SATURATED OIL-WATER SIMULATION, IMPES SOLUTION
Author: Professor Jon Kleppe
Assistant producers:
Farrokh Shoaei
Khayyam Farzullayev
NTNU
ENTER
Saturated Oil-Water Simulation, IMPES Solution
Two phase system (oil – water)
Two phase system
(oil – water)

The equations for multi-phase flow :

Oil-Water Relative
Permeabilities
and Capillary
Pressure

l ul    l Sl 
x
t
Where:
Discretization of
Flow Equations
ul  
Pcog  Pg  Po
S 1
l
l  o, w, g

flow equations for the two phases flow with substituting Darcy's equations:
  k.k rw Pw 
  S 

  qw   w 
x   w Bw x 
t  Bw 
Boundary
Conditions
IMPES Method
l  o, w, g
Pcow  Po  Pw
Upstream mobility
term
Expansion of
Discretized
equations
k .k rl Pl
 l x
  k.k ro Po 
  S 

  qo   o 
x   o Bo x 
t  Bo 
Where:
Pw  Po  Pcow
So  S w  1
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Oil-Water Relative Permeabilities
and Capillary Pressure
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Discretization of
Flow Equations

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:
Upstream mobility
term
Expansion of
Discretized
equations

SW = 1
oil
Kr
water
oil
Pcd
IMPES Method
QUESTIONS
Pc
Kr
oil
Limitations of the
IMPES method
SW =SWir
water
Pc
Boundary
Conditions
Imbibition process:
water
S
Swir
REFERENCES
Sw
ABOUT
1
Swir
1
w
Swir
Sw
1-Sor
Swir
1-Sor
Sw
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Discretization of Flow Equations
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure

We will use similar approximations for the two-phase equations as we did for one
phase flow.

Left side flow terms:
  k.k rw Pw 

  Txw i  1 ( Pw i 1  Pw i )  Txw i  1 ( Pw i 1  Pw i )
2
2
x   w Bw x  i
Discretization of
Flow Equations
Upstream mobility
term
Where:
Expansion of
Discretized
equations
Boundary
Conditions
IMPES Method

Limitations of the
IMPES method
QUESTIONS
  k .k ro Po 

  Txo i  1 ( Po i 1  Po i )  Txo i  1 ( Po i 1  Po i )
2
2
x   o Bo x  i
–
Oil transmissibility:
–
Oil mobility:
o 
Txoi  1 
2
2o i  1
2
 x
xi 

xi  i 1 
k
k
i 
 i 1
k ro
o Bo
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.
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Upstream mobility term
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure

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:
Discretization of
Flow Equations
QW
Upstream mobility
term
Sw
1
Expansion of
Discretized
equations
1-Sor
B.L with PC = 0
Boundary
Conditions
Swir
IMPES Method
x
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution

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:
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Discretization of
Flow Equations
–
–
Upstream mobility
term
Upstream selection:
weighted average selection:
 i   i
o
1
2
o
o i  
1
2
xi oi  xi1oi1 
xi  xi1 
QW
Sw
Expansion of
Discretized
equations
1
1-Sor
B.L (PC = 0)
Boundary
Conditions
Upstream
Weighted average
IMPES Method
Swir
x
Limitations of the
IMPES method
QUESTIONS
In reservoir simulation, upstream mobilities are normally used.
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Two phase system
(oil – water)

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.
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Sw
1
1-Sor
B.L (PC = 0)
Discretization of
Flow Equations
Small grid blocks
Large grid blocks
Upstream mobility
term
Expansion of
Discretized
equations
Swir
x

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.
Boundary
Conditions
IMPES Method
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Expansion of Discretized equations
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Discretization of
Flow Equations
Upstream mobility
term

The right hand side of the oil equation:
  So   So
 

 
 So  
t  Bo  Bo t
t  Bo 

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:
  So 

  Cpooi ( Poi  Poit )  Cswoi ( Swi  Swit )
t  Bo i
Expansion of
Discretized
equations
Boundary
Conditions
Where:
Cpooi 
IMPES Method
Cswoi  
Limitations of the
IMPES method
QUESTIONS
i (1  Swi )  cr d (1 / Bo) 
 Bo  dPo 
t
i
REFERENCES
ABOUT
i
B o i t i
EXIT
Saturated Oil-Water Simulation, IMPES Solution

The right hand side of the water equation:
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Discretization of
Flow Equations
  S w   Sw
 

 
 Sw  
t  Bw  Bw t
t  Bw 

By:
–
–
–
Upstream mobility
term
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:
Expansion of
Discretized
equations
Boundary
Conditions
  S w 

  Cpowi ( Poi  Poit )  Cswwi ( Swi  Swit )
t  Bw i
Where:
IMPES Method
Cpowi 
Limitations of the
IMPES method
C swwi 
QUESTIONS
REFERENCES
ABOUT
i Swi  cr

t  Bo

d (1 / Bw ) 

dPw  i
i
 dP 
  cow  C powi
Bwi t i  dS w  i
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Two phase system
(oil – water)
The discrete forms of the oil and water equations
 Oil equation:
Oil-Water Relative
Permeabilities
and Capillary
Pressure



Txoi  1 Poi 1  Poi   Txoi  1 Poi 1  Poi   qoi  Cpooi Poi  Poit  Cswoi Swi  Swit
2
2

Where:
Discretization of
Flow Equations
Txoi  1 
2
Upstream mobility
term
QUESTIONS
 x
x 
xi  i 1  i 
ki 
 ki 1
2
2oi  1
2
 x
x 
xi  i 1  i 
ki 
 ki 1
o i 1 if Po i 1  Po i
o i   
o i if Po i 1  Po i
1
2
1
2
 Water equation:
Txw i  1 Po i 1  Po i   Pcowi 1  Pcowi   Txw i  1 Po i 1  Po i   Pcowi 1  Pcowi   q wi 
2
IMPES Method
Limitations of the
IMPES method
Txoi  1 
2
oi 1 if Poi 1  Poi
oi   
 oi if Poi 1  Poi
Expansion of
Discretized
equations
Boundary
Conditions
2oi  1
2



 Cpowi Po i  Po it  Cswwi Swi  Swit

 Transmissibility and mobility terms are the same as for oil equation, except the
subtitles are changed from “o” for oil to “w” for water.
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Boundary Conditions
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
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:
Discretization of
Flow Equations
Upstream mobility
term
qwi 

Expansion of
Discretized
equations
At the end of a time step, the bottom hole injection pressure may theoretically
be calculated using the well equation:
Qwi  WCi oi Pwi  Pbhi 
Boundary
Conditions
where:
Well constant

IMPES Method
WC i 
Limitations of the
IMPES method
QUESTIONS
Qwi
Axi
REFERENCES
ABOUT
2ki h
r 
ln  e 
 rw 

Drainage radius
re 
yxi

EXIT
Saturated Oil-Water Simulation, IMPES Solution
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Discretization of
Flow Equations
Upstream mobility
term

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:
k ro k rwi 
Qwi Bwi  WCi  i 
(Pw i  Pbhi )


 oi
oi 
B

Qwi  WCi  oi oi  wi ( Pwi  Pbhi )
 Bwi

qinj
–
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.
Expansion of
Discretized
equations
Boundary
Conditions
or
Time
Pbh
Pmax
IMPES Method
Limitations of the
IMPES method
Pbp
QUESTIONS
REFERENCES
ABOUT
Time
EXIT
Saturated Oil-Water Simulation, IMPES Solution
2.
Injection at constant bottom hole pressure

Injection of water at constant bottom hole pressure is achieved by:
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
–
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.
Discretization of
Flow Equations
Upstream mobility
term

The well equation:
B

Qwi  WCi  oi oi  wi ( Pwi  Pbhi )
 Bwi

Expansion of
Discretized
equations
Capillary pressure
is neglected
Boundary
Conditions
IMPES Method

B

Qwi  WCi  oi oi  wi ( Po i  Pbhi )
 Bwi

At the end of the time step, the above equation may be used to compute the actual
water injection rate for the step.
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
3.
Constant oil production rate

qoi 
Discretization of
Flow Equations
Upstream mobility
term
for a constant surface oil production rate of Qoi (positive) in a well in
grid block i:
Qoi
Axi
in this case oil production will generally be accompanied by water production.
 The water equation will have a water production term given by:
Expansion of
Discretized
equations
 (Pwi  Pbhi )
qwi
  qoi wi
 oi (Po i  Pbhi )
Capillary pressure is neglected
Around the production well
qwi
  qoi
 wi
o i
Boundary
Conditions
IMPES Method
 the bottom hole production pressure for the well may be calculated using the well
equation for oil:
Qoi  WCi oi Poi  Pbhi 
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
qprod
Two phase system
(oil – water)
–
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Discretization of
Flow Equations
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
Upstream mobility
term
Expansion of
Discretized
equations
Pmin
WC(%) vs. Time(year)
Pbp
80
Time
70
Boundary
Conditions
60
–
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.
50
40
IMPES Method
30
Limitations of the
IMPES method
20
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.
10
0
0
QUESTIONS
2
REFERENCES
4
ABOUT
6
8
10
12
14
16
18
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Two phase system
(oil – water)
4.
Constant liquid production rate

Oil-Water Relative
Permeabilities
and Capillary
Pressure
Total constant surface liquid production rate of QLi (positive):
QLi  Qoi  Qwi
If capillary pressure is neglected:
Discretization of
Flow Equations
qoi 
Upstream mobility
term
oi
QLi
oi  wi Axi
and
qwi 
wi
QLi
oi  wi Axi
qprod
Total liquid
Expansion of
Discretized
equations
Oil
Water
Boundary
Conditions
0
IMPES Method
Time
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
5.
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
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  Qwi Bwi  QwinjBwinj
Discretization of
Flow Equations
If capillary pressure is neglected:
Upstream mobility
term

oi
qoi  
 oi Boi  wi Bwi
Expansion of
Discretized
equations
  Qwinj Bwinj 


A

x
i


and
Boundary
Conditions

wi
q wi  
 oi Boi  wi Bwi
IMPES Method
  Qwinj Bwinj 


A

x
i


Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
6.
Production at Constant bottom hole pressure
Two phase system
(oil – water)

Oil-Water Relative
Permeabilities
and Capillary
Pressure
Production well in grid block i with constant bottom hole pressure, Pbhi:
Qo i  WCi o i (Po i  Pbhi )
Discretization of
Flow Equations
and
Qwi  WCi wi (Pwi  Pbhi )
Substituting the flow terms in the flow equations:
Upstream mobility
term
qoi 
Expansion of
Discretized
equations
Boundary
Conditions
WCi
 oi(Po i  Pbhi )
Axi
and
qwi
 
WCi
 wi (Pwi  Pbhi )
Axi

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.
IMPES Method
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
IMPES Method
Two phase system
(oil – water)

2
2
2






 Cpowi Po i  Po it  Cswwi Swi  Swit
Where:

i=1, …, N
Upstream mobility
term

Expansion of
Discretized
equations
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.
t
t
Txot , Txwt , Csot , Csw
, C tpo , C tpw , PCow
Limitations of the
IMPES method
QUESTIONS
2
Txw i  1 Po i 1  Po i   Pcowi 1  Pcowi   Txw i  1 Po i 1  Po i   Pcowi 1  Pcowi   q wi 
Discretization of
Flow Equations
IMPES Method

Txoi  1 Poi 1  Poi   Txoi  1 Poi 1  Poi   qoi  Cpooi Poi  Poit  Cswoi Swi  Swit
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Boundary
Conditions
Discretized form of flow equations:
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Two phase system
(oil – water)

The two equations are combined so that the saturation terms are eliminated. The
resulting equation is the pressure equation:
ai Poi 1  bi Poi  ci Poi 1  di
Oil-Water Relative
Permeabilities
and Capillary
Pressure
i=1, …, N
–
Discretization of
Flow Equations
Upstream mobility
term
Expansion of
Discretized
equations

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:
Swi  Swit 
Boundary
Conditions
1
t
swoi
C
T
xo
t
i  12
Poi 1  Poi   Txoit Poi 1  Poi   qoi  Cpooit Poi  Poit 
1
2
IMPES Method
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Two phase system
(oil – water)

Oil-Water Relative
Permeabilities
and Capillary
Pressure
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:
Discretization of
Flow Equations
fws i 
Upstream mobility
term

Expansion of
Discretized
equations
qwi
qwi  qoi
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
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Limitations of the IMPES method
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure

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.
Discretization of
Flow Equations
Upstream mobility
term
Expansion of
Discretized
equations
Boundary
Conditions
IMPES Method
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
Questions
Two phase system
(oil – water)
1.
Make sketches of typical Kro, Krw and Pcow curves for an oil-water system, both for oildisplacement of water and water-displacement of oil, and label all relevant points.
Oil-Water Relative
Permeabilities
and Capillary
Pressure
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).
Discretization of
Flow Equations
Make also a sketch of the exact solution.
Upstream mobility
term
3.
Expansion of
Discretized
equations
Boundary
Conditions
Write the two flow equations for oil and water on discretized forms in terms ot
transmissibilities, storage coefficients and pressure differences.
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).
IMPES Method
Limitations of the
IMPES method
QUESTIONS
5.
List the assumptions for IMPES solutin, and outline briefly how we solve for pressures
and saturations.
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
References
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure
Discretization of
Flow Equations

Kleppe J.: Reservoir Simulation, Lecture note 6

Snyder and Ramey SPE 1645
Upstream mobility
term
Expansion of
Discretized
equations
Boundary
Conditions
IMPES Method
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT
Saturated Oil-Water Simulation, IMPES Solution
About this module
Two phase system
(oil – water)
Oil-Water Relative
Permeabilities
and Capillary
Pressure

Title: SATURATED OIL-WATER SIMULATION, IMPES SOLUTION (PDF)

Author:

Discretization of
Flow Equations
 Address:
NTNU
S.P. Andersensvei 15A
7491 Trondheim
Upstream mobility
term
 Website
Expansion of
Discretized
equations
Boundary
Conditions
Name: Prof. Jon Kleppe
 Email

Size: 750 Kb
IMPES Method
Limitations of the
IMPES method
QUESTIONS
REFERENCES
ABOUT
EXIT