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VA – GP - OA: Numerical Multiphase PTA
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2.3. Pseudo-relative permeabilities
Pseudo-relative permeabilities were originally developed to reduce the numerical dispersion
during multiphase simulations (Kyte and Berry, 1968). Later, they were also used for
numerical upscaling with limited success, as reviewed by Barker and Thibeau, 1996. In Ecrin,
the idea was to develop pseudo-Kr curves in order to approach the continuous model and
decrease the level of oscillations. Main features of these pseudos should be:
Linear total mobility versus S
w
Null Kr(S
w
) while S
w
<S
f
The implementation of the pseudo-functions is a complex task, because the resulting curves
must be continuous and the correction should apply only in cells where the front is present. As
shown on Figure 10, the released solution significantly decreases the amplitude of oscillations.
When used in conjunction with grid refinement, the pseudo-Kr correction can give very
satisfying results.
Figure 10: Oscillations damped by Pseudo-Kr and grid refinement
2.4. Test Case 2 – Interpretation of the LogLog plot
The objective of this section is to analyze the loglog plot, and to validate our numerical results
against analytical models. An injection test has been simulated, based on the case from
Levitan, 2002.
Test description
The reservoir is infinite (in fact, we simulate a circle with R=40,000ft) with thickness h=100ft.
The porosity is
Φ
= 0.20 and the permeability is k = 1000 md. The rock compressibility is
cr=5e-6 psi-1. The initial pressure of the reservoir is Pi = 5000 psi.
The well is localized at the center, with rw=0.357 ft. No wellbore storage is considered, in
order to show the early time behavior.
The initial fluid is oil, with constant properties co = 9e-6 psi-1 and μo = 0.3 cp.
The injected fluid is water, also with constant properties cw = 3e-6 psi-1 and μw = 0.25 cp.
Finally, Kr curves are power laws with exponent 2, and:
Swr = 0.20, Sorw = 0.25
Krw(1-Sor) = 0.20, Krow(Swr) = 0.80
0.01
0.1
1
10
100
1000
10000
Time [hr]
10
100
Corey 3
Corey 3 with pseudos
Corey 3 refined (gpr=1.2) with pseudos