Material Balance Equations
Author: Jon Kleppe, NTNU
Assistant producer: Vidar W. Moxness
The Statfjord area in the North Sea.
ENTER
Source: Statoil
Material Balance Equations
Introduction
INTRODUCTION
MODELLING
APPLICATION
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.
SUMMARY
Learning goals
• Basic understanding of material balance
The handout “Material Balance Equations” can be
downloaded from here:
matbal.pdf
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.
Introduction
Application
Modelling
Summary
Block
diagram
Saturation
Material
conservation
Equations
Graph A
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Graph B
Water
influence
Plot 1
Plot 2
Initial
gascap
Plot 3
HELP
Material Balance Equations
Block diagram of a producing reservoir
INTRODUCTION
MODELLING
Block diagram
Material conservation
Graph A B
Equations
Saturation
The essence of material balance is described in the
block diagram below.
Due to change in pressure, the pore volume as well as
the fraction of the volume occupied by gas, oil & water
will change.
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.
APPLICATION
SUMMARY
Click to display
symbols used
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Material Balance Equations
Principle of material conservation
INTRODUCTION
MODELLING
Block diagram
Material conservation
Graph A B
Equations
Saturation
From the block diagram we get the expression below, which is the basis for the material balance formulas.
Amount of fluids present  Amount of  Amount of fluids remaining

 
 

 in the reservoir initially  - fluids produced =  in the reservoir finally 

  (st. vol.)  

(st. vol.)
(st. vol.)

 
 

APPLICATION
SUMMARY
Note that “fluids produced” include all influence on the reservoir:
• Production
• Injection
• Aquifer influx
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Material Balance Equations
Formation Volume Factor in the Black Oil model
INTRODUCTION
MODELLING
Block diagram
Material conservation
Graph A B
Equations
Saturation
The formation volume factors (FVF) tell how much the
oil, gas and water is compressed at a given pressure.
The graphs below show how the FVF of oil, gas and
water develop vs pressure. Click on the buttons to show
the graphs.
Bo = reservoir volume of oil / standard volume of oil
APPLICATION
Bg = reservoir volume of gas / standard volume of gas
SUMMARY
Bw = reservoir volume of water / standard volume of
water
Bo vs. P
Bg vs. P
Bo
Bw vs. P
Bg
P
Bw
P
P
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symbols used
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Material Balance Equations
Solution Gas-Oil Ratio in the Black Oil model
INTRODUCTION
MODELLING
Block diagram
Material conservation
Graph A B
Equations
Saturation
APPLICATION
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.
Click on the button below to see the typical pressure
dependency of the solution gas-oil ratio in the black oil
model.
SUMMARY
Rs = standard volume gas / standard volume oil
Rso vs. P
Rso
P
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symbols used
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Material Balance Equations
The complete black oil material balance equation
INTRODUCTION
MODELLING
Block diagram
Material conservation
Graph A B
Equations
Saturation
The final material balance relationships is given below. How these expressions are derived can be
studied in the Material Balance pdf document.
matbal.pdf


F = N E o  mE g  E f ,w  Wi  We Bw2  Gi Bg2
APPLICATION
SUMMARY
Where:
production terms are


 
F = N p Bo2  R p - Rso2 Bg2  W p Bw2
oil and solution gas expansion terms are
E o = Bo2 - B o1  Rso1 - Rso2 B g2
gas cap expansion terms are
 B g2

E g = B o1 
- 1
 B g1

and rock and water compression/expansion terms are
E f ,w = -1  mBo1
Cr  Cw S w1
P
1 - S w1
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symbols used
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Material Balance Equations
Saturation and pressure development
INTRODUCTION
MODELLING
Block diagram
Material conservation
Graph A B
Equations
Saturation
APPLICATION
SUMMARY
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.
Click to display
symbols used
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Material Balance Equations
Application of Material Balance
INTRODUCTION
MODELLING
APPLICATION
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.
Initial gascap
Plot 1
Plot 2
Water influence
Plot 3
The animation below shows a producing reservoir with
gas and water injection.
SUMMARY
In the following pages ways of finding some of these
values will be explained.
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symbols used
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Material Balance Equations
Application of Material Balance
Initial gas cap (Havlena and Odeh approach)
INTRODUCTION
MODELLING
APPLICATION
Initial gascap
Plot 1
Plot 2
Water influence
Plot 3
SUMMARY
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.
General mass balance formula:
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)
Eg
F
= N  mN
Eo
Eo


F = N E o  mE g  E f ,w  Wi  We Bw2  Gi Bg2
Assuming no water influence, gas injection and rock
or water compression/expansion.
F = N Eo  mEg 
(2)
(3)
Large version
Plot 1
Large version
Plot 2
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symbols used
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(1)
HELP
Material Balance Equations
Application of Material Balance
Water influence (Havlena and Odeh approach)
INTRODUCTION
MODELLING
APPLICATION
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)
Initial gascap
Plot 1
Plot 2
Water influence
Plot 3
General mass balance formula:


F = N E o  mE g  E f ,w  Wi  We Bw2  Gi Bg2
(1)
Assuming no water or gas injection and Bw=1.
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.
SUMMARY
F = N Eo  mEg  E f , w   We
(4)
Neglecting Ef,w due to it’s small influence and assuming
no initial gascap.
F = NEo  We
(5)
W
F
=N e
Eo
Eo
(6)
Water influx model for radial aquifer shape:


We = cw  c f  re2 - ro2 fhp
(7)
Large version
Plot 3
Click to display
symbols used
FAQ REFERENCES
ABOUT
HELP
Material Balance Equations
Summary
INTRODUCTION
MODELLING
APPLICATION
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.
SUMMARY
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.
Block diagram
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 m*N.
Saturation & pressure
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.
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Material Balance Equations
References
INTRODUCTION
MODELLING
APPLICATION
Jon Kleppe. Material balance. http://www.ipt.ntnu.no/~kleppe/SIG4038/02/matbal.pdf
SUMMARY
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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Material Balance Equations
About this module
INTRODUCTION
MODELLING
APPLICATION
Title: Material Balance Equations
SUMMARY
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: --
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