Material Bala Balanc nce e Equ Equa ati tion ons s
By : Dr. Ir. Dedy Kristanto, M.Sc
Petroleum Petroleum Engi neering neering Department Department UPN UPN ” Vetera Veteran’ n’ Yogyakarta
Material Balance Equation s
INTRODUCTION
Introduction
MODELLING APPL ICATION SUMMARY
To illustrate the simplest possible model we can have
Learning goals
for analysis of reservoir behavior, we will start with derivation of so-called “Material Balance Equations”.
• Basic understanding of material balance
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.
The handout “Material Balance Equations” can be downloaded from here:
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
Graph B
Water influence
Plot 1
Plot 2
Initial gascap
Plot 3
Material Balance Equation s
INTRODUCTION
Block diagram of a produc ing reservoir
MODELLING Block diagram Material con servation Graph A B Equations Saturation APPL ICATION SUMMARY
Click to display symbols used
The essence of material balance is described in the
Due to change in pressure, the pore volume as well as
block diagram below.
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.
Material Balance Equation s
INTRODUCTION
Principle of material conservation
MODELLING Block diagram Material con servation Graph A B Equations Saturation APPL ICATION SUMMARY
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.) ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ Note that “fluids produced” include all influence on the reservoir: • Production • Injection • Aquifer influx
Material Balance Equation s
INTRODUCTION
Formation Volum e Factor in the Black Oil model
MODELLING Block diagram Material con servation Graph A B Equations Saturation APPL ICATION SUMMARY
The formation volume factors (FVF) tell how much the
The graphs below show how the FVF of oil, gas and
oil, gas and water is compressed at a given pressure.
water develop vs pressure. Click on the buttons to show the graphs.
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
Bo vs. P
Bg vs. P
Bo
Bg
P Click to display symbols used
Bw vs. P
Bw
P
P
Material Balance Equation s
INTRODUCTION
Solutio n Gas-Oil Ratio in the Black Oil model
MODELLING Block diagram Material con servation Graph A B Equations Saturation
The Rso plot shows how the solution gas ratio develops
Click on the button below to see the typical pressure
vs pressure. When the pressure reaches the bubblepointpressure, it is no longer possible to solve
dependency of the solution gas-oil ratio in the black oil model.
APPL ICATION
more gas into the oil. Thus the gradient of the curve becomes zero.
SUMMARY
Rs = standard volume gas / standard volume oil
R so vs. P
R so
P Click to display symbols used
Material Balance Equation s
INTRODUCTION
The complete black oil m aterial balance equation
MODELLING Block diagram Material con servation Graph A B Equations Saturation APPL ICATION
The final material balance relationships is given below. How these expressions are derived can be studied in the Material Balance.
F
=
(
N Eo
+ mE g +
E f ,w
)+ (W + W )B i
e
w2
+ Gi Bg2
SUMMARY
Where:
production terms are
F
= N p
[ +( Bo2
Rp
− R so2
)B ]+ W B g2
p
w2
oil and solution gas expansion terms are
E o
= (Bo2 − B o1 ) + (Rso1 −
)
Rso2 Bg2
gas cap expansion terms are
E g
⎛ B g2 ⎞ = B o1 ⎜⎜ − 1⎟⎟ ⎝ B g1 ⎠
and rock and water compression/expansion terms are
E f ,w Click to display symbols used
= −(1 + m)Bo1
Cr
+ Cw S w1 ∆P 1 − S w1
Material Balance Equation s
INTRODUCTION
Saturation and pressure developm ent
MODELLING Block diagram Material con servation Graph A B Equations Saturation
View the animations below to see how the pressure and
The plot to the left shows how the saturations and the
oil-, gas- and water-saturation typically develops in a reservoir initially above the bubblepoint develops versus
pressure in the reservoir develop vs time in a reservoir if there is small or no water injection.
APPL ICATION
time. Also included is how pressure might develop versus time.
The plot to the right shows the same for a reservoir with
SUMMARY
Click to display symbols used
large water injecton.
Material Balance Equation s
INTRODUCTION
Applicati on of Materi al Balanc e
MODELLING APPL ICATION Initial gascap Plot 1 Plot 2 Water infl uence Plot 3
In material balance calculations there are in most cases
The animation below shows a producing reservoir with
many uncertainties with regard to reservoir parametres. Uncertain values may for instance include the size of the
gas and water injection.
initial gascap, the initial amount of oil in the reservoir and the influx of the aquifer.
SUMMARY
In the following pages ways of finding some of these values will be explained.
Click to display symbols used
Material Balance Equation s
INTRODUCTION MODELLING
Applicati on of Materi al Balanc e Initi al gas cap (Havlena and Odeh approach)
APPL ICATION Initial gascap Plot 1 Plot 2 Water infl uence Plot 3 SUMMARY
For gascap reservoirs the value of m is in most cases
General mass balance formula:
uncertain. The value of N can however usually be defined well through producing wells. In this case a good
F
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)
Large version Plot 1 Large version Plot 2
Click to display symbols used
=
(
N Eo
+ mE g +
E f ,w
)+ (W + W )B i
e
w2
+ Gi Bg2
(1)
As su mi ng no wat er i nf lu ence, gas in jec ti on and ro ck or water compression/expansion.
F = N E o F E o
+ mE g E g
= N + mN
E o
(2)
(3)
Material Balance Equation s
INTRODUCTION MODELLING
Applicati on of Materi al Balanc e Initi al gas cap (Havlena and Odeh approach)
APPL ICATION Initial gascap Plot 1 Plot 2 Water infl uence Plot 3 SUMMARY
Return
Large version Plot 2
Click to display symbols used
For gascap reservoirs the value of m is in most cases
For a too large value of m, the plot will deviate down and
uncertain. The value of N can however usually be defined well through producing wells. In this case a good
for a too small value it will deviate up.
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.
As su mi ng no wat er i nf lu ence, gas in jec ti on and ro ck or water compression/expansion.
F = N E o
+ mE g
(2)
Material Balance Equation s
INTRODUCTION MODELLING
Applicati on of Materi al Balanc e Initi al gas cap (Havlena and Odeh approach)
APPL ICATION Initial gascap Plot 1 Plot 2 Water infl uence Plot 3 SUMMARY
Large version Plot 1 Return
Click to display symbols used
If both the value of m and N are uncertain one should
As su mi ng no wat er i nf lu ence, gas in jec ti on and ro ck
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
or water compression/expansion.
have a slope of mN. (eq. 3)
F E o
E g
= N + mN
E o
(3)
Material Balance Equation s
INTRODUCTION MODELLING
Applicati on of Materi al Balanc e Water infl uence (Havlena and Odeh appr oach)
APPL ICATION Initial gascap Plot 1 Plot 2 Water infl uence Plot 3 SUMMARY
In water drive reservoirs the biggest uncertainty is in
General mass balance formula:
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
F
known model. (e.g. eq. 7)
=
(
N Eo
+ mE g +
E f ,w
)+ (W + W )B i
e
w2
+ 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.
F = N E o
+ mE g + E f , w + W e
(4)
Neglecting Ef,w due to it’s small influence and assuming no initial gascap.
F = NE o
+ W e
(5)
F
W e
(6)
E o
= N +
E o
Water influx model for radial aquifer shape:
W e Large version Plot 3
Click to display symbols used
= (cw + c f )π (r e2 − r o2 ) fhφ ∆p
(7)
Material Balance Equation s
INTRODUCTION MODELLING
Applicati on of Materi al Balanc e Water infl uence (Havlena and Odeh appr oach)
APPL ICATION Initial gascap Plot 1 Plot 2 Water infl uence Plot 3 SUMMARY
Return
Click to display symbols used
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.
F E o
= N +
W e E o
(6)
Material Balance Equation s
INTRODUCTION
Summary
MODELLING APPL ICATION 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. 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. 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
plotting F/Eo vs We/Eo. In this plot We must be calculated with a known model.
Saturation & pressure
Material Balance Equation s
INTRODUCTION
References
MODELLING APPL ICATION SUMMARY
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.
Material Balance Equation s
INTRODUCTION
About this module
MODELLING APPL ICATION SUMMARY
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) Estimated time to co mplete: --
Material Balance Equation s
INTRODUCTION MODELLING
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APPL ICATION SUMMARY
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Material Balance Equation s
INTRODUCTION
Symbols used in m aterial balance equations
MODELLING APPL ICATION
Bg
Formation volume factor for gas (res.vol./st.vol.)
Sg
Gas saturation
SUMMARY
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.)
Ef,w
Rock and water expansion/compression term
We
Cumulative aquifer influx (st.vol.)
Eg
Gas cap expansion term
Wi
Cumulative water injected (st.vol.)
Eo
Oil & solution gas expansion term
Wp
Cumulative water produced (st.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.)
P
Pressure
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)
Click to return to calculation
