Numerical Simulation of Water Injection in Disconnected and Connected Fractured Media Using Jacobian-free Fully Implicit Control Volume Method

摘要

In this work we present an algorithm for water injection in 2Dand 3D connected and discrete fractured media using a fullyimplicit approach for the control-volume method. Thecrossflow equilibrium concept (that is, the discrete fractureapproach) is used to reduce the fracture dimension by one. Acongruent mathematical formulation is provided for thediscrete fracture approach, establishing a relationship betweenthe matrix and fracture main variables following the physics oftwo-phase flow at the matrix-fracture interface. The system ofpartial differential equations is discretized within the discretefractureframework using the control-volume method in afully-implicit formulation. The resulting system of non-linearequations is solved with a Jacobian-free Newton-Krylovsolver. In this solver, there is no need to assemble the Jacobianmatrix of the non-linear system of equations, which reducesmemory and computational effort requirement. We presentnumerical results for several examples with various degrees ofnon-linearity in capillary pressure and relative permeabilityusing both 2D and 3D discrete-and connected- fracturedmedia. The Jacobian-free Newton-Krylov methoddemonstrates to be very efficient for all the numericalexamples. The performance of the fully-implicit model iscompared with the IMPES formulation to determine the rangeof applicability of the proposed method.