Oil-water Simulation - Impes Solution

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


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


Exercise

Drive the partial differential Flow equation for oil-water system in a 1D-horizontal

block of reservoir rock.

Mass_in Mass_Out


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


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


Typical pressure dependencies

g

PP

w o

P

P

Bg Bo

P P

Rso

Pb Pb

Bw

P

B=


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


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


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

+


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


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

=


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:


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:


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:


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


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.


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


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


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


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:


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


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.


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


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

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


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


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


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

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


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


4. Constant liquid production rate

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

QLi Qoi Qwi

i

Li

wioi

oi

oixA

Qq

If capillary pressure is neglected:

i

Li

wioi

wi

wixA

Qq

and

Oil

Water

Total liquid

Time

qprod

0


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


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.


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


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


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

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 before we can proceed to the next time step.


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.

Documents

Oil-water Simulation - Impes Solution