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

TL;DR

This paper introduces a space-fractional diffusivity model for petroleum reservoirs that accounts for heterogeneity levels, using a generalized Taylor's formula and an implicit numerical scheme to analyze flow deviations.

Contribution

It develops a novel fractional radial-cylindrical diffusivity model that quantifies heterogeneity effects in petroleum reservoirs, extending classical models with fractional calculus.

Findings

The model captures the impact of heterogeneity on pressure drops.

Numerical simulations demonstrate deviations from homogeneous flow.

Higher fractional order correlates with increased localized pressure drops.

Abstract

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 heterogeneity..