Reservoir Simulation PCB 3053
Nov 14, 2015
Reservoir Simulation PCB 3053
Reservoir Simulation PCB 3053
Objective
At the end of this section students will be able to Understand the challenges in multiphase flow simulation
Develop the equations necessary to simulate a two phase oil-water reservoir system
Clearly identify the challenge in solving the equation.
Practice IMPES solution method for two phase oil-water reservoir system
Reservoir Simulation PCB 3053
Introduction
We are interested in flow of three phases in the reservoir.
Oil Phase - liquid hydrocarbons
Gas Phase -hydrocarbon vapor
Aqueous Phase - water
o
os
g
g
w
w
o
o
B
SR
B
Szyx
B
Szyx
B
Szyx
615.5)Scf(GasofVolume
615.5)STB(WaterofVolume
615.5)STB(OilofVolume
Free gasDissolved gas
Oil component only in
Oil phase
Water component only in
water phase
Gas component exists
In both oil and gas phases
Reservoir Simulation PCB 3053
Exercise
Drive the partial differential Flow equation for oil-water system in a 1D-horizontal
block of reservoir rock.
Mass_in Mass_Out
Reservoir Simulation PCB 3053
MULTI-PHASE SIMULATION (GENERAL)
Previously you have developed the one phase flow equation forone-dimensional, horizontal flow in a layer of constant crosssectional area. Similarly the continuity equation for multiphaseflow is:
and corresponding Darcy equations for each phase:
gwolSt
ux
llll ,,,
gwolx
Pkku l
l
rll ,,,
where
wocow PPP
ogcog PPP
1,,
gwoi
iS
Reservoir Simulation PCB 3053
OIL-WATER SIMULATION
Where:
flow equations for the two phases flow after substitution of Darcy's equations:
w
w
w
w
ww
rw
B
S
tq
x
P
B
kk
x
.
o
o
o
o
oo
ro
B
S
tq
x
P
B
kk
x
.
cowow PPP 1 wo SS
Relative permeability and capillary pressure are functions of water saturation, and
Formation volume factors, viscosity and solution gas-oil ratio are functions of pressures.
and
Reservoir Simulation PCB 3053
Typical pressure dependencies
g
PP
w o
P
P
Bg Bo
P P
Rso
Pb Pb
Bw
P
B=
Reservoir Simulation PCB 3053
Review of Oil-Water Relative Permeability and Capillary Pressure
most processes of interest, involve displacement of oil by water in a water wet environment, 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: Imbibition process:
SW = 1
Sw
Kr
1
oil
water
Swir Swir
Sw
Pc
1
oil
Sw
Kr
1-Sor
Pc
SwSwir 1-Sor
oil
water
Swir
SW =SWirwater
Pcd
Reservoir Simulation PCB 3053
Discretization of Flow Equations
where f ( x) includes permeability, mobility and flow area.
The right side of the flow equation is of the following form
+1/2
=
+/2
1!
+/2
2
2!
2
2
+
1/2
=
+/2
1!
+/2
2
2!
2
2
+
=
+1/2
1/2
+ 2
and
Which yields
Reservoir Simulation PCB 3053
As we can see, due to the different block sizes, the error terms for the last two approximations are again of first order only.
By inserting these expressions into the previous equation, we get the following approximation for the flow term:
Similarly we may obtain the following expressions
+1/2
=+1
+1 + /2+
1/2
= 1
+ 1 /2+ and
=2 +1/2
+1 +1 +
2 1/2 1
+ 1
+
Reservoir Simulation PCB 3053
The multiphase flow term, is of the form
Therefore
=
= ,,
=
2 +1/2
+1 +1 +
2 1/2
1 + 1
+
= +1/2 +1 1/2 1 +
Recall the definition of Transmissibility
Transmissibility in positive direction Transmissibility in minus direction
Reservoir Simulation PCB 3053
Using Txli +1/ 2 as example, the transmissibility consists of three groups of parameters
We therefore need to determine the forms of the latter two groups before proceeding to the numerical solution
2
+1 + =
+1/2 = =
+1/2
=
= ,
One Phase Two Phase
2
+1 + =
+1/2 = =
1
+1/2
=1
=
Reservoir Simulation PCB 3053
Starting with Darcy's equation: one phase
We will assume that the flow is steady state, i.e. q=constant, and that k is dependent on position. The equation may be rewritten as:
Reservoir Simulation PCB 3053
Permeability
integrating the equation in previous slide between block centers:
The left side may be integrated in parts over the two blocks in our discrete system, each having constant permeability:
Reservoir Simulation PCB 3053
which is the harmonic average of the two permeabilities. In terms of our grid block system, we then have the following expressions for the harmonic averages:
Reservoir Simulation PCB 3053
Fluid Mobility Term
Integrating the right hand side
Let
Assuming the pressure gradient between the block centers to be constant, we find that the weighted average of the blocks mobility terms is representative of the average
and
Reservoir Simulation PCB 3053
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:
)()(.
112
1
2
1 ioioixoioioixo
i
o
oo
ro PPTPPTx
P
B
kk
x
)()(.
112
1
2
1 iwiwixwiwiwixw
i
w
ww
rw PPTPPTx
P
B
kk
x
Where:
Oil transmissibility:
i
i
i
ii
io
xoi
k
x
k
xx
T
1
1
21
21
2
o krooBo
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.
Reservoir Simulation PCB 3053
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
Reservoir Simulation PCB 3053
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 oo
21
1
11
21
ii
ioiioi
ioxx
xx weighted average selection:
In reservoir simulation, upstream mobilities are normally used.
QW
Swir
x
Sw
1-SorB.L (PC = 0)
Upstream
Weighted average
1
Reservoir Simulation PCB 3053
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.
B.L (PC = 0)
Small grid blocks
Swir
x
Sw
1-Sor
1
Large grid blocks
Reservoir Simulation PCB 3053
Expansion of Discretized equations
The right hand side of the oil equation:
o
oo
oo
o
BtS
t
S
BB
S
t
io
o
o
riwiipoo
dP
Bd
B
c
t
SC
)/1()1(
iio
iiswo
tBC
)()( tiwiwiswot
ioioipoo
io
o SSCPPCB
S
t
By:
Replacing oil saturation by water saturation.( = 1 )
Use a standard backward approximation of the time derivative.
the right hand side of the oil equation thus may be written as:
Where:
Reservoir Simulation PCB 3053
The right hand side of the water equation:
w
ww
ww
w
BtS
t
S
BB
S
t
iw
w
w
riwiipow
dP
Bd
B
c
t
SC
)/1(
)()( tiwiwiswwt
ioioipow
iw
w SSCPPCB
S
t
By:
Expansion of 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
i
swwi CdS
dP
tBC
Reservoir Simulation PCB 3053
The discrete forms of the oil and water equations
tiwiwiswotioioipoooiioioixoioioixo SSCPPCqPPTPPT 1121
21
tiwiwiswwtioioipowwiicowicowioioi
xwicowicowioioixw
SSCPPC
qPPPPTPPPPT
11112
1
2
1
Oil equation:
i
i
i
ii
io
ixo
k
x
k
xx
T
1
1
21
21
2
i
i
i
ii
io
ixo
k
x
k
xx
T
1
1
21
21
2Where:
ioioio
ioioio
io
PPif
PPif
1
11
21
ioioio
ioioio
io
PPif
PPif
1
11
2
1
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.
Reservoir Simulation PCB 3053
BOUNDARY CONDITIONS
1. Constant water injection rate
2. Injection at constant bottom hole pressure
3. Constant oil production rate
4. Constant liquid production rate
5. Production at Constant reservoir voidage rate
6. Production at Constant bottom hole pressure
Reservoir Simulation PCB 3053
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
ibhiwoiiwi PPWCQ
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
2
i
e
xyr
Drainage radius
Reservoir Simulation PCB 3053
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:
QwiBwi WCikroioi
krwioi
(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.
)( ibhiwwioiwi
oiiwi PPB
BWCQ
or
Time
qinj
Time
Pbh
Pmax
Pbp
Reservoir Simulation PCB 3053
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 PPB
BWCQ
At the end of the time step, the above equation may be used to compute the actual water injection rate for the step.
If Capillary pressure
is neglected
)( ibhiowioiwi
oi
iwi PPB
BWCQ
Reservoir Simulation PCB 3053
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
q wi q oiwi(Pwi Pbhi)
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:
If Capillary pressure is neglected
Around the production wellq wi q oi
wioi
the bottom hole production pressure for the well may be calculated using the well equation for oil:
ibhiooiioi PPWCQ
Reservoir Simulation PCB 3053
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.
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.
Time
Pbh
Pmin
Time
qprod
Pbp
Reservoir Simulation PCB 3053
4. Constant liquid production rate
Total constant surface liquid production rate of QLi (positive):
QLi Qoi Qwi
i
Li
wioi
oi
oixA
If capillary pressure is neglected:
i
Li
wioi
wi
wixA
and
Oil
Water
Total liquid
Time
qprod
0
Reservoir Simulation PCB 3053
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
i
injwinjw
wiwioioi
oi
oixA
BQ
BBq
i
injwinjw
wiwioioi
wi
wixA
BQ
BBq
If capillary pressure is neglected:
and
Reservoir Simulation PCB 3053
6. Production at Constant bottom hole pressure
Production well in grid block i with constant bottom hole pressure, Pbhi:
andQoi WCioi(Poi Pbhi) Qwi WCiwi(Pwi Pbhi )
Substituting the flow terms in the flow equations:
andq oi WCiAxi
oi(Poi Pbhi) q wi WCiAxi
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.
Reservoir Simulation PCB 3053
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
t
Cow
t
pw
t
po
t
sw
t
so
t
xw
t
xo PCCCCTT ,,,,,,
Discretized form of flow equations:
tiwiwiswotioioipoooiioioixoioioixo SSCPPCqPPTPPT 1121
21
tiwiwiswwtioioipowwiicowicowioioi
xwicowicowioioixw
SSCPPC
qPPPPTPPPPT
11112
1
2
1
Where:
i=1, , N
Reservoir Simulation PCB 3053
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 such as Thomas Algorithm.
The saturations may be solved explicitly by using one of the equations.
Using the oil equation yields:
tioiotipoooiioiotixoioiotixotiswo
t
iwiw PPCqPPTPPTC
SS 1121
21
1
i=1, , N
Reservoir Simulation PCB 3053
Having obtained oil pressures and water saturations for a given time step, well rates or bottom hole pressures may be computed as qwi, qoi 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 before we can proceed to the next time step.
Reservoir Simulation PCB 3053
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.
Reservoir Simulation PCB 3053