Fractional radial-cylindrical diffusivity model for levels of heterogeneity in petroleum reservoirs

The generalized Taylor’s Formula is used to derive a fractional radial-cylindrical diffusivity model via a fractional conservation of mass in radial geometry in a petroleum reservoir. The result is a space-fractional generalization of the diffusivity model with an arbitrary order that explains the degree of heterogeneity of the medium (a continuous spectrum of zero to one where zero heterogeneity equates to homogeneity). An implicit unconditionally stable numerical difference scheme of the linear form of the derived model is used to illustrate deviations of the model from the homogeneous medium. The variation of the order produces additional localized pressure drops during flow which is congruous with fluid inhibition effect of heterogeneities.