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Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Petroleum Reservoir Simulation of Two-PhaseFlow
N.Manjunath ReddyDr. Anugrah SinghDr. Pankaj Tiwari
Indian Institute of Technology Guwahati
December 12, 2014
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Contents
1 Introduction
2 Mathematical Modeling
3 Boundary Conditions
4 Solution Methods
5 Results
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Introduction
Hydrocarbon reservoir holds hydrocarbons that are trappedunderground in porous and permeable rock.
At initial reservoir conditions, the fluids are in either asingle-phase or a two-phase state.
source http://writepass.com/journal/2012/12/the-effect-of-water-content-on-the-strength-of-rock/
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Recovery methodssource https://www.youtube.com/watch?v=pe71rV92GY8http://ecurrentaffairs.in/blog/gs-paper-iii-st-science-of-petroleum-extraction
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Governing equations
Continuity equation
∂
∂t(φραSα) +∇.(ρα ~Uα) = 0
Darcy’s law
~Uα = −kα
µα
∇(Pα − ραgz)
Effective permeabilities are no longer rock properties alone. Ifthe reservoir consists of oil and water then the effectivepermeability can be given as:
kw = kkrw (Sw )
ko = kkro(Sw )
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Two phase flow equations
Oil flow equation
−φ∂Sw∂t
= ∇.(kkro
Boµo
∇(Po − ρogz))± q′
o
Water flow equation
φ∂Sw∂t
= ∇.(kkrw
Bwµw
∇((Po − Pcow )− ρwgz))± q′
w
qi qp
→ So + Sw = 1 and Pcow = Po − Pw
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Boundary conditions
qp (or) Pbh
No �ow No �ow
qi (or) Pbh
→ For constant bottom-hole pressure the flow rate is given as :
q′
pi = WCiλi (Pi − Pbhi )/∆v
→ Where WCi is productivity index of the well
WCi =2πkih
ln( rerw)
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
IMplicit Pressure Explicit Saturation Method
IMPES method is splitting approach.
Eliminating saturation from oil and water equation givespressure equation
∇.(kkro
Boµo
∇(Po − ρogz)) +∇.(kkrw
Bwµw
∇((Po − Pcow )− ρwgz)) =
q′
o + q′
w
Once the pressure is solved, saturation can be calculated fromeither of oil and water equations.
−φ∂Sw∂t
= ∇.(kkro
Boµo
∇(Po − ρogz))− q′
o
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Implementation of IMPES method in OpenFOAM
∇.(λtk∇P)
−JλoP
−JλwP
== ∇.(λok∇(ρogh))
+∇.(λwk∇(ρwgh))
+∇.(λwk∇(Pcow )
−q′
i
−JλoPbh
−JλwPbh
fvScalarMatrix pEqn(fvm::laplacian(λt*k,p)- fvm::Sp(J*n*λo ,p)- fvm::Sp(J*n*λw ,p)==fvc::laplacian(λo*k*ρo ,gh)+ fvc::laplacian(λw*k*ρw ,gh)+ fvc::laplacian(λw*k,pcow)- qi- fvc::Sp(J*n*λo ,pbh)- fvc::Sp(J*n*λw ,pbh));
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
1D Case
Reservoir of dimensions 20mx1mx1m is discretized into100x1x1 cells in each direction.
Parameters Value
φ 0.30k (mD) 100µo , µw (cP) 1, 0.1ρo , ρw (kg/m3s) 700, 1000
Relative permeability curvesare
krw = S2w
kro = (1− Sw )2
qi qp
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Comparison with BUCKLEY -LEVERETT solution
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Sa
tura
tio
n (
Sw
)
Distance (m)
B-L solOpenFOAM (IMPES)
OpenFOAM (SS)MRST(IMPES)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Sa
tura
tio
n (
Sw
)
Distance (m)
100 cells150 cells200 cells
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Homogeneous reservoir case
Reservoir of dimensions 40mx40mx1m is considered anddiscretized into 1600 cells.
Water is injected at the rate of 0.5 m3/day and productionwell is maintained at constant bottomhole pressure of 1 bar.
Injection wellProduction well
x
y
z
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Homogeneous reservoir case
OpenFOAM MRSTt
= 1
00
days
t =
200
days
Figure : Saturation profiles
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Validation
0.1
0.2
0.3
0.4
0.5
0.6
0 200 400 600 800 1000 1200 1400 1600
Sa
tura
tio
n(S
w)
Time(days)
OpenFOAM solutionMRST solution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
200 400 600 800 1000 1200 1400 1600
Pro
du
ctio
n r
ate
(m3/d
ay)
Time(days)
Oil rateWater rate
Figure : Comparison with MRST
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Effect of viscosity
Displacement processes are sensitive to the properties of thefluids involved.
Different types of fluids and fluid mixtures exist in thereservoir ranging from heavy oils to lighter gases.
Viscosity of these fluids varies from 100 cP -0.25 cP.
In the recovery of these fluids the viscosity of the fluid that isselected for injection plays an important role in thedisplacement.
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Displacement with light fluid ( µinj < µoil )
Initially the reservoir is saturated with the heavy fluid andwater of viscoities 10 cP and 0.1 cP.
(a) t = 800 days (b) t = 1600 days
Figure : Saturation profiles
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Displacement with equal viscosity fluid ( µinj = µoil )
In this case equal viscosity of fluid is selected for the injection.
(a) t = 200 days (b) t = 400 days
(a) t = 500 days (a) t = 600 days
Figure : Saturation profiles
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Displacement with the heavy fluid ( µinj > µoil )
Heavy fluid of viscosity 10 cP is considered for thedisplacement of light oil.
(a) t = 200 days (b) t = 400 days
(a) t = 500 days (a) t = 600 days
Figure : Saturation profiles
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Oil recovery
Initial oil in place - 384 m3
1988
Oil recoveredOil left
21%
79%
(a) ( µinj < µoil )
1988
Oil recoveredOil left
84%
16%
(b) ( µinj = µoil )
1988
Oil recoveredOil left
96%
4%
(c) ( µinj < µoil )
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Heterogeneous reservoir
The real field oil reservoirs are heterogeneous in permeabilityand porosity distributions.
(a) Porosity (b) Permeability
Figure : Porosity and permeability distribution in heterogeneous reservoir
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Heterogeneous reservoir
O����OAM M���
t =
10
0 d
ays
t =
200
days
Figure : Comparison of saturation profiles with MRST
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Heterogeneous reservoir
0.1
0.2
0.3
0.4
0.5
0.6
0 200 400 600 800 1000 1200 1400 1600
Sa
tura
tio
n(S
w)
Time(days)
OpenFOAM (Hetero)MRST (Hetero)
0
0.1
0.2
0.3
0.4
0.5
200 400 600 800 1000 1200 1400 1600
Pro
du
ctio
n r
ate
(m3/d
ay)
Time(days)
Oil rate (Hetero)Water rate (Hetero)
Figure : Breakthrough and production rates
Introduction Mathematical Modeling Boundary Conditions Solution Methods Results
Conclusion and prospective
Conclusion
IMPES method is implemented in general purpose CFD solver,OpenFOAM.Validated the results with B-L solution and MRST solution.The effect of fluid and rock properties on the recovery isstudied.Studied the effect of heterogeneity.
Prospective
The solvers should be modified to account the compressibiltyand miscibility of fluids.Heat and mass transport equations can be added to the solversto account the heat and mass transfer phenomena.The solver can be updated to improved IMPES method.