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VA – GP - OA: Numerical Multiphase PTA
p 2/29
2. Water or gas injection
We start with the analysis of injection of one phase into another, without mass transfer
between the two phases. Although we illustrate the effects based on the results of simulations
of water injection into dead oil, the main features described below remain valid for other
injection schemes (e.g. water injection into gas, gas injection, etc...).
Two test cases are presented. Case 1 illustrates and explains the development of oscillations,
while Case 2 is a validation of the numerical model against analytical predictions.
2.1. Test Case 1
Test description
We consider a rectangular reservoir of dimensions 10,000
10,000 ft², with thickness 100 ft.
The injection well is located at the center of the reservoir. The formation compressibility is
cf=3e-6 psi
-1
, the permeability is k=1000 mD, and the porosity is
=0.2. The reservoir initially
contains dead-oil, with constant properties (
o = 0.3 cp, Bo=1). The initial water saturation in
the reservoir is Swi=0.2. Water is injected for 10,000 hr at constant rate Qw=10,000 stb/D,
followed by a 10,000 hr fall-off. Water is also characterized by constant properties (
w =
0.3cp, Bw=1). Equal viscosities between oil and water have been chosen so that the mobility
effects are only due to the choice of relative permeability curves. Relative permeabilities have
the following properties:
Kromax = 0.8; Krwmax = 0.5; Swr = 0.2; Sor = 0.25
The curve shape is a power curve (Corey type). 3 values of the Corey exponent have been
tested: 1 (called test “Corey 1”), 2 (“Corey 2”) and 3 (“Corey 3”), in order to show the
influence of the non-linearity of Kr curves on oscillations.
Results
During the injection phase, the pressure derivative exhibits oscillations on the loglog plot. The
level of these oscillations increases with the non-linearity of the relative permeability curves
(Figure 1). This can be related to an increase of the mobility contrasts with the non-linearity of
the curves.
Figure 1: Water injection for 3 values of the exponent of the relative permeability curves
0.01
0.1
1
10
100
1000
10000
Time [hr]
10
100
Corey 1
Corey 2
Corey 3