Section 5

CHAPTER 5 IMMISCIBLE DISPLACEMENT - WATERDRIVE

Gordon R. Petrie, Thru-u.com.

Contents:

Page

5.1

Introduction

3

5.2

General Waterdrive Method

5

5.3

Good Cross Flow - Segregated Flow

14

5.4

No Cross Flow – Stiles Method

20

5.5

No Cross Flow – Dykstra-Parsons Method

24

5.6

Waterdrive Production Capacity

29

A-5

Appendix 5 –

32

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5 -

IMMISCIBLE DISPLACEMENT - WATERDRIVE

The basics of immiscible displacement has already been discussed in the previous section, i.e. fractional flow relationship, Buckley-Leverett flow theory and the Welge graphical construction technique: -

The above analytical methods are readily applied to one-dimensional flow where the flow pattern can be described as diffuse.

-

Diffuse flow involves the separate movement of planes of constant water saturation through the reservoir section. Each plane moves at a particular velocity and, depending on the mobility ratio, a shock front can develop.

-

Diffuse flow occurs when the reservoir thickness is much less than the height of the capillary transition zone ( h << H ). Such an assumption is ideal, for instance, when flooding laboratory core plugs.

-

The question is how to retain 1-D flow theory but to apply it in the sort of practical situation outlined below:

NOTE -

-

Flow is not diffuse, but has segregated into distinct and separate oil and water layers.

-

Flow is no longer one-dimensional, but can flow in three spatial directions.

-

Reservoir section is not homogeneous, but is divided up into layers each with distinct properties.

-

In some cases cross-flow exists between layers in some cases these is no cross-flow.

Buckley-Leverett/Welge analysis techniques are only one-dimensional!

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5.1 -

INTRODUCTION The 1-D flow theory already developed will be retained, but care must be taken that the following fundamental factors are woven into the overall method.

-

-

Mobility Ratio of fluids.

-

Reservoir heterogeneity and vertical layering effects.

-

Effects of gravity in conjunction with above two.

The end-point mobility ratio ( M ) dramatically affects the efficiency of the displacement process. Mobility ratio (-) is defined as the mobility of the displacing phase divided by the mobility of the displaced phase.

M =

-

′ / μW krW kro′ / μ o

……….……………….(4.17)

If M ≤ 1 , then the oil will have the same or greater mobility than the water. Therefore, the oil is easily displaced ahead of the water without any tendency

for the water to by-pass or “finger” through the oil. -

In this case Piston-Like Displacement is observed in a microscopic scale all the oil is displaced ahead of the “piston” by an equal volume of water.

-

If M > 1 , then the oil will have a lower mobility than the water. Therefore, the oil moves through the rock less easily than the water and the latter tends to bypass or “finger” through the oil on a microscopic scale.

NOTE -

Waterdrive should not be used in cases where, due to large mobility ratios, there is a need for large quantities Wi per unit N P recovered.

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-

The mobility ratio is already intrinsically included in the fractional flow expression, equation (4.3), by virtue of the presence of μ o & μW terms in its denominator: -

As μ o decreases, M decreases and fractional flow moves to right:

-

1.0

High

fW

Medium Low 0 0

(1 −

SWC

μo

gives M ≤ 1 , waterdrive

becomes more efficient and fractional flow curve moves to right. For 1 M.O.V. of water injected, 1 M.O.V. of oil must be displaced.

μo

0.5

Low

μo

μo

Sor ) 1.0

-

High

μo

gives M > 1 , fractional

flow curve moves left, there is early water breakthrough into producers and more Wi is needed unit of N P recovered.

SW

-

Reservoir heterogeneity must also somehow be incorporated into the method. By heterogeneity it is meant not only vertical layering, but also the existence or otherwise of cross-flow between these layers. -

Heterogeneity should be incorporated into the method so that it affects the fractional flow curve; if heterogeneity improves efficiency, then fractional flow curve should move to the right and vice versa.

-

The effects of gravity must also somehow be incorporated into the method. Including the effects of gravity means more than simply including the “gravity term” - G in the fractional flow equation: -

Gravity should be incorporated into the method so that, in conjunction with heterogeneity, it affects the fractional flow curve; if gravityheterogeneity improves efficiency, then fractional flow curve should move to the right and vice versa.

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5.2 -

GENERAL WATERDRIVE METHOD The general method which should be used to include the effects of mobility ratio, heterogeneity and the influence of gravity is as follows: -

Split interval into N discernible layers.

-

′ krWi . Assign to each layer values for hi ki φi Sori SWCi k roi

-

Run RFT to check on cross-flow between layers.

-

Decide flooding order of the layers.

-

Calculate pseudo-relative permeability curves.

-

Use pseudo-curves to generate fractional flow curve.

-

Use Welge technique to determine Wid versus N pd versus t .

-

Convert Wid to Wi (MM STB), N Pd to N P (MM STB) and fWe to fWs (-).

-

Pseudo-curved reduce this 2-D problem to 1-D. Hence, Buckley-Leverett & Welge can be applied. Vertical Sweep has been modelled analytically.

-

Due to complexity, it is recommended that a numerical simulator be used to model areal sweep efficiency E A ; an analytical method is given in Appendix A5, but it can be difficult to set-up and apply to irregular well spacing: -

Recovery of STOIIP is:

EA

NP = f (E A , EI ) N AREAL SWEEP BETWEEN INJECTORPRODUCER (Five-Spot)

NOTE -

-

EI is Vertical Sweep Efficiency – fraction of oil recovered in a vertical cross-section by waterdrive.

-

E A is Areal Sweep Efficiency – fraction of oil recovered in an areal section by waterdrive

EI is input into simulator as a “known” using pseudo-curves.

Simulator is then used to adjust E A to get a good “history match” on production data - N P / N .

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5.2.1 -

Properties of Layers

The above approach shows how to analytically model Vertical Sweep Efficiency EI . The first step is to split the reservoir section into a number of discernible layers.

-

Dividing the interval into layers is accomplished by examining the permeability-depth data obtained from core samples: -

All appraisal wells should be cored and as many development wells as is necessary to assess the areal extent of any correlated permeability profile.

-

It is important not to use φ − k correlations, since they are not accurate or sensitive enough.

-

Any downdip wells drilled into the aquifer should also be cored.

-

Use linear scales for permeability-depth plots to define layers. Do not use

log k - depth plots, since they are not sensitive enough. -

-

In this way each layer ( i ) can be assigned particular permeability - ki .

Once the layers have been decided based on permeability, the average porosity of each layer can be found from core samples – assign φi to each layer.

-

The thickness of these layers can be found from cores – assign hi to each layer.

-

′ Sori k rWi ′ can be assigned to each From relative permeability data SWCi k roi

layer. Assign data to the layer from which the sample was taken: -

If no sample is available, use data from samples cut from similar rock type. Do not average relative permeability data!

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5.2.2 -

Establishing Cross-Flow Between Layers

In general once the layering has been fixed and properties assigned to each layer it is then necessary to establish the “degree of communication” between layers.

-

The best way to establish this fact is to run an RFT survey: Pressure

Depth

-

Survey in later Development Well

Survey in early Appraisal Well

-

RFT survey measures accurate pressure as function of depth in the open-hole.

-

The initial survey in the first appraisal well shows pressuredepth profile through reservoir.

-

Later in a development well RFT shows uniform pressure decline across entire section.

-

There is good communication between layers.

The above shows good “pressure communication” between layers. The layers are said to be in “pressure equilibrium” or that there is good “vertical equilibrium”; i,e, there is cross-flow between layers.

-

Consider an RFT survey which shows no cross-flow between layers: Pressure

Depth

Survey in later Development Well

Survey in early Appraisal Well

-

Again consider two open-hole RFT surveys.

-

The appraisal well shows initial pressure-depth profile through reservoir.

-

Later in a development well RFT shows non-uniform pressure decline.

-

There is no communication between layers. There is no cross-flow and an absence of pressure equilibrium.

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5.2.3 -

Pseudo-Curves and Recovery Equations

The flooding order of the layers must be decided upon before the pseudorelative permeability curves can be calculated. There are two cases: -

If there is good cross-flow between layers, then the system is in “vertical equilibrium”. Hence layers will flood from bottom to top sequentially.

-

If there is no cross-flow between layers, then the layers will flood in an order dependent on the properties assigned to each layer.

-

Consider the simplest case of good cross-flow between the layers. The water average saturation and relative permeabilities across any section, as each successive layer floods to water, is given by

n

SWn =

∑ hiφi (1 − Sori ) + i =1

N

∑ hφ S

i = n +1

N

∑ hφ i =1

i i WCi

…………(5.1)

i i

n

krWn =

∑ h k k′

i i rWi

i =1

………………………..(5.2)

N

∑h k i =1

i i

N

kron =

∑ h k k′

i = n +1 N

i i roi

…………………………(5.3)

∑h k i =1

i i

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-

Equations (5.1) through (5.3) are solved for each layer as it floods to water, from bottom to the top.

-

The results of these calculations can then be graphed k rW versus SW and kro versus SW . When plotted these functions are called “pseudo-relative permeability” curves: -

These pseudo-curves incorporate both heterogeneity and gravity effects and reduce 2-D problem to 1-D. This permits the use of Buckley-Leverett and Welge techniques to this now, 1-D normalised, waterdrive problem.

-

Just as with laboratory relative permeability curves, these pseudo-curves can be used to obtain a fractional flow curve, now fW versus SW : -

Once the fractional flow curve is obtained the effect of mobility ratio is included. Hence, mobility ratio, heterogeneity and gravity are now all included in this approach.

-

For waterdrive calculations all the expressions derived in Section 4, i.e. before breakthrough, at breakthrough and after breakthrough, still apply. It is customary, however, to use dimensionless pore volume as follows:

Wid =

5.615 × Wi ………………………………….(5.4) ALφ

N Pd =

5.615 × Bo N P ………………….………….(5.5) ALφ

where, Wid = Dimensionless pore volumes of water injected (-). N Pd = Dimensionless pore volumes of oil produced (-).

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(a) -

Breakthrough of Flood Front All the calculations before breakthrough are trivial, because N p d is simply equal to Wid . Moreover, the cumulative water injected, oil produced and time to breakthrough is readily found using the expression listed below.

-

At breakthrough of the flood front into the producing well, Wid is found from the tangent to the fractional flow curve, i.e. equation (4.11)a. This is easily expressed in terms of dimensionless quantities as

Wid =

1

N Pd = Wid

t=

-

………….….……..........(5.6)

⎛ dfW ⎞ ⎜ ⎟ dS W ⎠ S Wf ⎝

ALφWid 5.615 × qi

…………..................………...........(5.7)

.........................................................(5.8)

All the equations derived thus far for Wi and qi are derived so that that these quantities have units of (rb) and (rb/day) respectively. Note that the water F.V.F. (BW) is needed to convert to the more usual units of (STB) and B

(STB/day).

NOTE -

Using above equations Wid N Pd and t can be found at breakthrough.

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(b) -

After Breakthrough of Flood Front After breakthrough equation (5.6) still applies. However, at breakthrough equation (5.6) requires that the tangent to the fractional flow curve, be fixed or pass through the point, SW = SWC and fW = 0 .

-

On the other hand, after breakthrough Wid is found from equation (5.9) below. This looks similar to equation (5.6), but the slope of a tangent to the fractional flow curve is now found at the point SW = SWe .

Wid =

-

1

………….…………....(5.9)

⎞ ⎛ dfW ⎟ ⎜ dS W ⎠ SWe ⎝

After breakthrough equation (5.7) can no longer be used since water is bypassing the in-situ oil; therefore, N Pd is no longer equal to Wid .

-

As discussed in Section 4, the Welge equation can now used to find N Pd . Equation (4.16) is readily expressed in terms of N Pd and Wid using the definitions given by equation (5.4) and (5.5)

N Pd = [(SWe − SWC ) + (1 − fWe )Wid ]

NOTE -

…………....(5.10)

Equation (5.10) is simply the dimensionless form of (4.16), while equation (5.9) is just the dimensionless form of equation (4.14).

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-

After breakthrough a timescale can then be attached to the recovery calculations by the usual expression

t=

-

Wi qi

.............................................................(4.13)

Dimensionless pore volumes (-) of water injected can be converted to actual volumes (rb) by re-arrangement of equation (5.4)

Wi =

-

ALφ × Wid …………………………………...(5.4)a 5.615

Wi at this stage still has units of (rb) and should now be converted into Wi (MM STB) - use the water F.V.F as follows

Wi ( STB ) =

-

Wi ……………………………………(5.11) BW

Dimensionless pore volumes (-) of oil produced can be converted to actual volumes ( MM STB) by re-arrangement of equation (5.5)

NP =

-

ALφ × N Pd ………………….………….(5.5)a 5.615 × Bo

N P is returned by equation (5.5)a in units of (STB), without any further

conversion. Finally, the Recovery Factor ( N P / N ) can be found in terms of N Pd by dividing equation (5.5)a by N - STOIIP.

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(c) -

Recovery Factor & Surface Watercut The expression for N , STOIIP in the linear flooding symmetry block is

N=

-

ALφ × (1 − SWC ) …………………………………(A) 5.615 × Boi

Now dividing equation (5.5)a by equation (A) above leads to the recovery factor by waterflooding alone; waterdrive pressure is constant and F.V.F. is Bo .

N P Boi N Pd = × N Bo (1 − SWC )

-

…………………..(5.12)

The total recovery factor must include both primary and secondary recovery factors. Also, the surface and reservoir watercuts fWS & fWe are simply related.

⎛ N Pd ⎞ ⎞ N P Boi ⎛⎜ ⎜⎜ ⎟⎟ ⎟ = Δ + c p E EFF A ⎟ ( 1 − ) S N Bo ⎜⎝ WC ⎠ ⎠ ⎝

fWS =

NOTE -

1 ⎞ B ⎛ 1 1 + W ⎜⎜ − 1⎟⎟ Bo ⎝ fWe ⎠

………....(5.13)

……………….(5.14)

Recovery Factor can now be plotted versus watercut development.

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5.3 -

GOOD CROSS FLOW – SEGREGATED If good pressure communication (cross-flow) exists between layers and capillary transition zone is negligible, then fluids will tend to segregate into separate water and oil layers. Flow is then classified as segregated:

-

With segregated flow either water is flowing behind the flood front at saturation (1 − Sor ) or oil is flowing ahead of the flood front at saturation SWC .

-

-

In other words either oil or water is flowing. No intermediate water saturations are discernible at the front.

-

This is the exact opposite of Diffuse flow where planes of constant water saturation are moving separately through the reservoir section

For Segregated flow the height of the capillary transition zone is much smaller than reservoir thickness ( H << h ). Segregated flow is usually seen in thick sections of high permeability rock with good cross-flow between layers.

-

For diffuse flow the height of the capillary transition zone is much larger than the reservoir thickness ( H >> h) . Diffuse flow is normally seen in small core plugs, or very tight low permeability rock, which has a high PC .

NOTE -

Most candidate reservoirs for waterdrive are selected on basis of high permeability. Hence segregated flow usually is good assumption.

-

For segregated flow equations (5.1), (5.2) and (5.3) apply, since either oil or water are flowing at their respective k r′ .

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5.3.1 -

Effects of Vertical Heterogeneity

For good cross-flow between the distinct layers, the system will indicate a high degree of “vertical equilibrium”, flow will be segregated and the flooding order of the layers will be from bottom to top.

-

The permeability profile through the reservoir section affects the shape of the pseudo-curves and the shape of the fractional flow curve. This is how heterogeneity and gravity combine to affect the flood performance.

-

Consider the following typical permeability profiles often seen in reservoir sections:

(a) Coarsening Upwards k

k ro′ kr ′ k rW

Depth 0

SWC

-

SW

(1 − Sor )

In this type of reservoir the permeability is coarsening in the upward direction. Apply equations (5.1), (5.2) and (5.3) to find pseudo-curves – see above.

-

These pseudo-curves produce fractional flow curves displaced well to the right, even when M > 1 . This is because gravity is helping to “drag” water down from high permeability top layers. Gravity is counteracting adverse heterogeneity.

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-

In the next type of reservoir the permeability is coarsening in the downwards direction:

(b) Coarsening Downwards k

k ro′ kr ′ k rW

Depth 0

SWC

-

SW

(1 − Sor )

Again, apply equations (5.1), (5.2) and (5.3) to find the pseudo-curves - see typical set of curves alongside for this case.

-

Here the pseudo-curves have the effect of moving the fractional flow curve to the left; this has a strong negative effect on waterdrive. Gravity is actually working to amplify the adverse affect of heterogeneity - seen below:

(a) Coarsening UP

NOTE -

(b) Coarsening DOWN

Gravity assists the “coarsening up”, but gravity adversely affects the “coarsening down” case.

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-

In another type of reservoir the permeability profile is random. A random permeability profile can still be split into layers, but it acts effectively as a homogeneous reservoir with average permeability - k :

(c) Random Permeability Profile - Homogeneous k

k ro′ kr ′ k rW

Depth 0

SWC

-

SW

(1 − Sor )

Once again with good cross-flow all cases layers flood from bottom to top. Flow is segregated and the pseudo-curves can be found from equations (5.1), (5.2) and (5.3), given that either water or oil is flowing at their respective k r′ .

-

In this case, notice that the characteristic pseudo-curves produced by the random vertical heterogeneity are in fact linear.

-

Once again the fractional flow curve can be calculated from the pseudo-curves; the combined effect of mobility ratio, heterogeneity and gravity will then be included in the form that the fractional flow curve is forced to adopt.

NOTE -

Notice how the same method and the same recovery equations can be applied to all these different cases. All that changes are the pseudocurves and the fractional flow curve.

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5.3.2 -

Allowance for Initial Water

The preferred location for injection wells is within the aquifer. Better injection rates are obtained in the absence of relative permeability effects to oil: -

Occasionally the absolute permeability in the aquifer is very low due to changes in rock properties caused by the presence of water. Then injection well may have to be completed in the oil column.

-

However, assuming that the injection well is in the aquifer, there will be an initial water level filling the symmetry block as shown below: -

This static oil level will affect the recovery calculations.

-

Not as much water will need to be injected Wid .

-

Not as much oil will be produced - N Pd

L h

θ

-

Reservoir symmetry element has height, h , length, L , and width, W . PVIW is the water saturated pore volume divided by the total pore volume

PVIW

1/ 2 × h × h ×W ×φ tan θ = WLhφ

∴ PVIW =

NOTE -

h 2 × L tan θ

…………………….(5.15)

This is the fraction of the total pore volume saturated with water

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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Rev. 1.0

-

Recovery calculation using the method discussed and the equations presented are carried out assuming no initial water. A correction is subsequently made to * reduce Wid and N Pd to Wid* and N Pd respectively.

-

First, estimate how many dimensionless pore volumes of water would need to be injected into oil zone to achieve the same initial water level in reservoir block. It is PVIW times the moveable oil saturation, or

Reduced Injection Water Needed =

-

The actual water injected making allowance for initial water level Wid* is

Wid* = Wid −

-

h × (1 − Sor − SWC ) 2 × L tan θ

h × (1 − Sor − SWC ) 2 × L tan θ

……….(5.16)

Next estimate how many dimensionless pore volumes of oil cannot be recovered by the presence of an initial water layer, i.e. PVIW times the initial oil saturation

Reduced Oil Produced =

-

h × So 2 × L tan θ

* The actual oil produced making allowance for initial water level N Pd is

* = N Pd − N Pd

h × (1 − SWC ) 2 × L tan θ

………….(5.17)

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5.4 -

NO CROSS FLOW – STILES METHOD If there is no cross-flow between layers then there is no “pressure equilibrium” between adjacent layers. This means that fluids travel independently through each layer and the flooding order is no longer from bottom to top.

-

Flooding order now depends on the fluid/rock properties of the various layers. If M = 1 , then Stiles Method is used. If M ≠ 1 , then Dykstra-Parson’s Method is used.

-

If M = 1 and no cross-flow exists then fluid will travel independently down each layer with perfect “Piston-Like” displacement. -

Flow is no longer segregated but, as will be seen, similar methods to those discussed previously, in the segregated flow section, can be applied in this case.

-

Layers are defined in much the same way as before, except now there are distinct shale breaks between the layers to help identify them. For thick layers where cross-flow within the layer can be assumed, then distinguish between: k

k

SAND

SAND

Depth

Depth

NOTE -

Left hand case, coarsening-up, results in piston-like displacement across all three layers. Therefore they count as one layer.

-

Right hand case, coarsening-down, results in three separate layers.

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5.4.1 -

Flooding Order of Layers

Stiles Method assumes no cross-flow and mobility ratio of one. From Section 4 it was shown that the velocity at which a shock front moved through the i th layer of a reservoir section is

vi =

-

5.615 × qi ⎛⎜ dfW . ⎜ dS Aiφi ⎝ W

S Wf

⎞ ⎟ ⎟ ⎠i

If the M = 1 , then tangent to the fractional flow curve at breakthrough in any i th layer must have slope

⎛ df ⎜ W ⎜ dS ⎝ W

-

Substituting above equation into former and recognising that Ai = Whi leads to

vi =

-

SWf

⎞ 1 ⎟ = ⎟ 1− S − S ori WCi ⎠i

5.615 × qi Whiφi (1 − Sori − SWCi )

Now Darcy’s Law from Section 3, written for the reservoir flowrate of water entering any i th layer is

qi = −1.127 × 10−3 ×

-

′ Whi dpW ki krWi μW dl

The minus sign in the above equation can be dropped if dpW / dl is simply written as a pressure drop without regard to its implicit minus sign.

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-

Substituting the latter equation into the former gives the velocity of the advancing water flood front through any i th layer

′ dpW 6.328 × 10−3 × ki krWi vi = μW φi (1 − Sori − SWCi ) dl

-

Since displacement is stable, oil and water must have the same velocity. Hence, Darcy’s Law can be applied to the velocity either side of the flood front

M

-

dpW dpo = dl dl

Since M = 1 then the pressure gradient through water, behind the front, must equal the pressure gradient through oil, ahead of the front. Since these gradients are obviously linear, they must also both be equal to Δp / L .

-

Δp is overall pressure drop from injector to producer and L is the distance along the bedding plane between them. Now, substituting for dpW / dl gives

′ Δp 6.328 × 10−3 × ki krWi × vi = μW φi (1 − Sori − SWCi ) L

………………..(5.18)

where, vi = Velocity of “piston” flood front through i th layer (ft/day).

NOTE -

All other variables have normal field units. The quantity Δp / μW L will be the same for each layer. Hence (5.18) gives flooding order of layers.

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Rev. 1.0

-

(

)

Since the parameter 6.328 × 10−3 × .Δp / μW L is common to all layers. The order

that each layer will flood depends on the magnitude of the parameter ai which is unique to each layer. Notice that velocity of flood front, vi ∝ ai :

ai =

-

′ ki krWi φi (1 − Sori − SWCi )

……………………..(5.19)

The first step in Stiles method is to re-order the layers, so that layer with largest ai (first to food out) is placed at the bottom, next layer from bottom has the next

largest ai - so on until all layers have been re-ordered.

6 5 Layers ordered in terms of decreasing

4 3 2

ai

1 Layers are re-ordered in this sequence; original layering was different.

-

Once this has been done the situation is similar to segregated flow case. Equations (5.1), (5.2) and (5.3) can be used to calculate pseudo-curves. Then the fractional flow and Welge recovery calculations are completed just as before.

NOTE -

Theoretically successive points on the pseudo-curves, should be joined by step functions. However, if sufficient layers have been identified this is not necessary - smooth curves can be used.

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5.5 -

NO CROSS FLOW – DYKSTRA-PARSONS The only difference between the no cross-flow, Stiles method, and the crossflow method considered previously is as follows:

-

In the case of good cross-flow where flow is segregated, i.e. "coarseningup", "coarsening down" or "random" models, the flooding order is from bottom layer to top layer in all cases.

-

Stiles method assumes no cross-flow between the N layers with pistonlike displacement in each layer. In this case, the flooding order depends on the magnitude of ai .

-

In Stiles method the layers are re-ordered so the layer with highest ai is placed at the bottom, the layer with next highest ai second and so on, until finally the Nth layer, with smallest ai , is ordered on top.

-

In the case of good cross-flow where flow is segregated, water gradually breaks through at the producing well. However, with Stiles method, breakthrough in any given layer occurs in a stepwise fashion.

-

Although, given enough layers, this stepwise breakthrough tends to appear as a gradual breakthrough; a situation that is identical to ignoring “step pseudocurves” and drawing “continuous pseudo-curves” instead.

-

Hence, there is in fact little difference between the “cross-flow” and the “no cross-flow” cases - except for the "layer re-ordering". All methods use pseudocurves, Buckley-Leverett/Welge plus a common set of recovery equations.

-

The last case to be considered is where there is a “no cross-flow”, N-layer system, but where M ≠ 1 . The approach used is called Dykstra-Parsons method.

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-

In the case of Stiles method ( M = 1 ), the flood fronts in any layers maintain the same velocity between the injection and production wells - there is no tendency for any of the fronts to either speed up or slow down.

-

In the case of Dykstra-Parsons method ( M < 1 ), the flood fronts decelerate as they move along any layer. This has the effect of stabilising breakthrough between layers.

-

For Dykstra-Parsons method ( M > 1 ), the flood fronts accelerate as they move along any layer. This has the effect of destabilising breakthrough between layers.

-

Consider the case below ( M ≠ 1 ), where some layer (say i th layer) has flooded completely with water to some distance l along the layer (total length L ): l

i th layer

Water

OIL

Position of flood front in th

the i layer is a distance l from injection well.

L

-

The fractional distance travelled by flood front is x and x = l / L , so that dl = Ldx . Stable piston-like displacement is assumed, therefore oil and water

must have the same velocity. Applying Darcy’s Law either side of the flood front gives

M

-

dpW dpo = dx dx

This is the same equation used in Stiles method, except that the substitution has been made, dl = Ldx ; the “ L ” cancels on both sides of the equation. Notice, since M ≠ 1 , the pressure gradients are different through oil & water sections.

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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-

A full derivation of the displacement theory and equations associated with the Dykstra-Parsons method is given in Appendix A5.

-

First, the layers are ordered in terms of decreasing λ i , where this group is defined by equation (A5.1.4) and depends on the layer properties.

λi =

-

′ ki k rWi ……………………………….(A5.1.4) φi (1 − Sori − SWCi )

When layer 1 breaks through to water, the fractional position of the flood front in all other unflooded layers is found from equation (A5.1.6), where i = 1 and j = 2,3,4....N ; notice this is a quadratic expression in x .

∴

-

λ ⎛1 A 2 ⎞ x j + x j = j ⎜ A + 1⎟ ……………………….…(A5.1.6) 2 λi ⎝ 2 ⎠

Layer 2 will now be the layer with the next lower value of λ i . Again, solve equation (A5.1.6) for position of flood front in each of the remaining unflooded layers when layer 2 just floods out; now, i = 2 and j = 3,4....N :

LAYER FLOODING ORDER 1st Layer

2nd layer

( λ1 = ......... ) ( λ2 = ........ )

3rd Layer

4th Layer

5th Layer

( λ3 = ........ )

( λ4 = ........ )

( λ5 = ........ )

x1

1.0

x2

x2 =

1.0

x3

x3 =

x3 =

1.0

x4

x4 =

x4 =

x4 =

1.0

x5

x5 =

x5 =

x5 =

x5 =

1.0

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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-

Next step is to use equation (A5.1.7) to calculate the fractional flow as each layer in turn as it floods to water

n

fW ≈

-

hi ki

∑ A +1

i =1 N

hi ki ∑ i =1 Axi + 1

……………………………..…..(A5.1.7)

Taking five layers as an example, the calculation of the fractional flow is greatly assisted if the following table is constructed. Each element in the table is a numerical value for the group - hi ki / ( Axi + 1) :

hi ki / ( Axi + 1)

ki hi (md-ft) 1st Layer Floods Layer 1

k1h1 = ....

Layer 2

k2 h2 = ....

Layer 3

k3h3 = ....

Layer 4

k4 h4 = ....

Layer 5

k5h5 = ....

2nd Layer Floods

3rd Layer Floods

4th Layer Floods

5th Layer Floods

fW from equation (A5.1.7) SW from equation (5.1)

-

The values of hi ki / ( Axi + 1) are computed for each layer, when layer 1 floods to water ( x1 = 1 , but all other x-values are less than 1). The process is repeated when layer 2 floods to water (now x1 = x2 = 1 ), and so on until all layers flood.

-

The fractional flow is calculated, at the base of column 1, by summing all the elements above the blue line and dividing by the sum of all the elements in the column. The average water saturation is found as before from equation (5.1).

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-

Given sufficient number of layers (five would be too small in practice and was only used as a short illustration of the method), then a smooth fractional flow versus average water saturation curve can be drawn: fW

-

None of the fractional flow curves show any point of inflexion which is characteristic of this model.

-

When mobility ratio is favourable curve is displaced to the right as expected.

-

If mobility ratio is unfavourable then the curve is displced to the left.

Unfavourable Mobility Ratio

Favourable Mobility Ratio

SW

-

The fractional flow curve is obtained without need to find the psuedo-curves. Buckley-Leverett/Welge can now be applied. All the previously developed recovery equations can also be used.

-

The calculations thus far ignore the Areal Sweep Efficiency ( E A ). This approach is well justified since, not long after breakthrough, E A approaches one. This is particularly true in most well engineered waterdrive projects where M ≈ 1 .

-

As previously discussed, EI can be obtained from pseudo-curves. EI is then input into a numerical simulator as a “known” quantity. The simulator is then used to “history match” past production data using E A as adjustment parameter.

-

The Areal Sweep Efficiency ( E A ) is altered by installing faults and various noflow boundaries. Once a good history match is obtained, both E A & EI are fixed and the simulator can then be run in predictive mode.

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5.6 -

WATERDRIVE PRODUCTION CAPACITY In the case of Primary Recovery, production is sustained principally by the expansion of pore fluids and a decline in reservoir pressure.

-

In the case of Secondary Recovery by waterdrive, the reservoir pressure is usually maintained a constant level somewhere around the bubble-point. Production is now sustained simply by the displacement of oil by water.

-

The basic production equation of waterdrive states that the total rate of water injected into reservoir, qWi , must equal combined rates of oil, qo , and produced water, qWp , recovered from reservoir - all in units of (rb/d):

GAS

(qWi )SC (qo )SC qo

qWp

OIL

(q )

Wp SC

WATER

qWi

-

For a reservoir at constant pressure above its bubble-point, the displacement equation for an engineered waterdrive is given simply by:

(qWi )SC = (qo )SC Bo + (qWp )SC BW

……………….(5.20)

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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-

The subscript “ SC ” is used in equation (5.20) to empasise that these flowrates must have units of (STB/d). The term (qo )SC Bo is the same as qo - both have units of (rb/d). Also, (qWp )SC BW is the same as qW - both have units of (rb/d).

-

It will be noticed that (qWi )SC also has units of (STB/d) but there is no volume factor to correct this to (rb/d). This is because, as mentioned before, BW for injected water, which must be free of gas, is around one.

-

Assuming an estimate of recovery factor, N P / N , versus surface watercut, fWS , is available then equation (5.20) can be used as follows:

fWS -

This data may be available as a result of an initial numerical simulation study, based on appraisal well data.

-

It could also be available from actual field production information, taken from reservoirs producing in the same area.

NP / N

-

In addition to the above, a production schedule target, i.e. (qo )SC versus years, will normally be set by management.

-

Hence, knowing (qo )SC in (STB/d) from this financial target, a figure for N P may be estimated to year-end. Assuming that an estimate for N is available, then N P / N to year-end can be estimated.

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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-

The projected year-end surface watercut fWS can then be found from the targetted year-end recovery factor N P / N using the graph on previous page.

-

The surface watercut is defined as STB/d of water produced at surface divided by total STB/d of oil and water produced at the surface, or

fWS =

(q ) (q ) + (q ) Wp SC

o SC

-

……………………………….(5.21)

Wp SC

Knowing (qo )SC from the production target and fWS from the graph of fWS versus N P / N , an estimate can be made for the surface flowrate of produced water (qWp )SC from equation (5.21).

-

Knowing both (qo )SC and (qWp )SC then equation (5.20) can be used to predict the required surface water injection flowrate (qWi )SC which will be needed to sustain the targetted oil production flowrates.

-

A year-end list of the following surface flowrates can then be prepared: -

Total STB/d of oil plus water, (qo )SC + (qWp )SC . The maximum value appearing on the list determines the main oil/gas Separator Capacity.

-

Total STB/d of produced water, (qWp )SC . The maximum value appearing on the list determines the Oily-Water Treatment Capacity.

-

Total STB/d of injection water, (qWi )SC . The maximum value appearing on the list determines the Water Injection Capacity.

NOTE -

Water injection plant includes de-aerators, water treatment, highperformance filtration equipment and water injection pumps.

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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APPENDIX 5

Dykstra-Parsons Method

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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APPENDIX 5 (I) -

Dykstra-Parsons Method From the main section, the pressure gradient through the oil, ahead of the front, and the water, behind the front, was given by

∴

-

dpW 1 dpo = dx M dx

Now the oil pressure gradient is the overall Δp , minus the pressure drop through the water, divided by the distance from the front to the producing well

dpo = dl

-

dpW dl L−l

Δp − l

Changing from dl to dx , using dl = Ldx , leads to

dpo = Ldx

dp ∴ o = dx

-

dpW Ldx L−l

Δp − l

dpW dx 1− x

Δp − x

Substitute the above expression for dpo / dx into expression at top of page

dpW dp dx ∴ W = M (1 − x ) dx Δp − x

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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-

The two terms with dpW / dx can be brought over to the LHS, then the equation can be re-arranged to give Δp dpW M = ∴ dx Ax + 1 where,

∴A=

-

1 −1 M

………………………………………(A5.1.1)

Notice, the sign of “ A ” is a key factor, which depends on the mobility ratio. From Stiles method the velocity of the flood front can be expressed as

vi =

-

′ dl 6.328 × 10−3 × ki krWi dp × W = dt μW φi (1 − Sori − SWCi ) dl

Now substituting for dpW / dl and knowing that dl = Ldx gives

Δp ′ Ldx 6.328 × 10− 3 × ki krWi ML × = dt μW φi (1 − Sori − SWCi ) Ax + 1

∴

-

′ dx 6.328 × 10−3 × ki krWi Δp × = 2 dt μW φi ML (1 − Sori − SWCi ) Ax + 1

…..…(A5.1.2)

Notice that the velocity of the flood front changes with x . If M < 1 , A > 0 , as x increases, then velocity of front, dx / dt , decreases and vice versa. When

M = 1 , A = 0 and then dx / dt is constant for all x , which is Stiles result.

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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-

Equation (A5.1.1) and (A5.1.2) enables the flood front velocity to be found within any layer of the system. Notice that vi = f (M , layeri & x )

-

Now separate the variables in equation (A5.1.2), so that the fractional distance travelled by the flood front, x , after any time, t , can be found. The term outside the integral sign is a constant for any layer

xi

∴ ∫ ( Ax + 1)dx = 0

∴

-

′ × Δp t 6.328 × 10−3 × ki krWi dt μW φi ML2 (1 − Sori − SWCi ) ∫o

′ × Δp 6.328 × 10−3 × ki krWi A 2 ×t xi + xi = 2 2 μW φi ML (1 − Sori − SWCi )

Notice that xi = f (M , layeri & t ) . It is written for the i th layer and is a quadratic function of xi . Now re-arranging layer specific terms to one side leads to

−3 ⎞ 1 6.328 × 10 × Δp ⎛A 2 ∴ ⎜ xi + xi ⎟ = ×t = c×t μW ML2 ⎠ λi ⎝2

….…(A5.1.3)

where,

λi =

′ ki k rWi ……………………………….(A5.1.4) φi (1 − Sori − SWCi )

and, 6.328 × 10−3 × Δp ………………………………..(A5.1.5) c= μW ML2

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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-

Now, if at time t layer i has just flooded to water, the question is; where is the position of the flood front in any other, say j th layer?

-

To answer this question, recognise that the RHS of equation (A5.1.3) is identical for all layers at time t . Hence the LHS of equation (A5.1.3) is the same for both the i th layer and the j th layer, or

⎞1 ⎞1 ⎛A 2 ⎛A 2 ⎜ xi + xi ⎟ = ⎜ x j + x j ⎟ ⎠ λj ⎠ λi ⎝ 2 ⎝2

-

However if the flood front in the i th layer has just broken through to water, then xi = 1 and the above equation reduces to

∴

-

λ ⎛1 A 2 ⎞ x j + x j = j ⎜ A + 1⎟ λi ⎝ 2 2 ⎠

…………….…(A5.1.6)

The order in which the N-layers will flood will be that of decreasing λi and this parameter is found for each layer using equation (A5.1.4).

-

As each layer breaks through to water, the above equation enables the position of the flood front in all other layers to be calculated. Hence it is possible to calculate the fractional flow curve.

-

To derive an expression for fW start with equation (A5.1.2), expressed as

dx cλi = dt Ax + 1

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-

Now, the flowrate in any layer qi (rb/day) can be found by multiplying the above expression by the height of the layer hi and the section width W .

qi =

-

LWhi ⎛ cλi ⎞ ⎟ ⎜ 5.615 ⎝ Ax + 1 ⎠

fW is simply the total flow from all layers producing water (say n), divided by the total flowrate from all layers (say N) producing oil or water.

-

For the n layers already producing water, the fraction position of the flood front in these layers is one. However, some layers are still producing oil. For these layers x ≠ 1 . Hence fW becomes simply

n

n

fW =

∑ (q ) 1=1

n

∑ (q ) i =1

-

i Water

i W

+

=

N

∑ (q )

i = n +1

i Oil

⎡

LWhi cλi

⎤

∑ ⎢ 5.615 × ( A + 1)⎥

⎦ ⎣ ⎤ ⎡ LWhi cλi ∑ ⎥ ⎢ i =1 ⎣ 5.615 × ( Axi + 1) ⎦ i =1

N

Cancelling out all the common terms within each summation leads to

⎛ hi λi ⎞ ⎜ ⎟ ∑ A +1⎠ ≈ fW = Ni =1 ⎝ ⎛ hi λi ⎞ ⎜⎜ ⎟⎟ ∑ i =1 ⎝ Axi + 1 ⎠ n

NOTE -

n

hi ki

∑ A +1

i =1 N

hi ki ∑ i =1 Axi + 1

……………..(A5.1.7)

The fractional flow can be obtained directly from (A5.1.7)

Section 5 © Copyright: Thru-u.com Ltd. 2000.

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