Advanced Reservoir Characterization and Evaluation of CO{sub 2} Gravity Drainage in the Naturally Fractured Spraberry Trend Area Page: 92 of 190

                                
time level. n is the old time level for which we have a complete solution and all variables and
properties. n+1 is the new time level for which the solution is unknown. With this notation,
our finite-difference approximations to the 2D spontaneous imbibition equation is
Di,   1 / 2
Di-11/2, j  - Di-1/12,j - Di+1/2,j - Di   1/2 -Di,j+1/2    Di+1/2,j   Swi
Di,j-1/2
Shx h   [  n+1 n(13
D112,j and D i+12,]j are evaluated using arithmetic averaging as below
D i-1/2, j  =     2ij + D -,  ............................................(      4
Di + Di+1, j
D i+1/2, j  =     2       ........................................................................................(15)
Note that the relative permeabilities and capillary pressure are functions of water saturation;
thus, Eq. 13 must be solved iteratively. Once the new saturations are solved using the initial
guess, these new water saturations are compared with the old water saturations until
acceptance convergence has been reached.
The core plug is totally immersed in the water and therefore, boundary conditions are set to be
1-Sor. This implies that instantaneous imbibition occurs at the matrix-fracture interface. Initial
conditions are required to begin the time step sequence. In this study, initial conditions are
specified equal to initial water saturation.
2.2.6 Results and Discussions
In order to match the recovery from the imbibition experiments, only capillary pressure was
altered and other parameters were kept constant. The following relative permeability
correlations from Honarpour et al (1986) as inputs:
kro  = k,  (1-Sw  )3  ...........................................................................................(16)
k   = kr.(Sw )3..................................................(17)
For the base case, the maximum oil relative permeability (k,,,) was set to 1.0 and the
maximum water relative permeability (krw ) was set to 0.3. In these relationships, the S,* is
expressed as S.* = (Sw -Swi) .The residual oil saturation (Sor) was set to 0.5437 and
(I-Sor -Swi )
irreducible water saturation (Sw1) was set to 0.37. Those average values of Sor and Sw, values
are obtained from Table 2.2-1.

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