On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 31 of 188
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S injection well
0 O production well
- - -diagonal grid
Figure 1.8: Five-spot injection prodcution pattern used as an example calculation
stencil method  for the right-angled triangles, leading to grid orientation ef-
fects. As for the CVM, we have fy - + fjy,- < 0 and S increases; therefore, a
net flux exists in the diagonal direction.
Consider the obtuse triangle in Figure 1.7. For the CVFE formulation, as a
result of violating the positive transmissibility condition, we have fly -m+ fy- >
0, because Xzi - xJk > 0. The phase saturation Sj is, therefore, reduced to a value
less than Sir, which is physically impossible. In the CVM, we have fy -g = 0 and
fly < 0; therefore, the value of Sj increases.
1.9.2 Five-spot injection problem
The test problem for the proposed CVM and the CVFE is a five-spot water
injection problem. The arrangement of production and injection wells is shown in
Figure 1.9.2. The distance between any two adjacent production wells is 2000 ft.
One injection well is placed at the center of a square formed by four surrounding
production wells. The wetting and non-wetting phases considered in this problem
are water and oil, respectively. Oil and water are produced from production
wells, and water is injected through injection wells. Two types of grids are used
for testing the grid orientation effects. In the diagonal grid, the production and
injection wells are connected through the diagonal of the grid. In the parallel grid,
wells are connected through grid lines. These two subdomains within the larger
five-spot pattern are shown in Figure 1.9.2. Each subdomain is discretized into
20 by 20 square blocks. Each square block is further divided into two triangles. It
should be noted that the parallel grid domain is twice the size of the diagonal grid
domain. The water migration patterns are expected to be identical between the
injection and production wells in both cases. The oil production rates at different
water injection stages are also expected to be the same for the two grid systems.
Reservoir rock and fluid properties are listed in Table 1.1. Initially, oil pressure
is at 3000 psia and water saturation is 0.20. Because of the symmetric layout
of the wells, no flow boundary conditions are used for the two subdomains in
consideration. The total fluid rate for both injection and production wells is 40
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Deo, Milind D. On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs, report, August 31, 2005; Utah. (https://digital.library.unt.edu/ark:/67531/metadc877059/m1/31/: accessed May 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.