Use of Higher Moments for the Description of Upscaled, Process Independent Relative Permeabilities

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Abstract
Summary: Traditional approaches for the scale up of highly detailed reservoir descriptions to the coarser scales more appropriate for reservoir simulation often rely on the use of pseudo relative permeabilities. Two of the major limitations in the use of pseudo relative permeabilities are the process dependency inherent in the resulting curves and the need for a different set of curves in each coarse scale grid block. In this paper, these limitations are illustrated and quantified through numerical simulations of viscous dominated displacements in fine scale, heterogeneous, two dimensional systems. It is concluded that the functional form of traditional pseudo relative permeability descriptions is too limited to capture a wide variety of flow behavior and that an ‘extended description,’ in which the upscaled relative permeabilities depend on variables in addition to the phase saturations, is required. The additional variables we propose are higher moments of the fine scale variables, specifically the variance of saturation, the variance of pressure and the velocity-saturation covariance. It is then shown that these new variables are capable of accurately correlating upscaled relative permeabilities for a variety of displacements in a process independent manner; i.e., the extended description can account for varying inlet conditions and permeability realizations. The practical use of this methodology in an actual reservoir simulation will, however, require the derivation and solution of coarse scale evolution equations for the moments.