Author: Professor Jon Kleppe

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:


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








Boundary Conditions

IMPES Method


method

QUESTIONS


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








Boundary Conditions

IMPES Method


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.








Boundary Conditions

IMPES Method


method

QUESTIONS


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








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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:








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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.








Boundary Conditions

IMPES Method


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

Qq

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








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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

Qq

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








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


method

QUESTIONS

4. Constant liquid production rate

Total constant surface liquid production rate of QLi (positive):

QLi =Qoi + Qwi

i

Li

wioi

oioi xA

Qq

Δ+=′

λλλ

If capillary pressure is neglected:

i

Li

wioi

wiwi xA

Qq

Δ+=′

λλλ

and

Oil

Water

Total liquid

Time

qprod

0








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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.








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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








Boundary Conditions

IMPES Method


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

qq

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.








Boundary Conditions

IMPES Method


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.








Boundary Conditions

IMPES Method


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.








Boundary Conditions

IMPES Method


method

QUESTIONS

References

Kleppe J.: Reservoir Simulation, Lecture note 6

Snyder and Ramey SPE 1645








Boundary Conditions

IMPES Method


method

QUESTIONS

Title: SATURATED OIL-WATER SIMULATION, IMPES SOLUTION (PDF)

Author: Name: Prof. Jon Kleppe

Address:NTNU

S.P. Andersensvei 15A

7491 Trondheim

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Author: Professor Jon Kleppe