subsequent semi steady state flow period can be used to detect the reservoir limits. SI units will be used at reservoir conditions, therefore flowrates are in m3/s and the formation volume factor for oil is used to convert from stock tank to reservoir volumes. The pressure related items are in Pascal. The permeability and skin factor can be determined from the initial transient period using the line source solution:

Inspection of the equation shows that there is a linear relationship between bottomhole flowing pressure and logarithm of time such that Pwf = m lnt + c

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(3.20)

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Examining the data, the following are constant: initial pressure, Pi, permeability, k, porosity, , viscosity, , compressibility, c, wellbore radius, rw, and skin factor, s. Both permeability and skin factor are unknown (but they are known to be constant). Therefore in equation 3.20, there is a linear relationship between the bottom hole flowing pressure, Pwf and the logarithm of time, lnt, the slope of the relationship, m, is equal to

From this, the unknown value, i.e. the permeability, k, can be calculated. Once the permeability is known, equation 3.20 can be rearranged to determine the other unknown, the skin factor, as:

Any coherent set of data points can be used to determine the permeability and skin, however, it is not clear when the data represent the line source solution. Therefore all of the pressure data are plotted and a linear fit attached to those data which show the linear relationship between the bottom hole flowing pressure, Pwf and the logarithm of time, lnt. Therefore taking the logarithm of time, the data table becomes:

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And when these data are plotted, there is a linear section and a curved section. The linear section must contain the data where the line source solution is applicable (i.e. transient flow) and a therefore gradient can be determined.

Figure Ex16a

The plots of bottomhole flowing pressure show that the transient period (for which the logarithm approximation is valid) lasts for approximately 4 hours and from the plot, the slope, m, can be determined to be 1.98bar/log cycle. Substituting this into the equation gives:

(converting from stock tank cubic metres/day to reservoir cubic metres/second and from bar to Pascal producing a permeability in terms of m2 which is then converted to mD). Now that the permeability is known, the equation should now balance, i.e. the pressure drop should equal the right hand side of the equation. If it doesn't it means that there has been an extra pressure drop that is attributed to the skin factor, s around the well. To determine the skin factor, the slope, m, of the line is theoretically extrapolated to a

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convenient time. This is usually a time of 1 hour. The bottomhole pressure associated with this time is calculated and this is used to determine a pressure drop (Pi - Pwf ) during the time (t1 hour - t 0). This is then equal to the pressure drop calculated from the ln function plus an extra pressure drop caused by the skin. In this case, a real pressure measurement was recorded at time 1 hour. This is not necessarily the same number as calculated from the extrapolation of the linear section of the relationship since the real pressure recorded at time 1 hour may not be valid for use with the Ei function. Although it was recorded, it may have been too early for the Ei function to accurately approximate the reservoir flow regime. In this case P1 hour =201.2bar and therefore (by rearranging equation 3.20)

2s=20.25-13.02 = 7.23 s=3.6 This would indicate that there has been an extra pressure drop around the well giving a skin factor of 3.6. Analysis of the drilling and completion plan will give insight as to the reason (perhaps mud filtrate invasion, partial penetration of the reservoir section, etc.)

4

SUPERPOSITION

In the analyses so far, the well flow rate has been instantly altered from zero to some constant value. In reality, the well flowrates may vary widely during normal production operations and of course the wells may be shut in for testing or some other operational reason. The reservoir may also have more than a single well draining it and consideration must be taken of this fact. In short, there may be some combination of several wells in a reservoir and/or several flowrates at which each produce. The calculation of reservoir pressures can still be done using the previous simple analytical techniques if the solutions for each rate change, for example, are superposed on each other. In other words, the total pressure drop at a wellbore can be calculated as the sum of the effects of several flowrate changes within the well, or it may be the sum of the effects caused by production from nearby wells.

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There is also the possibility of using infinite acting solutions to mimic the effects of barriers in the reservoir by using imaginary or image wells to produce a pressure response similar to that caused by the barrier. Mathematically, all linear differential equations fulfill the following conditions: (i) If P is a solution, then C x P is also a solution, where C is a constant. (ii) If both P1 and P2 are solutions, then P1 + P2 is also a solution. These two properties form the basis for generating the constant terminal rate case. The solutions may be added together to determine the total effect on pressure, for example, from several applications of the equation. This is illustrated if a typical problem is considered: that of multiple wells in a reservoir.

4.1

Effects of Multiple Wells

In a reservoir where more than one well is producing, the effect of each well’s pressure perturbation on the reservoir is evaluated independently (i.e. as though the other wells and their flow rate/ pressure history did not exist), then the pressure drop calculated at a particular well at a particular time is the simple addition of all of the individual effects superimposed one effect upon the other. Consider 3 wells, X, Y and Z, which start to produce at the same time from an infinite acting reservoir (Figure 9).

Figure 9 The superposition of pressure changes from several wells

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Superposition shows that: (Pi-Pwf)Total at Well Y = (Pi -P)Due to well X + (Pi-P)Due to well Y + (Pi-P)Due to well Z Assuming unsteady state flow conditions, the line source solution can be used to determine the pressure in well Y. It is assumed here that the logarithm function can be used for well Y itself and that there will be a skin around the well. The effects of wells X and Z can be described by the Ei function. There is no skin factor associated with the calculation of pressure drop caused by these wells, since the pressure drop of interest is at well Y (i.e. even if wells X and Z have non-zero skin factors, their skin factors affect the pressure drop only around wells X and Z). The total pressure drop is then:

(4.1)

where qY is the flowrate from well Y qX is the flowrate from well X qZ is the flowrate from well Z rwY is the radius of well Y rXY is the distance of well Y from the X well rZY is the distance of well Z from the X well the rest of the symbols have their usual meaning This technique can be used to examine the effects of any number of wells in an infinite acting reservoir. This could be to predict possible flowing well pressures amongst a group of wells, or to deliberately use the interaction between wells to check reservoir continuity. These interference tests and other extended well tests are designed to characterise the reservoir areally rather than to determine only the permeability and skin factor around individual wells.

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4.2 Principle of Superposition and Approximation of Variable - Rate Pressure Histories The previous section illustrated the effect of the production from several wells in a reservoir on the bottomhole flowing pressure of a particular well. Of equal interest is the effect of several rate changes on the bottomhole pressure within a particular well. This is a more realistic situation compared to those illustrated previously where a well is simply brought on production at a constant flowrate for a specific period of time. For instance, a newly completed well may have several rate changes during initial cleanup after completion, then during production testing then finally during production as rates are altered to match reservoir management requirements (for example limiting the producing gas oil ratio during production). A simple pressure and flowrate plot versus time would resemble Figure 10.

Figure 10 Effect of flowrate changes on the bottomhole flowing pressure

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The well has been brought on production at an initial flowrate, q1. The bottomhole flowing pressure has dropped through time (as described by the appropriate boundary conditions and the flow regime) until at time t1, the flowrate has been increased to q2 and this change from q1 to q2 has altered the bottomhole flowing pressure (again as described by the boundary conditions and the flow regime). The total (i.e. the real bottomhole flowing pressure) is calculated by summing the pressure drops caused by the flowrate q1 bringing the well on production, plus the pressure drop created by the flowrate change q2 - q1 for any time after t1. During the first period (q1) the pressure drop at a time, t, (i.e. the time, t is less than time t1 when the rate changes) is described by

(4.2) For times, t, greater than t1 the additional pressure drop is added to give



(4.3)

This approach can be extended to many flowrate changes as illustrated in Figure 11.

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Figure 11 Multi rate pressure response in a wellbore

For the case where the well is shut in at a certain time and the pressure builds up, an additional term is added to reflect this.

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 - μcrw2 q1μ Pi -Pws = Ei  4πkh  4kt

  q 2  q1  μ  -μcrw2  Ei    4k  t-t shutin   4πkh   

where



Pws is the shut in bottomhole pressure tshutin is the total producing time before shut in t -tshutin is the closed in time from the instant of shut in q1 is the first drawdown rate q2 is the second buildup rate i.e. in this case q2 = 0

4.3

Effects of Rate Changes

The application of superposition to a well with several rate changes is illustrated as follows. A well is known to have the flowrate history as presented in Figure 12. It is seen that the well is brought onto production at a flowrate, q1 and this is maintained constant until time, t1 at which the flowrate is decreased to q2. This second flowrate continues until time t2 when the flowrate is increased to q3. In terms of the reservoir, it

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is assumed that the reservoir is in unsteady state flow regime and the line source can be used to describe the pressure drop caused by the flowrate changes. In this case, the first flow rate change is when the well is brought on production, so the change from zero to q1 causes the first pressure perturbation to move into the reservoir. It is the bottomhole flowing pressure, Pwf, that is of interest, and it can be calculated using the line source solution. There is the possibility of a skin zone around the well, so this must be accounted for. If no other flowrate change occurred, then eventually unsteady state would give way to either semi steady state or steady state conditions and the bottomhole flowing pressure would either decline at a steady rate or (if steady state) would remain constant at some level. Assuming that this did not occur and that unsteady state conditions still existed when the flowrate was changed to q2 then the change q2 - q1 would cause a second pressure perturbation that would move out into the reservoir, following the first one created when the well was put on production. The reservoir is still in unsteady state conditions i.e. the first pressure perturbation has not met any barriers so the reservoir fluid still reacts as if it were an infinite volume and this behaviour is still causing a decline in the pressure at the wellbore even though a second pressure perturbation has been created and is moving out into the reservoir. The pressure drop due to this flowrate change can be calculated by the line source solution and added to that produced by bringing the well onto production. Eventually at time t2, the flowrate is changed again. This time, the pressure perturbation caused by q3 -q2 follows the first and second perturbations into the reservoir, and again, as long as the reservoir fluid still behaves as if it were infinite in volume, the pressure drop created by this flowrate change can be added to the changes produced by the others to give the total pressure drop. This situation can be thought of as throwing 3 stones Into a large pool of water. The first stone creates waves that move off from the point of impact, then after a time, the second stone is thrown into the pool and it creates a series of waves that move off through the pool. So although the first stone has sunk to the bottom of the pool and is no longer creating waves, its effect when it was thrown into the pool still exists as the waves are moving outwards to the edge of the pool, now followed by the second set of waves from the second stone. And so on with the third stone. The main point is that the effect of the rate changes in the wellbore are still present in the form of the pressure disturbance they create moving out through the reservoir.

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Figure 12 The equivalence of flowrate changes in a reservoir

The pressure drop produced by bringing the well onto production is calculated by the logarithmic approximation of the Ei function (it is assumed that the checks have been made to the applicability of the Ei function and its logarithmic approximation).

The next pressure drop is that produced by the flowrate change q2 - q1 at time, t1. It is still the bottomhole flowing pressure that is to be determined, therefore any skin zone will still exist and still need to be accounted for. The second pressure drop is:

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And finally the third pressure drop is:

The total pressure drop at the wellbore caused by all of the flowrate changes is (Pi - Pwf )= P1 + P2 + P3

4.4

Simulating Boundary Effects (Image Wells)

One of the intriguing possibilities of the application of the principle of superposition to reservoir flow is in simulating reservoir boundaries. It is clear that when a well in a reservoir starts production, there will be a period where the flow regime is unsteady state while the reservoir fluid reacts to the pressure perturbation as if the volume of the reservoir was infinite (i.e. an infinite acting reservoir). Once the boundaries are detected, there is a definite limit to the volume of fluid available and the pressure response changes to match that of, for example, semi steady state or steady state flow. This assumes that the pressure perturbation reaches the areal boundary at the same time, i.e. if the well was in the centre of a circular reservoir, the pressure perturbation would reach the external radius at all points around the circumference at the same time (assuming homogeneous conditions). If the well was not at the centre then some parts of the boundary would be detected before all of the boundary was detected. This means that some of the reservoir fluid is still in unsteady flow whilst other parts are changing to a different flow regime. This would appear to render the use of the line source solution invalid, however, the effect of the nearest boundary in an otherwise infinite acting reservoir has the same effect as the interaction of the pressure perturbations of two wells next to each other in an infinite acting reservoir. Therefore if an imaginary well is placed at a distance from the real well equal to twice the distance to the boundary, and the flowrate histories are identical, then the principle of superposition can be used to couple the effect of the imaginary well to the real well in order to calculate the real well’s bottomhole flowing pressure. Figure 13 illustrates the problem and the effect of superposition. Figure 14 shows a simplification of the model.

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Figure 13 The pressure effect of the barrier in the real reservoir can be represented by an imaginary well

Figure 14 Representation of the boundary by a real well and an image well

This shows a plane-fault boundary in an otherwise infinite acting reservoir, as in the top of Figure 13. To determine the pressure response in the well, the line source solution can be used until the pressure perturbation hits the fault. Thereafter there are no solutions for this complex geometry. However, the reservoir can be modeled with an infinite acting solution if a combination of wells in an infinite-acting system that limit the drainage or flow around the boundary is found. The bottom of Figure 13 indicates 1 image well with the same production rate as the actual well is positioned such that the distance between it and the actual well is twice the distance to the fault of

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the actual well. No flow occurs across the plane midway between the two wells in the infinite-acting system, and the flow configuration in the drainage area of each well is the same as the flow configuration for the actual well. Pressure communication crosses the drainage boundary, but there is no fluid movement across it and the problem of the flow regime has been resolved: the real well can be thought of as reacting to the flowrate in it and to the pressure drop produced by the imaginary well on the opposite side of the fault. The pressure drop is therefore:

where the symbols have their usual meaning, and L is the distance from the real well to the fault. The skin factor is used in the actual well, but not in the other (image) well since it is the influence of this image well at a distance 2L from it that is of interest.

5

SUMMARY

Simple analytical solutions to the diffusivity equation can be used to determine the pressure in the reservoir In different flow regimes and for different times. The equations can be used to model quite complex rate changes and well interference.

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-Ei(-y),0.000<0.209,interval=0.001 y 0 1 2 3 0.00 + ¥ 6.332 5.639 5.235 0.01 4.038 3.944 3.858 3.779 0.02 3.355 3.307 3.261 3.218 0.03 2.959 2.927 2.897 2.867 0.04 2.681 2.658 2.634 2.612 0.05 2.468 2.449 2.431 2.413 0.06 2.295 2.279 2.264 2.249 0.07 2.151 2.138 2.125 2.112 0.08 2.027 2.015 2.004 1.993 0.09 1.919 1.909 1.899 1.889 0.10 1.823 1.814 1.805 1.796 0.11 1.737 1.729 1.721 1.713 0.12 1.660 1.652 1.645 1.638 0.13 1.589 1.582 1.576 1.569 0.14 1.524 1.518 1.512 1.506 0.15 1.464 1.459 1.453 1.447 0.16 1.409 1.404 1.399 1.393 0.17 1.358 1.353 1.348 1.343 0.18 1.310 1.305 1.301 1.296 0.19 1.265 1.261 1.256 1.252 0.20 1.223 1.219 1.215 1.210

4 4.948 3.705 3.176 2.838 2.590 2.395 2.235 2.099 1.982 1.879 1.788 1.705 1.631 1.562 1.500 1.442 1.388 1.338 1.291 1.248 1.206

5 4.726 3.637 3.137 2.810 2.568 2.377 2.220 2.087 1.971 1.869 1.779 1.697 1.623 1.556 1.494 1.436 1.383 1.333 1.287 1.243 1.202

6 4.545 3.574 3.098 2.783 2.547 2.360 2.206 2.074 1.960 1.860 1.770 1.689 1.616 1.549 1.488 1.431 1.378 1.329 1.282 1.239 1.198

7 4.392 3.514 3.062 2.756 2.527 2.344 2.192 2.062 1.950 1.850 1.762 1.682 1.609 1.543 1.482 1.425 1.373 1.324 1.278 1.235 1.195

8 4.259 3.458 3.026 2.731 2.507 2.327 2.178 2.050 1.939 1.841 1.754 1.674 1.603 1.537 1.476 1.420 1.368 1.319 1.274 1.231 1.191

9 4.142 3.405 2.992 2.706 2.487 2.311 2.164 2.039 1.929 1.832 1.745 1.667 1.596 1.530 1.470 1.415 1.363 1.314 1.269 1.227 1.187

-Ei(-y),0.000<2.09,interval=0.01 0.0 + ¥ 4.038 3.335 0.1 1.823 1.737 1.660 0.2 1.223 1.183 1.145 0.3 0.906 0.882 0.858 0.4 0.702 0.686 0.670 0.5 0.560 0.548 0.536 0.6 0.454 0.445 0.437 0.7 0.374 0.367 0.360 0.8 0.311 0.305 0.300 0.9 0.260 0.256 0.251 1.0 0.219 0.216 0.212 1.1 0.186 0.183 0.180 1.2 0.158 0.156 0.153 1.3 0.135 0.133 0.131 1.4 0.116 0.114 0.113 1.5 0.100 0.099 0.097 1.6 0.086 0.085 0.084 1.7 0.075 0.074 0.073 1.8 0.065 0.064 0.063 1.9 0.056 0.055 0.055 2.0 0.049 0.048 0.048

2.681 1.524 1.076 0.815 0.640 0.514 0.420 0.347 0.289 0.243 0.205 0.174 0.149 0.127 0.109 0.094 0.081 0.071 0.061 0.053 0.046

2.468 1.464 1.044 0.794 0.625 0.503 0.412 0.340 0.284 0.239 0.202 0.172 0.146 0.125 0.108 0.093 0.080 0.070 0.060 0.052 0.046

2.295 1.409 1.014 0.774 0.611 0.493 0.404 0.334 0.279 0.235 0.198 0.169 0.144 0.124 0.106 0.092 0.079 0.069 0.060 0.052 0.045

2.151 1.358 0.985 0.755 0.598 0.483 0.396 0.328 0.274 0.231 0.195 0.166 0.142 0.122 0.105 0.090 0.078 0.068 0.059 0.051 0.044

2.027 1.309 0.957 0.737 0.585 0.473 0.388 0.322 0.269 0.227 0.192 0.164 0.140 0.120 0.103 0.089 0.077 0.067 0.058 0.050 0.044

1.919 1.265 0.931 0.719 0.572 0.464 0.381 0.316 0.265 0.223 0.189 0.161 0.138 0.118 0.102 0.088 0.076 0.066 0.057 0.050 0.043

2.0
2 3.72x10-2 1.01x10-2 2.97x10-3 9.08x10-4 2.86x10-4 9.22x10-5 3.02x10-5 9.99x10-6 3.34x10-6

2.959 1.589 1.110 0.836 0.655 0.525 0.428 0.353 0.295 0.247 0.209 0.177 0.151 0.129 0.111 0.096 0.083 0.072 0.062 0.054 0.047

3 3.25x10-2 8.94x10-3 2.64x10-3 8.09x10-4 2.55x10-4 8.24x10-5 2.70x10-5 8.95x10-6 3.00x10-6

4 2.84x10-2 7.89x10-3 2.34x10-3 7.19x10-4 2.28x10-4 7.36x10-5 2.42x10-5 8.02x10-6 2.68x10-6

5 2.49x10-2 6.87x10-3 2.07x10-3 6.41x10-4 2.03x10-4 6.58x10-5 2.16x10-5 7.18x10-6 2.41x10-6

6 2.19x10-2 6.16x10-3 1.84x10-3 5.71x10-4 1.82x10-4 5.89x10-5 1.94x10-5 6.44x10-6 2.16x10-6

7 1.92x10-2 5.45x10-3 1.64x10-3 5.09x10-4 1.62x10-4 5.26x10-5 1.73x10-5 5.77x10-6 1.94x10-6

8 1.69x10-2 4.82x10-3 1.45x10-3 4.53x10-4 1.45x10-4 4.71x10-5 1.55x10-5 5.17x10-6 1.74x10-6

9 1.48x10-2 4.27x10-2 1.29x10-3 4.04x10-4 1.29x10-4 4.21x10-5 1.39x10-5 4.64x10-6 1.56x10-6

Table 1 Exponential Integral

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TOPIC 2: DRIVE MECHANISMS 1 DEFINITION 2 NATURAL DRIVE MECHANISM TYPE 2.1 Depletion Drive Reservoirs 2.2 Water Drive 2.3 Compaction Drive 2.4 Gravity Drainage 2.5 Depletion Type Reservoirs 2.5.1 Solution Gas Drive 2.5.2 Gas Cap Drive 2.6 Water Drive Reservoirs 2.7 Combination Drives 3 RESERVOIR PERFORMANCE OF DIFFERENT DRIVE SYSTEMS 3.1 Solution Gas Drive 3.1.1 Solution Gas Drive, Oil Production 3.1.2 Solution Gas Drive, Gas / Oil Ratio 3.1.3 Pressure 3.1.4 Water Production, Well Behaviour, Expected Oil Recovery and Well Location Gas Cap Drive 3.2 Gas Cap Drive 3.2.1 Oil Production 3.2.2 Pressure 3.2.3 Gas / Oil Ratio 3.2.4 Water Production, Well Behaviour, Expected Oil Recovery and Well Locations 3.3 Water Drive 3.3.1 Rate Sensitivity 3.3.2 Water Production, Oil Recovery 3.3.3 History Matching Aquifer Characteristics 3.3.4 Well Locations 4 SUMMARY 4.1 Pressure and Recovery 4.2 Gas / Oil Ratio

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LEARNING OBJECTIVES Having worked through this chapter the student will be able to: 

Define reservoir drive mechanism



Describe briefly with the aid of sketches a depletion drive reservoir.



Describe briefly with the aid of sketches a water drive reservoir.



Describe briefly with the aid a sketches gravity drainage.



Describe briefly with the aid of sketches solution gas drive distinguishing behaviour both above and below the bubble point.



Describe briefly with the aid of sketches gas cap drive.



Describe briefly with the aid of sketches the reservoir performance characteristics of a solution gas drive reservoir.



Describe briefly with the aid of sketches the reservoir performance characteristics of a gas drive reservoir.



Describe briefly with the aid of sketches the reservoir performance characteristics of water drive reservoir.



Describe briefly with the aid of sketches the rate sensitivity aspect of water drive reservoir.



Summarise the characteristics of solution gas drive, gas cap drive and water drive reservoirs.

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RESERVOIR DRIVE MECHANISMS In the previous chapter we have considered the physical properties of the porous media, the rock, within which the reservoir fluids are contained and the properties and behaviour of the fluids. In this chapter we shall examine the various methods used to calculate the performance of different reservoir types, we will introduce the various drive mechanisms responsible for production of fluids from a hydrocarbon reservoir. In this qualitative description of the way in which reservoirs produce their fluids we will see how the various basic concepts come together to give understanding to the various driving forces responsible for fluid production. One of the main preoccupations of reservoir engineers is to determine the predominant drive mechanism, for dependant on the drive mechanism, different recoveries of oil can be achieved.

1 DEFINITION A reservoir drive mechanism is a source of energy for driving the fluids out through the wellbore. It is not necessarily the energy lifting the fluids to the surface, although in many cases, the same energy is capable of lifting the fluids to the surface.

2 NATURAL DRIVE MECHANISM TYPES There are a number of drive mechanisms, but the two main drive mechanisms are depletion drive and water drive. Other drive mechanisms to be considered are compaction drive and gravity drive. These drive mechanisms are natural drive energies and are not to be confused with artificial drive energies such as gas injection and water injection.

2.1 Depletion Drive Reservoirs A depletion type reservoir is a reservoir in which the hydrocarbons contained are NOT in contact with a large body of permeable water bearing sand. In a depletion type reservoir the reservoir is virtually totally enclosed by porous media and the only energy comes from the reservoir system itself. Figures 1 and 2 illustrate the types of accumulations which can give rise to depletion drive characteristics. In Figure 1 the hydrocarbons are enclosed in isolated sand lenses which have been generated by a particular depositional environment. Over geological time the hydrocarbons have found their way into the porous media. The surrounding rocks may have permeability but it is so low as to prevent energy transfer from other sources.

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In Figure 2 is illustrated another depletion type reservoir where a mature reservoir has been subjected to faulting, resulting in the isolation of a part of the reservoir from the rest of the accumulation. In a total field system, such a situation can give rise to parts of the reservoir having different drive mechanism characteristics.

Figure 1 Depletion reservoir: No aquifer. Isolated sand lenses

Figure 2 Depletion reservoir: Aquifer limited by faults

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2.2 Water Drive

Figure 3 Water drive: Active aquifer

A water drive reservoir is one in which the hydrocarbons are in contact with a large volume of water bearing sand. There are two types of water drive reservoirs. There are those where the driving energy comes primarily from the expansion of water as the reservoir is produced, as shown in Figure 3 The key issue here is the relative size and mobility of the water of the supporting aquifer relative to the size of the hydrocarbon accumulation. Water drive may also be a result of artesian flow from an outcrop of the reservoir formation, Figure 4. In this situation either surface water or seawater feeds into the outcrop and replenishes the water as it moves into the reservoir to replace the oil. The key issues here are the mobility of the water in the aquifer and barriers to flow from the outcrop to the reservoir. It is not often encountered, and the water drive arising from the compressibility of an aquifer, Figure 3, is the more common.

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Figure 4 Reservoir having artesian water drive.

2.3 Compaction Drive Figure 5 illustrates another drive mechanism, compaction drive. Although not a common drive energy, the characteristics of its occurrence can be dramatic. Compaction drive occurs when the hydrocarbon formation is compacted as a result of the increase in the net overburden stress as the reservoir pore pressure is reduced during production. The nature of the rock or its degree of consolidation can give rise to the mechanism. For example a shallow sand deposit which has not reached its minimum porosity level due to consolidation can consolidate further as the net overburden stresses increase as fluids are withdrawn. The impact of the further consolidation can give rise to subsidence at the surface. This phenomenon of compaction with increasing net overburden stress is not restricted to unconsolidated sands, since chalk also demonstrates this phenomenon. One of the spectacular occurrences of compaction drive is that associated with the Ekofisk Field, in the Norwegian sector of the North Sea. This is a very undersaturated chalk reservoir. The field was developed on the basis of using depletion drive down to near the bubble point and then to inject sea water to maintain pressure above the bubble point. During this period of considerable pressure decline, the net overburden stress was increasing, causing the formation to compact to an extent that subsidence occurred at the seabed. In an offshore environment such uniform subsidence can go undetected, as was the case for Ekofisk. The magnitude of the subsidence has been such that major jacking up of the structures has been required.

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Figure 5 Compaction drive

2.4 Gravity Drainage Gravitational segregation or gravity drainage can be considered as a drive mechanism. Figure 6 illustrates a situation where the natural density segregation of the phases can be responsible for moving the fluids to the well bore. Gravity drainage is where the relative density forces associated with the fluids cause the fluids, the oil, to drain down towards the production well. The tendency for the gas to migrate up and the oil to drain down clearly will be influenced by the rate of flow of the fluids as indicated by their relative permeabilities. Gravity drainage is generally associated with the later stages of drive for reservoirs where other drive mechanisms have been the more dominant energy in earlier years. Gravity drainage can be significant and effective in steeply dipping reservoirs which are fractured. Of the drive mechanisms mentioned the major drive mechanisms are depletion drive, which are further classified into solution gas drive and gas cap drive and water drive. Gravity Drive typically is active during the final stages of a depletion reservoir.

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Figure 6 Gravity drive

2.5. Depletion Type Reservoirs In depletion drive reservoirs the energy comes from the expansion of the fluids in the reservoir and its associated pore space. There are two types of depletion drive reservoirs, solution gas drive reservoirs and gas cap drive reservoirs. In solution gas drive reservoirs there are two stages of drive mechanism where different energies are responsible for fluid production. 2.5.1. Solution Gas Drive In solution gas drive reservoirs the initial condition is where the reservoir is undersaturated, i.e. above the bubble point. Production of fluids down to the bubble point is as a result of the effective compressibility of the system. When considering pressure volume phase behaviour, in the chapter on phase behaviour, we observed a small increase in volume of the oil for large reductions in pressure, for oil in the undersaturated state. Associated connate water also has a compressibility as has the pore space within which the fluids are contained. This combined compressibility provides the drive mechanism for depletion drive above the bubble point. Perhaps this part of the depletion drive should be called compressibility drive. The low compressibility causes rapid pressure decline in this period and resulting low recovery. Of the three compressibilities, although it is the oil compressibility which is the larger, the impact of the other compressibility components, the water and the pores, should not be neglected.

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As pressure is reduced, oil expands due to compressibility and eventually gas comes out of solution from the oil as the bubble point pressure of the fluid is reached. The expanding gas provides the force to drive the oil hence the term solution gas drive. It is sometimes called dissolved gas drive (Figure 7). Gas has a high compressibility compared to liquid and therefore the pressure decline is reduced. Solution gas drive only occurs once the bubble point pressure has been reached.

Figure 7 Solution gas drive reservoir

2.5.2. Gas Cap Drive Another kind of depletion type is where there is already free gas in the reservoir, accumulated at the top of the reservoir in the form of a gas cap (Figure 8), as compared to the undersaturated initial condition for the previous solution gas drive reservoir. This gas cap drive reservoir, as it is termed, receives its energy from the high compressibility of the gas cap. Since there is a gas cap then the bottom hole pressure will not be too far away from the bubble point pressure and therefore solution gas drive could also be occurring. The gas cap provides the major source of energy but there is also the expansion of oil and its dissolved gas and the gas coming out of solution. The oil expansion term is very low and is within the errors in calculating the two main energy sources. The two significant sources of driving energy are ; (1)

Gas cap expansion

(2)

Expansion of gas coming out of solution

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Figure 8 Gas cap drive reservoir

2.6 Water Drive Reservoirs Water drive reservoirs are also of two types. There is an edge water drive reservoir. The reservoir is thin enough so that the water is in contact with the hydrocarbons at the edge of the reservoir (Figure 9). The other type of water drive reservoir is the bottomwater-drive reservoir; where the reservoir is so thick or the accumulation so thin that the hydrocarbons are completely underlain by water (Figure 10).

Figure 9 Edge water drive reservoir

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Figure 10 Bottom water drive reservoir

2.7 Combination Drives ‘Pure’ types of reservoirs are those reservoirs where only one drive system operates, for example, depletion drive only - no water drive or water drive only - no gas drive. It is rare for reservoirs to fit conveniently into this simple characterisation. In many of them a combination of drive mechanisms can be activate during the production of fluids. Such reservoirs are called combination drives, Figure 11. In the case in Figure 11, which is not unusual, we have a gas cap with the oil accumulation underlain by water providing potential water drive. So both free gas and water are in contact with the oil. In such a reservoir some of the energy will come from the expansion of the gas and some from the energy within the massive supporting aquifer and its associated compressibility.

Figure 11 Combination water and gas - cap drive

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Sometimes it may be only water drive in the above situations. If the hydrocarbons are taken out at a rate such that for every volume of oil removed water readily moves in to replace the oil, then the reservoir is driven completely by water. On the other hand there may be only depletion drive. If the water does not move in to replace the oil, then only the gas cap would expand to provide the drive.

3 RESERVOIR PERFORMANCE OF DIFFERENT DRIVE SYSTEMS Having considered the basic aspects of the drive types we will now examine their respective characteristics in relation to production, recovery and pressure decline issues. The performance of different types of reservoirs in relation to the daily production, gas/oil ratio and water production can give some indication of the type of drive mechanism operative in the reservoir.

3.1 Solution Gas Drive In the first part of solution gas drive, in what we termed compressibility drive, within the reservoir no production of gas occurs and the fluid moves as a result of decompression of the three components oil, water and pore space. The pressure reduction is rapid in relation to volumes produced. The gas to oil ratio produced at the surface is constant since the reservoir at this stage is above its bubble point pressure. Once the bubble point is reached gas comes out of solution. Initially the gas bubbles are small and isolated. The size and number of the bubbles increase until they reach a critical saturation when they form a continuous phase and become mobile. At this stage the gas has relative permeability. The impact of the first bubbles of gas on the oil is very significant. The relative permeability to the oil is reduced by the presence of the non wetting gas. As the increase in saturation of gas increases at the expense of oil saturation, the relative permeabilties move in the same directions giving rise to reduced well productivity to oil and increased productivity to gas, Figure 12. That is the oil relative permeability decreases and the gas relative permeability increases. The gas although providing the displacing medium is effectively leaking out of the system. Not only does the gas progress to the wellbore, depending on vertical permeability characteristics it will move vertically and may form a secondary gas cap. If this occurs it can contribute to the drive energy. Well location and rate of production can be used to encourage gas to migrate to form such a gas cap as against being lost through production from the wellbore.

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Figure 12 Schematic of solution gas drive.

We will now review the various production profiles, specific to the drive mechanisms but before doing so we will review the various phases of production.

Figure 13 Phases in production.

Production Phases (Figure 13) The first phase, production build up, which may exist or not depending on the drilling strategy is the increased production as wells are brought on stream. Clearly, as in some cases, wells might be predrilled through a template and then all brought on stream together when connected to production facilities, such a build up of production will, therefore, not occur. The next stage represents the period when the productivity of the production facility is at its design capacity and the wells are throttled back to limit their productivity. This period is called the plateau phase when production is maintained at the design capacity of the facilities. Typical production rates for the plateau period cannot be presented since it depends on the techno-economics of the field. Clearly for a field with a very large front loaded capital investment there is an incentive to have a high production rate during the plateau phase, say 20% of the STOIIP, whereas for a lower cost onshore field 5% might be acceptable. Governments will also impose their considerations on this aspect as well.

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A time will come when the reservoir is no longer able to deliver fluids to match the facilities capacity and the field goes into the decline phase. This phase can be delayed by methods to increase production. Such methods could include artificial lift, where the effort required to lift the fluids from the reservoir is carried out by a downhole pump or by using gas lift to reduce the density of the fluid system in the well. There comes a time when the productivity of the reservoir is no longer able to generate revenues to cover the costs of running the field, This abandonment time again is influenced by the size and nature of the operation. Clearly a single, stripper well, carrying very little operational costs, can be allowed to produce down to very low rates. A well, as part of a very high cost offshore environment however, could be abandoned at a relatively high rate when perhaps the water proportion becomes too high or the productivity in relation to all production is not sufficient to meet the associated well and production costs. We will now review the performance characteristics of the various mechanisms in light of the forgoing production phases.

3.1.1 Solution Gas Drive, Oil Production (Figure 14 ) After a well is drilled and production starts for a solution gas drive reservoir, the pressure drops in the vicinity of the well. The initial pressure drop is rapid as flow results from the low compressibility of the system above the bubble point. Pressure continues to decline and solution gas drive becomes effective as gas comes out of solution. Mobility of gas occurs and the reduced mobility to oil and resulting decreasing oil relative permeability further causes the pressure to decline and productivity to oil flow decrease. Initially when all wells are on stream the oil production is high but the production rapidly declines and there is a short plateau and decline phase until an economic limit is reached.

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Figure 14 Production for solution gas drive

A good analogy for this type of reservoir is a bottle of carbonated water when opened a short lived high production scenario followed by rapid decline!

3.1.2 Solution Gas Drive, Gas/Oil Ratio The distinctive characteristic of the solution gas drive mechanism is related to the producing gas to oil ratio. When the reservoir is first produced the GOR being produced may be low corresponding to the RSi value of the reservoir liquid. If the reservoir is highly undersaturated there will be a period when a constant producing GOR occurs (points 1-2) in Figure 15. When the bubble point is reached in the near well vicinity, the initial gas which comes out of solution is immobile and therefore oil entering the wellbore is short of the previous level of solution gas. Theoretically at the surface the producing GOR level is less than the original GOR (points 2-3) in Figure 15. As the pressure further reduces the released gas becomes mobile and moves at a velocity greater than its associated oil due to the relative permeability effects. Oil enters the well bore, with its below bubble point solution GOR value, but also gas enters the well bore from oil which has not yet arrived. The net effect is that at the surface the producing GOR increases rapidly as free gas within the reservoir, which has come out of solution, moves ahead of the oil (points 3-4) in Figure 15.

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As the pressure continues to decline the productivity of the well continues to decline from the combined impact of reducing relative permeability and drop in bottom hole pressure. The production GOR goes though a maximum as oil eventually is produced into the well bore with a low solution GOR and the associated gas which has come out of solution has progressed much faster to the well and contributed to earlier gas production 4-5 in Figure 15.

Figure 15 Producing GOR for solution gas drive reservoir

When the pressure drops below the bubble point throughout the reservoir a secondary gas cap may be produced and some wells have the potential of becoming gas producers.

3.1.3 Pressure At first the pressure is high but as production continues the pressure makes a rapid decline.

3.1.4 Water Production, Well Behaviour, Expected Oil Recovery and Well Location Since by definition there is little water present in the reservoir there should be no water production to speak of. Because of the rapid pressure drop artificial lift will be required at an early stage in the life of the reservoir. The expected oil recovery from these types of reservoirs is low and could be between 5 and 30% of the original oil-inplace. Abandonment of the reservoir will depend on the level of the GOR and the lack of reservoir pressure to enable production. Well locations for this drive mechanism are chosen to encourage vertical migration of the gas, therefore the wells producing zones are located structurally low, but not too close to any water contact which might generate water through water coning, Figure 16.

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Figure 16 Well location for solution gas drive reservoir.

3.2 Gas Cap Drive Whereas for a solution gas drive reservoir where we have a reservoir initially in an undersaturated state, for a gas cap drive reservoir, Figure 17, the initial condition is a reservoir with a gas cap. Since the gas oil contact will be at the bubble point pressure the pressures within the oil accumulation will not be higher than this only so far as relates to the density gradient of the fluid. It is the gas cap, with its considerable compressibility, which provides the drive energy for such fields, hence the name. To get flow in the wells it is likely that gas will come out of solution in the near well bore vicinity and therefore some degree of solution gas drive will also take place. A good analogy for this type of reservoir is the plastic chemical dispenser fitted with a pump to maintain gas pressure above the dispensed liquid. Note in figure 17 that there is aquifer water. The question is: why is this gas cap drive? As seen in the figure after production, the reservoir has produced oil and the gas cap has expanded, but the oil water contact remains at the same depth. In some reservoirs, the geological processes have reduced the permeability between the aquifer and the reservoir and the water cannot flow into the reservoir. This is clearly a major characteristic of the aquifer reservoir system and needs to be determined during exploration, however, companies generally do not test the aquifer and this is the reason that, operationally, one of the main objectives during the first few years after field start-up is to determine if the aquifer is active and will contribute to the drive mechanism.

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Figure 17 Gas-cap drive

3.2.1 Oil Production The producing characteristics for a gas cap drive reservoir are illustrated in Figure 18. Although the production may be high as in the solution gas drive, the oil production still has a significant decline but not as rapid as for solution gas drive. This decline in oil production is due to the reducing pressure in the reservoir but also from the impact of solution gas drive on the relative permeability around the well bore. If the well is allowed to produce at too fast a rate, the very favourable mobility characteristics of the gas, arising from its low viscosity compared to the oil, are such that preferential flow can cause gas breakthrough into the wells and the well is then lost to oil production. Indeed it is this condition which will determine well abandonment.

3.2.2 Pressure With an associated gas cap a loss of volume of fluids from the reservoir is associated with a relatively low drop in pressure because of the high compressibility of the gas. In solution gas drive much of the driving gas is produced, but with a gas cap the fluid remains till later in the life of the reservoir. The pressure drop for a gas cap system therefore declines slowly over the years. The decline will depend on the relative size of the gas cap to the oil accumulation. A small gas cap would be 10% of the oil volume whereas a large gas cap would be 50% of the volume.

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Figure 18 Reservoir performance gas - cap drive.

3.2.3 Gas/Oil Ratio During the early stages of replacement of oil by gas a 100% replacement takes place. Later on gas by-passes oil and causes a reduced displacement efficiency. In the early stages the GOR remains relatively steady increasing slowly as the impact of solution gas drive generates gas from oil still to reach the well bore. The increasing mobility of the gas is such that there is an increasing GOR both from dissolved gas and by-pass gas and eventually the well goes to gas as the gas cap breaks through.

3.2.4 Water Production, Well Behaviour, Expected Oil Recovery and Well Locations Like solution gas drive there should be negligible water production. The life of the reservoir is largely a function of the size of gas cap but it is likely to be a long flowing life. The expected oil recovery for such a system is of the order of 20 to 40% of the original oil-in-place. The well locations, similar to solution gas drive, are such that the production interval for the wells should be situated away from the gas oil contact but not too close to the water oil contact to risk water coning.

3.3 Water Drive The majority of water drive reservoirs predominantly get their drive energy from the compressibility of the aquifer system. The effectiveness of water drive depends on the ability of the aquifer to replace the volume of the produced oil. The key issues with a

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water drive reservoir are therefore the size of the aquifer and permeability. This is because the only way for a low compressibility system to be effective is for its relative size to the oil accumulation to be large, and the permeability of the aquifer to water to enable flow though the aquifer and into the oil zone. These key issues set a considerable challenge to the reservoir engineer since to predict water drive behaviour, requires such information, which in pre production periods can only be obtained from exploration activity to determine the extent and properties of the aquifer. It is difficult to obtain justification to expend such exploration costs in determining the size of a water accumulation!

3.3.1. Rate Sensitivity. The characteristic features of natural water drive reservoirs are strongly influenced by the rate sensitivity of these reservoirs. If oil production from the formation is greater than the replacement flow of the aquifer then the reservoir pressure will drop and another drive mechanism will contribute to flow, for example solution gas drive. Three sketches below illustrate the various types of production profiles for different aquifer types and the influence of rate sensitivity. In Figure 19 we have the artesian type aquifer where there is communication to surface water though an outcrop. In this case if oil is produced at a rate less than the aquifer can move water into the oil zone, then the reservoir pressure, as measured at the original oil water contact, remains constant. The producing gas-oil ratio also remains constant since the reservoir is undersaturated. These reservoirs will enable a plateau phase, however as in all water drive reservoirs the decline of the reservoirs is not due to productivity loss through pressure decline but the production of water. The encroaching aquifer with perhaps its favourable mobility will preferentially move through the oil zone and if there are high permeability layers will move through these. Eventually the water-cut, the proportion of water to total production becomes too high and the well is abandoned to oil production.

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Figure 19 Producing characteristics for artesian water drive.

Figure 20 illustrates a more typical water drive reservoir where the drive energy comes from the compressibility of the aquifer system. In this case if the oil withdrawal rate is less then the rate of water encroachment from the aquifer then the reservoir pressure will slowly decline, reflecting the decompression of the total system , the oil reservoir and the aquifer. Clearly this pressure decline is related to the size of the aquifer. The larger the aquifer the slower the pressure decline. As with all water drive reservoirs productivity of the wells remains high resulting from the maintained pressure, however the productivity of the well to oil reduces as water breakthrough occurs. So a characteristic of water drive reservoirs is the increasing water production alongside decreasing oil production.

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Figure 20 Producing characteristics for water drive (confined aquifer).

Figure 21 illustrates the rate sensitive aspect of water drive reservoirs. If the oil withdrawal rate is higher than the water influx rate from the aquifer then the oil reservoir pressure will drop at a rate greater than would be the case with aquifer support alone, as the compressibility of the oil reservoir supports the flow. If this pressure drops below the bubble point then solution gas drive will occur, as evidenced by an increase in the gas-oil ratio. Cutting back oil production to a rate to less than the water encroachment rate restores the system to water drive, with the gas-oil ratio going back to its undersaturated level. When two drive mechanisms function as above then we have what is termed combination drive ( water drive and solution gas drive). Water drive reservoirs have good pressure support. The decline in oil production is related to increasing water production as against pressure decline.

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Figure 21 Reservoir performance - Water drive.

3.3.2 Water Production, Oil Recovery Because there is a large aquifer associated with the oil reservoir unlike depletion drive systems, water production starts early and increases to appreciable amounts. This water production is produced at the expense of oil and continues to increase until the oil/water ratio is uneconomical. Total fluid production remains reasonably steady. The expected oil recovery from a water drive reservoir is likely to be from 35 to 60% of the original oil-in-place. Clearly these recovery factors depend on a range of related aspects , including reservoir characteristics for example the heterogeneity as demonstrated by large permeability variations in the formation.

3.3.3. History Matching Aquifer Characteristics. Predicting the behaviour of water drive reservoirs in particular the rate of water encroachment is not straightforward. The topic is covered in a later chapter, but a significant perspective as mentioned previously is that data is required of the aquifer to carry out the calculations. In particular the size and geometry of the aquifer and its permeability and compressibility characteristics. Since such information is generally not available during the exploration and development phase, the characteristics of the aquifer are only determined once production has been operational and the support from the aquifer can be calculated from production and pressure data. (History Matching). Getting such information may require producing a significant proportion of the formation say 5% of the STOIIP. RFT surveys have provided a very effective way of determining the aquifer strength as well as the communicating layers of the formation. Pressure depth surveys taken in an open hole development well after production has started will give indications of pressure support in the formation

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Because water drive, through pressure maintenance provides the most optimistic recoveries, artificial water drive is often part of the development strategy because of the uncertainties of the pressure support from the associated aquifer. In the North Sea for example many reservoirs have associated aquifers. The risk of not knowing either the extent or activity of the aquifers is such that many operators are using artificial water drive systems to maintain pressure so that solution gas drive does not occur with the consequent loss of oil production.

3.3.4. Well Locations Well locations for water drive reservoirs are such that they should be located high in the structure to delay water breakthrough.

4 SUMMARY The following summaries and tables give the main features associated with the various drive mechanisms.

4.1 Pressure and Recovery Water-drive -pressure declines slowly and abandonment occurs when the water cut is too-high at around 50% of recovery, but depends on local factors. Gas-cap drive - the pressure shows a marked decline and economic pressures are reached around 20% of the original pressure when about 30% of the oil is recovered. Solution- gas drive - the pressure drops more sharply and at 10% of the pressure, reaches an uneconomical level of recovery at about 10% of the oil-in-place.

4.2 Gas/Oil Ratio Water drive - the curve for a water drive system shows a gas/oil ratio that remains constant. Variations from this indicate support from solution gas drive or other drive mechanisms Gas-cap drive - for this drive the gas/oil ratio increases slowly and continuously. Solution- gas drive - the curve for a solution gas drive reservoir shows that the gas/oil ratio increases sharply at first then later declines.

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SOLUTION GAS DRIVE RESERVOIRS

Characteristics 1. Reservoir Pressure 2. Gas/Oil Ratio 3. Production Rate 4. Water Production 5. Well Behaviour 6. Expected Oil Recovery

Trend Declines rapidly and continuously First low then rises to a maximum and then drops First high, then decreases rapidly and continues to decline None Requires artificial lift at early stages 5-30% of original oil-in-place

GAS CAP DRIVE RESERVOIRS

Characteristics 1. Reservoir Pressure 2. Gas/oil ratio 3. Production Rate 4. Water Production 5. Well Behaviour 6. Expected Oil Recovery

Trend Falls slowly and continuously Rises continuously First high, then declines gradually Absent or negligible Long flowing life depending on size of gas cap 20 to 40% of original oil-in-place

WATER DRIVE RESERVOIRS

Characteristics 1. Reservoir Pressure 2. Gas/Oil Ratio 3. Water Production 4. Well Behaviour 5. Expected Oil Recovery

Trend Remains high Remains steady Starts early and increases to appreciable amounts Flow until water production gets excessive up to 60% original oil-in-place.

Figures 22 and 23 give the pressure and gas-oil ratio trends for various drive mechanism types

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Figure 22

Figure 23

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TOPIC 3 VAPOUR LIQUID EQUILIBRIA 1

INTRODUCTION – THE IMPORTANCE OF VAPOUR-LIQUID EQUILIBRIUM

2

IDEAL SOLUTIONS 2.1 Raoult's Law 2.2 Dalton's Law 2.3 Ideal Equilibrium Ratio

3

NON IDEAL SYSTEMS

4

EQUATIONS FOR CALCULATING EQUILIBRIUM RELATIONS 4.1 Vapour – Liquid Calculations 4.2 Dew – Point Calculation 4.3 Bubble Point Calculation

5

SEPARATOR PROBLEMS 5.1 Gas/Oil Ratio 5.2 Oil Formation Volume Factor 5.3 Optimum Pressure of Separator System 5.4 Example Of Separator Problem

LEARNING OBJECTIVES Having worked through this chapter the Student will be able to: •

Define equilibrium ratio.

•

Discuss the use of equations for vapour-liquid equilibrium calculations for real systems and explain the application of the equations.

•

Discuss and explain the use of equations to determine the dew point pressure and bubble point pressure of a fluid mixture.

•

Describe in general terms the impact of separator conditions on the gas-oil ratio and oil formation volume factor.

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1 INTRODUCTION - THE IMPORTANCE OF VAPOUR-LIQUID EQUILIBRIUM The multiphase perspectives of hydrocarbon mixtures in and produced from reservoir accumulations are an important aspect in reservoir management, well productivity, facility design and pipeline transport. Predicting the relative amount of the phases and their respective physical properties is an essential element in all of the above operations. The topic of vapour-liquid equilibria is also at the heart of many subsequent and other process operations and therefore there have been a range of approaches into the solution of the problem. The behaviour in reservoirs of multicomponent mixtures and their production to surface has provided one of the most rigorous challenges to design engineers because of the complex and unique nature of the fluids and in many cases their behaviour near the critical point. Figures 1, and 2 illustrate the complex nature of oil and gas production where, particularly in a major offshore province, as well as onshore, a number of reservoirs produce into a common transport line and associated treatment facilities (Figure 1). Each of the fields with their unique composition clearly contribute to a compositional blend entering a treatment facility (Figure 2), where further separation occurs.

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Figure 1 The total system.

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Figure 2 Complexity of allocation of produced oil to supply fields.

The allocation of revenue based on quality of product and oil injected into a common pipeline provides a considerable challenge to metering and compositional analysis. To the reservoir engineer, the main issues are the multiphase behaviour in the formation and the relationship between the fluids in the reservoir and those produced at surface conditions.

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The critical element in reservoir simulation is the grid block where the saturations and flow behaviour of the respective fluids, gas, oil and water, are required. The grid block therefore can be considered as a ‘separator’ and vapour-equilibrium calculations are required to determine the relative amounts of the phases which lead to saturation values and relative permeability of the phases, and the composition of these phases which lead to important physical property values of density, viscosity and interfacial tension. In the previous chapter the considerations of the relative amounts of gas and liquid were considered in the simplistic two component black oil approach. In this chapter we will consider approaches to vapour-liquid equilibrium from a compositional model consideration both from an ideal behaviour perspective and then the consideration of real systems. On a pressure temperature phase diagram of a multi-component mixture, the area bounded by the bubble point and dew-point curves defines the conditions for gas and liquid to exist in equilibrium. It is an over-simplification to describe the system as involatile oil with associated solution gas. The behaviour of the individual components and their influence on the composition of the mixture need to be considered. If a sample of bubble point fluid is brought to surface to separator conditions, the fluid enters the two phase region at a temperature and pressure much lower than reservoir conditions. In the separator the liquid and gas phases, in equilibrium, are withdrawn separately. Large volumes of gas are formed at these separator conditions, and the liquid volume shrinks substantially because of decreased temperature and conversion of some of the fluid into the gas phase. The separator liquid is collected in the stock tank, at which additional temperature and pressure drop may occur, more gas may be released depending on the separator conditions to stock tank conditions. If Vo is the volume of liquid at reservoir conditions and Vst is the volume of stock tank oil. The oil formation volume factor Bo is :

Bo =

Vo Vst

If Vg and Vst are the total volume of gas and oil collected from the separator and stock tank. The solution gas to oil ratio is :

Rs =

Vg Vst

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The volume factors can be determined directly in the laboratory or from equilibrium calculations. In addition to separator calculations, vapour liquid equilibrium data can be used for: •

Reservoir calculations

•

Two phase pipeline flow calculations

•

Process calculations

Although phase behaviour considerations are required throughout the production process from reservoir to refinery the context of this particular chapter is in relation to reservoir predictions. When reservoir fluids undergo phase alteration as a result of changes in pressure, temperature or composition it is considered that these changes are slow and therefore the resulting separate phases are in equilibrium, i.e the properties of the phases are not changing with respect to time. For multicomponent phase behaviour predictions, thermodynamic principles have been applied to provide the predictive tools. Many works have been written which provide the foundation of the topic. Danesh1 in his text provides a good review of the topic both with respect to the foundation principles and the equations used. Vapour-liquid equilibrium calculations have been somewhat restricted to analysis of behaviour with the separate areas, i.e. reservoir and wells, surface separation, pipelines, onshore treatment and refinery operations. Increasingly in the modern multidisciplinary approach to technical management there is an interest in the integrated perspective of vapour liquid equilibrium. For example, what is the impact on the quality of product exiting from a multifield oil transport line of a pressure change in field X. Such integrated perspectives provide a considerable technical and commercial challenge to the various technical disciplines which have been separately involved.

2 IDEAL SOLUTIONS Before we consider the behaviour of real systems we will first examine the behaviour of an ideal solution, where no chemical interaction occurs and where no intermolecular forces occur when mixing components.

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These ideal solutions result in no heating effects when ideal solutions are mixed and the volume of the mixture equals the sum of the volumes the pure components would occupy at the same pressure and temperature.

2.1 Raoult’s Law Raoult’s equation states that the partial pressure of a component in the gas is equal to the product of the mole fraction, xj in the liquid, multiplied by the vapour pressure of the pure component pvj. pj = xjpvj

(1)

where pj is the partial pressure of component j in the liquid with a composition xj and pvj is the vapour pressure of the pure component j.

2.2 Dalton’s Law Dalton’s law states that the partial pressure of a component pj is equal to the mole fraction of that component in the gas, yj times the total pressure of the system p, i.e. pj = yjp

(2)

where yj is the composition of the vapour and p is the pressure of the system

2.3 Ideal Equilibrium Ratio By combining the above two laws, yjp = xjpvj Equilibrium ratio, K j 

yj xj



pvj p

i.e. the ratio of the component in the vapour and liquid phases is given by the ratio of the vapour pressure of the pure component to the total pressure of the system. This ratio is termed the Equilibrium ratio, Kj . If n is the total number of moles of mixture and zj is the mole fraction of component j in the mixture. zjn = xjnL + yjng

(5)

where nL and ng are the moles of liquid and gas such that nL + ng = n

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From equation 4.

(6)

xj by definition = 1.0

(7)

Similarly: c

zjn = 1.0 p j =1 n + .nL g pvj c

å yj = å j =1

If a basis of one mole of mixture is used i.e. n g + n

(8)

L=

1.0

(9)

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(10)

Using these equations in a trial and error method the compositions of vapour and liquid streams in a flash separation can be determined. The equilibrium ratio Kj is defined as the ratio of the composition of j in the vapour to liquid phase, i.e.

Kj =

yj xj

(11)

Clearly Kj is defined at a particular pressure and temperature. Other names for Equilibrium ratio, include K-factors, K-values, equilibrium vapour liquid distribution ratios. Fugacity Lewis introduced the concept of fugacity, for use in equilibrium calculations, to extrapolate or correct vapour pressures. This is required since a pure component only has a vapour pressure up to its critical point. The fugacity is a thermodynamic quantity defined in terms of the change in free energy in passing from one state to another. For an ideal gas, the fugacity is equal to its pressure, and the fugacity of each component is equal to its partial pressure. The ratio of fugacity to pressure is termed the fugacity coefficient, . For a multicomponet system,

(12)

All systems behave as ideal gases at very low pressures, therefore  when P is a few tens of psia

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When fugacities are not 1 , then this gives an indication of non-ideality. Fugacity has been imagined (Danesh)1 as a measure of the escaping tendency of molecules from one phase to an adjacent phase. In multicomponent systems, if the fugacity of a component in adjacent phases is the same, the two phases will be in equilibrium with no net transfer of molecules from one phase to another. At equilibrium therefore fg = fL.

(13)

The fugacity coefficient,  of a pure component can be calculated from the following general equation (Danesh). (14)

The ratio of the fugacity at the state of interest to that at a reference state is called the activity i = fi/fio The activity is a measure therefore of the fugacity contribution or activeness of the component in a mixture. fi = ifio . The ratio of activity to concentration is called the activity coefficient i, where i = i/xi Therefore fi=ixifio

(15)

3 NON IDEAL SYSTEMS Ideal solution assumptions cannot be applied to the systems relevant to multicomponent hydrocarbon fluids in reservoir flow, transport and processing conditions. The ideal assumptions only apply to low pressures and moderate temperatures, chemically and physically similar components and behaviour below the critical point. Different methods have been developed for treating vapour-liquid equilibrium for non ideal systems.

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The previous K value is based on both ideal and non ideal solutions laws. To extend the principle of equilibrium ratio to multicomponent hydrocarbon mixtures to the pressures and temperatures relevant to petroleum engineers, methods of treating non ideal systems need to be established. The subject of non ideal equilibrium ratios are treated later in the text. We assume in this section that K values are available either from whatever source, experimental, NGPSA data charts, or from equations of state and other predictive methods.

4 EQUATIONS FOR CALCULATING EQUILIBRIUM RELATIONS 4.1 Vapour-Liquid Calculations The calculations for determining the amount of liquid and vapour present in a mixture when the pressure and temperature are known are obviously important, for example, in optimising the performance of a separator process. The equilibrium equations which are used for a process separator are the same as those within a grid block or element of a reservoir simulator. Figure 3 represents such a separation element.

Figure 3 Vapour-liquid separation in an element.

F L

= =

total moles of system both liquid and gas total moles of material within liquid phase

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V zj xj yj

= = = =

total moles of material within vapour phase mole fraction of jth component in the mixture mole fraction of jth component in the liquid mole fraction of jth component in the vapour

It is common to express the feed F as 1.0 or 100 moles and express L and V as fractions or percentages of F. i.e. F = 1 = L + V

(16)

For component j zjF = xjL + yjV For F = 1.0 mole zj = xjL + yjV

(17)

The equilibrium ratio:

Kj =

yj xj

(18)

By definition: m

m

m

j =1

j =1

j =1

å x j = å yj = å zj = 1

(19)

where m is the number of components. Replacing yj by Kjxj in (17) zj = xjL + xjKjV zj = xj (L + KjV) dividing both sides by L + KjV

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zj L + K jV

xj =

(20)

and: (21) m

m

zj

å x = å L + K V = 1.0 j

j =1

j =1

j

similarly: m

m

zj

åy = åV + L K

= 1.0

j

j =1

j =1

j

(21a)

by multiplying (21) by V we get: m

åL j =1

V

zj + Kj

=V (22)

and (21a) in the same way: m

å j =1

zj =V L +1 K jV

(22a)

These equations are the key equations in vapour-liquid equilibrium calculations and their use is the same whether in those calculations to determine phase behaviour in a separator or those which take place within the numerous grid blocks of a reservoir simulator. Clearly in the latter the amount of calculations is considerable since each grid block can be considered a separator. In a large compositional based simulation study, thousands of grid blocks will be used.

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The method of calculation is therefore as follows for each separation element: (1) Select Kj for each component at the temperature and pressure of the system; (For the determination of K see the later section.) (2) Assume a vapour liquid split i.e. V&L such that V + L = 1.0; (3) Calculate either V, L, ∑xj or ∑yj from equations 21, 21a, 22 and 22a; (4) Either: (i) check V&L calculated against assumed V or L; (ii) determine if ∑xj or ∑yj = 1.0; (5) Repeat the calculation until assumed value is calculated value or until ∑xj and ∑yj = 1.0. It can be understood therefore that in a compositional reservoir simulator a considerable amount of computational time is taken up because of these iterative calculations at each grid block. In a black oil simulator no such iteration takes place; the specific pressure and temperature provide the direct phase values either from a PVT report or an empirical black oil correlation. This phase equilibrium perspective can also be used to calculate the reservoir saturation pressures for a particular temperature, ie. the dewpoint and bubble point pressures.

4.2 Dew-Point Calculation The dew-point is when the first drop of liquid begins to condense. At this point the composition of the liquid drop is higher in heavier hydrocarbons whereas the composition of the vapour is essentially the composition of the system: Figure 4.

Figure 4 Conditions at the Dew Point.

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At the dew point therefore: zj = yj or:

(22)

zj = xjKj

The mixture at the dew-point is therefore in equilibrium with a quantity of liquid having a composition defined by the above equation. Clearly:

zj = 1.0 j =1 K j

m

m

å xj = å j =1

(23)

Similarly for the bubble point.

4.3 Bubble Point Calculation The bubble point is when the first bubble of gas appears. At this point the composition of this bubble of gas is higher in lighter hydrocarbons whereas the composition of the liquid is essentially the composition of the system. Figure 5.

Figure 5 Conditions at the Bubble Point.

At the bubble point therefore: zj = xj

(24)

or:

zj =

yj Kj

The mixture at the bubble point is therefore in equilibrium with a quantity of vapour having a composition defined by the above equation.

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Also: m

m

åy = åz K j

j =1

j

j

= 1.0

j =1

(25)

The dew-point and bubble point when either temperature or pressure are known are determined by trial and error techniques until the above relationships are satisfied. The dew-point pressure or bubble point pressure are estimated, K values obtained and equations 23 or 25 checked. If the summation ≠ 1, different pressure values are tried until convergence is reached. When convergence is reached the respective dew point or bubble point pressure has been obtained.

5 SEPARATOR PROBLEMS In a separator a stream of fluid is brought to equilibrium at separator temperature and pressure. Vapour and liquid are separated within the unit and continue as separate streams. Several separators can be operated in series each receiving the liquid phase from the separator operating at the next higher pressure. Each condition of pressure and temperature at which vapour and liquid are separated is called a stage-separation. Hence a process using one separator and a stock tank is a two stage process a three stage process has two separators and one stock tank. (Figure 6). Separator calculations are performed to determine the composition of products, the oil formation-volume factor and the volume of gas released per barrel of oil and to determine the optimum separator conditions for the particular conditions of fluid. Using equilibrium calculations already derived we can calculate the separation achieved at each stage, the composition of the phases separated, the gas/oil ratio, and the oil formation volume factor.

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Figure 6 Schematic drawing of separation process.

5.1 Gas/Oil Ratio Gas is removed from each stage so that the solution GOR can be calculated for each stage or combination of stages.

Total gas / oil ratio =

sum of gas volumes (SCF) = RT volume of stock tank oil (bbl)

(a) Calculation for Stock Tank Oil, STO. If n1 moles enter first stage, moles of liquid entering 2nd = n2 = n1L1. where L1 = separation in stage one based on basis of one mole feed. Number of moles entering third stage n3 = n2L2 = L2L1n1 If third stage is the stock tank then: nST = L3n3 = L3L2L1n1

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nST is the moles of liquid in stock tank for n1 moles into first separator:

(26)

m = number of stages Li = mole fraction of liquid off ith stage n1 = moles of feed to first stage  If n1 = 1 then:

(27)

= mole fraction of STO in the feed.

(b) Calculation of Total Gas ngi = number of moles off stage i ng1 = V1n1 ng2 = V2n2 = V2L1n1 ng3 = V3n3 = V3L2L1n or in general for total gas:

 If nj = 1

(28)

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ngT = mole fraction of total gas in the feed Total gas volume per mole of feed =ngT Vmcu ft where Vm is the molar volume

5.2 Oil Formation Volume Factor Volume of stock tank oil per mole of feed

(29)

MST = molecular weight of stock tank oil ST

= moles of STO pet mole

ST = density of STO at standard conditions lb/bbl Total gas to oil ratio RT =

(30)

where RT is the total gas - oil ratio. If the feed to the first stage is a single-phase liquid into its point of entry into the production stream then Bo can be calculated.  res. = density of feed (lb/bbl) Volume of reservoir oil per mole = Vres = Mres/res Oil formation volume factor Bo =

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(31)

where

Mres = molecular weight of reservoir fluid =

lb.res. fluid lb.mole. fluid

and

nST =

lb.mol. stock tank fluid lb.mol.res. fluid

5.3 Optimum Pressure of Separator System The operating conditions of pressure and temperature of a separator influence the amount of gas and stock tank oil produced. Changes in these valves will change the GOR and the Bo. In quoting these values therefore it is important to keep note of the associated separation conditions of pressure and temperature. A number of units in series also influence these parameters. It is the role of the process designers to optimise the operating conditions of such limits and the number of units required. It is the equilibrium characteristics of the individual components as a function of temperature, pressure and composition which influence these total separation characteristics for the mixtures at each separation stage.

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Figure 7 Effect of separator pressure in a two-stage separation process (Amyx, Bass & Whiting).

Figure 7 illustrates the influence of a change of pressure for a two-stage separation process on GOR, Bo and the density of stock tank oil. Equilibrium ‘flash’ calculations, which the above are called, are used in many other applications. In reservoir engineering, flash calculations are at the core of compositional simulation.

5.4 Example of Separator Problem (McCain) The following example is taken from McCain’s text on Petroleum Fluids and the values for K used in the calculations come from the GPSA sources. Calculate the gas-to-oil ratio, stock-tank oil gravity and formation-volume factor which will result from a two-stage separation of the hydrocarbon mixture below. Use separator conditions of 76˚F and 100 psig. Assume that the mixture is a liquid at its bubble point at reservoir conditions of 2,695 psig and 220˚F.

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Step 1: Calculate the composition and quantities of separator gas and liquid using equation 21.

zj

å x = å L + K V = 1.0 j

j

The summation equals 1.0 when V = 0.4291 and L = 0.5709 and the compositions of the separator gas and liquid are:

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Step 2: Calculate the compositions and quantities of stock tank and liquid using equation 21, noting that the composition of the feed to the stock tank is the composition of the liquid from the separator.

The summation equals 1.0 so VST = 0.1351 and LST = 0.8649 and the compositions of the stock tank gas and liquid are:

Step 3: Calculate the density and molecular weight of the stock tank oil.

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220.656 lb = 53.73 cu ft Density of propane plus = 4.1069

0.181 = 0.001 Weight fraction ethane in ethane plus = 220.837

0.026 = 0.0001 Weight fraction methane in STO = 220.863

(from the Reservoir Engineering A course, chapter of liquid properties)

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o

API =

141.5 - 131.5 = 32.8 0.861

Step 4: Calculate gas to oil ratio

Similarly:

RT = RST + RSP

RSP =

SCF (2130)(0.4291)(53.73) = 450 STB (0.5709)(0.8649)(220.9)

RSP =

SCF (2130)(0.4291)(53.73) = 450 STB (0.5709)(0.8649)(220.9)

RT = 531

SCF STB

Step 5: Calculate the density and molecular weight of the reservoir liquid at reservoir conditions.

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114.205 lb = 52.33 cu ft Density of propane plus = 2.1825

2.089 = 0.018 . 294 116 Weight fraction ethane in ethane plus =

5.405 = 0.044 . 699 121 Weight fraction methane in reservoir oil = From the Reservoir Engineering A course, chapter of liquid properties: po = 49.1 lb.cu ft. compressibility correction 49.1 + 0.8 = 49.9 at 60˚F and 2710 psia. thermal expansion correction 49.19 - 3.86 = 46.04 at 220˚F and 2710 psia. or = 46.04 lb/cu ft. Step 6: Calculate formation volume factor using equation:

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Bo =

(121.7)(53.73) ( 46.04)(220.9)(0.5709)(0.8649)

Bo = 1.302

res bbl STB

Integration of the Black-Oil and Compositional Approach The example above illustrates the combination of the compositional based prediction of phase volumes and associated properties and that based around the black-oil model, centred around parameters of oil formation volume factor and gas-oil ratio. By such a combination, the weaknesses of the simple two component black-oil model which is at the heart of describing oil field parameters, can be overcome by using compositional derived values rather than using perhaps inappropriate empirical correlations and charts.

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TOPIC 4 PVT ANALYSIS 1

SCOPE

2

SAMPLING 2.1 Subsurface Sampling 2.2 Surface Sampling

3

SAMPLING WET GAS AND GAS CONDENSATE SYSTEMS 3.1 Introduction 3.2 Phase Behaviour 3.3 Sampling Gas And Gas Condensate Reservoirs 3.4 Separator Sampling Points 3.5 Sample Details

4

APPARATUS

5

RECOMBINATION OF THE SURFACE OIL AND GAS SAMPLES

6

PVT TESTS 6.1 Flash Vaporisation 6.2 Differential Vaporisation 6.3 Separator Tests 6.4 Viscosity 6.5 Hydrocarbon Analysis

7

WAX AND ASPHALTENES 7.1 Wax Crystallization Temperature

8

SUMMARY OF RESULTS PROVIDED BY AN OIL SAMPLE PVT TEST

9

UNDERSTANDING PVT REPORTS

10 PURPOSE OF THE PVT REPORT 11 OIL SAMPLE PVT STUDY 11.1 Separator Tests of Reservoir Fluids 11.2 Fluid Properties at Pressures Above The Bubble Point Pressure 11.3 Total Formation Volume of Original Oil Below The Bubble Point Pressure

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11.4 11.5 11.6 11.7

Differential Liberation Tests Calculation of Gas - Oil Ratios Below The Bubble Point. Calculation of Formation Volume Below The Bubble Point Viscosity Data

12 MERCURY 13 Appendix – PVT Report

LEARNING OBJECTIVES Having worked through this chapter the Student will be able to: •

Describe briefly the scope of PVT analysis.

•

Describe the sampling options for oil systems.

•

Describe the impact of well flow interruption on the sampling of ‘wet’ and gas condensate systems.

•

List the main items of equipment in PVT analysis.

•

List and describe the five main PVT tests for oils systems and their application.

•

Be able to use an oil PVT analysis report to determine:

•

The gas-oil ratio and oil formation volume factor at the bubble point pressure.

•

Determine the bubble point pressure from a set of P vs. V relative volume test data.

•

Calculate oil formation volume factors above the bubble point.

•

Determine the total formation volume factors above and below the bubble point.

•

Determine the oil formation volume factors and gas-oil ratios for pressures below the bubble point pressure.

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1

SCOPE

Reservoir fluid analysis provides some of the key data for the petroleum engineer. The quality of the testing, therefore, is important to ensure realistic physical property values are used in the various design procedures. As important is the quality of the samples collected to ensure that the fluids tested are representative of the field. Clearly, any subsequent high quality testing is of little value if the sample is not representative. PVT-analysis of a reservoir fluid comprises the determination of: a The correlation between pressure and volume at reservoir temperature; b

Various physical constants that enter into reservoir engineering calculations, such as viscosity, density, compressibility etc;

c

The effect of separator pressure and temperature on oil formation volume factor, gas/oil ratio etc;

d

The chemical composition of the most volatile components.

The physical properties measured depend on the nature of the fluid under evaluation. For a dry gas, the key parameters are, the composition, the specific gravity, the gas formation volume factor, compressibility factor, z , and viscosity as a function of pressure. The isothermal gas compressibility can be determined from the z value with pressure variation. For a wet gas, then all of the above parameters for a gas are required. However there are some variations because of the production of liquids with gas. The formation volume factor used is the gas condensate formation volume factor, Bgc, which is the reservoir volumes of gas required to produce one stock tank volume (barrel or m3) of condensate. The composition, specific gravity and molecular weight of the produced condensates and produced gas are required. The composition and apparent specific gravity of the reservoir gas are obtained by recombining the values for the gas and condensates.

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For an oil system, the following information is required; the bubble point pressure at reservoir temperature, the composition of the reservoir and produced fluids, the formation volume factor, the solution gas to oil ratio, the total formation volume factor, and viscosity, all as a function of pressure. The coefficient of isothermal compressibility of oil. The impact of separation on the above properties. The impact of operating below the bubble point on the formation volume factor and solution GOR. For a gas condensate system, properties measured reflect those for both wet gas and oil analysis. The related property to the saturation pressure for the oil, the bubble point pressure, for a gas condensate is the dew point pressure. Above the dew point the compressibility characteristics of the gas are required, and the impact of allowing the reservoir to drop below the dew point is another important evaluation.

2

SAMPLING

The value to be attached to the laboratory determinations depends on whether the sample investigated is representative of the reservoir contents or at least of the drainage area of the well sampled. Therefore, the composition of the fluid entering the well and of the samples taken should be identical with that of the fluid at all other points in the drainage area. It is therefore desirable to take samples early in the life of the reservoir in order to minimise errors caused by differences in relative movement of oil and gas after solution gas has been liberated. The taking of samples can be accomplished either by sub-surface sampling or by surface sampling. There have been considerable advances in recent years in this area aimed at taking samples under down hole pressure conditions. A brief survey of these methods is given below, for detailed descriptions of the techniques, which is outside the scope of this section, service companies providing sample collection service should be consulted.

2.1 Subsurface Sampling In this case a subsurface sampler is lowered into the well and kept opposite the producing layer for a sufficiently long time, Figure 1. Many types of bottom hole samplers have been devised and described in the literature. Subsurface samples can only be representative of the reservoir contents when the pressure at the point of sampling is above or equal to the saturation pressure. If this condition is not fulfilled, one should take a surface sample. Even at pressures close to the saturation pressure there is a serious possibility of sample integrity being lost as a result of the system going two phase during transfer to the sample chamber.

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A simple test can be carried out in the field at the well site to find out whether a reliable sample has been obtained. In this test the sample cylinder is pressurised either by using a piston cell or using mercury as the displacing fluid. Whereas mercury was the most common fluid to be used as a pressure transfer and volume change fluid, because of toxic and other concerns its use is diminishing. From the relation between injection pressure and volume of mercury injected, the following properties are derived: 1 The pressure existing within the sampler when it is received at the surface; 2 The compressibility of the material within the sampler; 3 The bubble-point pressure of the contents of the sampler. If two or three samples taken at short time intervals show the same measured properties, it is highly probable that good samples have been obtained. In recent years there have been considerable advances on downhole fluid sampling. The contents of the sampler are transferred by means of a mercury pump into a high pressure shipping container. Two fillings of a sampler of 600 ml capacity are usually sufficient for a complete PVT analysis. In recent years mercury has been replaced using the application of piston cells by some companies. Subsurface sampling is generally not recommended for gas-condensate reservoirs nor for oil reservoirs containing substantial quantities of water.

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Figure 1 Subsurface Sampling.

2.2

Surface Sampling

A sample of oil and gas is taken from the separator connected with the well (Figures 2 - 5 give sketches of vertical and horizontal separators and the arrangement for collecting different fluid samples). The surface oil and gas samples are recombined in the laboratory on the basis of the producing GOR. Particular care, therefore, must be exercised in the field to obtain reliable samples and accurate measurement of the GOR and separator conditions. In the case of two or three stage separation the samples are taken from the high pressure separator.

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Figure 2 Vertical and Horizontal Separators.

Figure 3 Separator Gas Sampling.

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Figure 4 Separator Liquid Sampling by Gas Displacement.

Figure 5 Separator Liquid Sampling by Water Displacement.

3 3.1

SAMPLING WET GAS AND GAS CONDENSATE SYSTEMS Introduction

The use and value of any PVT study or other analysis of a reservoir fluid is dependant on the quality of the sample collected from the reservoir. Many PVT reports show a variation of results from fluids from the same well. Sampling wet gas and gas condensate fluids can give rise to errors. During the sampling procedure it is often possible to alter the conditions such that the fluids sample are not representative of those within the reservoir, the characteristics of which are being assessed during the PVT report. It is important, therefore, in sampling reservoir fluids to ensure that the conditions during which the samples are being taken are not altered to give rise to false samples.

3.2

Phase Behaviour

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The behaviour characteristics of fluids are uniquely described by a phase diagram, Figure 6. The hydrocarbon mixture with its own unique composition will have its own phase diagram and phase envelope. Within the phase envelope the system is in two phases, whereas outside the phase envelope a single phase exists.

Figure 6 Phase diagram for a reservoir fluid.

At a particular point within the envelope the composition of each component in each phase is constant. The separation of oil and gas as predicted by the phase diagram results in each phase itself having a phase diagram. These phase diagrams intersect at the separation temperature and pressure, the oil will exist at its bubble point and the gas at its dew-point. Therefore, for example, in the separator, gas will be separated and produce gas at its dew-point and the oil separated will be at its bubble point.

Figure 7 Conditions in a separator.

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For a given system, therefore, a change in the temperature or pressure within the phase envelope will result in alteration of the system and therefore alteration in the characteristics of the two phases produced. The behaviour just described, therefore, will have implications on the way samples are taken and on the techniques used to collect the sample, for example, from the separator.

3.3

Sampling Gas And Gas Condensate Reservoirs

The potential locations for sampling these reservoirs are shown in Figure 8; samples could possibly be taken in the reservoir, at the bottom of the well, at the wellhead or in the separator. The respective advantages and disadvantages are given in Table 1. The reservoir would be the most ideal sampling point but clearly this is impossible.

Figure 8 Locations for sampling.

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Table 1 Merits and disadvantages of sampling locations.

The bottom hole sample where the fluid is in single phase, probably, would be an ideal situation. Present technology, however, is such that it is often difficult to produce a single phase sample representative of the fluids bottom hole. The necessary pressure drop to get the fluid from the bottom hole into the sample container can often give rise to a two phase situation and an unrepresentative collection of these two phases. Bottom hole samples are also more costly to collect. The wellhead from a cost point of view could be the most suitable location point, however, again the question of the representative nature of the sample is a concern. As a result of the lower pressure and temperature it is likely that the single phase fluid at the bottom of the well has gone into the two phase region at the wellhead and therefore the relative proportions of liquid to gas would be unknown and their sampling would be difficult to produce a representative sample. The most common sample location is the separator. Considerable care, however, has to be taken to ensure that the samples taken from the separator are those representative of the reservoir from which the fluids derive. The well behaviour can significantly influence the nature and characteristics of the fluids which eventually arrive from the separator. For example, Figure 9 in a flowing well, gas condensate entering the wellbore as it travels to the surface will experience a drop in pressure likely to give rise to retrograde liquid behaviour in the wellbore. The flow must be sufficient to lift this uniform liquid and gas fluid to the surface. If the flow is slow it is possible that some liquid may fall back therefore altering the overall composition moving up the wellbore.

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Figure 9 Flowing well.

If the reservoir is shut in after flow then considerable changes can rise in the composition of the fluid in the wellbore. The reservoir gas flowing into the wellbore sets up a new equilibrium with condensed retrograde liquid which has rained down within the wellbore. This separation in the wellbore gives rise to a lean gas near the top of the well with a more than rich mixture at the bottom of the well. Figure 10.

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Figure 10 Shut well in after flow

When the well is flowing after a shut-in period, Figure 11 and sampling takes place there will be a variation in the compositions produced at the surface and therefore unrepresentative samples collected from the separator. For example, in the early period after shut-in the lean gas at the top of the well enters the separator producing a fluid with a GOR higher than that representative of the reservoir. As the fluids at the bottom of the well move to the surface, much richer as a result of the liquid having collected at the bottom of the well, they are produced with a GOR lower than that of the representative reservoir fluid. It is important, therefore, for the well to be flowed for a sufficient period for all the fluid within the well to have been displaced and also fluid in the near wellbore region which also could have been influenced by the pressure and compositional changes experienced during the shut-in period.

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Figure 11 Well flowed after shut in period

In assessing whether good samples have been taken it is important to know how long it will take for unrepresentative samples to be displaced from the separator, the wellbore and the near wellbore reservoir zone. For example, for a 12ft x 5ft diameter separator with a liquid flowrate of the order of 200 barrels per day, it could take 1 hour to displace an 8.4bbls. If a contaminant enters a separator then the time to reach 1% of the original concentration could be over 4.5 hours. For example, if the tubing is 4 inches in diameter with a length of 9000ft and an average tubing pressure of 5000psi and a temperature of 170°F and a gas flowrate of 5MM/scf/day, the volume in the tubing would be 0.23MM standard cubic feet and the time to displace this gas would be just over 1 hour.

3.4

Separator Sampling Points

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A very practical aspect is often ignored in the design of separators and in particular in relation to the sampling points associated with them. These sampling points are often located primarily in relation to accessibility rather than the representative nature of the fluids which can be extracted from them. For example, in the gas line, Figure 12a the sampling valve might be located on the lower portion of the valve. It is likely that entrained liquids in the gas stream could collect at this point giving rise to a very rich gas composition if these entrained liquids were collected in the sample containers. Similarly sampling valves in the liquid line Figure 12b could be located on the upper portion of the line any gas which is carried through the line again will collect in the dead volume of the sampling valve such that when samples are taken the gas will enter the gas bottle of the container giving rise to an unrepresentative sample. An alternative would be to locate both of these sampling valves on the side of the pipe rather than at the top or bottom of the line. Figure 12c.

Figure 12 Location of sampling points

3.5

Sample Details

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In taking samples there are some obvious aspects which are often ignored. Clearly details in relation to the sample should be taken in particular: the date and the time of a sample, the identification of the cylinder into which the sample is to be collected, the location of where the sample was taken, the temperature and pressure at which the sample was taken, details of any other aspects which will be important for those subsequently handling the sample, for example, the presence of any H2S etc. The details of the sample including, for example, the gas to oil ratio during the separation will be transmitted with a sample to the laboratory which will carry out the analysis. For example, if the sample identification cylinder has not been noted then if sample details become separated from the sample cylinder then the sample would be wasted. Prior to any liquid samples being transported to the laboratory it is important to reduce the pressure within the container to a value below the bubble point to ensure that a two phase mixture is transported. Very high pressures can occur as a result of a temperature rise on a single phase liquid sample. Such pressures could go over safe working pressures of the vessel!

4

APPARATUS

The apparatus required for PVT analysis consists of: a apparatus for the transfer and the recombination of separator oil and gas samples; b apparatus for measuring gas-volumes and for performing separator tests; c the PV cell and displacement pumps and dispensing cell; d high pressure viscometer; e gas chromatograph. Diagrams of the layouts for an oil system is shown in Figure 13.

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Figure 13 Equipment for the Study of Subsurface Samples

5

RECOMBINATION OF THE SURFACE OIL AND GAS SAMPLES

The GOR given by the field refers to the separator tank gas-oil ratio. In order to recombine the separator gas and the separator oil in the correct ratio, the volumetric ratio between stock tank oil and separator oil is determined in the laboratory 85% of the cylinder containing separator oil is occupied by oil. There is a gas cap on top. This precaution is taken in view of the long transport time and the risk of great fluctuations in temperature involved. After the gas cap has been dissolved by pressing water into the cylinder at a pressure higher than the separator pressure a given quantity of oil is then flashed at pressure and room temperature through a separator operating under the same conditions as the stock tank in the field i.e. at the same pressure and temperature. The collected stock tank oil is weighed and the density determined, after which the volume of oil is known. The volume and the density of the liberated gas (stock tank gas) are also determined. From the above measurements, the stock tank gas/stock tank oil ratio is also known. If the volume of stock tank oil is lower than the corresponding quantity of separator oil we speak of shrinkage, in the opposite case of expansion; shrinkage occurs when a large quantity of gas is produced and expansion occurs when a small quantity of gas is dissolved in the separator oil. When the shrinkage or the expansion is known, recombination can take place on a stock tank-oil basis.

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6

PVT TESTS

At this stage we should remind ourselves of the main applications of the PVT data. The three main application areas are; • To provide data for reservoir calculations, • To provide physical property data for well flow calculations • For surface facility design. Although all are cited as users of the data, the reservoir calculation requirement has provided the main driving force for the tests to be carried out. In surface facility design for example the more simplistic black oil approach around which the PVT analysis is structured is considered too limiting, and the main data for this application is the compositional analysis of the fluids. Over recent years, as the data is subsequently applied to computer based simulation tools, the ability to handle more complex descriptions of the fluids has led to more extended compositional analysis, beyond the C7+ limit which was the basis for many years. It is common practice in some PVT laboratories to measure co-position to C28 and then define a C29+ component In reservoir calculations the PVT tests and subsequent report provides the source of the reservoir engineering properties necessary to describe the behaviour of the reservoir over its development and production. The tests conducted therefore have to take into consideration the processes going on both above and below the saturation pressure. There are five main PVT tests for oil systems plus associated compositional analysis: i The flash vaporisation or relative volume tests. ii The differential test. iii The separator tests. iv Viscosity measurements v Compositional measurements. A simple layout of a PVT facility is given in Figure 14.

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Figure 14 Simple Schematic of PVT Facility for Oil Tests.

6.1

Flash Vaporisation (Relative Volume Test)

By flash vaporisation is meant the determination of the correlation between pressure and volume of a reservoir liquid at constant temperature (reservoir temperature) from high pressure to the lowest possible pressure. The gas liberated below the point of saturation remaining in equilibrium with the oil throughout the experiment. That is, the system remains constant (Figure 15). The vaporisation process occurring in the reservoir cannot be duplicated in the laboratory, since in the reservoir below the bubble point the system does not remain constant as the increased mobility of the gas causes it to move away from its associated oil. The flash vaporisation test gives the relationship between P & V of a reservoir liquid at constant (reservoir) temperature. Liberated gas remains in equilibrium with the oil.

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Figure 15 Flash Vapourisation. Determination of the relationship between P & V of a reservoir liquid at constant (reservoir) temperature. Liberated gas remains in equilibrium with the oil.

By plotting the volume of the system versus pressure a break is obtained in the slope. This occurs at the Bubble Point pressure. To carry out a relative volume test run, the PV cell is set up as in Figure 16.

Figure 16 PVT Set Up for Flash Vapourisation /Relative Volume Test

The PVT cell is filled with a certain quantity of reservoir liquid at a pressure above the estimated bubble point and room temperature. After the PV cell has been filled (about 90 ml), it is immersed in a temperature bath and heated to reservoir temperature.

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During heating, it is necessary to maintain pressure by increasing the content volume of the PV cell. When the pressure remains constant, the temperature in the cell is equal to that of the bath. The thermal expansion factor () can then be calculated from the volume withdrawn. It is equal to:

where: V2 = volume of the oil at reservoir temperature T2 V1 = volume of the oil at room temperature T1 The thermal expansion factor is expressed, for example, in ˚C-1. The pump reading taken at the moment when the pressure became constant is the first reading for the PV curve. The pressure is now reduced by gradually withdrawing small quantities of transfer fluid from the PVT cell and after each withdrawal equilibrium is established by shaking the cell. After each equilibration the pressure and the volume are read. By plotting the pressures against the volumes a curve is obtained showing a break at the bubble point (Pb). Figure 16. The saturation pressure or bubble point pressure is that pressure below which gas is liberated. Hence, a two-phase system is formed, whereas above the bubble point pressure a one phase system is present (undersaturated liquid). The compressibility of the oil phase above the bubble point can now be calculated from the graph.

c=

V2 - V1 V2 ( P2 - P1 )

where: V2 = volume at pressure P2 V1 = volume at pressure P1 The compressibility is expressed in reciprocal atmospheres, psi, etc. After the reservoir liquid has expanded to its maximum volume (dependent on the capacity of the PV cell), the gas cap is removed at constant pressure (lowest possible flash expansion pressure). Volume, density, gas expansion and gas compressibility factor of the liberated gas are then determined successively.

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The main objectives of the flash vaporisation test are to provide the reservoir bubble point pressure and together with the information from the separator test, the formation volume factors above the bubble point.

6.2

Differential Vaporisation

When the reservoir pressure falls below the bubble point the process of gas liquid separation in the reservoir is one of a constant changing system. A PVT process has been designed in an attempt to provide a means of in part simulating the changing systems as separation occurs within the reservoir below the bubble point. The differential vaporisation differs from the flash in that the liberated gas is removed from the cell stepwise. At each step below the bubble point the quantity of gas, oil volume, density, gas expansion and gas compressibility are determined. The objectives of the differential test therefore are to generate PVT data for conditions below the bubble point. Figure 17 below indicates the differential process.

Figure 17 Differential Vaporisation. In differential vaporisation. At each successive pressure drop liberated gas is removed.

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The bubble point Pb is the starting pressure for the differential test. The next step is to reduce the pressure in the PV cell by expansion of the PV cell volume. The reduction of pressure causes the system to go two phase. All the gas phase is removed at constant pressure by reducing the cell volume as gas is withdrawn. The volume of the remaining oil is then determined. The cell pressure is then again dropped by expansion of the PV cell and the above process repeated until the cell pressure has been dropped to atmospheric pressure. The pressure steps for the tests cover a range of around 8-10 steps. All the above steps have taken place at reservoir temperature. The final stage is to reduce the cell to 60°F keeping the pressure at atmospheric pressure. The final oil volume is measured. This remaining all is termed residual oil to distinguish it from stock tank oil which although at the same pressure and temperature conditions has got there by a different process. The cumulative weight of the amounts of gas withdrawn are used in the calculation of the densities of the oil phase in the differential vaporisation process. These densities can also be determined directly if a pressure pycnometer is available. Differential liberation is considered to be representative of the gas-liquid separation process in the reservoir below the bubble point pressure. Flash liberation is considered to take place between the reservoir and through the separator. Differential liberation tests are carried out therefore to obtain oil formation volume factors and GOR’s that can be used to predict the behaviour of a reservoir when the pressure has dropped below the bubble point pressure.

6.3

Separator Tests

The object of these tests are to examine the influence of separator pressure and temperature on formation volume factor, gas/oil ratio, gas density and tank-oil density. These tests are not driven by those who will be responsible for the optimised separation process if the field ultimately is developed. They are carried out to give an indication of the oil shrinkage and GOR which occurs when the fluids are produced to surface conditions. It should be emphasised at this stage that there is not a unique value for the formation volume factor and solution gas-oil ratio. It depends on the stages and conditions of separation through which the fluids pass. With the equipment available, a single test or a multiple separation test can be carried out.

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The system is set up as shown in the schematic below, Figure 18.

Figure 18 Schematic of a Two Stage Separator Test

The procedure for the separation test is as follows. The starting point is oil in the PVT cell at its reservoir bubble point, ie. the same starting condition as the differential test. Fluid is displaced from the PVT cell ensuring that the PVT cell contents remain at bubble point pressure. The gas and liquids are collected from the separation stage(s) and their respective properties measured. The final stage is at stock tank conditions. The resulting fluid is termed stock tank oil. A single separator test is carried out by flashing reservoir liquid at bubble point pressure and reservoir temperature through the separator operating at the average annual temperature and at pressures which may be expected in the field. The difference in results when using a single or a double separator is that in the former case the total gas/oil ratio is higher, the shrinkage is greater and the density of the tank oil is higher than in the latter case. The main objectives of the separator test are in combination with the flash vaporisation and differential tests to provide formation volume factor and solution gas-oil ratios over a full pressure range above and below the bubble point. In quoting these values it is important to recognise that the values are separator condition specific. In the sketch Figure 18 oil formation volume factor Bob is equal to Vres/L2 reservoir volumes/stock tank volumes. The solution to gas-oil ratio, Rsb, is equal to (V1 + V2)/L2 Standard cubic volume (SCF or SM3)/Stock Tank volumes (STB or STM3).

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6.4

Viscosity

The viscosity is measured at reservoir temperature and at different pressures both above and below saturation pressure. It is important for viscosity measurements below the bubble point to generate the fluid for study by a differential mode to simulate the nature of the fluid that would exist at these conditions.

Viscosity measurements were largely carried out with a “rolling ball” high pressure viscometer consisting of a highly polished-steel capillary of 1/4" dia. which can be closed at the top by means of a plunger and is provided at the bottom with a contact which is connected with an amplifier. A steel or platinum ball rolls in the capillary, its diameter hence slightly smaller than that of the capillary. When the ball reaches the bottom, it makes contact between the wall of the capillary and the point of contact, as a result of which a circuit is closed and a whistling sound is heard. The time of rolling is a measure of the viscosity. In recent years the pressure drop along a capillary tube of known length and internal dimensions has been used. The viscosity being calculated using the Poiselle equation, the laminar flow pressure drop equation for a pipe of a particular diameter and length. Although being used it is clearly restricted by operating under a fixed flow regime, laminar and velocities to ensure that there is a sufficient pressure drop to measure and not too large to influence physical properties.

6.5

Hydrocarbon Analysis

A hydrocarbon analysis is made of the methane to an upper paraffin fraction of the recombined surface (or subsurface) sample. Historically this upper limit was C 6 and the remainder lumped as a C7+ fraction. Higher C numbers are now used as analytical methods enable even higher levels of characterisation. The plus component having been separated in a distillation column needs to be characterised by its specific gravity and its apparent molecular weight. The latter is achieved using a depression of freezing point method. The different components are determined by means of gas/liquid chromatography. If the description is based solely on paraffins then non paraffinic components are added to the next higher paraffin. The value of a higher characterisation is particularly helpful to process engineering considerations, say up to C30 because at lower temperatures long chain hydrocarbons will form a solid phase (such as wax) and adhere to surfaces. At reservoir conditions they are in the liquid phase and therefore do not effect the reservoir flow process.

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In some cases the fluids produced in the final separation stage are identified to a higher C number together with aromatic and napha components. Compositions are also made of the produced gases from the various tests.

7

WAX AND ASPHALTENES

The formation of solid deposits during oil production is a concern. Some heavy hydrocarbons as mentioned above at low temperatures can form solid phases waxes and in transfer lines and process facilities. The wax formation temperature is therefore an important measurement. Ashphaltene, is another solid phase of concern. Asphaltenes are large molecules largely of hydrogen and carbon with sulphur, oxygen or nitrogen atoms. Asphaltenes do not dissolve in oil but are dispersed as colloids in the fluid.

7.1

Wax Crystallization Temperature

Different techniques can be used for this. In the following procedure a sample of separator oil is transferred to a vessel and then pre filtered by passing through a 0.5 micron filter. The crystallisation temperature (WCT) is measured by carrying out a series of flow experiments where the oil is flowed through a fine filter (15 microns) across which the pressure differential is measured. Prior to reaching the filter the oil is flowed through a temperature equilibrium coil at the same temperature as the filter at a temperature of 140°F. The temperature of the bath is gradually lowered and the pressure difference measured. A sudden increased in pressure indicates the onset of wax crystals building up on the filter giving an indication of the WCT. From this rough indication constant temperature flows are taken at temperatures just above and below the indicated WCT to give a more precise value. A plot of differential pressure vs. flow indicates a more precise value of the WCT. Figures 19 & 20 below from a PVT report provided by Core Laboratories (UK) Ltd gives the plots used to determine the WCT for a separator oil sample. The appearance temperature is considered to be affected by super cooling, whereas the disappearing temperature is considered to be the equilibrium value. There is a concern that the appearance temperature may not be so accurate as would be the case for the disappearance temperature.

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Figure 19 Graph of differential pressure v temperature during constant cooling. (Core laboratories)

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Figure 20 Graph of Cumulative Volume Flow Across Filter v Differential Pressure. (Core laboratories)

8

SUMMARY OF RESULTS PROVIDED BY AN OIL SAMPLE PVT TEST

The PVT analysis as described above furnishes the following data: 1 Saturation pressure. ie. bubble point. 2

Compressibility coefficient

c =-

3

1 dV (specify unit of pressure change) V dP

Coefficient of thermal expansion

(specify unit of temperature change)

4

Relative total volume of oil and gas: vt

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5

Cumulative relative gas volume: vg

6

Relative oil volume: vo

7

Gas formation or gas expansion factor: Bg or E, i.e. volume occupied under sc (standard conditions) by that amount of gas which occupies unit volume at reservoir temperature and pressure.

8

Gas compressibility factor: z defined by Pv = zn RT.

9

Specific gravity of gases (air = 1)

10 Liquid density:  11 Viscosity of the liquid phase at different pressures:  12 Oil Formation Volume Factor or Shrinkage factor C, where C = ratio of the volume of tank oil produced under specified separator conditions and then measured under sc to the volume occupied by that quantity of oil and its dissolved gas under reservoir conditions. depends on separator conditions. 13 Solution gas to Oil Ratio or Gas-solubility factor D = that volume of gas liberated under separator conditions and measured under sc which is held in solution at reservoir temperature and any particular pressure by that quantity of oil which will occupy unit volume when produced; depends on separator conditions. 14 Shrinkage of separator oil to tank oil. 15 Tank-gas/tank-oil ratio. 16 Hydrocarbon analysis of the reservoir and produced fluids. Figure 21 below illustrates the volume relationship of fluids in an oil PVT tests to the black oil description of volumes in a reservoir. In PVT analysis the basis of reference is the bubble point, Figure 21a whereas for the black oil system the reference state is surface or stock tank conditions, Figure 21c. The relationships between the two are given in Figure 21b.

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Figure 21 Nomenclature PVT SYMBOLS

a b c

9

All volumes relative to oil volume at bubble point Same as (a) using black oil notations All volumes relative to oil volume at standard surface conditions

UNDERSTANDING PVT REPORTS

Having considered the various aspects of PVT analysis we will now consider the PVT report and examine how we can generate the various reservoir engineering parameters of interest. We will remind ourselves of the reason for the report and then using a PVT report go through the main tests and interpret the detail.

10

PURPOSE OF THE PVT REPORT

Although the PVT report can be a source of information for a variety of applications from reservoir, through well to surface facility calculations, the reservoir engineering application has provided the main basis and structure of the report. The report is structured to provide the much of the black oil model information together with limited compositional data. The material balance equation which is covered in a latter chapter also provides a basis for the PVT report. It is the PVT report which is the source of much of the data embodied in the material balance equation. Thus, some of the tabular information is set up to satisfy that need.

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The PVT report can be used for a range of purposes from its use in determining the potential prospects of a hydrocarbon accumulation to history matching a reservoir which has been on production for some time. The report should therefore cover all past, present and future situations which may require calculations. To do this with a minimum of tables and curves, the data are normalised to a reference state and only data for the reference state given. In PVT data reporting as indicated in earlier the reference state is the bubble point. The petroleum engineer must then “work back” from the reference state to the particular situation. As described previously, the laboratory tests are carried out in an attempt to simulate the processes which take place in a reservoir and through the production system. These will include the flash equilibrium separation of gas and oil in the surface traps during production and for an oil below the bubble point the differential equilibrium separation of gas and oil in the reservoir during pressure decline. In interpreting the data the engineer needs to use both sets of data to provide the information for reservoir calculations. The PVT report is clearly specific to a particular fluid, collected from a specific well under specific conditions. This sample may not be representative of the total field system and therefore using subsequent reports it may be necessary to to adjust the data for field application. In a PVT report therefore detail is given as to the manner of obtaining the sample and the conditions that existed at the sampling time. Also, the compositional analysis of the sample is given so that equilibrium calculations can be made for conditions other than studied in the laboratory. In this area there has been considerable progress in compositional analysis and although the report used in this section only goes up to C7+ it is now common practise to characterise to higher C numbers, even as high as C29. We will now examine a PVT report for an oil sample and is commonly used in textbooks to illustrate the interpretation of a PVT report. The report is provided by Core Laboratories. Ltd.

11

OIL SAMPLE PVT STUDY

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The first report is for an undersaturated oil from Texas field attributed to the Good Oil Company. The report is given at the end of this chapter. Although not covered in this report there may be a number of data sheets reporting the validation of the samples used and selected in the PVT report. These sheets would include the various gas to oil ratios when taking the samples and other information as part of the sample validation. A text description is usually given to the report describing the various tests conducted and the principal observations. The source of these principal observations and calculations will be covered as in the following sections. The pages of the report often include processed data. To determine black oil parameters of oil formation volume factor and gas to oil ratios as a function of pressure, a combination of tables of the report is required. We will first look at the separator tests the results of which provide the basis for the Bo and GOR values.

11.1

Separator Tests of Reservoir Fluids

In the separator test, page 7 of 15 of the report, oil at reservoir temperature and the bubble point pressure in a PVT cell has been carefully displaced from the cell through a series of pressure and temperature steps. The tests show what quantity of surface gases and stock tank oil results when one barrel of bubble point oil is flashed through a certain surface trap sequence. The tabulation also gives the ˚API gravity of the stock tank oil and, in some instances, the gravity of gas coming from the primary trap. There are four separate tests reported. The first where the first separation is at 50 psig and 75°F and the tank separation at 0 psig and 75°F, the other tests where the first stage conditions are at 100 psig, 200 psig and 300 psig. The temperatures and conditions for the final stage are the same for each test. Column 1 and 2 give the pressure-temperature conditions of the surface trap tests that were investigated. These should be specified by the reservoir engineer at the time the test is planned. A difficulty here is that the engineer specifying to the PVT service company these separation conditions is unlikely to be involved in the ultimate optimised surface separation conditions if the field ultimately is developed.

EXERCISE PVT 1. What is the solution gas-oil ratio and formation volume factor resulting from the separator test, at first stage of 300 psig and 2nd stage 0 psig and both at 75oF?

SOLUTION

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This exercise illustrate the results using one of these tests, a two-stage separation; a primary trap operating at 300 psig and 75˚F followed by a stock tank operating at 14.7 psia (0 psig) and 75˚F. When one barrel of bubble point oil; defined as oil saturated at 2620 psig and 220˚F in footnote 3, on page 7 of 15 is flashed (processed) through this separation arrangement, the stock tank has a quality of 40.1˚API (column 5). The formation volume factor of the bubble point oil, Bob = 1.495 B/BSTO (column 6). This would have been the volume of oil displaced from the PVT cell divided by the volume collected at the final stage and then corrected for the thermal reduction from 75˚F to 60˚F. The source of this bubble point pressure value will be indicated later. Columns 3 and 4 show the surface gas-oil ratio from the first stage and the tank. The first stage ratio of 549 ft3/BSTO ( column 4) and the tank stage gas amount to 246 ft3/BSTO. It is important to read the footnotes of the report. Column 3 gives the results in relation to the volumes at indicated P & T whereas column 4 gives the volumes with respect to stock tank conditions of 14.65 psia and 60˚F. The solution gas-oil ratio at bubble point conditions (2620 psig and 220˚F), is therefore Rsb is 549 + 246 = 795 ft3/ BSTO when flashed through this surface trap arrangement. If we compare these results for the 50psig, 0psig arrangement we obtain a Bob of 1.481 B/BSTO and a solution GOR of 778 ft3/ BSTO. Clearly therefore Rsb, Bob, ˚API all vary with the separation pressure-temperature situation. There is not one unique result. When reporting Bo and GOR data for a reservoir therefore it is important to report that these are for a specific separation route or averaged for a series of tests. The latter is not so useful since it is not so straightforward to calculate the result for a different separation route using VLE methods. The results from the separation test are based on the bubble point condition and to obtain volumetric information at other pressures we require the results from other tests.

11.2

Fluid Properties at Pressures Above The Bubble Point Pressure

The source of information for calculations for conditions above the bubble point is a combination of two tests, the flash vaporisation test (relative volume test ) and the separation tests. The results of the flash test are presented in page 4 of 15, titled Pressure -Volume Relations at 220˚F.

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Remember in this test, the contents of the cell at reservoir temperature have been expanded and the volumes measured. None of the contents has been removed, the system has remained constant. In the style presented here the expansion of the fluid as measured has been plotted and then the intersection of the two slopes of the liquid phase expansion and the two phase, gas/liquid phases, has been interpreted as the bubble point pressure and bubble point volume. All the volumes have then been normalised to this bubble point condition and presented as a relative volume. (column 2). The first and second columns of the Reservoir Pressure Volume Relations Data on page 4 give the pressure volume relations of the original fluid at 220˚F. Note that the data are presented in terms of a unit volume at the bubble point condition. This flash vaporisation test gives us therefore the reservoir temperature bubble point pressure, which in this case is 2620psig. i,e the point where the relative volume is 1.0. Column 2 gives the volume of the system at pressure per unit system volume at 2620 psig and 220˚F. These are listed as relative volumes, relative to the bubble point. Column 3 presents what is called the Y function, this function should provide a straight line or a slight curve and can be used to pick out anomalous data. We will now see how we can use the relative volume data to provide us with some formation volume factors above the bubble point.

EXERCISE PVT 2. What is the formation volume factor and the density of the oil at the last reservoir pressure measured.

SOLUTION The well characteristics give the last reservoir pressure as 3954 psig. @ 8500 ft: We obtain the oil formation volume at 3954 psig by multiplying the formation volume factor at the bubble point by the relative volume (to the bubble point). Why multiply? Because:

Bo =

vol reservoir oil vol bubble point oil vol reservoir oil = ´ vol stock tank oil vol stock tank oil vol bubble point oil

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and the reference bubble point oil volume cancels out. Therefore Boi, the initial formation volume factor is 1.495 x 0.9778 = 1.4618 when the 300 psig primary trap is involved. It is a different value if another separation pressure is used. The 0.9778 was obtained by interpolation between 3500 and 4000 psig in column 2. Reservoir oil density at pressures greater than 2620 psig also make use of the relative volume data of column 2, page 4. The added information we have is the density of the bubble point oil. This is given in the summary data on page 3 of the report. We see here that the specific volume at the bubble point, vb = 0.02441 ft3/lb. This comes from direct weight-volume measurements on the sample in the PVT cell. We can now calculate the density, oi, of the initial reservoir oil as:

The compressibility of the oil above the bubble point can also be obtained from the relative volume test. The definition of compressibility is:

It makes no difference whether the volume units in the equation are relative volumes to the bubble point, formation volumes, or specific volume values. To evaluate CO at pressure p it is only necessary to graphically differentiate the p-vrel data in columns 1

¶v and 2 to get ¶p at the pressure and divide by vrel. A less accurate value can be obtained by the assumption:

For example, to get Co at 4500 psig using relative volume values of 500 psi on each side of 4500 psig:

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co =

1 (0.9639 - 0.9771) 0.9639 + 0.9771 (5000 - 4000 ) 2

co =

1 0.0219 = 13.6(10 -6 )vol / vol / psi 0.9639 1000

The report also lists some compressibility numbers on page 3. These are not the same as indicated above because they are changes in volume (in the pressure interval indicated) per unit volume at the higher pressure. For example, the value of 13.48 (106 ) for the 5000 and 4000 psi interval is obtained as:

-

1 ( 0.9771 - 0.9639) 0.9639 (5000 - 4000)

The compressibility data on page 2 are set up in this manner because of the way they are used in one form of the material balance.

11.3 Total Formation Volume of Original Oil Below The Bubble Point Pressure. In the liquid properties chapter we introduced the total formation volume factor, Bt. This factor is of little practical significance since it describes the volume of an oil and its associated gas both above and below the bubble point, when the system does not change. In reality below the saturation pressure the system changes as gas and oil have different mobilities. In some forms of the material balance equation Bt is used however to express oil volumes. We have just seen that to calculate the formation volume factor of the oil above the bubble point we multiply the bubble point formation volume by the relative volume given in column 2, page 4. If we multiply Bob by vrel at pressures less than pb, we also get a formation volume factor, the total formation volume B t, of the original system. That is at p < pb we will have two phases and the Bt is the volume in relation to both gas and liquid phases in equilibrium at pressure p. One form of the material balance equation makes use of the expansion of the original oil between the initial system pressure and any subsequent pressure. This expansion is given by the term: Eo = N(Bt - Boi)

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where N is the initial stock tank barrels in the reservoir and (Bt - Boi) is the expansion per unit stock tank oil. Eo is, therefore, the expansion (bbl) of the original oil system. Sometimes we see the expansion equation written: Eo = N(Bt - Bti) Figure 28 below illustrates the change of Bt and Bo with pressure over the total pressure range.

Figure 28 Shape of Total Formation Volume Factor Bt and Oil Formation Volume Factor Bo

Above pb it makes no difference whether we consider the formation volume to be a total formation volume or an oil formation volume.

11.4

Differential Liberation Tests.

Previously we have considered what happens when reservoir fluid comes to the surface and is separated into surface gas and oil products. In determining what happens in the separator test we consider it as flash equilibrium conditions because we believe that the action going on in the separator is essentially one in which the whole system entering the trap immediately separates into two components – separator gas and liquid. This constitutes the elements of a flash separation. However as we discussed in section 6 in the reservoir the separation below the bubble point is different and the differential test has been devised to enable calculation of the appropriate volume factors and GOR’s.

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The standard PVT report includes data referred to as “the differential data”. These are gas solubility and phase volume data taken in a manner to model what some people believe happens to the oil phase in the reservoir during pressure decline. Basically, the argument that differential liberation tests model the subsurface behaviour comes primarily from two things: 1 the reservoir pressure changes are not violent and large as are the pressure changes in entering surface separators. The subsurface changes are more gradual and might be considered to be a series of infinitesimal changes. 2

because of the relative permeability characteristics of reservoir rock-fluid systems, the gas phase moves toward the well at a faster rate than the liquid phase. Of consequence, the overall composition of the entire reservoir system is changing. These two ideas promote the idea that a test procedure modelled on a differential process should be used to study subsurface behaviour. Because of experimental limitations and time-cost considerations the laboratory cannot perform a true differential procedure. Instead, they perform a series, usually about ten, of stepwise flashes at the reservoir temperature, commencing at the bubble point. Of course, the greater the number of steps, the closer the true differential process is modelled.

The differential data are reported on page 5. Note that the table is headed by the title “Differential Liberation at 220˚F”. Probably the best way to understand these data is to explain again the manner of obtaining the values. To begin with, the laboratory starts with a known volume of the original system in the PVT cell. This may be of the order of 100-200cm3. The volume at the bubble point pressure (2620 psig in this instance) is determined accurately as it is a reference for all subsequent measurements.

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Figure 29 Volume Changes During Differential Liberation

Referring to page 5, we see that the first pressure step was to 2350 psig. At this pressure the original system will be in two phases. Its volume would be at b on the adjoining sketch. Figure 29. The first step in altering the overall system composition is made at 2350 psig by removing the gas phase from the PVT cell while maintaining constant pressure. The quantity of gas removed is determined by collecting it in a calibrated container. The volume that the gas phase occupied in the cell is determined by the amount of mercury or non contacting fluid injected during the removal process. Also, the gas gravity is measured on the sample bled off. The volume of liquid remaining in the cell is shown at b' in the sketch. The above procedure is repeated by taking the 2350 psig saturated liquid to 2100 psig (point c) and removing a second batch of gas at that pressure. Again the volume of the displaced gas in the cell at 2100 psig is determined as is the gravity of the removed gas. The volume of liquid phase remaining after the second gas removal step is illustrated by point c' in the sketch.

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This process of removing batches of equilibrium gas continues until the cell pressure at the last displacement is 0 psig. As indicated by the differential data on page 5, there were ten equilibrium removals, all at 220˚F. The final volume of liquid phase remaining in the cell at 0 psig and 220˚F is corrected by thermal expansion tables (or by cooling the cell) to 0 psig and 60˚F. This 0 psig/60˚F liquid is called residual oil. Note that residual oil and stock tank oil are not the same fluids. They are both products of the original oil in the system but are generated by different pressuretemperature routes. Having now got to residual oil the data obtained are recalculated and presented on the basis of a unit barrel of residual oil. By the time 0 psig and 220˚F had been reached, the original system had liberated 854ft3/B residual oil. Column 2 expresses the amount of gas in solution at the various pressures. This is the difference of the 854ft 3 total liberated and the amount liberated between the original bubble point pressure and that pressure. It is important to understand why the solution gas-oil ratio determined from surface flash by taking oil at its bubble point directly to separator and surface conditions compared to differential removal will be different, although the starting and finishing conditions are the same. It is because the process of obtaining residual oil and stock tank oil from bubble point oil are different. The first is a multiple series of flashes at the elevated reservoir temperature ( the differential test); the second is generally a one or two-stage flash at low pressure and low temperature (flash tests). The quantity of gas released will be different and the quantity of final liquid will be different because the changing composition of remaining liquid at each stage will influence the distributions of components between the phases. Also, the quality (gravity) of the products will be different (compare ˚API of residual oil vs ˚API of stock tank oil). The only thing that will be the same for the two processes is the total weight of end products. Column 3 are the relative volumes of the liquid phase measured during the differential liberation of gas. Note that (per the footnote) these volumes at pressure p are expressed per unit volume of residual oil. Again, these relative volumes must not be confused with formation factor volumes because formation factor volumes are specified per barrel of stock tank oil. Note on page 5 that relative volumes start at 1.000 at 0 psig/60˚F and that the value of 1.075 at 0 psig/220˚F is the thermal expansion of 35.1˚API residual oil from 60˚F to 220˚F.

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Above 2620 psig, the original bubble point, the system remained constant in composition. Therefore, the relation of the relative oil volume at p to the bubble point value, 1.600, must be the same as the relative volume in numbers in column 2, page 4 of the report. The other data on page 5 are differential liberation that refer to the oil and gas phases in the reservoir at 220˚F. Column 8 shows that the gravity of the gas liberated between 2620 psig and 2350 psig was 0.825. The next batch between 2350 psig and 2100 psig was 0.818. The gas deviation (compressibility) factor of the first liberated gas was 0.846 at 2350 psig. The oil density at 2350 psig/220˚F was 0.6655 gm/cc. Now we understand the basic difference between flash and differential data as given in the standard PVT report, we can calculate flash solubilities and oil formation volume factors below the bubble point from a combination of the differential and flash data. It is important to appreciate that there are two separation stages in separating the oil from its original solution gas when the fluid in the reservoir has dropped below the bubble point. The drop of the reservoir pressure from the bubble point pressure to a lower pressure is considered to be by a differential process. The separation of gas from the reservoir pressure to the surface is then by the flash process.

11.5

Calculation of Gas – Oil Ratios Below The Bubble Point.

If we examine the separator and differential tests there seems to be a confusing result since the initial and final pressures are the same. a Differential solubility data at the bubble point state (2620 psig/220˚F) and eleven pressures below the bubble point pressure gives a bubble point value at 854ft3/B residual oil. All fluids at pressures above pb have this amount of gas. b

The flash solubility of the bubble point oil for four different surface trap situations, where these vary from 778ft3/B stock tank oil to 795ft3/B stock tank oil for a 300 psig primary trap-tank situation. These are shown on the sketch. Figure 30. The 59 ft3/B difference in values is not experimental error but is a result of the total differential process of the test. In reality there is only a small differential element in the early stages of depletion.

The pressure depletion starts at the bubble point and the solution GOR is that from the flash separator tests. We now need to develop the GOR curve below this value. We will use the 300 psig primary - 0 psig tank situation and will examine the GOR for the reservoir pressure of 1850 psig.

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Figure 30 Comparison of data from flash and differential tests.

EXERCISE PVT 3 Calculate the solution GOR at 1850 psig using the 300/0psig separator data.

SOLUTION. Looking at the differential liberation data in column 2, page 5, we see that 242ft 3 of gas has come out of solution, per barrel of residual oil, when the pressure declined from 2620 psig to 1850 psig. 854 - 612. In other words, we can say that the 1850 psig saturated oil contains less gas by this amount. If this liquid were taken to the surface and processed through the traps, it would also show somewhat less gas solubilities than the 795ft3/B stock tank oil that the bubble point oil shows; but it would not be 242ft3 less because we now have a different oil base. If we let (∆Rs)diff be liberated gas-oil ratio by differential vaporisation, we can convert this to a (∆Rs)flash as follows: (∆Rs)diff = ft3/B Residual Oil

(1)

(2)

(3)

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ft 3 = ( Rs ) flash B Stock Tank Oil

(4)

B Residual Oil 1 = In equation (2) B Bubble Point Oil 1.600

B Residual Oil = 1.495 In equation (3) B Stock Tank Oil

Therefore:

and

Solution gas to oil ratio at 1850 psi = 569 scf/STB. For those who prefer equations, this can be generalised as:

11.6

Calculation of Formation Volume Below The Bubble Point

Examination of the formation volume factors between the bubble point pressure and surface pressures also show a distinct difference between the flash tests and the differential data. This is illustrated by the Figure (Figure 31). The bubble point state has a relative oil volume of 1.600 B/B residual oil. We also have the formation volume factor at the bubble point state, Bob, with a value of 1.495 B/B stock tank oil (300/0 separator combination).

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Figure 31 Comparison of differential data with flash for volume factor volumes.

The result from the separator test is the correct value, since it is based on stock tank volumes. The differential data is used to calculate the change in this separator value below the bubble point. We can see that the relative oil volume and the formation volume factor at pressure p can be related by transferring to the common point – the bubble point. Let: V/VR = relative oil volume at pressure p, B/B residual oil. Then:

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Therefore:

Bo = V VR .

Bob Vb VR

EXERCISE PVT 4 What is the oil formation volume factor at 1850 psig.

SOLUTION At 1850 psig we would have: from page 5 relative volume of 1.479 B/B residual oil

Bo1850 = 1.479

B1850 1 B residual B Bubble point x x 1.495 Bresidual 1.600 B at bubble point B stock tank oil

Bo1850 = 1.3819 B/B stock tank oil 11.7

Viscosity Data

Page 6 of 15 presents the viscosity data for the fluid measured for the oil and calculated for the gas. It should be noted that the pressure for the data below the bubble point are the same as those for the differential tests, since the viscosity is also measured below the bubble point having generated the fluid pressure through a differential process.

12

MERCURY

Historically the transfer fluid in PVT tests has been mercury. It has proved to be a very effective fluid to generate variable volumes in PVT apparatus as well as being non contamination with respect to the hydrocarbon fluids. Unfortunately health and safety concerns with respect to personnel exposed to increased levels of mercury, and its incompatibility with certain materials e.g. Aluminium, are such that mercury is being replaced by alternate systems. Such alternate systems are not as simple to replace the "flexible metal" which mercury has proved to be. Although in these notes we refer to mercury, the principles are the same, where for example the mercury is replaced by a rigid piston driven by a safe fluid e.g. Water.

13

APPENDIX PVT REPORT

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TOPIC 5: MATERIAL BALANCE EQUATION 1. INTRODUCTION 2. LIST OF SYMBOLS 3. MATERIAL BALANCE FOR GAS RESERVOIRS 3.1 Dry gas, no water drive 3.2 Dry gas reservoir with water drive 3.3 Graphical Material Balance 4. MATERIAL BALANCE FOR OIL RESERVOIRS 5. MATERIAL BALANCE AS A STRAIGHT LINE 6. LINEAR FORM OF MB EQUATION 6.1 Short Hand Version of MB Equation 6.1.1 No Water Drive and No Original Gas Cap 6.1.2 Gas Drive Reservoirs, No Water Drive and Known Gas Cap 6.1.3 Gas Drive Reservoirs with No Water Drive, N and G Are Unknown 6.1.4 Water drive systems 7. DEPLETION DRIVE OR OTHER

LEARNING OBJECTIVES Having worked through this chapter the Student will be able to: •

Present a material balance (MB) equation for a dry gas reservoir

•

Demonstrate the linear form of the MB equation for an oil reservoir with and without a gascap

•

Given the equation be able to identify the component parts of the MB equation, eg. gas cap expansion etc

•

Apply the material balance to calculate STOIIP, production etc. for a given pressure decline

•

Comment briefly on the assumptions, significance , use, data and limitations of

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the MB equation.

•

Given the MB equation be able to present it in a short hand form as a basis for use in linear forms.

•

Using the various linear forms with sketches illustrate the MB equation for use for:

•

Reservoir with no water drive or gas cap.

•

No water drive but with known gas cap.

•

Comment with the aid of sketches the impact of water drive on the application of MB equation in linear and other forms.

1. INTRODUCTION In the chapter on Drive Mechanisms we reviewed qualitatively the various drive energies responsible for hydrocarbon production from reservoirs. In this and subsequent chapters we will introduce some reservoir engineering tools used in calculating reservoir behaviour. The petroleum engineer must be able to make dependable estimates of the initial hydrocarbons in place in a reservoir and predict the future reservoir performance and the ultimate hydrocarbon recovery from the reservoir. In this chapter the material balance equation is presented. The material balance equation is one of the basic tools in reservoir engineering. Practically all reservoir engineering techniques involve some application of material balance. Although the principle of conservation of mass underlies the material balance equation, custom has established that the material balance be written on a volumetric basis, because oilfield measurements are volumetric and significant factors can only be expressed volumetrically. The principle of conservation underpins the equation: Mass of fluids originally in place = fluids produced + remaining reserves.

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The equation was first presented by Schilthuis1 in 1936 and many reservoir engineering methods involve the application of the material balance equation. Since the equation is a volumetric balance, relating volumes to pressures, it is limited in its application because of any time dependant terms. The equation provides a relationship with a reservoir’s cumulative production and its average pressure. However when combined with fluid flow terms, we have a basis to carry out predictive reservoir modelling, for example to put a time scale to production figures. Over recent years, as increasingly powerful computers have enabled the application of large numerical reservoir simulators, some have looked down on the simple material balance equation and the tank model of the reservoir which it represents. Reservoir simulators however apply the material balance approach within each of their multi-dimensional cells. The value of this classical tool is that it enables the engineer to get a ’feel’ of the reservoir and the contribution of the various processes in fluid production. A danger of blind application of reservoir simulators is that the awareness of the various components responsible for production might be lost to the engineer using the simulation output in predictive forecasting. The basic ‘material balance’ equation is presented as a volumetric reservoir balance as follows: The reservoir volume of original fluids in place = reservoir volume of fluids produced + volume of remaining reserves. When fluids (oil, gas, water) are produced from an oil reservoir, which may or may not have a primary gas cap, the pressure in this reservoir will drop below the original value. As a consequence of this pressure drop, a number of things will happen: • • • • • •

the pore volume of the reservoir will become smaller the connate water will expand oil, if still undersaturated, will expand oil, if at or already below bubble point, will shrink while gas will come out of solution free gas, if present, will expand water may start flowing into the reservoir, for instance, across the original oil/water contact (OWC).

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The question is now: if we start off with a given reservoir, and after some time we have produced certain quantities of oil, gas and water, what can we say about the average pressure in the reservoir, and what can we say about the average saturation distribution? The answer to these questions can be obtained by considering our reservoir at two stages: (a) at the initial pressure pi, (b) when we have produced certain amounts of oil, gas and water, by which time the average pressure has declined to p (to be calculated). Besides these natural phenomena the equation also has to be capable of handling other factors affecting behaviour, for example injecting gas and or water. There are a number of ways of developing the equation. We will look at two approaches, the first examining the equation as applied to specific reservoir types and then a simple volumetric expansion approach. The nomenclature to be used for the various terms is given below: NOTE: In the following derivations, volumes at standard conditions will be converted into subsurface volumes and vice versa. Remember that to convert a volume from standard conditions to reservoir conditions, one must multiply by a formation volume factor (B) and to convert from reservoir into standard conditions one must divide by a formation volume factor.

2. LIST OF SYMBOLS Symbols

Units

Units SI

Bg Bo Bt Bw cf cw G Gp Gps

bbl/SCF bbl/STB bbl/STB bbl/STB vol/vol/psi vol/vol/psi SCF SCF SCF

M3/SCM M3/STM3 M3/STM3 M3/STM3 vol/vol/MPa vol/vol/MPa SCM SCM SCM

Gas formation volume factor Oil formation volume factor Total formation volume factor Water formation volume factor Pore compressibility Water compressibility Initial gas-cap volume Cumulative gas produced = Gps + Gpc Cumulative solution gas produced

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Gpc Cumulative gas cap produced m Ratio initial reservoir free gas volume to initial reservoir oil volume N Stock tank oil initially in place Np Cumulative tank oil produced p Average reservoir pressure pi Initial reservoir pressure Rp Cumulative gas/oil ratio Rs Solution gas/oil ratio Sw Average connate water saturation We Cumulative water influx Wp Cumulative water production

i b

SCF

SCM

bbl/bbl STB STB psi psi SCF/STB SCF/STB fraction bbl or STB bbl or STB

M3/M3 STM3 STM3 MPa MPa SCM/STM3 SCM/STM3 fraction M3 or STM3 M3 or STM3

Other subscripts at initial conditions at bubble point

Bw may be unknown. In this case assume it equals 1.0. Bw may be omitted from material balance equations and the reader must assume its value equals 1.0.

3. MATERIAL BALANCE FOR GAS RESERVOIRS The simplest material balance equation is that applied to gas reservoirs. The compressibility of gas is a very significant drive mechanism in gas reservoirs. Its compressibility compared to that of the reservoir pore volume is considerable. If there is no water drive and change in pore volume with pressure is negligible (which is the case for a gas reservoir), we can write an equation for the volume of gas in the reservoir which remains constant as a function of the reservoir pressure p, the volume of gas produced SCF, the original volume of gas, SCF, and the gas formation volume factor. A representation of the equation for a gas drive reservoir with no water drive is given below.

3.1 For a dry gas reservoir - no water drive: Figure 1 G.Bgi = (G-Gp) Bg

(1)

Bgi - based on zi, pi, Ti

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Bg - based on z, p, T

Figure 1 Material Balance For a Dry Gas Reservoirs no Water Drive.

N.B: pV = znRT If the gas reservoir is supported by water drive then as gas is produced water will encroach into the gas pore space, and some of this water may be also be produced. Figure 2 below illustrates the contact with a supporting aquifer. Because the mobility of gas is far greater than water, evidence in the form of produced water may be delayed as the water keeps to the gas water contact. The support from the water would be evidenced however by the pressure support given to the reservoir. In earlier years this may not be so easy to detect.

3.2 For a dry gas reservoir with water drive With water drive water will enter pore volume originally occupied by gas and some water may be produced. Figure 2

Figure 2 Material Balance For a Dry Gas With Water Drive.

GBgi = (G-Gp)Bg + We - Wp

(2)

3.3 Graphical Material Balance One can use a graphical form of the material balance equation to analyse a gas reservoir and predict its behaviour especially if no water drive is present.

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(3) hence plot of Gp vs p/z should give a straight line

Figure 3 Gp vs. p/z.

If gas was ideal a plot of Gp vs p would be a straight line. It is often practice to do this and get a relatively straight line, but caution has to be taken, since deviation from a straight line could indicate additional energy support. - when p/z = 0

Gp = G the original gas in place

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- when Gp = 0

p/z = pi/zi

This procedure is often used in predicting gas reserves. Often the influence of water drive is ignored resulting in a serious error in reserves. This simple analysis method for gas reservoirs has gained wide acceptance in the industry as a history matching tool, to determine for example an estimate of initial gas reserves based on production data. This figure, (Figure 3), can then be compared to estimates from exploration methods. It can also give indications of gas to be produced at abandonment pressures.

4 MATERIAL BALANCE FOR OIL RESERVOIRS Material balance is a reservoir engineering technique which is used to investigate the driving mechanisms in a producing reservoir, to investigate the initial oil in place and size of any associated gas cap, if it is present, and for predicting the future reservoir performance. It is one of the many applications of the law of conservation of mass to engineering. In its simplest form, the material balance depends on the following ideas: 1.

The reservoir is regarded as a tank in which no pressure differences occur. The reservoir pressure, P, may depend on time and cumulative production but not on location. This is a global or zero dimensional analysis. (More sophisticated techniques of reservoir engineering simulation can subdivide the reservoir into 1-D, or 3-D analyses with, essentially, material balances for each block of the reservoir).

2.

The production of all of the wells is added together. production rates are not part of the calculation.

3.

The calculation compares the state of the reservoir initially with the state after some time. During any time interval, if the pressure drops from its initial

Individual well

value, Pi, to a lower value, P every phase as oil, gas, water and rock will expand. The total expansion must be equal to the underground withdrawal since the total reservoir volume is practically fixed. There is no need to investigate intermediate steps, i.e. this is an analysis of cumulative production. Occasionally, pressure decrements and corresponding volume changes must be considered - differential material balance analysis. [Reservoir simulators use changes in material balances as checks on simulation stability].

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Consider the Following Situation A reservoir is represented by a cylinder with a moveable piston. The cylinder is filled with N stock tank units of oil kept at a constant temperature, which assumes the reservoir to be vast enough to make no difference to the overall temperature (a large heat source/sink). The piston exerts a pressure Pi on the liquid. The initial volume of oil is therefore NBoi (Figure 4a). The pressure is then released from Pi to a lower pressure P which is above the bubble point pressure, Pb. Consequently, the oil expands to NBo (Figure 4b), and the total expansion is NBo-NBoi or N(Bo-Boi). Finally, oil is withdrawn to return the piston to its original position (because in the reservoir, the volume is fixed therefore the only way to reduce the pressure is to remove some of the fluid volume and allow th pressure to decline). If Np is the stock tank volume of the produced oil, NpBo is the volume at the lower pressure, P .

NBo

NBoi

Figure 4a

Figure 4b

Therefore, for an isolated, undersaturated reservoir with constant hydrocarbon filled pores: N(Bo - Boi) = NpBo If N is known (stock tank oil initially in place, STOIIP) then Np may be calculated from PVT data. This is production forecasting. If the cumulative production at

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pressure P is known, the STOIIP may be estimated. This is the analysis of past performance. Material Balances based on the above ideas have the following drawbacks: 1.

In general, it is not possible to forecast the production of both oil and gas. A simple volume balance is insufficient to predict the behaviour of a reservoir if several phases occur simultaneously. In that case more data are needed.

2.

A one tank model cannot take account of the well pattern, the inhomogeneities of the reservoir and the possibility that different parts of the pay zone may be produced by different mechanisms.

3.

The calculations rest on the assumption of differential liberation of gas in the pay zone. In an actual reservoir, the liberation process is exceedingly complex and depends on the production process itself. The actual process generally varies from point to point in the reservoir and changes with time.

These disadvantages may be partly overcome by splitting the reservoir into many blocks and setting up differential material balances for each block to take account of the flow of oil and gas and water to the adjacent blocks. The block parameters are chosen so that the past performance of each producing well can be matched closely (history matching). The available data may be insufficient to permit a unique determination of the relevant parameters. This may be the case for one tank material balance history matching as well as for multi-tank models. The history match may fail because the model is inadequate or too detailed, e.g. a one tank MBE for a reservoir with a long production history may yield reasonable results, whereas the corresponding multi-tank MBE fails. Sensitivity Some parameters may be sensitive to changes in other parameters, others may be insensitive to even large changes in others. Sensitivity may be analysed mathematically, or it may involve a sensitivity study in which all of the equations are solved for the expected variations in parameter values. Material balance by one tank model is important in furnishing basic data for complex simulation studies using modern computer techniques. In many cases, global analysis provides quite reliable values for oil and gas initially in place and allows prediction of future water influx into the reservoir from an adjacent aquifer.

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COMBINATION DRIVE RESERVOIR

oil, gascap gas & solution gas

Gas Cap

oil & solution gas oil, solution gas & water

original GOC

gas cap expansion

Oil Zone

original WOC water inf lux Aquif er

The general material balance equation is simply a volumetric balance which states that since the volume of a reservoir is constant, the algebraic sum of the volume changes of the oil, free gas, water and rock volumes must be zero, e.g. if both the oil and the gas volumes decrease, the sum of these two decreases must be balanced by an increase of equal magnitude in the water volume. If the assumption is made that complete equilibrium is attained at all times in the reservoir between oil and its solution gas, it is possible to write a generalised material balance expression relating the quantities of oil, gas and water produced, the average reservoir pressure, the quantity of water which may have encroached from the aquifer and the initial oil and gas content of the reservoir. The following production, reservoir and lab data are required. 1.

The initial reservoir pressure and average reservoir pressure at successive time intervals after start of production.

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2.

The stock tank volume of oil produced at any time period or during any production interval.

3.

The total standard volume of gas produced. With gas injection facilities, injected gas must be taken into account.

4.

The ratio m =

initial hydrocarbon volume of the gascap initial hydrocarbon volume of the oil

If this can be determined with reasonable accuracy, there is only one unknown, N, in the material balance of volumetric gascap reservoirs and two, N & We, (We = cumulative water influx from the aquifer) in water drive reservoirs. The value of m is determined from log and core data and from well completion data which helps locate the GOC and OWC. The value, m, is often known more accurately that the absolute values of gascap and oil zone volumes. 5.

The gas and oil formation volume factors and the solution gas oil ratios. These are obtained as functions of pressure by lab measurements on samples by differential and flash liberation methods.

6.

The quantity of water produced. With water injection facilities, the injected water must be taken into account.

7.

The quantity of water which has encroached from the aquifer.

In general, the material balance is a linear relationship between three basic variables and may be stated as: EXPANSIONS - WITHDRAWALS + INFLUX = 0 Considering the combination drive reservoir previously mentioned, a material balance is as follows: (pressure change from Pi to P ). System Expansion = [oil + associated gas] + [gascap gas] + [connate water] - [pore volume] 1.

Oil and Associated Gas

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Oil: N stm3 will occupy NBoi rm3 at the initial pressure. At the lower pressure, P , the oil volume is NBo, Bo is the formation volume factor at P . The expansion is therefore N (Bo - Boi) rm3 Gas: Initially, the oil is in equilibrium with the gascap, therefore the oil is at bubble point pressure. Reduction in the pressure will liberate solution gas. Rsi is the initial solution gas oil ratio (st.m3/s.t.m3) at Pi, Rs the solution GOR at P . Therefore, the N stm3 of oil has NRsi stm3 in it at Pi and NRs at P . The gas volume liberated from Pi to P = N (Rsi - Rs) Bg rm3. Total expansion oil and associated gas = N[(Bo - Boi) + (Rsi - Rs) Bg] rm3 2.

Gas Cap Gas Original voluem of gascap gas = mNBoi rm3 = mNBoi/Bgi stm3 At P , this volume will occupy

mNBoi Bg rm3 Bgi

Therefore, the expansion is

(

mNBoiBg ) - (mNBoi) Bgi

= mNBoi( 3.

Bg - 1) rm3 Bgi

Connate Water HCPV Total pore volume = (1 - S ) wc HCPV Swc Connate water volume = (1 - S ) wc

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The total HCPV = (1 + m) NBoi at Pi If Cw is the connate water compressibility and ∆P = Pi - P , then the expansion of the water is given by (1 + m)NBoi 1 V (1 - Swc) . Swc . Cw . P (Compressibility = V P at constant Temperature)

4.

Pore Volume Contraction (1 + m)NBoi Original pore volume = (1 - S ) wc

 Contraction = -

(1 + m)NBoi (1 - Swc) . Cf . P

where Cf is the formation compressibility. The reduction in volume which can be occupied by the hydrocarbons at the lower pressure P must correspond to an equivalent amount of production expelled from the reservoir. It should be added to the fluid expansion terms. Withdrawals Withdrawals = [oil] + [associated and free gas] + [water] 1.

Oil: associated with NpBo rm3 of oil is

2.

Gas: NpRp stm3 of gas which incorporates NpRs stm3 of solution gas. Therefore the volume of liberated gascap gas = Np(Rp - Rs) Bg rm3 (at the reduced pressure P ) Therefore, withdrawals = NpBo + Np(Rp - Rs) Bg rm3

Influx

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Influx = Injected gas + encroached water + injected water - produced water = GiBg + (We + Wi - Wp)Bw Therefore equating the terms: Bg Np(Bo + (Rp - Rs ) Bg )= N(Bo - Boi) + N(Rsi - Rs)Bg + mNBoi(B - 1) gi CwSwc + Cf + (1 + m)NBoi( (1 - S ) ) P + (We + Wi - Wp)Bw + GiBg wc That is,

Np(Bo + (Rp - Rs ) Bg )= NBoi [

(Bo - Boi) + (Rsi - Rs )Bg Bg + m( Boi Bgi - 1)

CwSwc + Cf + (1+ m )( (1 - S ) ) P ] wc + (We + Wi - Wp)Bw + GiBg

This is the general material equation which can be modified for specific cases. The main cases are: 1.

Undersaturated Reservoirs Above bubble point P > Pb. No gascap, m = 0 and gas oil ratio is constant, i.e. Rsi = Rs = Rp The general MBE becomes:

NpBo = NBoi [

(Bo - Boi) CwSwc + Cf + ( (1 - S ) ) P] + (We - Wp)Bw Boi wc

assuming that there is no water injected and that there is no gas injected.

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Analysis of Past Performance Np v’s P is known a.

Both N and compressibility are known. We v’s P may be found to estimate the future natural water influx adjoining aquifers.

b.

Compressibility is known. We is negligible. STOIIP may be estimated.

From NpBo = NBoi [

(Bo - Boi) CwSwc + Cf + ( Boi (1 - Swc) ) P]

or Y = AX

N=

N p Bo Y P or A = X C S  Cf ( Bo  Boi )  Boi ( w wc (1  S wc )

In rectangular coordinates, Y = AX is the equation of a straight line, the slope of which is A(N). If Np v’s p is known for times t = t1, t2, t3 ... tn, a system of the form Y1 = AX1, Y2 = AX2, Y3 = AX3 .. Yn = AXn can be set up. If all of the data are correct, the system should have the unique solution: Y1 Y2 Y3 Yn A = X1 = X2 = X3 = ... Xn This is seldom the case, but the best fit straight line through the data will yield A. This can be done graphically (plot points and eyeball best line) or by least - AXj)2 becomes a minimum. This leads to the equation dS(Yj - AXj)2 = 0 with the solution: dA SXjYj A = XjXj

e.g. j 1 2

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Xj 1 3

Yj 3 11

XjYj 3 33

XjXj 1 9

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3

8

20

160 196

64 74

i.e.

196 A = 74

c.

Both N and We are unknown - this is natural water influx.

Forecast of Future Performance Both N and compressibility are known.

If We is negligible then the

cumulative production can be calculated for any reservoir. 2.

Undersaturated Reservoirs Below Pb (Solution Gas Drive) If the pressure of an undersaturated reservoir drops below the bubble point then gas will be liberated in the reservoir and some of the gas will be produced. It is assumed that m = 0, no initial gas cap, negligible water influx and the compressibility terms may be neglected once a significant free gas saturation develops in the reservoir. The general MBE reduces to: Np(Bo + (Rp - Rs) Bg) = N[(Bo - Boi) + (Rsi - Rs)Bg] dGp Rp is the cumulative produced GOR. R is the producing GOR ( dNp .) Occasionally, a 2-phase formation volume factor is used: Bt = Bo + (Rsi - Rs) Bg; Bti = Boi The producing GOR changes as the reservoir pressure declines from above P b to below as the following figure shows:

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Pb R

Pressure and Gas Oil R atio

Pi

R si Time

Provided the reservoir pressure remains above Pb, R will equal Rsi since no gas is liberated in the reservoir. If P drops slightly below Pb, R may be smaller than Rsi because some of the liberated gas may build up an immobile phase until the critical gas saturation is reached. After this, R will build rapidly as the liberated gas moves faster to the wells than the oil. Ultimately, R will decrease sharply since most of the originally dissolved gas has been produced. The shape of the R graph will depend on several factors. If the production rates are low and the reservoir is steeply dipping, part of the liberated gas may flow up dip to form a secondary gas cap at the top of the pay zone. This process of gravity segregation may be prevented by inhomogeneities and by capillary trapping. Therefore closing a well in temporarily to allow gas oil separation to occur may have little effect in reducing the producing gas oil ratio. Again, STOIIP’s and water influx may be calculated by the appropriate choice of data. Checks for solution gas drive can be made to ensure neither gascap nor waterdrive mechanisms are in effect. Future performance may be analysed in terms of reducing gas production to improve recovery. 3.

Gas Cap Drive (Saturated Reservoirs) The presence of an initial gas cap means that the compressibility functions of the general MBE may be neglected since the gascap is much more compressible than the formation or connate water. Therefore the MBE is: (Bo - Boi) + (Rsi - Rs)Bg Bg Np (Bo + (Rp - Rs) Bg) = NBoi [ + m( - 1)] Boi Bgi

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The expansion term will involves the solution gas drive since it is still active. Often in gascap drive reservoirs, the ratio, m, is the lest precisely determined value, and plots of withdrawals against expansions can reflect the value of m.

Pressure and Producing Gas Oil Ratio

The graph of producing GOR against time is as follows:

Pi

Producing GOR

R =Rsi Time

The expansion of the gascap as the pressure declines is less severe than in solution gas reservoirs. The oil recovery is better, 25-35%. The peaks in the GOR are caused by controls on the GOR (production rate etc). As the gascap expands the updip wells will become “gaswells” and as such will be shut in or completed as gas injectors.

5. MATERIAL BALANCE AS A STRAIGHT LINE In recent times the computing power behind other reservoir engineering tools like numerical simulation, has cast a shadow of a lack of confidence in the old material balance approach. One reason for perhaps a lack of appreciation of the equation might be the immediate impression of complexity through its many terms. A significant step forward in the equation which had been originally presented by Schilthuis was by Odeh and Havlena, who examined the equation in its various linear forms.

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6. LINEAR FORM OF MB EQUATION 6.1 Short Hand Version of MB Equation The material balance equation in itself is not a difficult concept to understand, the difficulties lie in the application of the equation to real reservoir problems. The problem which generally faces the engineer is the inadequate understanding of the reservoir preventing knowing the extent of the driving mechanism or mechanisms. Havlena and Odeh presented a method that consists of re-arranging the material balance equation to result in an equation of a straight line. The method requires the plotting of a variable group versus another variable group with the variable group selection depending on the drive mechanism. Their technique is useful in that if a linear relationship does not exist for a particular interpretation of the reservoir, then this deviation from linearity suggests that the reservoir itself is not performing as anticipated and other mechanisms are involved. Once linearity has been achieved, based on matching pressure and production data then a mathematical model has been produced. This technique is referred to as history matching, and the application of the model to the future enables predictions of the reservoir’s future performance to be made. The material balance equation can be written in the following form:

(1) In some instances water formation volume factors are not included, i.e. Wp, We and in some they are WiBw Havlena and Odeh simplified the equation into a short hand form: F = NEo + NmEg + NEfw +We

(2)

The left hand side of equation 2 represents the production terms in reservoir volumes and are denoted by F, i.e.

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F = Np[Bo + Bg (Rp - Rs)] + Wp...bbl

(3)

The right hand side includes: (i) the expansion of the oil and its originally dissolved gas, Eo, where: Eo = Bo - Boi + (Rsi -Rs) Bg ....bbl/STB

(4)

(ii) the expansion of the pores and connate water Efw where: (5)

(iii) expansion of the free gas Eg where:

(6) With the above terms the material balance equation can be written: F = NEo + NmEg + NEfw + We

(7)

This equation as presented above neglects water or gas injection terms. Using this equation as a basis, Havelena and Odeh manipulated the equation making different assumptions to produce a linear function.

6.1.1 No Water Drive and No Original Gas Cap In this condition We and Nm are zero and the equation becomes: F = NEo

(8)

i.e. a plot of F vs Eo should produce a straight line through the origin (Figure 5). This is the simplest relation and is just a plot of observed production against determined PVT parameters. The slope of the line gives the oil in place N.

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Figure 5 F vs. Eo No Water Drive and No Gas Cap.

6.1.2 Gas Cap Drive Reservoirs, No Water Drive and Known Gas Cap Although We is zero, the gas cap has a volume as given by m, and the equation 7 becomes: F = N(Eo + mEg)

(9)

A plot of F vs (Eo + mEg) should produce a straight line through the origin with a slope N. Figure 6. If m is not known then by making assumptions for m a number of plots can be generated with the linear slope being the correct value for m.

Figure 6 F vs (Eo +mEg) Gas Drive, With Known Gas Cap, But No Water Drive.

6.1.3 Gas Drive Reservoirs with No Water Drive, N and G Are Unknown If there is uncertainty in both the size of the oil and gas accumulation then Havlena and Odeh suggest the following form of the material balance equation, by dividing both sides by Eo.

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E F = N +G g Eo Eo

(10)

where:

G = Nm

Boi Bgi

A plot of F/Eo vs Eg/Eo should be linear with an intercept of N and a slope of mN. Figure 6.

Figure 6 F/Eo vs. Eg/Eo

6.1.4 Water Drive Systems Water influx is discussed in greater detail in the chapter on natural water influx and we will examine this linearisation of the MB equation in that context then.

7 DEPLETION DRIVE OR OTHER Determining the drive energies responsible for production is important in understanding and predicting the future behaviour of a reservoir. The material balance equation can be used in this short hand form to get an indication of whether a field is depleting volumetrically or there is other energy supporting the system for example, water drive or gas cap expansion. If we consider a reservoir which does not have an aquifer but there is the possibility of a supporting aquifer. In this case the short hand version of the MB equation is;

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F = N(Eo + Efw) + We.....bbl

(11)

If both sides are divided by Eo +Efw Then the following equation results:

F We =N+ ....STB Eo + E fw Eo + E fw

(12)

The right hand side has two unknowns, N and We, and the MB in this form is a powerful tool in assessing whether there is a supporting aquifer or not by plotting F/(Eo+Efw) vs Np, or time or pressure drop, p. The plot will take different shapes depending on the energy support . Figure 7 illustrates this.

Figure 7 Determining Drive Mechanism.

The examples above in Figure 7 give various scenarios as a result of plotting regular production Figures. in curve A, the horizontal line, indicates that the left and right hand side of the equation are constant, ie. We =0. and the pore compressibility is constant. This is a solely depletion drive where the energy comes from the compressibility of the oil and its originally dissolved gas. The intercept on the y axis is also the constant term, N, the oil in place, the STOIIP. The other plots indicate that pressure support is also coming from elsewhere, water drive, or abnormal compaction or a combination of both. Perhaps in C there is a water drive from an infinite aquifer, where the aquifer boundary has still to feel the pressure. Curve B might be for a finite

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aquifer, where later in production, there is less support from the aquifer. Another feature of this presentation is that back extrapolation of the B and C curves also gives the STOIIP volume, N.

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TOPIC 6: NATURAL WATER INFLUX 1.

DRIVING FORCE FOR WATER DRIVE

2.

MODELS FOR WATER ENCROACHMENT 2.1 The Diffusivity Equation 2.2 States of Flow 2.3 Schilthuis Steady State 2.4 Hurst Modified Steady State 2.5 Van Everdingen and Hurst unsteady-state

3.

RESERVOIR PERFORMANCE PREDICTION 3.1 Unsteady-State Model - Van Everdingen & Hurst 3.2 Application to a Declining Pressure 3.3 What are the correct values ∆p?

4.

HISTORY MATCHING WATER INFLUX 4.1 Water Drive, No Gas Original Cap 4.2 Water Drive, Gas Cap of Known Size

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LEARNING OBJECTIVES Having worked through this chapter the Student will be able to: •

Calculate the total water influx resulting from a known aquifer volume in terms of total aquifer compressibility and pressure drop over the aquifer.

•

Sketch and describe the Schiltuis steady state model and the Van Everdingen and Hurst Unsteady State Model for Water.

•

Sketch the progressive pressure profile for a constant boundary pressure.

•

Explain how a constant boundary pressure profile solution can be used for declining pressure aquifer/ reservoir pressure.

•

Calculate given prerequisite equations the water influx as a function of time for a declining pressure profile.

•

Describe and sketch the short hand linear forms of the MB equation for water drive reservoirs for: • Water drive and no gas cap • Water drive and gas cap • Describe the above for very small aquifers.

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RESERVOIR PERFORMANCE PREDICTION - WATER INFLUX In the preceding chapters on drive mechanisms and material balance we identified the positive characteristics of water drive. In this chapter we will examine the various methods which can be used to predict the amount of natural water drive. A large proportion of the world’s hydrocarbon reservoirs have an associated aquifer which depending on production rates can provide a major part of the energy for producing the oil. The consideration is that the original reservoir system was occupied with water and hydrocarbons have migrated in, displacing some of the water. The hydrocarbon and aquifer are therefore part of the same reservoir system responding to the various pressure changes resulting from the production of fluids. As pointed out during the drive mechanism chapter if the aquifer bounds the edges of the oil zone it is called edge water drive and if the water bounds the base of the oil reservoir it is called bottom water drive. The characteristics of water drive are usually the most efficient displacing agent naturally available in the reservoir. The most significant characteristics of a water drive system are: (1) (2) (3) (4)

Pressure decline is very gradual. Excess water production occurs in structurally low wells. The gas-oil ratio normally remains steady during the life of the reservoir. A good recovery of oil can be anticipated

1. DRIVING FORCE FOR WATER DRIVE The driving force for water drive comes from the response to pressure being lowered as a result of oil production, and since the aquifer is part of this system it also responds to this declining pressure. As pointed out in the material balance equation chapter, fluid production is a response to the compressibility of the oil reservoir and the same is true in most cases for aquifer water drives. The porous system representing the hydrocarbon reservoir and the aquifer are compressible. All its elements: hydrocarbon, water and rock expand as pressure declines. It is on the basis of this compressibility that water encroachment is understood and calculated. Water encroaches (moves ) into a reservoir in response to pressure reduction resulting from well production. This pressure reduction causes: (a) Expansion of the water due to pressure drop within the aquifer

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(b) Expansion of hydrocarbons in the aquifer, if any (c) Expansion of rock, which decreases porosity (d) Artesian flow, if any, where the outcrop is located structurally higher than the hydrocarbon accumulation, and the water is replenished at the surface Clearly the amount of expansion or fluid encroachment is proportional to: (a) The size of the aquifer (b) The porosity and permeability of the rock (c) The presence of any artesian water support The amount of water flowing into the hydrocarbon reservoir is also influenced by other factors: (a) The cross sectional area between the water zone and the hydrocarbon accumulation (b) The permeability of the rock in the aquifer (c) The viscosity of the water It is clear therefore in predicting the performance of an aquifer a whole range of aquifer reservoir characteristics are required. The decline in pressure resulting from oil or gas production moves with a finite velocity ( related to fluid flow) into and through the aquifer. The reduction in pressure causes the aquifer, water and rock to expand. As long as this moving pressure disturbance has not reached the external limits of the aquifer, the aquifer will continue to provide expansion water to the hydrocarbon reservoir. In describing the size of aquifers we refer to infinite and finite aquifers. Clearly there is not an aquifer which extends to an infinite extent! The terminology indicates, where in the time considerations of the analysis, the pressure disturbance has not reached the external limits of the aquifer.

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Although natural water drive provides very effective recovery characteristics “there are still more uncertainties attached to this subject in reservoir engineering, than to any other. This is simply because one seldom drills wells into an aquifer to generate reservoir characteristics. Instead these properties have frequently to be inferred from what has been observed in the reservoir. Even more uncertain is the geometry and the areal continuity of the aquifer itself. The reservoir engineer should therefore consult both production and exploration geologists. Due to these inherent uncertainties the aquifer fit obtained from history matching is seldom unique and the aquifer model may require frequent updating as more production and pressure data becomes available.” Dake 1978. In relation to artesian aquifer support it is considered that their occurrence is probably rare, although there are some reservoirs which purportedly have this type of drive. Due to faulting or pinchouts, most hydrocarbon reservoirs do not communicate to a surface outcrop. In addition, for artesian flow to contribute substantially to water influx into a reservoir there must be sufficient ground water moving into the outcrop to replace the fluid withdrawals from the reservoir, and this water must move through the entire distance from outcrop to reservoir at this same rate. Thus it is believed that water influx is usually the result of expansion as a result of pressure drop. The compression of the void spaces in the reservoir and aquifer rock as a result of pressure decline in the pore spaces can affect reservoir performance and contribute to water influx from an aquifer. The compression of the void spaces results in a reduction in the pore volume of the reservoir as withdrawals continue. From a practical standpoint it is usually difficult to separate the water expansion from the rock compression. Therefore, these two effects, which are additive, are usually combined into one term which, for convenience, is referred to as effective water compressibility. The compressibility of water, as well as the compressibility of other liquids, will vary slightly, according to the pressure and temperature imposed on the water. Increasing the pressure will reduce the compressibility of water and increasing the temperature will increase the compressibility of water. The compressibility of fresh water at one atmosphere pressure and 60ºF is 3.3 x 10-6 bbl/bbl/psi. Effective water compressibilities which have been used in reservoir engineering calculations with good results vary from 1.0 x 10-6 bbl/bbl/psi to 1.0 x 10-4 bbl/bbl/psi. The Figure below illustrates an aquifer supported oil reservoir.

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Figure 1 Aquifer supported oil reservoir.

Assuming no restrictions due to permeability etc. the maximum water influx associated with an aquifer system, We, the water influx from the aquifer, can therefore be related to the volume of the aquifer, its effective compressibility and the pressure drop over it. We = cWi(pi-p)

(1)

where: Wi = pi = p = c =

initial volume of water in aquifer (function of geometry) initial aquifer/reservoir pressure current reservoir pressure. Generally assumed to be pressure at original oil water contact. total aquifer compressibility. cf +cw where cf = pore compressibility cw = water compressibility

The main problem facing the reservoir engineer is determining the characteristics of the aquifer; its geometry, size and flow characteristics. The following example illustrates the water influx impact of a relatively low compressibility oil reservoir aquifer system.

2. MODELS FOR WATER ENCROACHMENT

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As we have indicated water influx arises as a result of pressure changes decompressing the aquifer. If the pressure in an aquifer can be calculated then the resulting volumetric changes can be determined from the pressure / volume compressibility relationship The Figure 2 below illustrates the pressure profiles in a reservoir aquifer system. It is important to appreciate that these profiles are generated as a result of decompression of the oil and aquifer as a result of well bore production.

Figure 2 Pressure distribution in an oil reservoir aquifer system.

Before examining the different models we will review the development of equations which enable the pressure, time and distance solution to be obtained

2.1. The Diffusivity equation The diffusivity equation results from a combination of the continuity equation and Darcy’s law.

Figure 3 Radial flow segment

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The flow rate at any radius r + dr is q, Figure 3. The rate of flow at radius r will be larger by the amount dq caused by: (i) Expansion of the fluid q due to pressure drop dp over element dr (ii) Expansion of the fluid in the element due to pressure changing with time dp/dt The expansion of (i) is too small and can be neglected. Volume of element V = 2rh dr

(2)

Change in volume dV due to pressure drop dp is: dV = - cV dp = - c 2rh dr dp

(3)

(4)

(5) Darcy's law for radial flow is

(6) Differentiating

(7) Equating equation 5 and 7 gives

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Dividing by r gives

(8)

This is the diffusivity equation and describes the flow of a slightly compressible fluid in porous media. The pressure with respect to distance and time is related by the parameters c and k. This group of terms is referred to as the diffusivity constant, where

(9) The equation becomes:

(10)

The name diffusivity equation comes from its application to the flow or diffusion of heat. The equation is also applied in range of flow systems, heat, electricity as well as flow in porous media as is the application in a reservoir situation. In the radial diffusivity equation when applied to an aquifer hydrocarbon reservoir system the inner boundary is the extent of the hydrocarbon reservoir and the outer boundary is the limit of the aquifer. In the analysis of the flow and pressures in the hydrocarbon reservoir the inner and outer boundaries are the radius of the well bore and the radius of the reservoir.

2.2. States of flow The diffusivity equation developed above shows that the pressure is a function of time. As long as this exists, the pressure change with time p/t is not constant, and the flow is termed unsteady state. All states of flow at the start are unsteady state. During this period we need to analyse the pressure at elements across the radial symmetry and from that determine the resulting expansion After a time period, p/t becomes

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constant; when this occurs the system is termed pseudo steady state and fluid expansion can be obtained from a tank model concept. All aquifers are finite in size, however there is a period of time when a pressure disturbance created by production from a well has not travelled far enough and reached the boundary of the aquifer. During this time the aquifer behaves as being infinite and unsteady state flow applies. After the boundary influences the behaviour of the system pseudosteady state flow starts. The diffusivity equation demonstrates that the states of flow are influenced by the initial conditions and the boundaries, the outer boundary having a significant influence. In analysing behaviour, the two boundary conditions must be specified: the inner boundary the oil-water interface, and the outer boundary the limit of the aquifer. Conditions may be constant pressure, constant rate, closed boundary etc. The initial condition describes the condition of the system at time, t=0, where a uniform pressure distribution exists. To solve the equation for water encroachment we need to specify the boundary and initial conditions. In general for water influx calculations, the most common conditions are a closed system, no flow at the outer boundary of the aquifer and constant rate or constant pressure at the inner boundary. In general constant pressure is used in aquifer modelling, whereas in reservoir behaviour constant rate is assumed at the inner, well bore boundary. We will now consider the various aquifer models in light of the above discussion.

2.3. Schilthuis steady-state model The simplest model is that due to Schilthuis. The assumptions associated with this model are that the aquifer is very large such that the pressure remains constant, and it has a high permeability such that there is no pressure gradient across the aquifer. Craft and Hawkins present a useful hydraulic analogue of this steady state model as shown in Figure 4 below.

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Figure 4 Hydraulic analogue of the Schilthuis steady state model.

In this model, the aquifer tank pressure remains constant, and could represent an artesian type aquifer recharged with water or an aquifer large compared to the reservoir. The reservoir is considered to be relatively small in size with high permeability such that a flat pressure profile exists. The relative sizes should be at least 10-20:1 . Initially both aquifer and reservoir tanks are at the original reservoir pressure and as the reservoir is produced at constant rate the pressure in the reservoir drops. At any instant, when the reservoir pressure has dropped to a value p, the rate of water influx by Darcy’s law, will be proportional to the permeability of the sand in the pipe, the cross-sectional area of the pipe, and the pressure drop (pi - p); and inversely proportional to the water viscosity and the length of the pipe, provided the pressure of the aquifer tank remains constant. The maximum rate of influx occurs when p=0. If this rate is greater than the reservoir production rate then at some intermediate pressure the rate of influx will be equal to the rate of production and the pressure will stabilize, at a steady-state value. This is an analog of steady-state water influx into a reservoir as expressed analytically by the Schilthuis equation in which the constant, C, depends upon the permeability and dimensions of the aquifer rock and the average viscosity of the water in the aquifer. Schilthuis equation for this is:

(11) C is the aquifer constant and contains the unchanging components of Darcy’s Law, (units vol/time/pressure).

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Expressed in terms of rate of water influx:

dWe = C( pi - p ) dt

(12)

(pi-p) is the boundary pressure drop . Hurst in 1943 proposed an equation which recognised that at least part of the aquifer flow was transient.

2.4. Hurst Modified Steady State The analogue of this is if the tank is neither very large nor replenished, then as production proceeds, the level in the aquifer tank falls and the potential of the aquifer declines. If this decline is exponential then it may be represented by Hurst’s following equation in which the declining value of C is represented by C1/log at, and a is a time conversion constant.

(13)

2.5. Van Everdingen and Hurst unsteady-state The following equation which we will develop later is the Van Everdingen and Hurst unsteady-state equation, which is a model which has become generally accepted in water influx modelling. Before developing the equations we will consider the hydraulic analogue of Craft & Hawkins2, Figure 5.

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Figure 5 Hydraulic Analogue of unsteady State Water Influx2

where B is the water influx constant in barrels per pounds per day per square inch, ∆p is the pressure decrement in pounds per square inch, and Q(t) is the dimensionless water influx, which is a function of the dimensionless time. This equation will be discussed later. The unsteady-state hydraulic analog is shown above where the reservoir tank on the right is connected to a series of tanks of increasing size which are connected by sandfilled pipes of constant diameter and sand permeability, but of decreasing length between the larger tanks. Initially all tanks are at a common level or pressure Pi representing the original pressure across the system As production occurs, the pressure in the reservoir tank drops causing water to flow from aquifer tank 1, and a resulting lower pressure in tank 1. This pressure drop in tank 1 in turn generates flow from tank 2, and so on. Clearly the pressure drops in the aquifer tanks are not uniform but vary with time and production rate, and are progressive across the reservoir. An illustration of these pressure profiles in a radial aquifer are shown in Figures 6 and 7 for a constant rate of water influx and for a constant boundary pressure. Even if there is an infinite number of aquifer tanks, it is evident that reservoir pressure can never fully stabilise at constant production rate, because an ever-increasing portion of the water influx must come from an ever-increasing distance.

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Figure 6 Pressure Distributions in an aquifer at several time periods, for a constant rate of water influx at Rw

Figure 7 Pressure distributions in an aquifer at several time periods for a constant boundary pressure Rw

The water analog can also be illustrated by the series of concentric circles in the Figure 8 below. The Figure represents the cylindrical elements in an aquifer surrounding a circular reservoir. An analysis of the pressure in each element will enable the amount of expansion of water each element can produce as a result of effective compressibility in a pressure decline from pi to zero.

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Figure 8 Cylindrical element in an aquifer surrounding a reservoir.

3. PERFORMANCE PREDICTION Although a consideration of the nature of various aquifers lends support to the analytical expressions presented above, there is no certainty beforehand that any one of the three will adequately represent the water influx into a particular reservoir, and studies must be made to determine the most suitable expression. Examination of the mechanics of water expansion into a hydrocarbon reservoir shows that it must be an unsteady state process. However, for combination drive reservoirs, where the water influx rate is small compared to the other driving forces, the use of the Schilthuis steady state equation can usually be used with reliable results. The water influx constant, C, can be determined from past production data, and then this same value of C can be used to aid in predicting reservoir performance. For active water drive reservoirs, the use of steady state water influx equation will not usually result in reliable predictions of reservoir performance. As the pressure drop due to water expansion moves out further into the aquifer, the expanding water will not move into the hydrocarbon reservoir at the same rate, because for a given pressure drop the water has to move a greater distance in order to enter the oil or gas zone. The favoured approach of analysis is the unsteady state model of Van Everdingen & Hurst.

3.1. Unsteady-State Model - Van Everdingen & Hurst

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From the analog above it is clear the exact solution for water influx is the unsteady state solution, where to access the water production we need to determine the pressure time distance profile across the aquifer. The diffusivity equation in radial form expresses the relation between pressure and radius and time for a radial system such as drainage from an aquifer, where the driving potential of the system is the water expandability and the rock compressibility :

(8)

This diffusivity equation is the same basic equation as has been used to calculate heat flow and electrical flow, as well as fluid flow through porous media. The term is usually defined as the diffusivity constant () and will be essentially constant for any given reservoir. where:

An exact analytical solution of the diffusivity equation for specified boundary and initial conditions define this pressure time profile and therefore will allow the calculation of the rate of water influx into a reservoir, provided the proper data are available. Van Everdingen & Hurst did this in 1949. Their analysis was for two cases: (a) The Pressure case, where the pressure at the inside boundary is known and the outside boundary is closed, or the reservoir is infinite; and we want to calculate the water influx. (b) The Rate case, where the rate is known at the inside boundary. At the outside boundary there is no flow or the pressure is constant or the reservoir is infinite, and we want to calculate the total pressure drop. To enable their analysis to be applicable for different reservoirs they produced a more general solution of the diffusivity equation by generating dimensionless functions. Dimensionless time, tD, in place of real rime, t, and dimensional radius, rD, which is re/ro where re is the radius of the aquifer and ro is the radius of the oil reservoir.

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The dimensionless form of the diffusivity equation is

(14) where: (15)

Since only basic equations have been utilised, the units for the above quantities are:

 

tD t k   c ro

= = = = = = =

time, dimensionless time, seconds permeability, darcy viscosity, centipoise porosity, fraction effective aquifer compressibility, vol/vol/atmosphere reservoir radius, centimetres

Converting equation 15 to more commonly used units of t = days; k = millidarcies; = centipoises;  = fraction; c = vol/vol/psi and r = feet; results in:

(16)

The solution of equation 14 with constant terminal rate boundary conditions is used in well testing. Hurst and Van Everdingen also derived the constant terminal pressure solution which is used in water influx calculations.

(17) qD(tD) = dimensionless influx rate at rD=1. It is the change in rate from zero to q due to a pressure drop ∆p applied at the outer hydrocarbons reservoir boundary, ro , at time t=0.

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Examining equation (17) from t = 0 to t

(18)

(19)

Where: We Q(t)

= cumulative water influx = dimensionless water influx

(20) This equation gives the cumulative water influx for a fixed pressure drop p. The equation applies in Darcy units. However, when oilfield units are used the following equation applies.

(21)

where; We = cumulative water influx, barrels ∆p = pressure drop, psi Q(t) = dimensionless water influx 5.615 = conversion factor, cubic feet to barrels. Equation (21) may also be expressed as:

(22) where:

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B = 1.119cr2oh

(23)

This term B can be considered to be an aquifer characteristic, where the terms do not change during the decline. Van Everdingen and Hurst's paper presented the solution of equation 8 in the form of dimensionless time, tD, and dimensionless water influx Q(t). Their solution of the diffusivity equation can therefore be applied to any reservoir where the flow of water into the reservoir is essentially radial in nature. They provided solutions for external boundaries of an infinite extent and for those of limited extent. Tables 1 and 2 show the tabulated form of their solutions. In addition to being presented in tabular form the dimensionless water influx as a function of dimensionless time, Dake1 also reproduced Van Everdingen & Hurst data solutions in graphical form. The graphs are presented in Figures, 9a-e

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Dimensionless water influx and dimensionless pressures for infinite radial aquifers (courtesy of SPE)3 tD 1.0 x 10-2 5.0 x 10-2 1.0 x 10-1 1.5 x 10-1 2.0 x 10-1

Qt 0.112 0.278 0.404 0.520 0.606

pD 0.112 0.229 0.315 0.376 0.424

tD 1 . 5 x 1 03 2.0 x 103 2.5 x 103 3.0 x 103 4.0 x 103

Qt 4.136 x 102 5.315 x 102 6.466 x 102 7.590 x 102 9.757 x 102

tD 1.5 x 107 2.0 x 107 2.5 x 107 3.0 x 107 4.0 x 107

Qt 1.828 x 106 2.398 x 106 2.961 x 106 3.517 x 106 4.610 x 106

tD 1.5 x 1011 2.0 x 1011 2.5 x 1011 3.0 x 1011 4 . 0 x 10 1 1

Qt 1.17 x 1010 1.55 x 1010 1.92 x 1010 2.29 x 1010 3.02 x 1010

2.5 x 10-1 3.0 x 10-1 4.0 x 10-1 5.0 x 10-1 6.0 x 10-1

0.689 0.758 0.898 1.020 1.140

0.469 0.503 0.564 0.616 0.659

5.0 x 103 6.0 x 103 7.0 x 103 8.0 x 103 9.0 x 103

11.88 13.95 15.99 18.00 19.99

102 103 103 103 103

5.0 x 107 6.0 x 107 7.0 x 107 8.0 x 107 9.0 x 107

5.689 x 106 6.758 x 106 7.816 x 106 8.866 x 106 9.911 x 106

5.0 x 1011 6.0 x 1011 7 . 0 x 10 1 1 8.0 x 1011 9.0 x 1011

3.75 x 1010 4.47 x 1010 5.19 x 1010 5.89 x 1010 6.58 x 1010

7.0 x 10-1 8.0 x 10-1 9.0 x 10-1 1.0 1.5

1.251 1.359 1.469 1.570 2.032

0.702 0.735 0.772 0.802 0.927

1 . 0 x 1 04 1 . 5 x 1 04 2.0 x 104 2.5 x 104 3.0 x 104

21.96 x 102 3.146 x 103 4.679 x 103 4.991 x 103 5.891 x 103

1.0 x 108 1.5 x 108 2.0 x 108 2.5 x 108 3.0 x 108

10.95 x 106 1.0 x 1012 1.604 x 107 1.5 x 1012 2.108 x 107 2.0 x 1012 2.607 x 107 3.100 x 107

7.28 x 1010 1.08 x 1011 1.42 x 1011

2.0 2.5 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 x 101

2.442 2.838 3.209 3.897 4.541 5.148 5.749 6.314 6.861 7.417

1.020 1.101 1.169 1.275 1.362 1.436 1.500 1.556 1.604 1.651

4.0 x 104 5.0 x 104 6.0 x 104 7.0 x 104 8.0 x 104 9.0 x 104 1.0 x 105 1.5 x 105 2.0 x 105 2.5 x 105

7.634 x 103 9.342 x 103 11.03 x 104 12.69 x 104 14.33 x 104 15.95 x 104 17.56 x 104 2.538 x 104 3.308 x 104 4.066 x 104

4.0 x 108 5.0 x 108 6.0 x 108 7.0 x 108 8.0 x 108 9.0 x 108 1.0 x 109 1.5 x 109 2.0 x 109 2.5 x 109

4.071 x 107 5.032 x 107 5.984 x 107 6.928 x 107 7.865 x 107 8.797 x 107 9.725 x 107 1.429 x 108 1.880 x 108 2.328 x 108

1.5 x 101 2.0 x 101 2.5 x 101 3.0 x 101 4.0 x 101

9.965 1.229 x 101 1.455 x 101 1.681 x 101 2.088 x 101

1.829 1.960 2.067 2.147 2.282

3.0 x 105 4.0 x 105 5.0 x 105 6.0 x 105 7.0 x 105

4.817 x 104 6.267 x 104 7.699 x 104 9.113 x 104 10.51 x 105

3.0 x 109 4.0 x 109 5.0 x 104 6.0 x 109 7.0 x 109

2.771 x 108 3.645 x 108 4.510 x 108 5.368 x 108 6.220 x 108

5.0 x 101 6.0 x 101 7.0 x 101 8.0 x 101 9.0 x 101

2.482 x 101 2.860 x 101 3.228 x 101 3.599 x 101 3.942 x 101

2.388 2.476 2.550 2.615 2.672

8.0 x 105 9.0 x 105 1.0 x 106 1.5 x 106 2.0 x 106

11.89 x 105 13.26 x 105 14.62 x 105 2.126 x 105 2.781 x 105

8.0 x 109 9.0 x 109 1.0 x 1010 1.5 x 1010 2.0 x 1010

7.066 x 108 7.909 x 108 8.747 x 108 1.288 x 109 1.697 x 109

1.0 x 102 1.5 x 102 2.0 x 102 2.5 x 102 3.0 x 102

4.301 x 101 5.980 x 101 7.586 x 101 9.120 x 101 10.58 x 101

2.723 2.921 3.064 3.173 3.263

2.5 x 106 3.0 x 106 4.0 x 106 5.0 x 106 6.0 x 106

3.427 x 105 4.064 x 105 5.313 x 105 6.544 x 105 7.761 x 105

2.5 x 1010 3.0 x 1010 4.0 x 1010 5.0 x 1010 6.0 x 1010

2.103 x 109 2.505 x 109 3.299 x 109 4.087 x 109 4.868 x 109

4.0 x 102 5.0 x 102 6.0 x 102 7.0 x 102 8.0 x 102 9.0 x 102 1. 0 x 1 03

13.48 x 101 16.24 x 101 18.97 x 101 21.60 x 101 24.23 x 101 26.77 x 101 29.31 x 101

3.406 3.516 3.608 3.684 3.750 3.809 3.860

7.0 x 106 8.0 x 106 9.0 x 106 1.0 x 107

8.965 x 105 10.16 x 106 11.34 x 106 12.52 x 106

7.0 x 1010 8.0 x 1010 9.0 x 1010 1.0 x 1011

5.643 x 109 6.414 x 109 7.183 x 109 7.948 x 109

x x x x x

Table 1 Dimensionless Qt and PD vs. tD for infinite reservoir.

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Table 4-7 Dimensionless water influx for finite outcroping radial aquifer (courtesy of SPE)3 r D = 1. 5 tD

r D = 2. 0 Qt

5.0 x 10-2 0.276

tD

rD = 2.5 Qt

tD

rD = 3.0 Qt

5.0 x 10-2 0.278

1.0 x 10-1 0.408

6.0 x 10-2 0.304

7.5 x 10-2 0.345

7.0 x 10-2 0.330

1.0 x 10-1 0.404

8.0 x 10-2 0.354

tD

rD = 3.5 Qt

tD

r D = 4. 0 Qt

tD

rD = 4.5 Qt

tD

Qt

3.0 x 10-1 0.755

1.00

1.571

2.00

2.442

2.5

2. 83 5

1.5 x 10-1 0.509

4.0 x 10-1 0.895

1.20

1.761

2.20

2.598

3.0

3.196

2.0 x 10-1 0.599

5.0 x 10-1 1.023

1.40

1.940

2.40

2.748

3.5

3.537

1.25 x 10-1 0.458

2.5 x 10-1 0.681

6.0 x 10-1 1.143

1.60

2.111

2.60

2.893

4.0

3.859

9.0 x 10-2 0.375

1.50 x 10-1 0.507

3.0 x 10-1 0.758

7.0 x 10-1 1.256

1.80

2.273

2.80

3.034

4.5

4.165

1.0 x 10-1 0.395

1.75 x 10-1 0.553

3.5 x 10-1 0.829

8.0 x 10-1 1.363

2.00

2.427

3.00

3.170

5.0

4.454

1.1 x 10-1 0.414

2.00 x 10-1 0.597

4.0 x 10-1 0.897

9.0 x 10-1 1.465

2.20

2.574

3.25

3.334

5.5

4.727

-1

1.2 x 10

-1

-1

0.431

2.25 x 10 0.638

4.5 x 10

0.962

1.00

1 . 56 3

2.40

2.715

3.50

3.493

6.0

4.986

1.3 x 10-1 0.446

2.50 x 10-1 0.678

5.0 x 10-1 1.024

1.25

1 . 79 1

2.60

2.849

3.75

3.645

6.5

5. 23 1

1.4 x 10-1 0.461

2.75 x 10-1 0.715

5.5 x 10-1 1.083

1.50

1 . 99 7

2.80

2.976

4.00

3.792

7.0

5.464

1.5 x 10-1 0.474

3.00 x 10-1 0.751

6.0 x 10-1 1.140

1.75

2.184

3.00

3.098

4.25

3.932

7.5

5.684

1.6 x 10-1 0.486

3.25 x 10-1 0.785

6.5 x 10-1 1.195

2.00

2. 35 3

3.25

3.242

4.50

4.068

8.0

5.892

1.7 x 10-1 0.497

3.50 x 10-1 0.817

7.0 x 10-1 1.248

2.25

2. 50 7

3.50

3.379

4.75

4.198

8.5

6.089

-1

-1

1.8 x 10

-1

0.507

3.75 x 10 0.848

7.5 x 10

1.229

2.50

2. 64 6

3.75

3.507

5.00

4.323

9.0

6. 27 6

1.9 x 10-1 0.517

4.00 x 10-1 0.877

8.0 x 10-1 1.348

2.75

2. 77 2

4.00

3.628

5.50

4.560

9.5

6. 45 3

2.0 x 10-1 0.525

4.25 x 10-1 0.905

8.5 x 10-1 1.395

3.00

2.886

4.25

3.742

6.00

4.779

10

6.621

2.1 x 10-1 0.533

4.50 x 10-1 0.932

9.0 x 10-1 1.440

3.25

2.990

4.50

3.850

6.50

4.982

11

6.930

2.2 x 10-1 0.541

4.75 x 10-1 0.958

9.5 x 10-1 1.484

3.50

3.084

4.75

3.951

7.00

5.169

12

7.208

2.3 x 10

-1

-1

0.548

5.00 x 10 0.982

1.0

1.526

3.75

3.170

5.00

4.047

7.50

5.343

13

7.457

2.4 x 10-1 0.554

5.50 x 10-1 1.028

1.1

1.605

4.00

3.247

5.50

4.222

8.00

5.504

14

7.680

2.5 x 10-1 0.559

6.00 x 10-1 1.070

1.2

1.679

4.25

3.317

6.00

4.378

8.50

5.653

15

7.880

2.6 x 10-1 0.565

6.50 x 10-1 1.108

1.3

1.747

4.50

3.381

6.50

4.516

9.00

5.790

16

8.060

2.8 x 10-1 0.574

7.00 x 10-1 1.143

1.4

1.811

4.75

3.439

7.00

4.639

9.50

5.917

18

8.365

3.0 x 10-1 0.582

7.50 x 10-1 1.174

1.5

1.870

5.00

3.491

7.50

4.749

10

6.035

20

8.611

8.00 x 10 1.203

1.6

1.924

5.50

3.581

8.00

4.846

11

6.246

22

8.809

3..4 x 10-1 0.594

9.00 x 10-1 1.253

1.7

1.975

6.00

3.656

8.50

4.932

12

6.425

24

8.968

3.6 x 10-1 0.599

1.00

1. 2 9 5

1.8

2.022

6.50

3.717

9.00

5.009

13

6.580

26

9. 09 7

3.8 x 10-1 0.603

1.1

1. 3 3 0

2.0

2.106

7.00

3.767

9.50

5.078

14

6.712

28

9. 20 0

4.0 x 10-1 0.606

1.2

1. 3 5 8

2.2

2.178

7.50

3.809

10.00

5.138

15

6.825

30

9.283

-1

0.613

1.3

1. 3 8 2

2.4

2.241

8.00

3.843

11

5.241

16

6.922

34

9.404

5.0 x 10-1 0.617

1.4

1. 4 0 2

2.6

2.294

9.00

3.894

12

5.321

17

7.004

38

9.481

6.0 x 10-1 0.621

1.6

1. 4 3 2

2.8

2.340

10.00

3.928

13

5.385

18

7.076

42

9.532

7.0 x 10-1 0.623

1.7

1. 4 4 4

3.0

2.380

11.00

3.951

14

5.435

20

7.189

46

9.565

8.0 x 10-1 0.624

1.8

1. 4 5 3

3.4

2.444

12.00

3.967

15

5.476

22

7.272

50

9. 58 6

2.0

1.468

3.8

2.491

14.00

3.985

16

5.506

24

7.332

60

9. 61 2

2.5

1.487

4.2

2.525

16.00

3.993

17

5.531

26

7.377

70

9.621

3.0

1.495

4.6

2.551

18.00

3.997

18

5.551

30

7.434

80

9.623

4. 0

1.499

5.0

2.570

20.00

3.999

20

5.579

34

7.464

90

9. 62 4

5. 0

1.500

6.0

2.599

22.00

3.999

25

5.611

38

7.481

100

9.625

7.0

2.613

24.00

4.000

30

5.621

42

7.490

8.0

2.619

35

5.624

46

7.494

9.0

2.622

40

5.625

50

7.497

10.0

2.624

-1

3.2 x 10

4.5 x 10

0.588

-1

Table 2 Dimentionless Qt Vs. tD for finite reservoir.

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Figure 9a Graphical form of Qt vs. tD for infinite and finite reservoirs. Dake1.

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Figure 9b Graphical form of Qt Vs. tD for infinite and finite reservoirs. Dake.1

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Figure 9c Graphical form of Qt Vs. tD for infinite and finite reservoirs. Dake.1

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Figure 9d Graphical form of Qt Vs. tD for infinite and finite reservoirs.

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Figure 9e Graphical form of Qt Vs. to for infinite and finite reservoirs. Dake1.

Although the solutions are for a radial system the solution can be applied where the influx is not full radial but can be considered a segment of such. One of the simplest modifications which can be made is to determine the fraction of a circular area through which water is encroaching, and the equation is modified to:

B = 1.119fcr2ohf

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where: f = fraction of the reservoir periphery into which water is encroaching, Figure 10

Figure 10 Segment of radial water influx.

The graphical solutions demonstrate clearly the finite time it takes for a pressure disturbance to reach the limit of the aquifer, when in the Figures Q(t) becomes constant. Dake' has indicated that this maximum value of Q(t) depends on the size of the aquifer and is equal to: Q(t) max = 0.5 (reD2-1)

for radial systems

(25)

Q(t) max = 1

for linear systems

(26)

and

It is significant to note that when these values for Q(t) are put in equation 20 for a full radial system the following expression results.

(27)

Examination of this equation indicates that it is the total water influx resulting from from the ∆p being instantly communicated throughout the aquifer.

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Clearly for infinite acting radial aquifers there is no maximum Q(t) value since the effect of the pressure drop is continually moving out into the aquifer. For an infinite linear aquifer there is no plot of Q(t). The water influx can be directly calculated using the equation below: Dake1.

(ccs)

(28)

in field units this is:

(bbls)

(29)

t in the above equation is in hours. Table 3 lists the summary of expressions for Hurst and Van Everdingen for both radial and linear system1.

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Figure 11 Parameters for radial and linear geometry.

Table 3 Summary of equations and constants for Van Everdingen and Hurst water influx model

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3.2. Application to a Declining Pressure The application of the Van Everdingen & Hurst model to water influx modelling is by application of their constant terminal pressure solution; that is the boundary at the reservoir aquifer contact is constant. For this constant pressure solution the rate and the cumulative water influx is calculated. In the example above this was the condition and the effects of a fixed pressure drop were determined. In reality however the pressure at the reservoir / aquifer boundary is declining continuously. How can we apply this fixed terminal pressure drop solution to a situation where there is a declining pressure? Figure 12 below gives such a declining pressure at the aquifer /reservoir boundary

Figure 12 Declining pressure profile at the oil / aquifer boundary.

Van Everdingen & Hurst proposed a method of calculating the results of a series of successive pressure drops and adding the solutions together. By superimposing the effects of a series of fixed pressure drops a steady declining pressure can be simulated. The method is illustrated in the Figure 13 where the progressive impact of a series of fixed pressure drops is illustrated.

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Figure 13 Progression of fixed pressuredrops through the aquifer.

In order to use the unsteady state method it is necessary to assume that the boundary reservoir pressure declines in a series of steps. For example in Figure 13 above it is assumed that at the end of the first time period T1 the pressure at the reservoir aquifer boundary drops suddenly from pi to p1. It is further assumed that the pressure stays constant for another time period, at the end of which it again drops suddenly throughout at the reservoir aquifer boundary to p2. These stepwise decreases in reservoir pressure are continued for the length of time desired in the water influx calculations. If the boundary pressure in the reservoir is suddenly reduced from pi to p1, a pressure drop, will be imposed across the aquifer. Water will continue to expand and the new reduced pressure will continue to move outward into the aquifer. Given a sufficient length of time the pressure at the outer edge of the aquifer will finally be reduced to p1.

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If some time after the boundary pressure has been reduced to p1 a second pressure p2 is suddenly imposed at the boundary, a new pressure wave will begin moving outward into the aquifer as a result of the decompression resulting from the second pressure drop, decompressing further that decompressed from the first pressure drop. This new pressure wave will also cause water expansion and therefore encroachment into the reservoir. However, this new pressure drop will not be pi - p2 but will be p1 - p2. This second pressure wave will be moving behind the first pressure wave. Just ahead of the second pressure wave will be the pressure at the end of the first pressure drop, p1. Since these pressure waves are assumed to occur at different times, they are entirely independent of each other. Thus, water expansion will continue to take place as a result of the first pressure drop, even though additional water influx is also taking place as a result of one or more later pressure drops. In order to determine the total water influx into a reservoir at any given time, it is necessary to determine the water influx as a result of each successive pressure drop which has been imposed on the reservoir and aquifer. In calculating cumulative water influx into a reservoir at successive intervals, it is necessary to calculate the total water influx from the beginning. This is required because of the different times during which the various pressure drops have been effective. The aquifer term, B is usually a constant for a given reservoir. Thus where the water influx must be calculated for several different pressure drops, each of which has been effective for varying lengths of time, instead of calculating the water influx for each pressure step, the total water influx as a result of all the pressure steps can be calculated as follows: We1 = B x ∆p1 Q(t)1 We2 = B x ∆p2 Q(t)2 Wen = B x ∆p3 Q(t)n Combining the above equations:

(30) This equation is the form usually used to calculate water influx.

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3.3. What are the correct values ∆p? By considering the gradual pressure drop to be a series of series of fixed pressure drops we need a method which will go towards representing this. Clearly shorter time periods are an advantage together with pressure drops which overlay the curve as shown in the Figure 14 below.

Figure 14 Calculation of time period pressure drops.

Rather than use the entire pressure drop for the first period a better approximation is to consider that one half of the pressure drop, eg. 1/2 (pi - p1), is effective during the entire first period. For the second period the effective pressure drop then is one-half of the pressure drop during the first period, 1/2 (pi - p1) plus one-half of the pressure drop during the second period, 1/2 (p1 - p2), which simplifies to: 1

/2 (pi - p1) + 1/2 (p1 - p2) = 1/2(pi - p2)

Similarly, the effective pressure drop for use in the calculations for the third period would be one-half of the pressure drop during the second period, 1/2(p1 - p2) plus one-half of the pressure drop during the third period, 1/2 (p2 - p3), which simplifies to 1 /2 (p1 - p3). The time intervals must all be equal in order to preserve the accuracy of these modification.

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4. HISTORY MATCHING WATER INFLUX In carrying out water influx calculations it has become clear that a number of parameters have a big impact on the pressure support from an aquifer. Not least the relative size and geometry of the aquifer when compared to the hydrocarbon reservoir. In reservoir predictions many of these parameters are not available to the reservoir engineer. Clearly it is not until production has commenced and the reservoir reacts to fluid production can one determine the pressure support coming from various drive energies. It is during this phase that the aquifer characteristics can be determined. In the material balance equation chapter we discussed the Havelena and Odeh approach to history matching using a linearisation approach. We will now continue this in the context of reservoirs with a water drive, using their basic equation: F = NEo + NmEg + NEfw + We We have seen that water influx is due to expansion of the aquifer water and rock as a result of a decline in pressure. Simply, a drop in reservoir pressure due to fluid production is transmitted through the aquifer and the compressibility of the water albeit small causes the water to expand and flow into the hydrocarbon reservoir. Water Influx = Initial water volume x Pressure Drop x Aquifer Compressibility We = (cw +cf ) Wi x ∆P

(31)

As we have seen this equation is generally not sufficient to describe water influx behaviour, in particular for reasonably sized aquifers, because of the finite time required for the pressure effect to be felt throughout the aquifer. In water influx calculations therefore it is necessary to include this time dependency as a result of fluid flow. In Havlena and Odeh’s paper they recognise this time dependant water influx perspective, where they use the dimensionless water influx term to express We, ie: We = B∑∆pQt

(30)

They then apply their short hand MB equation to examine different reservoir senarios as follows:

4.1. Water Drive, No Gas Original Cap Havlena and Odeh’s equation can be re-arranged as:

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(32)

A plot of F/Eo vs.

should give a straight line, as shown in Figure 15.

Figure 15 Plot of F/Eo vs .

This line will be straight if the aquifer characteristics, B, and the radius of the aquifer are correct. The intercept will be the oil in place, N, the slope B. Havalena and Odeh suggest four other plots. Complete scatter, suggesting the calculations or basic data is in error. A systematically upward or downward curve suggesting that ∑∆pQtDis too small or too large (this means that re/ro and/or tD is too small or too large). An S shaped curve indicates that a better fit might be obtained by assuming linear water influx.

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Once the assumed values give realistic behaviour then the model, which has been obtained by history matching, can be applied in predicting future reference reservoir performance. The assumption is taken in such a situation that the reservoir and associated aquifer continue to behave as before. Because of this large assumption and that no aquifer model is likely to be unique the validity of the model should be updated as more pressure and production data becomes available. This linearisation approach has been used as a means of determining the extent of a supporting aquifer. In Figure 16 below the curves show the results for a range of dimesionless radii for a field in the North Sea. The results show a straight line fit with an infinite aquifer.

Figure 1 Havalena and Odeh approach applied to Piper Field.

Very small Aquifers For small aquifers, the assumption might be made that steady state flow exists as the pressure drop is quickly transmitted through the aquifer. In this case; We = B ∆p

(33)

where: ∆p = pi - p

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and: B = Wicw

(34)

Wi is the water volume in the aquifer. The equation now becomes

(35)

A plot of F/Eo vs ∆p'/Eo should give a straight line of slope B and intercept N. Havlena and Odeh point out that the points on this graph will plot backwards as in Figure 17 below. This is because Eo increases faster than ∆p, therefore ∆p/Eo decreases as the pressure decreases. In some situations a steady state water influx sets in after a certain period. In this case the points plotted for the unsteady state period will plot in forward sequence but when the steady state exists then the plotted points reverse the plot backwards.

Figure 17 Plot of F/Eo Vs. p'/Eo for small aquifers.

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4.2. Water Drive, Gas Cap of Known Size Using a similar approach to the treatment of We as before and applying it to a gas cap drive reservoir Havlena and Odehs’ equation yields:

(36) where We = B∑∆pQt Again this gives a straight line function ,Figure 18, if the geometry of the aquifer and time are assumed correct. If the line is not straight then assumptions regarding the aquifer need to be modified as for water drive systems without a gas cap.

Figure 18 Havalena and Odeh plot for water drive and known gas cap.

Small Aquifers, Gas Cap of Known Size For small aquifers, as for the system without a gas cap, a plot of the lefthand side and the B term of the following equation should result in a straight line, the points plotting in reverse sequence. Figure 19.

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(37)

Figure 19 Havlena and Odeh plot for small aquifers and known gas cap.

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TOPIC 7 IMMISCIBLE DISPLACEMENT 1

INTRODUCTION

2

THE REASON FOR WATER INJECTION 2.1 Zone Isolation 2.2 Permeability 2.3 Oil Viscosity 2.4 Undersaturated Reservoirs 2.5 Overpressured Reservoirs 2.6 Reservoir Depth 2.7 Facility Design 2.8 Thermal Fracturing 2.9 Water Handling

3

BASIC WATERDRIVE THEORY 3.1 Introduction 3.2 Water-Oil displacement at Microscopic and Macroscopic levels 3.3 Relative Permeability 3.4 Fractional Flow

4

DISPLACEMENT THEORIES 4.1 Introduction 4.2 Buckley- Leverett Theory

LEARNING OBJECTIVES Having worked through this chapter the Student will be able to: • • • • • •

Describe briefly the various reasons for water injection. Present a simple equation for the fractional flow of water in terms of water and oil flow rate. (equation 9) Comment briefly on the impact of angle of dip, capillary pressure, and velocity on the fractional flow. Plot a set of relative permeabilties and identify end-point relative permeabilities. (Figure 11) Be able to define mobility ratio and present an equation for it and calculate its value given relative permeability and viscosity data. Generate a fractional flow curve given relative permeability and viscosity data

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• •

1

for injected and displaced fluids. Understand the significance of the Buckley-Leverett Frontal Advance equation. (not expected to derive the equation ) Show the shape of the fractional flow curve and its associated derivative curve and the progressive saturation displacement profile for the following three types of displacement; • Water displacing viscous oil • Water displacing very light oil. • Water displacing medium denisty oil • Determine the breakthrough fractional flow, saturation and time for a diffuse flow displacement process given the necessary equations

INTRODUCTION

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In previous chapters we have examined the various fundamental properties associated with the behaviour of fluids when subjected to pressure and temperature changes and the characteristics of reservoir porous media in relation to its pore volume and transmission characteristics. At another extreme scale we have reviewed the various drive mechanisms responsible for providing the energy to move hydrocarbons in a reservoir. We have also examined the various volumetric methods used to relate the volumes of fluids produced in relation to the overall pressure decline of the reservoir and the original volumes in place and energy support provided by attached water and gas. The topic of water drive in the chapter on drive mechanisms showed that this drive mechanism provided the highest recovery factor in relation to reservoir depletion. For this reason therefore water drive provided by intervention, that is when water is injected into the reservoir through injection wells, is common practise in oilfield operations. Most of the reservoir engineering texts cover this topic. The author considers that Dake and the text of Chierici provide excellent detailed analysis of the topic. In the next sections we will review some of the reasons for using water injection, then review some of the basic properties used in prediction, derive the fractional flow equation and then examine procedures used to determine the movement and displacement of fluids within a reservoir.

2

THE REASON FOR WATER INJECTION

Water injection is the main intervention method used in reservoir development, primarily because of the associated recovery achieved and also the availability of the injection fluid. Historically it is termed a secondary recovery process in recognition of the application of using it after a reservoir has been depleted by its natural energy, and the pressure has dropped below the bubble point. The perspective here is using the injected water to displace some of the remaining oil and thereby recover more oil. Water injection can have two benefits, and for this reason it being termed as a secondary recovery process causes some confusion. Allowing a reservoir to fall below the bubble point leads to solution gas drive and resulting low recoveries. Keeping the reservoir above its saturation pressure by the injection of another fluid maintains the energy of the process providing good well productivity and more important keeps the reservoir fluid in single phase. This voidage replacement, pressure maintenance process using water injection has been common practise in major offshore oil sectors where there is plentiful supply of clean injection material.

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2.1

Zone Isolation

Although a field might be supported by an active aquifer providing natural pressure support, in some cases faulting within the reservoir structure, can result in zones being isolated from pressure support, Figure 1. If no intervention was used the zone would produce by its natural energy with a rapid loss of pressure and resulting poor recovery resulting from solution gas drive

Figure 1 Zone isolation.

2.2

Permeability

A characteristic of a number of offshore oil producing regions, for example the North Sea, is the moderate to high permeabilities, which enable production wells to be very productive reducing the required number of wells. Since the major cost in offshore production is the offshore structures then minimising well slots results in minimising the number of platforms. Maintaining high productivity through pressure maintenance can be obtained through water injection when good injectivities can be achieved.

2.3

Oil Viscosity

The displacing characteristics at pore scale are influenced by the relative mobility of the two fluids, the fluid being displaced and the fluid displacing. The ratio of the mobility of the displacing fluid to the displaced fluid is a ratio of Darcy’s law as applied to the system. The different parameters being the permeability of the one fluid in the presence of the irreducible saturation of the other, the end point relative permeability, and the viscosity of the fluids. The mobility ratio for water displacing oil is expressed as, M where:

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(1)

In sectors like the North Sea, Dake points out that relatively low oil viscosities lead to high flow rates and the favourable oil viscosity compared to water gives a mobility ratio for some North Sea reservoirs of less than 1. This means that at least at microscopic level the water cannot move faster than the oil and therefore displaces the oil in a piston like manner. If M is greater than one, the case where oil viscosities are higher, then the higher velocity of the water causes an increasing instability and water fingers through the oil and breaks through early compared to piston like behaviour. The behaviour is illustrated in the sketch below, Figure 2. As pointed out this behaviour only relates to the microscopic scale, and at reservoir scale the various heterogeneities and the influence of gravity will have a big impact on the reservoir flooding behaviour.

Figure 2 Impact of mobility ratio on horizontal displacement.

2.4

UNDERSATURATED RESERVOIRS

When a reservoir is above its bubble point, i.e. it is undersaturated, then if there is no pressure support from an aquifer the pressure declines rapidly. This pressure decline can be detected using pressure surveys in an open hole well as the dynamic behaviour of the reservoir is reflected in the various layers making up the formation. This is demonstrated in the context of the Montrose field, where following production the pressure depth profile was determined for successive development wells through the oil zone and basal aquifer. The pressure profile did not follow the original water pressure gradient established during the evolution of the field but reflected the

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permeability variation and communication between the various sand layers, Figure 3. As more development wells are drilled pressure surveys continue to confirm the layering of the formation (Figure 4). This powerful application of pressure surveys to determine the communication characteristics of a reservoir enables waterfloods to be planned and simulated much more effectively.

Figure 3 Pressure depth profile for Montrose Field well.

2.5

Overpressured Reservoirs

In those areas where reservoirs are overpressured, the overpressure provides extra energy support. This additional pressure enables high production rates in the early time period and also information from the reservoir of the dynamics of the various units making up the system particularly if pressure depth surveys are carried out during this

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period.

2.6

Reservoir Depth

The cost of offshore production facilities are such that it is important to maximise the functionality of each platform. If waterflooding is carried out then the water flooding wells are generally at the extremities of the formation. The well slots on the platform therefore have to be capable of reaching these limits, Figure 4. The deeper accumulations provide the drillers with an easier task to reach the outer limits of the reservoir using deviated and vertical wells. The application of horizontal wells in recent years also enables shallower accumulations to be reached.

Figure 4 Application of deviated wells from one structure to reach limits in the reservoir.

2.7

Facility Design

In planning water injection, at least two important considerations are required. The injection perspective: where should injection take place in relation to the various zones of the formation and the ability to inject in relation to formation characteristics? Secondly and equally important, the time and associated cost of handling the water when it eventually arrives at the production wells.

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2.8

Thermal Fracturing

It is beyond the scope of this text to go into detail, but in recent years, as experience in large waterflood operations has been obtained, new insight is developing on how reservoirs have reacted. Of great significance is the phenomenon of thermal fracturing. In waterfloods where large injection flow rates are required large pumps are utilised which can handle the necessary capacity. In many offshore zones where the injection water is cold, the reduction in temperature around the injection well reduces the natural fracture gradient and the pumps capable of overcoming the resistance of flow through the formation generate a pressure greater than the fracture gradient causing the formation to successively fracture. This generates a high surface area for flow and therefore injectivities have been maintained compared to those expected from predictions using simple radial flow around a well. This thermal fracturing phenomenon enabling good injectivities to be maintained is causing some companies to consider using forced fracturing associated with water injection where temperature gradients in warmer regions or with warm injection fluids will not reduce the natural fracture gradient. Such could be the case when reinjected produced water is being used. Although good injectivities can be achieved due to fracturing a greater understanding of the stress sensitivity of the formation is required. The fracture will follow the natural direction according to the natural stresses and strength characteristics of the formation. A concern is that such fractures will cause the injected water to by-pass the desired flood front and cause premature water breakthrough. ( Figure 5).

Figure 5 Impact of fracture on water injection flood profile.

2.9

Water Handling

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The handling of water is a major technical challenge in the oil industry particularly in offshore operations, where many operators as fields mature find themselves handling more water than oil. This technical challenge is also increasing as water disposal options in relation to reducing oil emissions become more limited. Those involved in providing associated production and treating facilities require important information from the reservoir engineer. A schematic layout of a typical offshore water injection scheme is shown in Figure 6. Some key information required is: when will water breakthrough to the producing wells, and how much water will be increasingly be produced? The water handling facilities required are not insignificant and therefore good forecasts are important. A more demanding challenge to the reservoir engineer and outside the scope of this text is how can we manage the reservoir to reduce water production.

Figure 6 Schematic of offshore facilities for Water Injection.

Dake points out an equation which links the reservoir engineer to the production engineer which is:

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qwi = qoBo + qwpBw (rb/d) where

(2)

qwi = injection rate (assume Bw = 1) qo + qwp = produced fluids requiring separation qwp = produced water for disposal or reinjection

He points out that this simple equation is fundamental to the process. In water injection, the injection rate, qwi is maintained constant, since this is the drive in water drive, and is therefore under engineering control. The right hand side, the fluids requiring facilities for treatment are under the control of the reservoir. Dake points out that this equation is not just a statement of material balance but it can be regarded as a ‘platform equation’ since it contains the key elements associated with topside capacities. If water breaksthrough prematurely then, since the water injection rate has to be maintained to maintain the reservoir pressure, there is an inevitable reduction in oil production. The injected water in the reservoir provides two functions maintaining pressure and displacing oil. Until breakthrough, only oil is produced, after water breakthrough an increasing watercut occurs. This watercut or fractional flow is defined as:

fws =

qwp qo + qwp

where ‘s’ denotes surface conditions.

(3)

Expressing the equation in terms of water production and substituting in (2) gives:

qwp = qo

fws 1 - fws

(4)

and

(5)

Behind these commercial calculations is the importance for the reservoir engineer to predict the producing watercut as a function of oil recovery.

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3

BASIC WATERDRIVE THEORY

3.1

Introduction

Before examining the various methods used in predicting the behaviour of reservoirs under a constant injection process, such as water drive or gas injection, we will review some of the important basic properties relevant to the application. The method presented is applicable to both water injection and gas injection where an immiscible displacement process occurs. An immiscible displacement process is where there is no mixing of the respective injection and displaced phases at the pore level through mass transfer of components. This is distinguished from a miscible displacement process where the injected phase mixes with the displaced phase by mass transfer of the components from the respective phases, for example in a CO2 enhanced oil recovery process. As in many reservoir engineering processes we are combining properties, measurements and application over a huge range of physical scales. Such an example of this is in immiscible displacement calculations in oilfield oil recovery predictions. It is important to keep this relative scale perspective in mind so as not to make an unrealistic "jump" in application of data beyond its significance. In water-oil displacement considerations we are dealing with a process which takes place at a range of scales. At pore level or the microscopic scale, where the isolation and movement of the respective phases is dependant on fundamental properties such as interfacial tension, wettability, viscosity, pore size and shape to name the obvious. At a significant larger scale, the macroscopic scale, we measure behaviour and generate properties at the laboratory level where fluid movement and displacement are examined at core plug scale, such as permeability, relative permeability and capillary pressure. The field scale,or behavioural scale, where the impact of characteristics at another quatum leap level of scale will impose behaviour on those measured at microscopic and macroscopic scale. For example the heterogeneous characteristics of the various layers of the formation giving rise to different mobilties within the layers and the large thicknesses of the layers resulting in vertical segregation perspectives. An illustration of these different perspectives is shown in Figure 7, where the oil water displacement process is illustrated at microscopic and reservoir behaviour scales. This scale up perspective is considerable and should not be forgotten, if not "giant leaps of faith” might be made using data beyond its range of applicability. The engineering of sub surface behaviour such as a water injection process can be compared to the engineering of an oil refining plant. In the later, the process takes place in vessels and pipes of centimetres and metres size over an area of a some hectares. In a reservoir, the pipes and vessels, "the pores” are of micron dimensions and are considerable in number to cover depths of hundreds of metres with an area perhaps of tens of square kilometres.

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Over recent years, considerable effort has been put into scale-up considerations in relation to reservoir simulation, where rock properties at microscopic level can be combined with geological characteristics at various scales to provide greater confidence in field scale predictions. This topic is covered in the Geoscience and Reservoir Simulation modules.

3.2

Water-Oil displacement at Microscopic and Macroscopic levels

Figure 8(a) illustrates the remaining oil at microscopic level following displacement by water. This remaining or residual oil is held by the competition between the interfacial tension forces and the viscous flow forces associated with fluid flow. As explained by the pore doublet model where the continuous phase of oil is broken leaving oil ganglia held by capillary forces. The residual oil saturation, Sor, in the water swept rock can be in a range of 10-40% of the pore space (Figure 7(a)). At the field displacement level the nature of the reservoir formation and well locations causes some of the rock to be unswept by the water. This leads to two residual oil saturations, in the swept portions oil at residual oil saturation and in the unswept portions oil at original oil saturation.

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Figure 7 (a) microscopic displacement (b) Residual oil remaining after a water flood.

This microscopic behaviour illustrates its effects in macroscopic properties of relative permeability, and capillary pressure curves.

3.3

Relative Permeability

Permeability is a macroscopic property of the rock describing its resistance to flow in terms of fluid velocity, fluid viscosity, pore size and shape, and pressure gradient. This flow resistance term comes from Darcy’s law:

(6)

in relation to Figure 8.

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Figure 8 Darcys' law for permeability

This equation is for single phase flow only and does not apply to flow resistance when two phases (for example oil and water) are present. For this purpose the concept of relative permeability is used, which is a measure of the permeability of one of the phases and is a function of the phase saturations. For example the relative permeability of water, krw is expressed as follows;

krw =

kew k

(7)

where kew is the effective permeability to water calculated from Darcy’s law when oil and water are present, and k is the absolute permeability (single phase). Darcy’s Law in linear flow for the two fluids allowing for gravity effects in an inclined configuration, Figure 9, is

(8)

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Figure 9 Configuration of water injection in a reservior

The relative permeabilities are a function of saturation and reflect the surface, and wettability forces of the fluid-rock system. An example of relative permeabilitiy curves for a water oil rock is given in Figure 10.

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Figure 10 Relative permeability curves for an oil-water system

Identified on the curves are the two conditions at the limiting saturation of the respective phases, the end point relative permeabilties for oil and water k'ro and k'rw. k'ro – the relative permeability to oil in the presence of irreducible water saturation and k'rw– the relative permeability to water in the presence of residual oil saturation. Dake reminds his readers that rock relative permeabilities are obtained from one dimensional core flooding experiments, where often a cleaned core is flooded with oil and then the oil displaced with water. Two types of experiments are then used. A viscous displacement of oil with water or a steady state experiment with co-injection of water and oil at increasing ratios of water to oil. Dake also notes that the relative permeability data, used in subsequent reservoir engineering calculations are unlikely to be representative of field characteristics. They have probably been carried out at flow rates orders of magnitude higher than in the reservoir, often using a synthetic oil not necessarily representative of the reservoir fluid, and with wetting characteristics probably different than in the reservoir. In the viscous displacement experiment the injected water, starting at the irreducible connate water level, Swc, where the water is immobile, generates increasing saturations in the core as a result of displacing oil. This increases until the saturation in the core, where there is no more oil mobile in the core, to water is 1-Sor, where Sor is the residual oil saturation.

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If we express the volume of the pores in the core plug as the pore volume, PV, then the oil displaced from the core flood experiment, is the movable oil volume, MOV, which is; MOV = (1- Sor - Swc) PV The importance of end point relative permeabilities was presented earlier in this chapter in the context of mobility ratio, M, where ;

(1) At the end point conditions this represents the maximum velocity of the water flow compared to the maximum velocity of the oil.

3.4

Fractional Flow

Considering flow in a core plug or a reservoir, the ratio of the flow of water at any point is termed the fractional flow, fw ,where:

fw =

qw qw + qo

(9)

The oil rate qo can also be expressed as; qo = qt - qw

(10)

If the Darcy equations for water and oil are subtracted and rearranged (using field units P in atmos.) the equations become;

(11) where;

¶Pc ¶po ¶pw = ¶x ¶x ¶x which is the capillary pressure variation in the direction of flow and,

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 = w - o is the density difference of water and oil. If values for flow rates using Darcy’s Law are now substituted in fraction flow equation (equation 9) it becomes;

(12)

Dake has also simplified this equation as;

fw =

1- G m k 1 + w ro mo krw where G is a positive gravity number;

in field units.

(13)

(14)

where  is the specific gravity difference relative to water. The above term not only considers gravity effects but also includes a velocity term v, which is qt/A The impact of the various components of this equation is worthy of consideration. The angle of dip. If water is being injected up dip then the gravity term, gsin/1.0133x106 will be positive, reducing the fractional flow of water and it would be positive for gas being injected downdip in a gas displacing oil senario. The density difference in gas displacing oil systems is larger and therefore the significance is greater. If the dip angle is zero, ie. horizontal flow, then the gravity term is zero.

The impact of capillary pressure, is illustrated from the slope of the capillary pressure and saturation with distance curves, Figure 12 since;

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Figure 11 Capillary pressure curve and saturation distribution as a function of distance

i.e. the capillary pressure term is also positive increasing the fractional flow, for a water displacing oil system as the two function gradients are negative. The capillary pressure term is often neglected because the saturation with distance profile is unknown being the objective of the displacement calculation, which we will consider later. Velocity. This velocity is the superficial velocity, the rate divided by the cross sectional area,A. The actual velocity is larger because of the impact of porosity. The impact of velocity is small. Dake notes that the value for G for an edge water drive, typical of the North Sea, is 0.22kro and a comparative bottom water drive is 10.29kro. This demonstrates the stability of the bottom water drive, where piston like displacement will inevitably occur. If both the angle of dip, and capillary pressure effects are neglected the fractional flow equation becomes;

fw =

1 m k 1 + w ro mo k rw

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The fractional flow equation enables a fractional flow versus saturation curve to be generated from relative permeability data. This curve is influenced by a number of parameters not least the viscosity of the respective phases. Its shape varies but can have a shape as given by Figure 12 below.

Figure 12 Fractional flow curve

An interesting presentation was given by Mayer-Gurr as illustrated in Figure 13 where the capillary pressure, relative permeability and fractional flow curves are presented. The impact of various well locations are considered.

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Figure 13 The relationship between capillary pressure, relative permeability and fractional flow in a reservoir

The capillary pressure curve represents the transition zone saturation profile associated with the advancing imbibition process as a result of water injection. If a well is located at A, the well will only produce oil since although the water saturation is 10%, the relative permeability to water is zero. At B, the 45% saturation level the well will produce both water and oil with a water cut of 50%. At location C, the advancing water has isolated an irreducible oil saturation and the well produces only water.

4

DISPLACEMENT THEORIES

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4.1 INTRODUCTION To model the displacement process a number of theories have been successfully applied. These theories are aimed at providing the important predictions of reservoir performance including the proportion of hydrocarbons recovered. In the methods presented there are a number of assumptions. The displacement is incompressible, which implies that steady state conditions exist, that is the pressures within the reservoir at any point remain constant. This will occur if the following reservoir flows exists; qt=qo+qw=qi where qt = the total flow rate in reservoir volumes/time. qo = the oil flow rate in reservoir volumes/time. qw = the water flow rate in reservoir volumes/time. qi = the water injection flow rate in reservoir volumes/time. Diffuse flow conditions exist. Diffuse flow means that the saturations at any point in the direction of linear displacement are uniformly distributed over the thickness. This diffuse flow assumption enables a one dimensional simple analysis to be used for the displacement modelling. In a simple core flooding relative permeability test such an assumption is not unreasonable. Diffuse flow can also be encountered in a reservoir where the injection rates are high preventing the establishing of vertical equilibrium and for low injection rates where the thickness of the reservoir is small compared to the thickness of the transition zone.

4.2

Buckley- Leverett Theory

The theory that has established itself in reservoir engineering for displacement calculations is that by Buckley and Leverett in 1942. Their theory is for linear, immiscible, one dimension displacement, in which the total flow rate is constant in every cross section, (incompressible). The theory determines the velocity of a plane of constant water saturation moving through a linear system, such as a core in a water flood test, Figure 14. The theory is well founded on the conservation of mass principle.

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Figure 14 Mass flow through a linear core.

Consider the linear system in which water is displacing oil. The systems has a porosity of  and we are considering the principle of conservation of mass around a volume element of length, dx. Therefore; Mass flow rate in –mass flow rate out =rate of increase of mass in the volume.

(16) or (16b)

This becomes (17)

Since we are assuming incompressible flow, w is a constant. Therefore;

(18)

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The differential of water saturation is

dSw =

¶Sw ¶S dx + w dt ¶x t ¶t x

(19)

We are examining the advancement of a particular saturation value. Since Sw is constant dSw=0. Then

¶Sw ¶S dx =- w ¶t x ¶x t dt sw

(20)

Also

(21)

Inserting equations 20 and 21 in equation 18 gives; (22)

For incompressible flow, the total injection rate, qt is constant, and the water flow rate is the total rate times the fractional flow, qw=qt x fw. Rearranging equation 22 therefore gives:

(23)

where vSw is the velocity of the plane of saturation, Sw.

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This is the Buckley-Leverett equation, and is also the equation of characteristics. It indicates the velocity of a plane of saturation moving through the linear system. It enables the calculation of Sw as a function of time and distance and indicates its dependance on the derivative of the fractional flow curve.

Chierici has presented a very thorough analysis of the displacement process for three fractional flow curves. In understanding the use of the equation it is important to appreciate the initial boundary conditions, for our injection process. These are; Sw = Swi for 0 < x ≤ L,t = 0 Sw = 1 - Sor for x = 0, t ≥ 0

(24)

That is the system is at its initial connate water saturation If the initial conditions at t=0 are applied to the general equation;

and the equation is then integrated a general solution to the displacement process is obtained which enables the calculation of Sw in terms of x and t.

(25)

This equation describes a series of straight lines, the characteristics, with an initial ordinate value of x0(Sw) and a slope of

Chierici considers three cases

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Case 1 For viscous oils In this case the viscosity of the displaced phase, the oil, is considerably greater than the injected water phase. The fractional flow curve has a concave downward shape, Figure 15A and its gradient fw increases from Sw=1-Sor to a maximum value at Sw=Swi+Swi. Figure 15B

Figure 15 Displacement of viscous oil by water. A = Concave downwards fractional flow curve B = Velocity of water saturation C = Characteristics of water saturations Sw

The velocity of saturation is therefore maximum where Sw is just greater than Swi and decreases to a minimum at Sw=1-Sor, Figure 15C. The progression of water profiles are shown in Figure 16 and shows the fraction of water at breakthrough at the producing end. As can be seen the breakthrough saturation is just greater than Swi and explains why for a very viscous oil breakthough occurs with low water saturations and then gradually increases until the saturation reaches an unacceptable level.

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Figure 16 Progressive saturation profile for a concave downwards fractional flow curve

Case 2, Very Light Oils In this case when the oil is very light with a low relative viscosity and large gravitational effects for example with a highly dipping structure, and with very low velocity, a concave upward fraction flow curve is generated, Figure 17A, resulting in a fw curve decreasing from its value at Sw=1-Sor to a minimum value at Sw=Swi, Figure 17B.

Figure 17 Displacement of oil by water for a concave upwards fractional flow curve (light oil displacement). A = concave upwards fractional flow curve. B = velocity of water saturation. C = characteristics of water saturations Sw.

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The implications of this are that the highest velocity is for the highest water saturation, Sw=1-Sor and that saturations less than this cannot exist since they would be overtaken by the Sw=1-Sor saturation, Figure 17C There is therefore a quick build up of a shock front with a saturation, Swf=1-Sor. The producing characteristics are shown in Figure 18, where , until the shock front arrives water-free oil is produced and thereafter only water is produced. The oil remaining in the reservoir with a saturation of Sor.

Figure 18 Progressive saturation profile for a concave upwards fractional flow curve.

Case 3 Typical medium density oils. Figure 19A presents the fractional flow curve for a medium density and viscosity oil, where the displacement velocities are not unlike field values. The S shaped curve generates the two curvatures we have considered in case 1 & 2. With the corresponding derivative values, fw'. The slope in the fw curve increases from its starting value, Sw=1-Sor and then decreases. Figure 19B

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Figure 19 Displacement of oil by water for a rock with an S-shaped fractional flow curve (light oil displacement). A = S shaped fractional flow curve. B = velocity of water saturation. C = characteristics of water saturations Sw.

The development of the saturation would be such that there would be a steady increase in the velocity of the increasing saturation, but this would reach a maximum at a saturation Swf, where Swi<Swf<(1-Sor). Behind this the velocities would decrease with decreasing Sw, Figure 19C. The impact on the process is such that a shock front is developed, at the value S wf, the saturations greater than this moving at a lower velocity; behind this shock front there is a steady increase in the saturations moving at decreasing velocity. This process is illustrated in Figure 20, which shows that water free oil is produced until breakthrough at a saturation of Swf, and a breakthrough fractional flow of fwbt. The saturation then climbs until it reaches the irreducible oil saturation level when only water is produced.

Figure 20 Progressive Saturation for an S-shaped Saturation Curve.

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