Least-Squares Formulations Applied to Miscible Flow Problems

Abstract

Incompressible miscible flows in porous media which characterize tertiary recovery process
in oil reservoir are mathematically modeled by a coupled non-linear partial differential equation system with appropriate boundary and initial conditions.
This system can be solved by an implicit sequential method breaking it in a elliptic sub-system involving pressure and velocity fields coming from mass conservation equation and Darcy’s law together with a transport equation predominantly convective for the concentration, which is the most important variable.
In this work, after rewriting these equations as first order differential equation systems, finite element method, with least-squares variational formulations is applied to solve this elliptic subsystem as well as the transport equation.
We also consider and discuss the approximation improvement for the vector variable when adding to our system the non rotational flux condition.
The formulations here considered besides of being mixed formulations are symmetric and equal order interpolations can be used for the elliptic sub-system involved fields as well as for the concentration and its derivative in the transport equation.
Numerical simulations are presented for the tracer injection problem with varied mobility ratios showing the good stability of the proposed formulations.