5- Material Balance

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By : Dr. Ir. Dedy Kristanto, M.Sc

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Introduction

INTRODUCTION MODELLING APPL ICATION

Learning goals

• Basic understanding of material balance

The handout “Material Balance Equations” can be downloaded from here:

To illustrate the simplest possible model we can have for analysis of reservoir behavior, we will start with derivation of so-called “Material Balance Equations”. This type of model excludes fluid flow inside the reservoir, and considers fluid and rock

expansion/compression effects only, in addition, of course, to fluid injection and production.

This module is meant to be an extra help to the lectures in “Reservoir recovery techniques” by giving examples to the curriculum covered by the handout “Material Balance Equations”.

The structure of the model is shown below. SUMMARY

Introduction

Application

Modelling

Summary

Saturation

Block

diagram

Water

influence

Material

conservation

Graph A

Graph B

Equations

Initial

gascap

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Block diagram of a produc ing reservoir

INTRODUCTION MODELLING Block diagram Material con servation Graph A B

Equations Saturation

Due to change in pressure, the pore volume as well as the fraction of the volume occupied by gas, oil & water will change.

The essence of material balance is described in the block diagram below.

From the initial stage oil, gas & water is produced. At the same time gas & water is (re)injected into the reservoir to maintain pressure. There is also an influx from the aquifer below the reservoir.

APPL ICATION SUMMARY

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Principle of material conservation

From the block diagram we get the expression below, which is the basis for the material balance formulas.

Amount of fluids present in the reservoir initially

(st. vol.)

Amount of fluids produced

(st. vol.)

Amount of fluids remaining in the reservoir finally

(st. vol.)

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

−

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

=

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

Note that “fluids produced” include all influence on the reservoir: • Production

• Injection • Aquifer influx

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Formation Volum e Factor in the Black Oil model

The graphs below show how the FVF of oil, gas and water develop vs pressure. Click on the buttons to show the graphs.

The formation volume factors (FVF) tell how much the oil, gas and water is compressed at a given pressure. Bo= reservoir volume of oil / standard volume of oil Bg= reservoir volume of gas / standard volume of gas Bw= reservoir volume of water / standard volume of water APPL ICATION SUMMARY

B

_o

vs.

P

B

_g

vs.

P

B

_w

vs. P

P

B

_o

P

B

_g

P

B

_w

Click to display

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Solutio n Gas-Oil Ratio in the Black Oil model

The Rso plot shows how the solution gas ratio develops vs pressure. When the pressure reaches the

bubblepointpressure, it is no longer possible to solve more gas into the oil. Thus the gradient of the curve becomes zero.

Rs= standard volume gas / standard volume oil

Click on the button below to see the typical pressure dependency of the solution gas-oil ratio in the black oil model. APPL ICATION SUMMARY

R

_so

vs. P

R

_so

P

Click to display symbols used

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The complete black oil material balance equation

The final material balance relationships is given below. How these expressions are derived can be studied in the Material Balance.

(

)

(

)

F N E mE

=

_o

+

_g E _f_,_w

+

W W B _i

+

_{e w2}

+

G B_{i g2}

Where: production terms are

(

)

[

]

F N B

=

_p _o2

+

R R B _p

−

_so2 _g2

+

W B_{p w2}

oil and solution gas expansion terms are

(

)

(

)

E _o

= − +

B _o2 B _o1 R _so1

−

R _so2B_g2

gas cap expansion terms are

E B B B 1 g o1 g2 g1

=

⎛

−

⎝

⎜⎜

⎞

_⎠

⎟⎟

and rock and water compression/expansion terms are

(

)

E 1 m B C C S 1 S P f w o1 r w w1 w1 ,

= − +

+

−

∆

APPL ICATION SUMMARY Click to display

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Saturation and pressure development

View the animations below to see how the pressure and oil-, gas- and water-saturation typically develops in a reservoir initially above the bubblepoint develops versus time. Also included is how pressure might develop versus time.

The plot to the left shows how the saturations and the pressure in the reservoir develop vs time in a reservoir if there is small or no water injection.

The plot to the right shows the same for a reservoir with large water injecton.

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Applicati on of Material Balance

In material balance calculations there are in most cases many uncertainties with regard to reservoir parametres. Uncertain values may for instance include the size of the initial gascap, the initial amount of oil in the reservoir and the influx of the aquifer.

In the following pages ways of finding some of these values will be explained.

The animation below shows a producing reservoir with gas and water injection.

Initial gascap Plot 1 Plot 2

Water infl uence Plot 3

SUMMARY

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Applicati on of Material Balance

Initi al gas cap (Havlena and Odeh approach)

For gascap reservoirs the value of m is in most cases uncertain. The value of N can however usually be defined well through producing wells. In this case a good approach will be to plot F as a function of (Eo+mEg) for an assumed value of m. (eq. 2) For the correct value of m the slope will be a straight line passing through origo with a slope of N. For a too large value of m, the plot will deviate down and for a too small value it will deviate up. If both the value of m and N are uncertain one should plot F/Eo as a function of Eg/Eo. This plot should be linear and will intercept the y axis at a value of N and have a slope of mN. (eq. 3)

General mass balance formula: Initial gascap

Plot 1

Plot 2 F N E mE

=

(

_o

+

_g E _f_,_w

)

+

(

W W B _i

+

_{e w2}

)

+

G B_{i g2} (1)

Water infl uence

Plot 3 _{Assu mi ng no wat er i nf lu ence, gas in jec ti on and ro ck}

or water compression/expansion. SUMMARY g o

mE

E

N

F

=

+

(2) o g o

E

mN

N

E

F

₌

₊

(3) Large version Plot 1 Large version Plot 2 Click to display symbols used

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Applicati on of Material Balance

Initi al gas cap (Havlena and Odeh approach)

For gascap reservoirs the value of m is in most cases uncertain. The value of N can however usually be defined well through producing wells. In this case a good approach will be to plot F as a function of (Eo+mEg) for an assumed value of m. (eq. 2) For the correct value of m the slope will be a straight line passing through origo with a slope of N.

For a too large value of m, the plot will deviate down and for a too small value it will deviate up.

Assu mi ng no wat er i nf lu ence, gas in jec ti on and ro ck or water compression/expansion.

Initial gascap Plot 1 Plot 2

Water infl uence Plot 3 SUMMARY g o

mE

E

N

F

=

+

(2) Return Large version Plot 2 Click to display

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Applicati on of Material Balance

Initi al gas cap (Havlena and Odeh approach)

If both the value of m and N are uncertain one should plot F/Eo as a function of Eg/Eo. This plot should be linear and will intercept the y axis at a value of N and have a slope of mN. (eq. 3)

Assu mi ng no wat er i nf lu ence, gas in jec ti on and ro ck or water compression/expansion. Initial gascap Plot 1 Plot 2 o g o

E

mN

N

E

F

+

=

Water infl uence Plot 3 (3) SUMMARY Large version Plot 1 Return Click to display symbols used

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Applicati on of Material Balance

Water influence (Havlena and Odeh appr oach)

In water drive reservoirs the biggest uncertainty is in most cases the water influx, We. To find this we plot F/Eo vs We/Eo. In this plot We must be calculated with a known model. (e.g. eq. 7)

For a correct model of We we will get a straight line. For the wrong model the plot will deviate from a straight line as shown in plot 3.

General mass balance formula: Initial gascap

Plot 1

Plot 2 _{F N E mE}

(

_E

)

(

_{W W B}

)

_{G B}

o g f w i e w2 i g2

=

+

_,

+

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Water infl uence

Plot 3 _{Assuming no water or gas injection and B}_w_=1.

SUMMARY e w f g o

mE

E

W

E

N

F

=

+

_,

+

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Neglecting Ef,wdue to it’s small influence and assuming no initial gascap. e o

W

NE

F

=

+

(5) Click to display o e o E W N E F

₌

₊

₍₆₎

(

c

)

(

r

)

fh

p

W

_e

=

_w

+

_f

π

_e2

−

_o2

φ

∆

Water influx model for radial aquifer shape:

(7) Large version

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Applicati on of Material Balance

Water influence (Havlena and Odeh appr oach)

For a correct model of We we will get a straight line. For the wrong model the plot will deviate from a straight line

as shown in plot 3. o e o E W N E F

₌

₊

₍₆₎ Initial gascap Plot 1 Plot 2

Water infl uence Plot 3

SUMMARY

Return

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Summary

MODELLING:

Block diagram:Material balance equations are based on a model with a know start- and end-point. Between the two stages oil, gas & water is produced and gas & water is (re)injected into the reservoir to maintain pressure. There is also an influx from the aquifer below the reservoir. Due to change in pressure, the pore volume as well as the fraction of the volume occupied by gas, oil & water will change.

Material conservation: Amounts of fluids in the reservoir at stage one is equal to the amount of fluids at stage two plus the amount of fluids produced.

Graph A:The formation volume factors (FVF) tell how much the oil, gas and water is compressed at a given pressure.

Graph B:The Rso plot shows how the solution gas ratio develops vs pressure. When the pressure reaches the bubblepointpressure, it is no longer possible to solve more gas into the oil. Thus the gradient of the curve becomes zero.

Equations:The material balance equations consist of a general part, oil and solution gas expansion terms, gas cap expansion terms and rock and water compression/expansion terms

Saturation:Pressure and saturations change versus time, depending on production/injection. See figure to the right.

APPL ICATION:

Initial gascap:In a gas drive reservoirs m may be calculated by plotting F as a function of (Eo+mEg). For the correct value of m the plot will be a straight line. Alternatively m & N may be calculated by plotting F/Eo vs Eg/Eo. The curve will intercept the y axis at a value of N and have a slope of m

∗

N.

Water in fluence:In a water drive reservoir the water influx, We, can be recovered by SUMMARY

Block diagram

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References

Jon Kleppe. Material balance. http://www.ipt.ntnu.no/~kleppe/SIG4038/02/matbal.pdf L.P. Dake 1978. Fundamentals of reservoir engineering, Elsevier, Amsterdam, 443 pp. L.P. Dake 1994. The practice of reservoir engineering, Elsevier, Amsterdam, 534 pp. Svein M. Skjæveland (ed.) & Jon Kleppe (ed.) 1992. SPOR monograph : recent

advances in improved oil recovery methods for North Sea sandstone reservoirs Norwegian Petroleum Directorate, Stavanger. 335 pp.

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About this module

Title: Material Balance Equations Au th or : Pro f. J on Kl epp e

As si st ant pr od uc er: Vidar W. Moxness Size: 0.8 mb

Publication date: 24. July 2002

Ab st rac t: The module describes the basics of material balance calculations. Software required: PowerPoint XP/XP Viewer

Prerequisites: none

Level: 1 – 4 (four requires most experience) SUMMARY

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Symbols used in m aterial balance equations

INTRODUCTION MODELLING

APPL ICATION Bg Formation volume factor for gas (res.vol./st.vol.) Sg Gas saturation

Ef,w Rock and water expansion/compression term We Cumulative aquifer influx (st.vol.)

Eg Gascapexpansionterm Wi Cumulative water injected (st.vol.)

Eo Oil & solution gas expansion term Wp Cumulative water produced (st.vol.)

P Pressure

Bo Formation volume factor for oil (res.vol./st.vol.) So Oil saturation

Bw Formation volume factor for water (res.vol./st.vol.) Sw Water saturation

Cr Pore compressibility (pressure-1) T Temperature

Cw Water compressibility (pressure-1) Vb Bulk volume (res.vol.)

∆

P P2-P1 Vp Pore volume (res.vol.)

Gi Cumulative gas injected (st.vol.) R Density (mass/vol.)

Gp Cumulative gas produced (st.vol.)

φ

Porosity

m Initial gas cap size (res.vol. of gas cap)/(res.vol. of oil zone) N Original oil in place (st.vol.)

Np Cumulative oil produced (st.vol.)

Pb Bubblepoint Pressure

Rp Cumulative producing gas-oil ratio (st.vol./st.vol.) = Gp/Np Rso Solution gas-oil ratio (st.vol. gas/st.vol. oil)

SUMMARY