Geoscience Reference

In-Depth Information

Upscaled Relative Permeability

1.0

0.8

k
rox

0.6

k
rwx

0.4

k
rwz

k
roz

0.2

0

Example Rock Curves

0

0.2

0.4

0.6

0.8

1.0

2.0

Sw

Pc Rock 1

krw Rock 1

kro Rock 1

Pc Rock 2

krw Rock 2

kro Rock 2

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

Upscale

0.2

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Sw

Fig. 4.11
Capillary equilibrium steady-state upscaling method applied to a simple layered model

applied resulting in a vertical trend in the

saturation at any chosen pressure reference.

The GCE solution should tend towards the

CE solution as

4. Find the phase permeabilities, e.g. k
o1
¼

k
1
*

k
ro1
;

5. Calculate the upscaled permeability for each

direction and for each phase using the arith-

metic and harmonic averages;

6. Invert back to upscaled relative permeability,

e.g. k
ro1
¼

the length-scale becomes

increasingly small.

All three steady-state methods involve a series

of independent single-phase flow calculations and

therefore can employ a standard single-phase

pressure solver algorithm. The methods can there-

fore be rapidly executed on standard computers.

The capillary equilibrium method can be eas-

ily calculated for a simple case, as illustrated in

Figure
4.11
by an example set input functions for

a regular layered model. Upscaled relative per-

meability curves for this simple case can be cal-

culated analytically (using a spreadsheet or

calculator). The method uses the following

steps (refer to Fig.
4.9
):

1. Chose a value for pressure, P
c1
;

2. Find the corresponding saturation value, S
w1
;

3. Determine the relative permeability for oil

and water for each rock type, k
ro1
,k
rw1
,k
ro2
,

k
rw2
;

k
o1
/k
upscaled
(once again the arith-

metic and harmonic averages are used to

obtain the upscaled absolute permeability);

7. Repeat for next value pressure, P
c2
.

Note that the upscaled curves are highly aniso-

tropic, and in fact sometimes lie outside the range

of the input curves. This is because of the effects

of capillary forces - specifically capillary trapping

when flowing across layers. Capillary forces result

in preferential imbibition of water (the wetting

phase) into the lower permeability layers, making

flow of oil (the non-wetting phase) into these low

permeability layers even more difficult.

These somewhat non-intuitive effects of capil-

lary pressure in laminated rocks can be demon-

strated experimentally (Fig.
4.12
). In the case of

two-phase flow across layers in a water-wet