Page 338 - Principles of Applied Reservoir Simulation 2E

Part V: Technical Supplements 323

a$
- - -
'•
°
-
iK —- " + jK — + kK " - - (32,4)
By z dz
have used the dyadic notation K to signify that permeability is a tensor of rank
two. The expanded form of Eqs. (32.2) through (32.4) employs the common
assumption that the coordinate axes of our reference system are aligned along
the principal axes of K. As discussed in Chapters 7 and 8, and associated

references, this assumption impacts the ability of the simulator to accurately
model fluid flow.
Combining Eqs. (4.27) through (4.29) with Eqs. (32.2) through (32.4)
gives
.
(32 5)
St B

(32.6)

and

(32.7)

Equations (32.5) through (32.7) are equivalent to Peaceman's [1977] Eqs. (1-
105) through (1-107) for a three-dimensional system, except we have also
allowed gas to dissolve in the water phase. Our rate and coordinate system sign
conventions also differ. If these differences are taken into consideration, the
formulations are seen to be equivalent.

32.2 Introduction of the Capillary Pressure Concept

The presence of oil-, water-, and gas-phase pressures in Eqs. (32.5)
through (32.7) complicates the problem. We simplify the handling of the phase