Geoscience Reference
In-Depth Information
Fig. 3.11
Periodic
pressure boundary
conditions applied to a
periodic permeability field,
involving an inclined layer.
Example boundary cell
pressure conditions are
shown
P(i,0)=P(i,nz)
x
z
P(nx+1,j)=P(1,j)-
D
P
P(0,j)=P(nx,j)+
D
P
P(i,nz+1)=P(i,1)
permeability. Some key papers on the calculation
of permeability for heterogeneous rock media
include White and Horne (
1987
), Durlofsky
(
1991
) and Pickup et al. (
1994
).
To illustrate the numerical approach we take an
example proposed by Pickup and Sorbie (
1996
)
shown in Fig.
3.11
. Assuming a fine-scale grid of
permeability values, k
i
, we want to calculate the
upscaled block permeability tensor, k
b
.An
assumption on the boundary conditions must be
made, and we will assume a period boundary con-
dition (Durlofsky
1991
) - where fluids exiting one
edge are assumed to enter the opposite edge - and
apply this to a periodic permeability field (where
the model geometry repeats in all directions). This
arrangement of geometry and boundary conditions
gives us an exact solution.
First a pressure gradient
Usually, at the fine scale we assume the local
permeability is not a tensor so that only one value
of k is required per cell.
We then wish to know the upscaled block per-
meability for the whole system. This is a relatively
simple step once all small scale Darcy equations
are known, and involves the following steps:
1. Solve the fine-scale equations to give pressures,
P
ij
for each block.
2. Calculate inter-block flows in the x-direction,
using Darcy's Law.
3. Calculate total flow, Q, by summing individual
flows between any two planes.
4. Calculate k
b
using Darcy's Law applied to the
upscaled block.
5. Repeat for the y and z directions.
For the upscaled block this results in a set of
terms governing flow in each direction, such that:
P is applied to the
boundaries in the x direction. For the boundaries
parallel to the applied pressure gradient, the peri-
odic condition means that P in cell (i, 0) is set as
equal to P in cell (i, nz), where n is the number of
cells. A steady-state flow simulation is carried
out on the fine-scale grid, and as all the
permeabilities are known, it is possible to find
the cell pressures and flow values (using matrix
computational methods).
We then solve Darcy's Law for each fine-
scale block:
ʔ
0
1
P
P
P
∂
1
ʼ
k
xx
∂
k
xy
∂
k
xz
∂
@
A
u
x
¼
x
þ
y
þ
z
∂
∂
0
@
1
A
1
ʼ
P
P
P
∂
k
yx
∂
k
yy
∂
k
yz
∂
u
y
¼
x
þ
y
þ
ð
3
:
11
Þ
z
∂
∂
0
@
1
A
1
ʼ
P
∂
P
P
∂
k
zx
∂
k
zy
∂
k
zz
∂
u
z
¼
x
þ
y
þ
z
∂
For example, the term k
zx
is the permeability in
the z direction corresponding to the pressure gradi-
ent in the x direction. These off-diagonal terms are
intuitive when one looks at the permeability field.
Take the vertical (x, z) geological model section
shown in Fig.
3.12
. If the inclined orange layers
have lower permeability, then flow applied in the
+x direction (to the right) will tend to generate a
flux in the -z direction (i.e. upwards). This results
!
¼
=ðÞ
k
:
∇
P
ð
:
Þ
1
3
10
where
!
is the local flow vector
ʼ
is the fluid viscosity
k
is the permeability tensor
