A unified analysis of fully mixed virtual element method for wormhole propagation arising in the petroleum engineering

Abstract

Wormhole propagation, arising in petroleum engineering, is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under the dissolution of injected acid and plays a very important role in the product enhancement of oil and gas reservoirs. In this paper, a fully mixed virtual element method (VEM) is employed to discretize this problem, in which mixed VEM is used not only for the Darcy flow equations but also for approximation the concentration equation by introducing an auxiliary flux variable to guarantee full mass conservation. The stability, existence and uniqueness of solution of the associated mixed VEM are proved by fixed point theory. Also, we obtain unconditionally optimal error estimate for concentration and auxiliary flux variable of convection-diffusion equation, as well as for the velocity and pressure of Darcy equations in the L 2 norm. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the applicability for solving actual problems.