Material Balance Equations

Material Balance Equations

Author: Jon Kleppe, NTNU

Assistant producer: Vidar W. Moxness

The Statfjord area in the North Sea. Source: Statoil


REFERENCES ABOUT HELPFAQ

INTRODUCTION Introduction

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.

Learning goals• Basic understanding of material balance

The handout “Material Balance Equations” can be

downloaded from here:

MODELLING

APPLICATION

SUMMARY

SaturationBlockdiagram

Materialconservation

Graph A Graph B

Equations

Waterinfluence

Initialgascap

Introduction

Modelling Application

Summary

matbal.pdf

Plot 1 Plot 2 Plot 3



Block diagram of a producing reservoir

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.

Due to change in pressure, the pore volume as well as

the fraction of the volume occupied by gas, oil & water

will change.

INTRODUCTION

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SUMMARY

Block diagramMaterial conservationGraph A BEquationsSaturation

Click to display symbols used



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

Principle of material conservationINTRODUCTION


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

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.)

APPLICATION

SUMMARY

MODELLING



P

Bo

Bo vs. P

P

Bg

Bg vs. P

P

Bw

Bw vs. P

Formation Volume Factor in the Black Oil model


INTRODUCTION


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

The graphs below show how the FVF of oil, gas and

water develop vs pressure. Click on the buttons to show

the graphs.

APPLICATION

SUMMARY

MODELLING



P

Rso

Rso vs. P

Solution Gas-Oil Ratio in the Black Oil modelINTRODUCTION



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.

APPLICATION

SUMMARY

MODELLING



F N E mE E W W B G Bo g f w i e w2 i g2 ,

Where: production terms are

F N B R R B W Bp o2 p so2 g2 p w2

oil and solution gas expansion terms are

E B B R R Bo o2 o1 so1 so2 g2

gas cap expansion terms are

E BB

B1g o1

g2

g1

and rock and water compression/expansion terms are

E 1 m BC C S

1 SPf w o1

r w w1

w1,

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 pdf document.

INTRODUCTION



matbal.pdf

APPLICATION

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MODELLING



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.

INTRODUCTION


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Application 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.

INTRODUCTION

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Initial gascap Plot 1 Plot 2Water influence Plot 3



Application of Material BalanceInitial gas cap (Havlena and Odeh approach)


General mass balance formula:


go mEENF

o

g

o E

EmNN

E

F

(1)

(2)

(3)

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)

Assuming no water influence, gas injection and rock

or water compression/expansion.

Large version Plot 1


INTRODUCTION

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Application of Material BalanceWater influence (Havlena and Odeh approach)


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.

eo WNEF


ewfgo WEmEENF ,

o

e

o E

WN

E

F

General mass balance formula:

Assuming no water or gas injection and Bw=1.

Neglecting Ef,w due to it’s small influence and assuming

no initial gascap.

(1)

(4)

(5)

(6)


pfhrrccW oefwe 22(7)

Water influx model for radial aquifer shape:

INTRODUCTION

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SUMMARY



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.

APPLICATION:

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 mN.

Water influence: In a water drive reservoir the water influx, We, can be recovered by

plotting F/Eo vs We/Eo. In this plot We must be calculated with a known model.

Block diagram

INTRODUCTION

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Saturation & pressure



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.

ReferencesINTRODUCTION

MODELLING

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About this module

Title: Material Balance Equations

Author: Prof. Jon Kleppe

Assistant producer: Vidar W. Moxness

Size: 0.8 mb

Publication date: 24. July 2002

Abstract: The module describes the basics of material balance calculations.

Software required: PowerPoint XP/XP Viewer

Prerequisites: none

Level: 1 – 4 (four requires most experience)

Estimated time to complete: --

INTRODUCTION

MODELLING

APPLICATION

SUMMARY

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Material Balance Equations