Mixed finite element-based fully conservative methods for simulating wormhole propagation

Abstract

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. MIXED FINITE ELEMENT-BASED FULLY CONSERVATIVE METHODS FOR SIMULATING WORMHOLE PROPAGATION *JISHENG KOU † , SHUYU SUN ‡ , AND YUANQING WU § Abstract. Wormhole propagation during reactive dissolution of carbonates plays a very important role in the product enhancement of oil and gas reservoir. Because of high velocity and nonuniform porosity, the Darcy-Forchheimer model is applicable for this problem instead of conventional Darcy framework. We develop a mixed finite element scheme for numerical simulation of this problem, in which mixed finite element methods are used not only for the Darcy-Forchheimer flow equations but also for the solute transport equation by introducing an auxiliary flux variable to guarantee full mass conservation. In theoretical analysis aspects, based on the cut-off operator of solute concentration, we construct an analytical function to control and handle the change of porosity with time; we treat the auxiliary flux variable as a function of velocity and establish its properties; we employ the coupled analysis approach to deal with the fully coupling relation of multivariables. From this, the stability analysis and a priori error estimates for velocity, pressure, concentration and porosity are established in different norms. Numerical results are also given to verify theoretical analysis and effectiveness of the proposed scheme.

A semi-analytic porosity evolution scheme for simulating wormhole propagation with the Darcy–Brinkman–Forchheimer model

“…We now focus on the discrete schemes of porosity. Based on the explicit treatment for the concentration and implicit treatment for the porosity in the porosity equation, a semi-implicit (SIM) scheme has been presented in [13]…”

“…For the cut-off operator for velocity defined in (3.6), we assume that the constant M is chosen as M > 2 u (L ∞ (J×Ω)) d . Under this assumption, using the similar approach in [13], we deduce that…”

“…It is worth accounting for both the porous media and clear fluid area, and the Darcy-Brinkman-Forchheimer framework exactly deals with the problems [11,24]. In [13], the Darcy-Forchheimer model is used to model the flow of high velocity in nonuniform porosity instead of the conventional Darcy framework. In this work, for the purpose of retrieving more reasonable results, numerical modeling is based on the Darcy-Brinkman-Forchheimer framework, which is more capable of handling with the multiple complication factors in wormhole propagation, including the boundary-layer development, macroscopic shear stress, microscopic shear stress and inertial force.…”

“…Numerical methods for simulating matrix acidization in carbonate reservoirs have been extensively studied in [4, 6, 9, 13-17, 19, 21, 23-25]. In [13], a mixed finite element scheme based on the Darcy-Forchheimer model has been suggested for the numerical simulation of wormholes, in which mixed finite element methods are used not only for the Darcy-Forchheimer flow equations but also for the solute transport equation by introducing an auxiliary flux variable to guarantee full mass conservation. The block-centered finite difference methods for simulating wormhole propagation have been developed and analyzed in [15,16].…”

“…Therefore, the time schemes of porosity have a significant effect on accuracy and stability of the numerical simulation for wormhole propagation. A semi-implicit time scheme has been presented in [13], in which the porosity is implicitly treated in the equation of porosity. In this paper, we propose a semi-analytic time scheme, and its great virtue is that the equation of porosity is analytically solved at each time step for given acid concentration.…”