Section 2

CHAPTER 2 MATERIAL BALANCE & OIL RESERVOIRS

Gordon R. Petrie, Thru-u.com. Contents:

Page

2.1

Introduction

2

2.2

Volume Correction Factors

3

2.3

Gas Solubility

8

2.4

Stock Tank Oil Initially In Place (STOIIP)

10

2.5

Material Balance Equation - Schilthuis Formulation

11

2.6

Material Balance Equation - Havlena & Odeh Formulation

17

2.7

Drive Mechanism Identification

19

2.8

Volumetric Depletion Drive

20

2.9

Natural Water Drive

26

2.10

Gascap Drive

28

A-2

Appendix 2 – Water Influx Models

31

Section 2 © Copyright: Thru-u.com Ltd. 2000.

1 Rev. 1.0

2 -

MATERIAL BALANCE & OIL RESERVOIRS

One of the major problems in Reservoir Engineering is that the object of study, i.e. the reservoir itself, is far removed from direct observation: -

If variables cannot be directly observed, they cannot be measured.

-

Hence, there are usually too few independent equations linking together a larger number of unknown variables.

-

By assuming values for these unknown variables a solution can be obtained. However, there is a distinct lack of “uniqueness” – meaning many variations in the initial assumptions can lead to the same result.

-

A principle objective, therefore, must be to limit the number of assumptions made.

2.1 -

INTRODUCTION Modern reservoir engineering work revolves around the construction of computerised reservoir simulators. These simulators require a very large amount of input data, which means that many assumptions must be made. The results obtained, therefore, often simply reflect the input data.

-

Material balance methods are simple analytical techniques for finding G (GIIP) or N (STOIIP), while at the same time helping to determine and quantify reservoir drive mechanisms.

-

Material balance results can then be used to fix some reservoir simulator input parameters. Hence, reducing number of assumptions needed.

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2 Rev. 1.0

2.2 -

VOLUME CORRECTION FACTORS The material balance equation has already been discussed in Section 1 in relation to gas reservoirs. This discussion will now be extended to oil reservoirs.

-

The material balance equation treats the oil (or gas) reservoir as a single unit, i.e. the reservoir properties can vary with time, but are uniform throughout the reservoir at any given instant.

-

Hence no detailed geological model of the reservoir is needed. All that is required are pressure-production and p − v − T data.

-

For oil reservoirs a number of volume factors must be defined to switch between surface and reservoir volumes. The gas expansion factor, E , which has already been discussed, is easily calculated from equation below

E = 35.37

p ..............................................................(1.9) ZT

where, p = Pressure of gas at reservoir conditions (psia). T = Temperature of gas at reservoir conditions (R). Z = Compressibility factor of gas at reservoir conditions (-).

NOTE -

The gas expansion factor E (SCF/CF) is the volume occupied by one cubic foot (CF) of gas at reservoir conditions, when brought to the surface at Standard Conditions (SCF).

-

The “Standard” in SCF means the volume is measured at Standard Conditions – usually 14.7 psia and 60oF (520oR).

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3 Rev. 1.0

2.2.1 -

Gas Formation Volume Factor - Bg B

In oil reservoirs the pore volume occupied by free gas is expressed in reservoir barrels (rb) and not (CF). The Gas Formation Volume Factor Bg is therefore related to the gas expansion factor E through the expression

Bg =

1 5.615 × E

...................................................(2.1)

where,

Bg = Gas Formation Volume Factor (rb/SCF). E = Gas Expansion Factor (SCF/CF). 5.615 = Conversion factor (CF/rb).

1 SCF of Surface gas

Reservoir T&P

© Copyright: Thru-u.com Ltd. 2000.

The Gas Formation Volume Factor Bg :

-

Take 1 SCF of surface gas.

-

What volume does it occupy, in reservoir barrels (rb), as free gas in the reservoir?

-

Bg is the latter quantity divided by the former. It is typically around 0.001 rb/SCF

Bg rb of free gas

Section 2

-

-

E is typically around 170 SCF/CF.

4 Rev. 1.0

2.2.2 -

Solution Gas Oil Ratio – RS

In the definition of Bg the emphasis is on the ratio between the volume of free

gas (rb), under reservoir conditions, to the volume of 1 SCF of gas at surface conditions: That is Bg in no way accounts for the volume of gas which was once

-

dissolved in the oil and has since been liberated as it passes through wellbore and surface equipment.

-

The Solution Gas Oil Ratio ( RS ) is the number of SCF of gas, at standard conditions, that will dissolve in 1 STB of oil – when taken down to the reservoir and subjected to reservoir temperature and pressure:

1 STB of oil

Reservoir T&P

NOTE

-

+

RS SCF of Surface gas

Oil plus Dissolved gas

Bg (rb/SCF) and E (SCF/CF) can be determined analytically using equation (1.9) and equation (2.1).

-

RS (SCF/STB) cannot be calculated – can only measured in p-v-T experiments.

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5 Rev. 1.0

2.2.3 -

Oil Formation Volume factor - Bo

The Oil Formation Volume Factor Bo (rb/STB) is the volume, under reservoir conditions (rb), occupied by 1 STB of oil plus all the gas ( RS SCF/STB) it is capable of dissolving at reservoir T & p:

1 STB of oil

Reservoir T&P

+

RS SCF of Surface dissolved gas

Bo rb of oil plus dissolved gas

-

The more dissolved gas, the larger the oil formation volume factor: -

Low values of Bo ≈ 1.05 rb/STB suggests low volatility oil with little dissolved gas, i.e. RS < 60 SCF/STB.

-

For medium volatility oil Bo ≈ ≈ 1.25 rb/STB, RS ≈ 300 - 600 SCF/STB.

-

For high volatility oil Bo ≈ 2.5 - 3.0 rb/STB, RS > 2500 SCF/STB.

NOTE

-

As before, the oil volume factor Bo (rb/STB) cannot be calculated it can only be measured in p-v-T experiments.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

6 Rev. 1.0

2.3 GAS SOLUBILITY -

Gas solubility in oil is a complex function of temperature, pressure and composition - both crude oil and gas phases. For a particular composition, it is generally true that the amount of gas that will dissolve in the oil increases:

-

-

As the pressure increases, keeping temperature constant.

-

As the temperature decreases, keeping the pressure constant.

Gas is infinitely soluble in crude oil. Only the prevailing temperature and pressure and the quantity of gas which is available limit the ultimate solubility.

-

As far as gas solubility is concerned, the oil can be described as being

undersaturated or saturated: -

Undersaturated oil is one that is capable of dissolving more gas at prevailing T & p than it presently contains.

-

Saturated oil is one that cannot dissolve any more gas at the prevailing T & p .

-

The way to determine whether or not an oil is saturated or undersaturated at prevailing T & p is to decrease the pressure keeping the temperature constant: -

No gas will be released from solution if oil is undersaturated at these conditions.

-

Gas will be released from solution if the oil is saturated at these conditions.

-

If the pressure of undersaturated oil sample is gradually dropped eventually it will become saturated, i.e. oil reaches its bubble-point. Further expansion below the bubble-point pressure will cause dissolved gas to break out of solution.

NOTE

-

If oil is undersaturated then p > pbp , i.e. oil is above bubble-point.

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7 Rev. 1.0

-

An undersaturated oil implies that not enough gas has percolated through the oil, during geological time, to saturate it at prevailing T & p :

-

-

This implies no free gas in contact with the crude oil.

-

Which, in turn, implies that no gascap has accumulated updip.

Consider the shape of RS vs. p and Bo vs. p plots, both above the bubblepoint and below the bubble-point:

p < pbp

p > pbp

600

-

Above the bubblepoint the value of RS is constant.

-

Above bubble-point no gas is liberated from solution.

-

Above the bubblepoint oil is undersaturated.

-

Below the bubblepoint gas is liberated and RS decreases.

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Above the bubblepoint the oil expands due to compressibility of oil.

-

Above bubble-point no gas is liberated.

-

Below the bubblepoint gas is liberated and oil volume shrinks.

RS = RSi

RS (SCF/STB)

300

Bubble Point Pressure

0 0

1000

2000

3000

4000

Pressure (psia)

1.4

BO (rb/STB) 1.2

1.0 0

1000

2000

3000

4000

Pressure (psia)

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8 Rev. 1.0

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For an undersaturated reservoir, the instantaneous gas oil ratio (R) is the same as the initial gas oil ratio, i.e. R = RSi ; that is, there is no free gas in the reservoir – only liquid oil with its dissolved gas is produced into the wellbore.

-

After some initial depletion, the pressure of the reservoir drops to its bubble point and the oil becomes saturated. A further drop in reservoir pressure causes gas to be released in the reservoir as free gas; i.e. it comes out of solution.

-

Gas then accumulates in the reservoir. When the gas saturation exceeds some critical gas saturation, it becomes mobile and begins to flow as free gas into the wellbore. A typical plot of the instantaneous gas oil ratio, R is shown below:

-

When p > pbp then

R = RSi and instantaneous GOR is same as initial GOR. -

R (SCF/STB)

When p < pbp then free gas accumulates in reservoir.

-

Since free gas mobility greatly exceeds oil mobility

R >> RSi

R = RSi -

Bubble-point Pressure

As gas becomes depleted R peaks then starts to decline.

Reservoir pressure (psia)

-

Critical gas saturation can occur somewhere around 10% PV.

NOTE

-

Solution gas usually does not form a secondary gascap. It cannot usually be controlled, to any significant extent, by workovers.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

9 Rev. 1.0

2.4 STOCK TANK OIL INITIALLY IN PLACE -

The expression for Stock Tank Oil Initially In Place, STOIIP, at the time of discovery is similar in form to the expression for Gas Initially In Place, G

N =

Vbφ (1 − SWC ) Boi

……………………....(2.2)

where, N = Stock Tank Barrels of Oil Initially in Place (STB).

Vb = Net bulk volume of the reservoir (rb).

φ = Porosity - fraction of bulk volume which is pore space (-). SWC = Connate Water Saturation - fraction of PV occupied by water (-). Boi = Oil FVF - volumetric conversion factor (rb/STB).

-

Stock Tank Barrels (STB) is the volume of oil in barrels at Standard stock tank Conditions - taken to be 14.7 psia and 60oF.

-

The Pore Volume, PV, is the volume of the pore space, in reservoir barrels (rb), which can be fluid filled – PV = Vbφ = NBoi /(1 − SWC ) .

-

The Oil Hydrocarbon Pore Volume, HCPV, is the volume occupied by oil only, in reservoir barrels (rb) – Oil HCPV = Vbφ (1 − SWC ) = NBoi .

NOTE -

SWC is the water left behind in the reservoir after the accumulation of hydrocarbons. φ & SWC are determined from logs/cores. Boi , is obtained from p − v − T studies on oil samples at time of discovery.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

10 Rev. 1.0

2.5 MATERIAL BALANCE EQUATION - Schilthuis Formulation -

The oil material balance equation is derived in much the same way as the gas material balance equation.

-

Total cumulative volumetric production of oil & dissolved gas, plus production of free gas, LH cylinder - lower diagram, must be equal to total volumetric expansion of pore fluids from initial conditions to the current stage of depletion (lower volume - both cylinders, minus upper volume - RH cylinder only):

PRODUCTION

-

Consider initial condition with reservoir charged at initial pressure pi .

-

At this stage no fluids have been produced.

-

At later stage, reservoir pressure has depleted to lower pressure p .

-

Cumulative production must equal expansion of original pore fluids from pi to lower pressure

pi RESERVOIR

PRODUCTION

p RESERVOIR

p.

NOTE -

However, production must be expressed in terms volume occupied by hydrocarbons at current reservoir pressure p .

-

All that are known are measured surface production volumes. Hence, these must be converted into reservoir volumes at current pressure p .

Section 2 © Copyright: Thru-u.com Ltd. 2000.

11 Rev. 1.0

-

Surface production is measured as cumulative surface oil produced N P (STB) and cumulative surface gas produced GP (SCF). These surface volumes produced must be converted to reservoir volumes at current pressure p .

-

In gas reservoirs all the surface gas produced existed in the reservoir as free gas at reservoir pressure p . For oil reservoirs, some of the surface gas is dissolved gas and some, depending on pressure, can be free gas at reservoir pressure p .

-

Hence, cumulative surface production of oil and gas must be converted to cumulative reservoir production of the following: oil + its dissolved gas at reservoir pressure p ; plus free gas at reservoir pressure p , or

(oil+dissolved gas) + (rb)

N P Bo

+

(free gas) (rb)

N P ( RP − RS ) Bg ....................…....(2.3)

where, N P = Cumulative oil produced - surface conditions (STB).

Bo = Oil FVF - referred to pressure p (rb/STB). RP = Cumulative gas oil ratio GP/NP - surface conditions (SCF/STB).

RS = Dissolved GOR - referred to pressure p (SCF/STB). Bg = Free gas FVF - referred to pressure p (rb/SCF).

NOTE -

Equation (2.3) mathematically represents the volume of oil, its dissolved gas and the free gas within LH cylinder- lower diagram.

-

This is cumulative production referred to current reservoir pressure p . Notice only surface volumes and p − v − T parameters are needed.

-

All the terms in equation (2.3) are known quantities.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

12 Rev. 1.0

-

There are several effects that simultaneous lead to production of pore fluids: -

(A) Expansion of oil plus expansion of liberated gas.

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(B) Expansion of gascap gas.

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(C) Expansion of immobile connate water.

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(D) Reduction in pore volume as pore pressure decreases.

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(E) Natural water influx into space occupied by hydrocarbons.

(A) Expansion of Oil & Expansion of Liberated Gas - dVo -

By re-arranging equation (2.2) the oil HCPV, including dissolved gas, at initial pressure pi is NBoi (rb). Once reservoir pressure drops to p the notional expanded oil HCPV, including dissolved gas, would be NBo (rb).

-

Hence, notional expansion of all this original oil plus dissolved gas is the difference between the notional final volume minus the actual initial volume (rb) N ( Bo − Boi )

-

In addition, free gas may also be released if the pressure drops below pbp . Hence, expansion as free gas is liberated from solution is N ( RSi − RS ) SCF. Therefore, the expansion caused by liberated gas (rb) must be (rb) N ( RSi − RS ) Bg

-

(rb)

Hence, expansion of oil (and dissolved gas) plus expansion of liberated gas is (rb) dVo = N ( Bo − Boi ) + N ( RSi − RS ) Bg …….............….…..(A)

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13 Rev. 1.0

(B) -

Expansion of Gascap Gas - dVg Define the ratio of the HCPV of gascap to the HCPV of the oil as “ m ”

m=

-

HCPV .of .Gascap HCPV .of .Oil

Initial HCPV of gascap, referred to initial volume of oil, must be mNBoi

(rb) mNBoi

-

Notional expanded volume of original gas HCPV to lower reservoir pressure p

(rb) mNBoi

-

Bg Bgi

Therefore, the expansion of the gascap dVg after some decline in pressure is the notional final gas volume (rb) minus the actual original gas volume (rb):

(rb) ⎛ Bg ⎞ − 1⎟ ….……………………………….(B) dVg = mNBoi ⎜ ⎜B ⎟ ⎝ gi ⎠

NOTE -

These expansion terms (A) and (B) are both equal to the notional expanded volume minus actual original volume.

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14 Rev. 1.0

(C) -

Expansion of Immobile Connate Water - dVCW Expansion connate water dVCW (rb) from initial reservoir pressure pi to any

lower pressure p , at some future stage of depletion, is therefore given by

(rb) dVCW = cWVW Δp = cW

(1 + m )NBoi S Δp ………………....(C) (1 − SWC ) WC

where, cW = Isothermal compressibility of water (psi-1).

Δp = Pressure drop from pi to p (psi).

⎛ (1 + m )NBoi ⎞ VW = ⎜⎜ SWC ⎟⎟ or connate water volume (rb). ⎝ (1 − SWC ) ⎠

(D) -

Reduction in Pore Volume – dVf As pore pressure declines, overburden pressure remains constant and grain pressure increases; the result is a shrinkage in Pore Volume dV f (rcf). This “squeezing” of the PV has the same effect as expansion of pore fluids.

(rb) dV f = c f VPV Δp = c f

(1 + m)NBoi Δp …………………...(D) (1 − SWC )

where, c f = Isothermal compressibility of formation (psi-1).

Δp = Pressure drop from pi to p (psi). ⎛ (1 + m )NBoi ⎞ ⎟⎟ or Pore Volume (rb). V f = ⎜⎜ ⎝ (1 − SWC ) ⎠

Section 2 © Copyright: Thru-u.com Ltd. 2000.

15 Rev. 1.0

(E) -

Decrease in HCPV due to Natural Water Influx: Water encroachment also decreases HCPV and must be added to RHS of material balance equation, just like an in-situ fluid expansion term (rb) dVW = We − WP BW …………..…………………………………(E)

where, We = Cumulative water influx from adjoining aquifer (rb). WP = Cumulative water produced at the surface (STB).

BW = Water Formation Volume Factor (rb/STB).

-

Now production given by equation (2.3) must be equal to (A) +(B)+(C)+(D)+(E). Hence, general material balance equation is

[

]

N P Bo + ( RP − RS ) Bg = N ( Bo − Boi ) + N ( RSi − RS ) Bg ⎛B ⎞ ⎡ c + SWC cW ⎤ + mNBoi ⎜ g − 1⎟ + (m + 1) NBoi ⎢ f ⎥ Δp + We − WP BW ⎜B ⎟ ⎣ (1 − SWC ) ⎦ ⎝ gi ⎠

………………..(2.4)

-

All the volume factors are functions of pressure so the material balance may be solved at any stage so long as p is known. The results obtained from material balance equation then become inputs for computer simulation runs.

NOTE -

Equation (2.4) is zero dimensional and treats the reservoir as a single unit. It is used to find, N , m and/or history match aquifer models.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

16 Rev. 1.0

2.6 MATERIAL BALANCE EQUATION - Havlena & Odeh Formulation

-

Havlena and Odeh showed that the material balance equation could be simplified and represented graphically: -

These graphs are used to check the value of estimated quantities.

-

Assumptions concerning drive mechanisms can be tested. The reservoir simulator input file can be upgraded leading more accurate predictions.

-

First, Havlena & Odeh defined F in much the same way as for gas reservoirs F = N P (Bo + ( RP − RS ) Bg ) + WP BW ....................…..….(2.5)

-

Notice F is cumulative reservoir production from the formation into the wellbore (rb); produced water is also accounted for as a production term. Next, Eo is defined in a similar fashion as for a gas reservoir

Eo = ( Bo − Boi ) + ( RSi − RS ) Bg ..................……......…..(2.6)

-

Notice Eo is net expansion of oil plus dissolved gas & free gas (rb/STB), between pi and lower pressure p . Next, define Eg as per gas reservoir

⎛ Bg ⎞ − 1⎟ ………………………………..…..(2.7) E g = Boi ⎜ ⎜B ⎟ ⎝ gi ⎠

-

Notice Eg is the expansion of gascap gas (rb/STB) between pi and lower pressure p ; by itself, expansion, Eg represents a unit gascap, where m = 1 .

Section 2 © Copyright: Thru-u.com Ltd. 2000.

17 Rev. 1.0

-

Expansion , E fW , is again defined in much the same way as for a gas reservoir

⎡ c f + SWC cW ⎤ E fW = (m + 1) Boi ⎢ ⎥ Δp ….……....……..(2.8) ( 1 ) − S WC ⎣ ⎦

-

E fW is the reduction in PV and the expansion of connate water (rb/STB),

between pi and lower pressure p .

-

Now (2.5), (2.6), (2.7) and (2.8) are substituted into equation (2.4), to give the general Havlena & Odeh formulation of the oil material balance equation

F = N ( Eo + mEg + E fW ) + We

-

…………….…..(2.9)

Notice multiplying “ Eg ” by “ m ” represent expansion of the actual gascap gas (not unit gascap gas, where Eg appears by itself).

NOTE -

The LHS of equation (2.9) should always be known, assuming that good production records are kept.

-

The unknowns on the RHS depend very much on the drive mechanism.

-

However, in general situation for reservoir with both gascap and natural waterdrive, unknowns can be N , m and We .

-

Estimates of N & m may be available from Volumetric Method. Solving for We requires an independent aquifer model equation.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

18 Rev. 1.0

2.7 DRIVE MECHANISM IDENTIFICATION -

The analysis below is quite similar to that given for a gas reservoir. Assume reservoir has no gascap then equation (2.9) reduces to, after re-arrangement:

F We =N+ (Eo + E fW ) (Eo + E fW ) ……………..……..…..(2.10)

-

Now using production-pressure data, plot F / (Eo + E fW ) versus N P . Notice, all the terms in the group F / (Eo + E fW ) should be known, as should N P . -

This plot helps to qualitatively identify the drive mechanism. Even though no gascap has been assumed, the approach still works with a gacap.

-

The no gascap assumption is made simply to keep calculation of ordinate values as assumption-free as possible:

B

⎛ F ⎜ ⎜E +E fW ⎝ o

⎞ ⎟ ⎟ ⎠

and pressure-production data will plot as a horizontal straight line as shown.

A

N

-

Moreover, the intercept will be N .

-

Curve A and B qualitatively indicate that production has been boosted by some sort of internal reservoir drive mechanism.

-

Extrapolation of curves A or B back to ordinate to find N is unreliable.

Volumetric Depletion

0 0

NOTE -

For true volumetric depletion mechanism, We m and E g are zero

(N P )

Above plot helps to identify a true volumetric depletion drive reservoir. Curves A or B can be caused either by water influx, gascap expansion, variable compaction, or various combinations of drive mechanism.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

19 Rev. 1.0

2.8 VOLUMETRIC DEPLETION DRIVE -

If, after plotting pressure-production data on the previous graph a reasonable horizontal straight line is obtained, then reservoir production-pressure history matches that of a simple volumetric depletion drive mechanism.

-

There are two cases: one, either the reservoir is at a pressure above the bubblepoint; or, two the reservoir pressure is below bubble-point, where a Solution Gas Drive mechanism is operative:

2.8.1 -

Depletion Above Bubble-Point

In this case oil is undersaturated and normally there is no gascap. If there is also negligible natural water drive, then the assumptions are: m = 0

We = WP = 0 RP = RS = Rsi

-

The last assumption is a consequence of the oil being above its bubble-point. Hence, all the gas at the surface must have been dissolved gas; that is there is no free gas present. Recall above pbp the GOR remains constant at RSi .

-

Such a simple case does not require Havlena & Odeh plot, so simplifying the general material balance equation (2.4) leads to

⎡ c f + S WC cW ⎤ N P B o = N ( B o − B oi ) + NB oi ⎢ ⎥ Δp ⎣ (1 − S WC ) ⎦

-

The pore volume and connate water expansion term cannot be ignored.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

20 Rev. 1.0

-

Re-arranging the above expression gives

⎡ ( B − Boi ) c f + SWC cW ⎤ + N P Bo = NBoi ⎢ o ⎥ Δp (1 − SWC ) ⎦ ⎣ Boi Δp

-

While, the compressibility of oil co can also be written as

co =

-

1 ΔV Bo − Boi = Vo Δp Boi Δp

Hence, substituting the above equation into the previous one leads to

c f + SWC cW ⎤ ⎡ N P Bo = NBoi ⎢co + ⎥ Δp (1 − SWC ) ⎦ ⎣

-

Which, in turn, can be simplified to

N P Bo = NBoi cEFF Δp

……………………………(2.11)

where,

cEFF = co + c f

NOTE -

1 SWC + cW (1 − SWC ) (1 − SWC )

……………(2.12)

Equation (2.11) simply expresses the relation dV = cVΔp recognising c is in fact cEFF .

Section 2 © Copyright: Thru-u.com Ltd. 2000.

21 Rev. 1.0

-

Notice that the cEFF weights compressibility of each component, be it oil, pore space or connate water, in accordance with that fraction of the HCPV, which the component in question actually occupies.

-

Normally production-pressure history would allow N to be found between equations (2.11) and (2.12). The values of N so found between initial conditions and some later stage should be reasonable constant.

-

The value of N found by applying the material balance equation to the pressure-production history may then be compared with the value of N found using the volumetric method: -

The values must be compared and reasons found for any discrepancies.

-

Remember, so long as good records are kept and sound average reservoir pressures have been observed, the material balance equation provides active STOIIP, which is contributing to pressure-production history.

-

It may be that, for instance, there are errors in geologic maps, or missing faults, or the presence of tight zones, which are not contributing to production.

-

Once the value for N is settled and agreed, the reservoir simulator’s input file can be updated and better prediction made. At this stage the recovery factor can be estimated from the expression:

N P Boi = cEFF Δp N Bo

NOTE

-

Low recoveries are generally obtained from undersaturated oil reservoirs above their bubble-point, typically much less than 10%, this is due principally to a small value for cEFF .

Section 2 © Copyright: Thru-u.com Ltd. 2000.

22 Rev. 1.0

2.8.2

Depletion Below Bubble-Point – Solution Gas Drive

-

Few reservoirs are now produced in this way. Although, it was the preferred method when oil prices were low. It is a highly wasteful drive mechanism: -

High producing GOR serves to deplete reservoir of high compressibility gas, which leads to in a sharp decline in reservoir pressure.

-

Also, since gas acts as the non-wetting phase, the relative permeability to oil becomes quite small. Hence, a sharp decline in well PI often occurs.

-

Occasionally tight, highly fractured, carbonate reservoirs are developed in this manner, for fear of early water breakthrough and large-scale by-passing of oil will occur, in the event that secondary recovery be implemented too soon.

-

Assuming the reservoir was initially undersaturated, the first step is to examine the pressure-production history, above and below the bubble-point, in order to determine whether or not the field is a volumetric depletion type.

-

As before, ignore gascap expansion, so that Havlena & Odeh formulation gives

F = N ( Eo + E fW ) + We ……………..…………….…..(2.9)

or F We =N+ (Eo + E fW ) (Eo + E fW ) ……………..……..…..(2.10)

NOTE -

Plot F / (Eo + E fW ) versus N P . If a horizontal straight line is obtained, then reservoir is a volumetric depletion type and N is easily found.

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23 Rev. 1.0

-

For a volumetric depletion reservoir above the bubble-point, the relevant Havlena & Odeh and material balance expression are

F = N ( Eo + E fW ) ……………..…………….…..(2.13)

and ⎡ c f + S WC cW N P B o = N ( B o − B oi ) + NB oi ⎢ ⎣ (1 − S WC )

-

⎤ ⎥ Δp …..(2.14) ⎦

The last expression is Schilthuis formulation of material balance and identical to equation (2.11).

-

For volumetric depletion drive reservoir below the bubble-point, there is free gas present and, in addition, E fW may be ignored, which results in

F = NEo

……………..…………….…….....(2.15)

and

[

]

N P Bo + ( RP − RS ) Bg = N ( Bo − Boi ) + N ( RSi − RS ) Bg ….(2.16)

NOTE -

The modern approach is to allow reservoir pressure to fall to around, or just below pbp . -this helps to identify reservoir drive mechanism.

-

Water or gas drive is then usually implemented to maintain reservoir pressure and to displace oil into producing wells.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

24 Rev. 1.0

-

Equation (2.16) shows that the recovery factor ( N P / N ) at any pressure p , by solution gas drive, is inversely proportional to the cumulative gas oil ratio RP : -

However, controlling RP can be difficult in solution gas drive situation. Closing in wells with high GOR, or instituting production restrictions to promote gas segregation, all too often have little long-term affect.

-

However, in some reservoirs conditions do permit the development of an artificial gascap by a process of gravitational segregation.

-

Equation (2.17) below, again shows that the more gas remaining in the reservoir, s g , the higher will be the recovery factor N P / N

⎡ ⎛ N ⎞B ⎤ S g = ⎢1 − ⎜1 − P ⎟ o ⎥ (1 − SWC ) …………………….(2.17) N ⎠ Boi ⎦ ⎣ ⎝

-

A typical production profile for a solution gas drive reservoir is shown below:

Bubble-point Pressure

pi

-

pb R the instantaneous GOR

R = RSi

% Watercut

-

Above the BP, pressure declines rapidly, recovery is low and GOR constant. Recovery factor < 10%

-

Below B.P, pressure declines less rapidly, GOR peaks and recovery improves.

-

However, recovery factor is still low overall, usually ≈ 10 − 25% .

-

With small aquifer watercut is low and steady.

Time

-

The best ways to improve recovery are by water and/or gas injection into the reservoir. Often recovery factor can be improved to around 50%.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

25 Rev. 1.0

2.9 -

NATURAL WATER DRIVE Take the case of an undersaturated oil reservoir with no gascap. If there is natural water influx into the reservoir, then general Havlena & Odeh expression, equation (2.9), reduces to

F = N ( Eo + E fW ) + We

-

……………..……………...(2.18)

The general Schilthuis material balance expression, equation (2.4), can also be simplified and written term by term in exactly the same way as (2.18) above

⎡ ⎛ c f + SWC cW N P Bo + WP BW = N ⎢( Bo − Boi ) + (m + 1) Boi ⎜⎜ ⎝ (1 − SWC ) ⎣⎢

-

⎞ ⎤ ⎟⎟Δp ⎥ + We .(2.19) ⎠ ⎦⎥

Re-arranging equation (2.18) which is the simpler of the two equations to deal with, gives

We F ……………..……………….(2.20) =N+ ( Eo + E fW ) ( Eo + E fW )

-

The next step is to take pressure-production data and to plot F / (Eo + E fW ) versus We / (Eo + E fW ) . Both axes will have units of (MM STB)

NOTE -

To do this an aquifer model is needed to generate values for We as reservoir pressure declines. See Appendix 2 for such a model.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

26 Rev. 1.0

-

Given a suitable mathematical model for the aquifer, Appendix 2, then We values can be calculated against reservoir pressure data. The Havlena & Odeh plot, shown below, may then be constructed:

-

We - too small

data plot as a curve above the straight line case, then the aquifer model is too weak.

We - Correct

F ( Eo + E fW " ) o

45

-

If the data plots as a curve below the straight line, then the aquifer model is too strong.

-

The correct aquifer model will produce results, which plot along the straight line with slope 45o.

We - too large

N

If the pressure-production- We

We ( Eo + E fW )

-

Once the correct aquifer model has been identified, backward extrapolation of the straight line to the ordinate will yield, as intercept, STOIIP ( N ), units for N will be (MM STB). A natural water drive reservoir is shown below:

pi Reservoir Pressure

-

Pressure decline is moderated by water influx and may eventually stabilise.

-

If water drive is sufficient then oil pressure will not drop too far below saturation level.

-

Hence, GOR is nearly constant at R = RSi .

-

Watercut will rise as water influx breaks through into down-dip producers.

% Watercut

GOR R ≈ RSi

Time

Section 2 © Copyright: Thru-u.com Ltd. 2000.

27 Rev. 1.0

2.9 -

GASCAP DRIVE Consider a gascap drive reservoir without any natural water influx then, in the presence of a highly compressible gascap, the generally small pore shrinkage and water expansion effects can be ignored.

-

In the presence of a gascap the oil is generally at or near its bubble-point so that any drop in pressure generally leads to free gas accumulating in the oil column. This can cause problems associated with measuring reservoir pressure.

-

The general Havlena & Odeh expression, equation (2.9), and the general Schilthuis material balance expression, equation (2.4) become in this case

F = N ( Eo + mEg )

…………….……………...(2.21)

and

⎛ Bg ⎞ − 1⎟ .(2.22) N P Bo + ( RP − RS ) Bg = N ( Bo − Boi ) + N ( RSi − RS ) Bg + mNBoi ⎜ ⎜B ⎟ ⎝ gi ⎠

[

-

]

There are two Havlena & Odeh plots, both of which are recommended in view of the uncertainty associated with average reservoir pressure determination. These plots should be as follows: -

First, plot F versus ( Eo + mEg )

-

Second, plot F / Eo versus E g / Eo

Section 2 © Copyright: Thru-u.com Ltd. 2000.

28 Rev. 1.0

-

The first plot should be constructed as follows:

2

Plot F vs. ( Eo + mE g ) .

1 -

If m is correct then data will plot as a straight line of slope N passing through origin.

-

If curve 2 is produced then m is too small.

-

If curve 3 is produced then m is too large.

3

F

( Eo + mEg )

-

The second plot is then constructed as follows:

-

By plotting F/Eo versus Eg/Eo then data should plot as a straight line.

-

The intercept should be N and the slope of the line should be (mN).

-

Hence, it should be possible to independently find both m and N.

Slope - mN

F

Eo

N Eg Eo

NOTE -

-

Both plots should be constructed and results compared.

Notice the major use is to verify estimated quantities and discover reservoir drive mechanism. These become known inputs into numerical simulator.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

29 Rev. 1.0

-

A typical plot showing the production features of a gascap drive reservoir with little or no natural water influx is given below:

pi Reservoir Pressure

R the instantaneous GOR R = RSi % Watercut

-

The pressure decline is less steep than for a solution gas drive case.

-

Watercut is low and steady – assuming little water influx.

-

The shape of the gas oil ratio R curve reflects updip producers being closed-on gas breakthough.

-

Recoveries will be higher than solution gas drive reservoirs without gascap - Generally around 30- 40% STOIIP depending on reservoir geology.

Time

-

The recovery depends on many factors such as: gascap size; permeability distribution; potential for gravity segregation of gas.

-

GOR development for gascap reservoir is less pronounced than for volumetric solution gas drive reservoir. Pressure maintenance is better and higher pressures tend to keep more gas in solution.

-

Working over high GOR wells which breakthrough to gas up-dip is generally more successful for gascap drive than solution gas drive. Recompleting up-dip wells and concentrating oil production from down-dip wells can facilitate high oil production rates at moderate GOR.

NOTE -

For gascap reservoirs with strong water drive Havlena & Odeh can be used if some of the potential unknowns ( N , m,We ) can be fixed.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

30 Rev. 1.0

APPENDIX 2

Water Influx Calculations

Section 2 © Copyright: Thru-u.com Ltd. 2000.

31 Rev. 1.0

APPENDIX 2 (I) -

Steady-State Water Influx Calculations The very simplest possible type of aquifer response occurs when the aquifer is approximately the same size as the reservoir itself. It is can then be assumed that any Δp in the reservoir will be instantaneously transmitted throughout the aquifer; unsteady-state behaviour of the aquifer can be ignored, leading to

We = c WΔp where, c = Total Compressibility of aquifer, cW + c f (psi-1). W = Total volume of water in aquifer (rb).

Δp = Total pressure drop in reservoir since start of production (psi).

-

Taking a radial geometry the total volume of water in the aquifer is then

W =

πf (re2 − ro2 )hφ 5.615

ro -

θ

f =

re

Section 2 © Copyright: Thru-u.com Ltd. 2000.

The encroachment factor is that fraction of a circle that the aquifer is encroaching into the gas reservoir along.

θ 360

32 Rev. 1.0

-

Solving for We leads to the aquifer model below

(

)

We = 0.560 × c f re2 − ro2 hφΔp

-

Adjusting the parameters of this model to get a satisfactory match with production-pressure data is a process called “aquifer fitting”.

(II) Unsteady-State Water Influx Calculations -

The above is the very simplest type of aquifer where the pressure is transmitted everywhere throughout the aquifer instantaneously. It is rarely applicable except where the aquifer is small – approximately the same size as the reservoir.

-

More realistically the aquifer is much larger than the reservoir so that pressure response is unsteady-state.

-

As the reservoir pressure depletes the pressure wave gradually travels through the aquifer, so that is never experiences as low a pressure as the reservoir itself.

-

The most realistic method of calculating water influx from a large active aquifer is that of Hurst and van Everdingen. However, the method is time-consuming to apply and simpler techniques have evolved which produce similar answers.

-

One popular method was developed by Carter & Tracy. This method, which is very easy to use, is described here. In Module-2, the Hurst and van Everdingen approach is presented with worked examples.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

33 Rev. 1.0

-

The Carter & Tracy Method uses the Constant Terminal Rate (CTR) solution of the diffusivity equation. The basic expression for We is given below

⎡U × Δp j − (We ) j −1 × Pj′ ⎤ ⎥ (t D ) j − (t D ) j −1 ……..(I) ⎣⎢ Pj − (t D ) j −1 × Pj′ ⎦⎥

[

(We ) j = (We ) j −1 + ⎢

]

where,

(We ) j

= Cumulative water influx to the (t D ) j time period (rb).

(W e ) j − 1

= Cumulative water influx to the (t D ) j −1 time period (rb).

U = Radial geometry aquifer constant (rb/psi).

Δp j = Total drop in pressure since start of production, pi − p j , (psi). Pj = CTR solution of diffusivity equation to (t D ) j time period (-). Pj′ = Derivative of above function, dP / dt D , at (t D ) j (-).

(tD ) j

= Dimensionless time at the jth time period (-).

(tD ) j −1 = Dimensionless time at the (jth-1) time period (-).

-

Now, the radial flow aquifer constant is given by the expression

U = 1.119 fφhcro2 ………………………………………...….(II) where, f = Fractional encroachment angle, θ / 360 , (-).

φ = Aquifer porosity (-). h = Aquifer thickness (ft). c = Effective aquifer compressibility, c f + cW , (psi-1).

ro2 = Reservoir radius (ft).

NOTE -

Parameters in equation (II) must be determined in association with geoligists amd petrophysicists.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

34 Rev. 1.0

-

Now, the expression for dimensionless time t D (-) can be evaluated at any time period. For equation (I) it must be evaluated twice, first at the (jth-1) time period, then at the subsequent jth time period. For any time period t D is

⎛ 0.00634 × k ⎞ ⎟⎟ × t …………………………………..(III) t D = ⎜⎜ 2 ⎝ φμcro ⎠

where, 0.00634 = Dimensionless time constant where t is expressed in days. 2.309 = Dimensionless time constant where t is expressed in years.

k = Aquifer permeability (md).

φ = Aquifer porosity (-).

μ = Aquifer water viscosity (cP). c = Effective aquifer compressibility, c f + cW , (psi-1).

ro2 = Reservoir radius (ft). t = Time period, where t is expressed in (days or years) - depending on constant used in equation (III).

-

The CRT solution of the diffusivity equation, evaluated at dimensionless time

(tD ) j

is Pj . These functions are often presented graphically of in tabular form.

However, the functions can be regressed and expressed as polynomials.

-

This gives more accurate values using a spreadsheet or a calculator than interpolating graphs by eye or interpolating tabular values which may be nonlinear functions. The expression for Pj is given below

[

]

Pj = ao + a1 (t D ) j + a2 ln (t D ) j + a3 ln (t D ) j ………….….(IV)

-

2

The constants ao a1 a2 a3 are functions of dimensionless radius reD .

Section 2 © Copyright: Thru-u.com Ltd. 2000.

35 Rev. 1.0

-

Pj′ is the derivative of above function, i.e. dP / dt D , at (t D ) j . Hence, differentiating Pj with respect to t D results in

Pj′ = a1 +

-

2 × a3 ln (t D ) j a2 + ………………………….(V) (t D ) j (t D ) j

The values of the coefficients ao a1 a2 a3 are functions of dimensionless radius reD . These values were determined by Fanchi and are listed below

-

reD

ao

a1

a2

a3

1.5 2.0 3.0 4.0 5.0 6.0 8.0 10.0 ∞

0.10371 0.30210 0.51243 0.63656 0.65106 0.63367 0.40132 0.14386 0.82092

1.66657 0.68178 0.29317 0.16101 0.10414 0.06940 0.04104 0.02649 − 3.68 × 10−4

-0.04579 -0.01599 0.01534 0.15812 0.30953 0.41750 0.69592 0.89646 0.28908

-0.01023 -0.01356 -0.06732 -0.09104 -0.11258 -0.11137 -0.14350 -0.15502 0.02882

Solving equations (IV) and (V) at the required (t D ) j , where (t D ) j is obtained from equation (III), then finding a value for the aquifer constant U from equation (II), the results are substituted into equation (I) to find (We ) j .

-

This is the cumulative water influx (rb) into the reservoir system up to the jth time increment, assuming that the cumulative water influx up to the (jth-1) time interval, (We ) j −1 , has been found for the previous calculation cycle.

Section 2 © Copyright: Thru-u.com Ltd. 2000.

36 Rev. 1.0