SATURATED OIL-WATER SIMULATION, IMPES SOLUTION Author: Professor Jon Kleppe NTNU Assistant producers: Farrokh Shoaei Khayyam Farzullayev
SATURATED OIL-WATER SIMULATION, IMPES SOLUTION
Author: Professor Jon Kleppe
NTNU
Assistant producers:
Farrokh Shoaei
Khayyam Farzullayev
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Where:
Where:
Two phase system (oil – water)
The equations for multi-phase flow :
( ) ( )llll St
ux
φρρ∂
∂=
∂
∂− gwol ,,=
wocow PPP −=
ogcog PPP −=
Sl
l=o,w,g∑ =1
x
Pkku l
l
rll ∂
∂−=
μ.
flow equations for the two phases flow with substituting Darcy's equations:
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
=′−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
w
ww
w
ww
rw
B
S
tq
x
P
B
kk
x
φ
μ
.⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
=′−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
o
oo
o
oo
ro
B
S
tq
x
P
B
kk
x
φ
μ
.
cowow PPP −=1=+ wo SS
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Oil-Water Relative Permeabilities and Capillary Pressure
most processes of interest, involve displacement of oil by water, or imbibition.
the initial saturations present in the rock will normally be the result of a drainage process at the time of oil accumulation.
Drainage process:
SW = 1
Sw
Kr
1
oilwater
Swir Swir
Sw
Pc
1
oil
Imbibition process:
Sw
Kr
1-Sor
Pc
SwSwir 1-Sor
oilwater
Swir
SW =SWir water
Pcd
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Discretization of Flow Equations We will use similar approximations for the two-phase equations as we did
for one phase flow.
Left side flow terms:
)()(.
1121
21 ioioixoioioixo
i
o
oo
ro PPTPPTx
P
B
kk
x−+−≈⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
−−++μ
)()(.
1121
21 iwiwixwiwiwixw
i
w
ww
rw PPTPPTx
P
B
kk
x−+−≈⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂
∂∂
−−++μ
Where:
– Oil transmissibility:
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ+ΔΔ=
+
+
+
+
i
i
i
ii
io
xoi
kx
kx
x
T
1
1
21
21
2λ
λo =kro
μoBo– Oil mobility:
The mobility term is now a function of saturation in addition to pressure. This will have significance for the evaluation of the term in discrete form.
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Upstream mobility term
Because of the strong saturation dependencies of the two-phase mobility terms, the solution of the equations will be much more influenced by the evaluation of this term than in the case of one phase flow.
Buckley-Leverett solution:
QW
Swir
x
Sw
1-Sor
B.L with PC = 0
1
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
In simulating this process, using a discrete grid block system, the results are very much dependent upon the way the mobility term is approximated.
Flow of oil between blocks i and i+1:
– Upstream selection: ii ooλλ =+ 2
1
( )( )1
11
21
+
+++ Δ+Δ
Δ+Δ=
ii
ioiioiio xx
xx λλλ– weighted average selection:
QW
Swir
x
Sw
1-Sor
B.L (PC = 0)
Upstream
Weighted average
In reservoir simulation, upstream mobilities are normally used.
1
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
The deviation from the exact solution depends on the grid block sizes used.
For very small grid blocks, the differences between the solutions may become negligible.
The flow rate of oil out of any grid block depends primarily on the relative permeability to oil in that grid block.
If the mobility selection is the weighted average, the block i may actually have reached residual oil saturation, while the mobility of block i+1 still is greater than zero.
For small grid block sizes, the error involved may be small, but for blocks of practical sizes, it becomes a significant problem.
Swir
x
Sw
1-Sor
B.L (PC = 0)
Small grid blocks
1
Large grid blocks
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Expansion of Discretized equations The right hand side of the oil equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
oo
o
oo
o
BtS
t
S
BB
S
t
φφφ
io
o
o
riwiipoo
dP
Bd
B
c
t
SC ⎥⎦
⎤⎢⎣
⎡ +Δ−
=)/1()1(φ
iio
iiswo
tBC
Δ−=
φ
)()( tiwiwiswo
tioioipoo
io
o SSCPPCB
S
t−+−≈⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂ φ
By:
– Replacing oil saturation by water saturation.
– Use a standard backward approximation of the time derivative.
the right hand side of the oil equation thus may be written as:
Where:
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
The right hand side of the water equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
w
ww
ww
w
BtS
t
S
BB
S
t
φφφ
iw
w
o
riwiipow
dP
Bd
B
c
t
SC ⎥
⎦
⎤⎢⎣
⎡+
Δ=
)/1(φ
)()( tiwiwisww
tioioipow
iw
w SSCPPCB
S
t−+−≈⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂ φ
By:
– expression the second term– Since capillary pressure is a function of water saturation only– Using the one phase terms and standard difference approximations for
the derivatives
the right side of the water equation becomes:
Where:
powi
iw
cow
iwi
iswwi C
dS
dP
tBC ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
Δ=
φ
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
The discrete forms of the oil and water equations
( ) ( ) ( ) ( )tiwiwiswotioioipoooiioioixoioioixo SSCPPCqPPTPPT −+−=′−−+− −−++ 11
21
21
Oil equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ+ΔΔ=
+
+
+
+
i
i
i
ii
io
ixo
kx
kx
x
T
1
1
21
21
2λ
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ+ΔΔ=
−
−
−
−
i
i
i
ii
io
ixo
kx
kx
x
T
1
1
21
21
2λ
Where:
⎩⎨⎧
<≥
=+
+++
ioioio
ioioio
ioPPif
PPif
1
11
21
λ
λλ
⎩⎨⎧
<≥
=−
−−
−ioioio
ioioio
io
PPif
PPif
1
11
21
λ
λλ
Water equation:
Transmissibility and mobility terms are the same as for oil equation, except the subtitles are changed from “o” for oil to “w” for water.
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Boundary Conditions
1. Constant water injection rate
the simplest condition to handle
for a constant surface water injection rate of Qwi (negative) in a well in grid block i:
i
wiwi xA
Δ=′
( )ibhiwoiiwi PPWCQ −= λ
At the end of a time step, the bottom hole injection pressure may theoretically be calculated using the well equation:
where:
Well constant
⎟⎟⎠
⎞⎜⎜⎝
⎛=
w
e
ii
r
r
hkWC
ln
2π
πi
e
xyr
ΔΔ=
Drainage radius
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
The fluid injected in a well meets resistance from the fluids it displaces also.
As a better approximation, it is normally accepted to use the sum of the mobilities of the fluids present in the injection block in the well equation.
Well equation which is often used for the injection of water in an oil-water system:
Qwi Bwi =WCikroi
μoi+
krwi
μoi
⎛
⎝ ⎜
⎞
⎠ ⎟(Pwi −Pbhi )
– Injection wells are frequently constrained by a maximum bottom hole pressure, to avoid fracturing of the formation.
– This should be checked, and if necessary, reduce the injection rate, or convert it to a constant bottom hole pressure injection well.
Time
qinj
Time
Pbh
)( ibhiwwioiwi
oiiwi PP
B
BWCQ −⎟⎟
⎠
⎞⎜⎜⎝
⎛+= λλo
r
Pmax
Pbp
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
2. Injection at constant bottom hole pressure
Injection of water at constant bottom hole pressure is achieved by:
– Having constant pressure at the injection pump at the surface.
– Letting the hydrostatic pressure caused by the well filled with water control the injection pressure.
The well equation:
)( ibhiwwioiwi
oiiwi PP
B
BWCQ −⎟⎟
⎠
⎞⎜⎜⎝
⎛+= λλ
At the end of the time step, the above equation may be used to compute the actual water injection rate for the step.
Capillary pressure
is neglected)( ibhiowioi
wi
oiiwi PP
B
BWCQ −⎟⎟
⎠
⎞⎜⎜⎝
⎛+= λλ
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
3. Constant oil production rate
for a constant surface oil production rate of Qoi (positive) in a well in grid block i:
i
oioi xA
Δ=′
′ q wi = ′ q oiλ wi (Pwi − Pbhi )
λ oi (Poi − Pbhi )
in this case oil production will generally be accompanied by water production.
The water equation will have a water production term given by:
Capillary pressure is neglectedAround the production well
′ q wi = ′ q oiλ wi
λoi
the bottom hole production pressure for the well may be calculated using the well equation for oil:
( )ibhiooiioi PPWCQ −= λ
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
– Production wells are normally constrained by a minimum bottom hole pressure, for lifting purposes in the well. If this is reached, the well should be converted to a constant bottom hole pressure well.
Time
Pbh
Pmin
Time
qprod
– If a maximum water cut level is exceeded for well, the highest water cut grid block may be shut in, or the production rate may have to be reduced.
0
10
20
30
40
50
60
70
80
0 2 4 6 8 10 12 14 16 18
WC(%) vs. Time(year)
As the limitation for water cut was 75%, so at this point gridblocks that exceeded allowable water cut had been closed in order to keep the limit.
Pbp
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
4. Constant liquid production rate
Total constant surface liquid production rate of QLi (positive):
QLi =Qoi + Qwi
i
Li
wioi
oioi xA
Δ+=′
λλλ
If capillary pressure is neglected:
i
Li
wioi
wiwi xA
Δ+=′
λλλ
and
Oil
Water
Total liquid
Time
qprod
0
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
5. Production at Constant reservoir voidage rate
A case of constant surface water injection rate of Qwinj in some grid block.
total production of liquids from a well in block i is to match the reservoir injection volume so that the reservoir pressure remains approximately constant.
QoiBoi +QwiBwi =−QwinjBwinj
⎟⎟⎠
⎞⎜⎜⎝
⎛
Δ
−⎟⎟⎠
⎞⎜⎜⎝
⎛
+=′
i
injwinjw
wiwioioi
oioi xA
BQ
BBq
λλ
λ
⎟⎟⎠
⎞⎜⎜⎝
⎛
Δ
−⎟⎟⎠
⎞⎜⎜⎝
⎛
+=′
i
injwinjw
wiwioioi
wiwi xA
BQ
BBq
λλ
λ
If capillary pressure is neglected:
and
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
6. Production at Constant bottom hole pressure
Production well in grid block i with constant bottom hole pressure, Pbhi:
and
Qoi =WCiλoi (Poi −Pbhi ) Qwi =WCiλwi(Pwi −Pbhi )
Substituting the flow terms in the flow equations:
and′ q oi =WCi
AΔxi
λ oi(Poi − Pbhi ) ′ q wi =WCi
AΔxi
λ wi (Pwi − Pbhi )
The rate terms contain unknown block pressures, these will have to be appropriately included in the matrix coefficients when solving for pressures.
At the end of each time step, actual rates are computed by these equations, and water cut is computed.
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
the primary variables and unknowns to be solved for equations are:
– Oil pressures Poi, Poi-1, Poi+1
– Water saturation Swi
Assumption:
– All coefficients and capillary pressures are evaluated at time=t.
IMPES Method
tCow
tpw
tpo
tsw
tso
txw
txo PCCCCTT ,,,,,,
Discretized form of flow equations:
( ) ( ) ( ) ( )tiwiwiswotioioipoooiioioixoioioixo SSCPPCqPPTPPT −+−=′−−+− −−++ 11
21
21
Where:
i=1, …, N
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
The two equations are combined so that the saturation terms are eliminated. The resulting equation is the pressure equation:
iiiiiii dPcPbPa ooo =++ +− 11
– This equation may be solved for pressures implicitly in all grid blocks by Gaussian Elimination Method or some other methods.
The saturations may be solved explicitly by using one of the equations.
Using the oil equation yields:
( ) ( ) ( )[ ]tioio
tipoooiioio
tixoioio
tixo
tiswo
tiwiw PPCqPPTPPT
CSS −−′−−+−+= −−++ 11
21
21
1
i=1, …, N
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Having obtained oil pressures and water saturations for a given time step, well rates or bottom hole pressures may be computed as q’wi, q’oi and Pbh.
The surface production well water cut may be computed as:
oiwi
wiiws
qf
′+′′
=
Required adjustments in well rates and well pressures, if constrained by upper or lower limits are made at the end of each time step, before all coefficients are updated and we can proceed to the next time step.
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Limitations of the IMPES method
The evaluation of coefficients at old time level when solving for pressures and saturations at a new time level, puts restrictions on the solution which sometimes may be severe.
IMPES is mainly used for simulation of field scale systems, with relatively large grid blocks and slow rates of change.
It is normally not suited for simulation of rapid changes close to wells, such as coning studies, or other systems of rapid changes.
When time steps are kept small, IMPES provides accurate and stable solutions to a long range of reservoir problems.
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Questions
1. Make sketches of typical Kro, Krw and Pcow curves for an oil-water system, both for oil-
displacement of water and water-displacement of oil, and label all relevant points.
4. Write an expression for the selection of the upstream mobility term for use in the
transmissibility term of he oil equation for flow between the grid blocks (i-1) and (i).
2. Show how the saturation profile (Sw vs. x), if calculated in a simulation model,
typically is influenced by the choice of mobilities between the grid blocks (include
simulated results with upstream and average mobility terms) (neglecting capillary
pressure). Make also a sketch of the exact solution.
3. Write the two flow equations for oil and water on discretized forms in terms ot
transmissibilities, storage coefficients and pressure differences.
5. List the assumptions for IMPES solutin, and outline briefly how we solve for
pressures and saturations.
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
References
Kleppe J.: Reservoir Simulation, Lecture note 6
Snyder and Ramey SPE 1645
Saturated Oil-Water Simulation, IMPES Solution
REFERENCES ABOUT EXIT
Two phase system (oil – water)
Oil-Water Relative Permeabilities and Capillary Pressure
Discretization of Flow Equations
Upstream mobility term
Expansion of Discretized equations
Boundary Conditions
IMPES Method
Limitations of the IMPES
method
QUESTIONS
Title: SATURATED OIL-WATER SIMULATION, IMPES SOLUTION (PDF)
Author: Name: Prof. Jon Kleppe
Address:NTNU
S.P. Andersensvei 15A
7491 Trondheim
Website
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About this module