presentation_1

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


Contents

1 Introduction

2 Mathematical Modeling

3 Boundary Conditions

4 Solution Methods

5 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/


Recovery methodssource https://www.youtube.com/watch?v=pe71rV92GY8http://ecurrentaffairs.in/blog/gs-paper-iii-st-science-of-petroleum-extraction


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 )


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


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)


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


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));


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


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


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


Homogeneous reservoir case

OpenFOAM MRSTt

= 1

00

days

t =

200

days

Figure : Saturation profiles


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


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.


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



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



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



Oil recovery

Initial oil in place - 384 m3

1988

Oil recoveredOil left

21%

79%

(a) ( µinj < µoil )

1988


84%

16%

(b) ( µinj = µoil )

1988


96%

4%

(c) ( µinj < µoil )


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



O��OAM M��

t =

10

0 d

ays

t =

200

days

Figure : Comparison of saturation profiles with MRST



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


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.


Thank you...

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