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Abstract
This paper examines the sensitivity to gridding of first-contact miscible and three-component multicontact miscible condensing-gas drive predictions made with an upstream differenced simulator. These cross-section simulations are made for various reservoir descriptions and for models of increasing numbers of grid blocks. The largest models range from 5000 to 24,000 grid blocks. The paper examines gridding sensitivity when the coefficients of convective terms are evaluated by the single-point upstream method, and it also examines the utility of two-point upstream weighting and selectively refined initial gridding for moderating gridding sensitivity.
These simulations show the following behavior. For some problems with single-point upstream weighting, ultimate recovery changes monotonically with an increasing number of grid blocks. The change in recovery with increasing grid blocks can either be a decrease or an increase depending on the particular reservoir description, although decreasing recovery was the most common behavior for the reservoir descriptions examined here. Two-point upstream weighting and selectively refined initial gridding reduced, but didn't eliminate, this sensitivity. Some predictions appear to converge almost linearly with predictions appear to converge almost linearly with grid size to approximately the same answer in the limit of zero grid block size (i.e., infinite grid blocks) when the problem is worked in different ways, e.g., single-point weighting, two-point weighting, selectively refined initial gridding. Other problems, however, do not appear to be converging for the largest models that were feasible.
Addition of physical diffusion/dispersion of a magnitude that might occur in reservoirs didn't affect the gridding sensitivity for two of the three reservoir descriptions examined, at least not for grid sizes that were feasible. However, for one reservoir description, recoveries computed with and without physical diffusion/dispersion appear to be diverging physical diffusion/dispersion appear to be diverging with grid refinement.
Introduction
Miscible flood predictions and designs are cormonlv made with finite-difference simulators in which the l (Ppxpikrp)/up coefficients of the convective terms are evaluated at the upstream grid block. It is well known that this single-point weighting -auses truncation error that results in sensitivity to grid block size. The truncation error from single-point upstream differencing causes an artificial dispersion! that in one-dimensional problems is equivalent to a physical dispersivity of magnitude Ax/2 and is physical dispersivity of magnitude Ax/2 and is qualitatively similar to physical dispersion in two-dimensional problems. As grid block size decreasesli the influence of this artificial dispersion decreases, i although in reservoir-scale problems, grid block sizei would have to be intractably small for this artificiall dispersion to be on the order of realistic physical dispersion. Also, with a five-point differencing scheme, such a formulation resuits in sensitivity to grid orientation as well as to the number of grid blocks.
Young gives an example of gridding sensitivity for ani upstream differenced simulator and for a first-contact miscible, unfavorable mobility ratio, homogeneous, two-dimensional areal five-spot calculation. He used a nine-point difference approximation to reduce gridi orientation sensitivity but with single-point upstreami weighting and with no physical dispersion. He shows that as grid size decreases, recovery decreases monotonically, and an unchanging answer is not approached. He concludes that in the absence of physical dispersion, the problem is not well posed andl physical dispersion, the problem is not well posed andl such a result is not unreasonable. In Young's problem the influence of gravity is not a factor.
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