Methods for performing reservoir simulation

Details Methods for performing reservoir simulation Methods for performing reservoir simulation

1999-11-16

2003-12-09

Jones, Hugh (Department: 2123)

Data processing: structural design, modeling, simulation, and em

Simulating nonelectrical device or system

Fluid

C703S002000, C703S009000

Reexamination Certificate

active

06662146

ABSTRACT:

FIELD OF THE INVENTION
The present invention relates to reservoir simulation, and in particular, to methodologies for performing reservoir simulation by solving an implicit matrix equation or an implicit-IMPES matrix equation.
BACKGROUND OF THE INVENTION
In an attempt to understand and predict the physical behavior of reservoirs (such as petroleum reservoirs), reservoir engineers and scientists have generated various mathematical descriptions of reservoirs and the fluids they contain. These mathematical descriptions are often expressed as coupled sets of differential equations. Since it is quite often impossible to obtain solutions of the differential equations in all but the simple cases, the differential equations are discretized in space and time, and the resulting difference equations are solved using various numerical simulation techniques. For example, the following difference equations represent the volumetric accumulation of oil and water in a particular cell (i.e. cell i) over the course of a timestep from time index n to n+1 assuming rock and fluid incompressibility in a one-dimensional reservoir:
(
λ
o
)
i
+
1
/
2
β
⁡
[
(
p
o
)
i
+
1
α
-
(
p
o
)
i
α
]
-
(
λ
o
)
i
+
1
/
2
β
⁡
[
(
p
o
)
i
α
-
(
p
o
)
i
-
1
α
]
+
(
q
o
)
i
β
=
φ
⁢
 
⁢
V
i
B
o
⁢
Δ
⁢
 
⁢
t
⁡
[
(
S
o
)
i
n
+
1
-
(
S
o
)
i
n
]
,
⁢

(
B1
)
(
λ
w
)
i
+
1
/
2
β
⁡
[
(
p
w
)
i
+
1
α
-
(
p
w
)
i
α
]
-
(
λ
w
)
i
+
1
/
2
β
⁡
[
(
p
w
)
i
α
-
(
p
w
)
i
-
1
α
]
+
(
q
w
)
i
β
=
φ
⁢
 
⁢
V
i
B
w
⁢
Δ
⁢
 
⁢
t
⁡
[
(
S
w
)
i
n
+
1
-
(
S
w
)
i
n
]
,
(
B2
)
where &Dgr;t is the timestep size;
V
i
is the volume of cell i;
&phgr; is porosity, i.e. pore volume per cell volume;
(S
o
)
i
is the saturation of oil at cell i, i.e. the fraction of the pore volume occupied by oil in cell i;
(S
w
)
i
is the saturation of water at cell i, i.e. the fraction of the pore volume occupied by water in cell i;
B
o
and B
w
are the formation volume factors (FVF) for oil and water respectively;
(p
o
)
i−1
, (p
o
)
i
, (p
o
)
i+1
are oil pressures at cell i−1, cell i, and cell i+1 respectively;
(p
w
)
i−1
, (p
w
)
i
, (p
w
)
i+1
are water pressures at cell i−1, cell i, and cell i+1 respectively;
(q
o
)
i
is the rate of oil injection into cell i, and takes the value zero at most cells and takes a negative value at cells which interact with a depletion well;
(q
w
)
i
is the rate of water injection into cell i, and typically takes a zero value except at cells which interact with an injection or depletion well;
(x)
n
and (x)
n+1
represent a quantity x evaluated at time indices n and n+1 respectively, where the former is known information, having been determined from previous computations, and the later is an unknown to be solved for by some computational method; and
(x)
&agr;
and (x)
&bgr;
represent quantities which are to be evaluated at time index n or n+1 subject to user selection.
The oil transmissibility-mobility factors (&lgr;
o
)
i+½
and (&lgr;
o
)
i−½
are defined as
(
λ
o
)
i
+
1
/
2
=
(
Ak
x
i
+
1
-
x
i
)
⁢
(
M
o
)
i
+
1
/
2
,
(
B3
)
(
λ
o
)
i
-
1
/
2
=
(
Ak
x
i
-
x
i
-
1
)
⁢
(
M
o
)
i
-
1
/
2
,
(
B4
)
where A is the area normal to the axis of the one-dimensional reservoir;
(M
o
)
i+½
is the mobility of oil in transit between cell i and cell i+1;
(M
o
)
i−½
is the mobility of oil in transit between cell i and cell i−1;
x
k
is the position of the k
th
cell along the one-dimensional axis.
Similar definitions apply for the water transmissibility-mobility products (&lgr;
w
)
i+½
and (&lgr;
w
)
i−½
. The difference equations (B1) and (B2) above are augmented with several auxiliary relations as follows:
S
o
+S
w
=1,  (B5)
p
w
−p
o
=P
c
(
S
o
),  (B6)
M
o
=M
o
(
p
o
,S
o
),  (B7)
M
w
=M
w
(
p
w
,S
w
).  (B8)
Relation (B5) follows from the definition of saturation. Capillary pressure P
c
which is defined as the difference in pressure between water and oil is a known function of oil saturation. Oil mobility M
o
is a known function of oil pressure and oil saturation. Water mobility M
w
is a known function of water pressure and water saturation.
Since oil mobility M
o
is a function of oil pressure p
o
and oil saturation S
o
, and these later variables are defined at cell centers, a question arises as to the proper means of evaluating the in-transit oil mobilities (M
o
)
i+½
and (M
o
)
i−½
. According to the midpoint weighting scheme, the in-transit oil mobility is defined to be the average of the mobilities at the two affected cells. For example,
(
M
o
)
i+½
=½(
M
o
)
i
+½(
M
o
)
i+1
,  (B9)
where (M
o
)
i
is evaluated using the oil saturation (S
o
)
i
and oil pressure (p
o
)
i
prevailing at cell i, and (M
o
)
i+1
is evaluated using the oil saturation (S
o
)
i+1
and oil pressure (p
o
)
i+1
prevailing at cell i+1. Alternatively, according to the upstream weighting scheme, the in transit mobility may be defined as the oil mobility at the upstream cell of the two affected cells, where the upstream cell is defined as the cell with higher pressure (since fluids flow from high pressure to low pressure). For example,
(
M
o
)
i
+
1
/
2
=
{
(
M
o
)
i
,
if
⁢
 
⁢
(
p
o
)
i
≥
(
p
o
)
i
+
1
(
M
o
)
i
+
1
,
otherwise
.
(
B10
)
If the pressure variables and transmissibility-mobility factors in Equations (B1) and (B2) are evaluated at the new time index, i.e. &agr;=&bgr;n+1, Equations (B1) and (B2) take the form
(
λ
o
)
i
+
1
/
2
n
+
1
⁡
[
(
p
o
)
i
+
1
n
+
1
-
(
p
o
)
i
n
+
1
]
-
(
λ
o
)
i
-
1
/
2
n
+
1
⁡
[
(
p
o
)
i
n
+
1
-
(
p
o
)
i
-
1
n
+
1
]
+
(
q
o
)
i
n
+
1
=
φ
⁢
 
⁢
V
i
B
o
⁢
Δ
⁢
 
⁢
t
⁡
[
(
S
o
)
i
n
+
1
-
(
S
o
)
i
n
]
,
(
B11
)
(
λ
w
)
i
+
1
/
2
n
+
1
⁡
[
(
p
w
)
i
+
1
n
+
1
-
(
p
w
)
i
n
+
1
]
-
(
λ
w
)
i
-
1
/
2
n
+
1
⁡
[
(
p
w
)
i
n
+
1
-
(
p
w
)
i
-
1
n
+
1
]
+
(
q
w
)
i
n
+
1
=
φ
⁢
 
⁢
V
i
B
w
⁢
Δ
⁢
 
⁢
t
⁡
[
(
S
w
)
i
n
+
1
-
(
S
w
)
i
n
]
.
(
B12
)
The transmissibility-mobility factors and the phase injection rates are functions of saturation and pressure, and are evaluated at the new time level n+1. Thus, Equations (B11) and (B12) are non-linear in the unknown variables
(
p
o
)
i−1
n+1
,(
p
o
)
i
n+1
,(
p
o
)
i+1
n+1
,
(
p
w
)
i−1
n+1
,(
p
w
)
i
n+1
,(
p
w
)
i+1
n+1
,
(
S
o
)
i−1
n+1
and (
S
w
)
i−1
n+1
,
(
S
o
)
i
n+1
and (
S
w
)
i
n+1
,
(
S
o
)
i+1
n+1
and (
S
w
)
i+1
n+1
.  (B13)
Equations (B11) and (B12) may be expressed in terms of a reduced set of unknown variables using relations (B5) and (B6). For example, the variable (S
w
)
i
n+1
may be replaced by 1−(S
o
)
i
n+1
. Similarly, (p
o
)
i
n+1
may be replaced by (p
o
)
i
n+1
+P
c
[(S
o
)
i
n+1
]. Thus, Equations (B11) and (B12) may be expressed in terms of the following reduced set of unknown variables:
 (
p
o
)
i−1
n+1
,(
p
o
)
i
n+1
,(
p
o
)
i+1
n+1
, (
S
o
)
i−1
n+1
, (
S
o
)
i
n+1
,(
S
o
)
i+1
n+1
  (B14)
Assuming that there are N cells in the reservoir being modeled, Equations (B11) and (B12) describe a coupled non-linear system of 2N equations (two equations per cell) with 2N unknowns—each cell contributes an unknown pressure (p
o
)
i
n+1
and an unknown saturation (S
o
)
i
n+1
. An iterative method such as Newton's method is generally required to solve such systems.
Let vector X be the vector of 2N unknowns for the

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