Thermodynamically consistent Darcy–Brinkman–Forchheimer framework in matrix acidization

Abstract

Matrix acidization is an important technique used to enhance oil production at the tertiary recovery stage, but its numerical simulation has never been verified. From one of the earliest models, i.e., the two-scale model (Darcy framework), the Darcy–Brinkman–Forchheimer (DBF) framework is developed by adding the Brinkman term and Forchheimer term to the momentum conservation equation. However, in the momentum conservation equation of the DBF framework, porosity is placed outside of the time derivation term, which prevents a good description of the change in porosity. Thus, this work changes the expression so that the modified momentum conservation equation can satisfy Newton’s second law. This modified framework is called the improved DBF framework. Furthermore, based on the improved DBF framework, a thermal DBF framework is given by introducing an energy balance equation to the improved DBF framework. Both of these frameworks are verified by former works through numerical experiments and chemical experiments in labs. Parallelization to the complicated framework codes is also realized, and good scalability can be achieved.

A field-based general framework to simulate fluids in parallel and the framework’s application to a matrix acidization simulation

“…The acid injection velocity is increased from 1.04 × 10 −7 m/s to 1.04 × 10 −4 m/s, and an acid-efficiency curve can be drawn from the simulation results. From the results, it is concluded that the two acid-efficiency curves of works [ 42 , 45 ] match each other well. In addition, both works indicate that the optimal injection velocity is 1.04 × 10 −6 m/s.…”

“…A linear system can be constructed from one or more PDEs. For example, after discretization, the concentration equation in the DBF model ([ 37 – 42 ]): which is a PDE, forms a linear system, by which the acid concentration can be achieved. However, another linear system in the DBF model, which is the velocity-pressure linear system, is constructed from two PDEs, the momentum conservation equation and the mass conservation equation.…”

“…In Section 5.3 of [ 42 ], the authors used Masor to perform simulations and compared the numerical results of Masor with the work [ 45 ]. In that case, a matrix of 0.04 m × 0.04 m × 0.1 m, and a grid of 36 × 36 × 90 cells are used to discretize the matrix domain.…”

“…In addition, both works indicate that the optimal injection velocity is 1.04 × 10 −6 m/s. Moreover, five different dissolution patterns in the work [ 45 ] can be reproduced by Masor in [ 42 ]. Thus, the correctness and effectiveness of Masor’s method are proven by [ 42 ].…”

“…Moreover, five different dissolution patterns in the work [ 45 ] can be reproduced by Masor in [ 42 ]. Thus, the correctness and effectiveness of Masor’s method are proven by [ 42 ]. It is acknowledged that the other works ([ 37 – 42 , 46 ]) have also done many verification works, and we will not repeat them here.…”