Sandra M. C. Malta

Numerical analysis of a stabilized finite element method for tracer injection simulations

To solve a system of partial differential equations that models tracer injection processes in oil reservoirs we use a post-processing technique to compute an accurate approximation for the velocity field and a stabilized finite element method (SUPG) for the spatial approximation of the tracer concentration combined with a backward finite difference discretization in time. Stability, convergence and error estimates are achieved. Numerical results for homogeneous and heterogeneous media are presented illustrating the performance of the proposed methodology.

Numerical Analysis of Space–Time Finite Element Formulations for Miscible Displacements

A finite element formulation is proposed to approximate a nonlinear system of partial differential equations, composed by an elliptic subsystem for the pressure–velocity and a transport equation (convection–diffusion) for the concentration, which models the incompressible miscible displacement of one fluid by another in a rigid porous media. The pressure is approximated by the classical Galerkin method and the velocity is calculated by a post-processing technique. Then, the concentration is obtained by a Galerkin/least-squares space–time (GLS/ST) finite element method. A numerical analysis is developed for the concentration approximation. Then, stability, convergence and numerical results are presented confirming the a priori error estimates.

Finite-element simulations of miscible fingering problems

We present finite-element approaches to investigate the dynamical evolution of two-dimensional miscible porous media flows in the quarter five-spot arrangement. This takes into account the appearance of viscous fingers and its influence on the breakthrough time of the injected fluid and on the reservoir sweep. Then, two viscosity-concentration relationships for larger values of mobility ratios (the rate between the viscosities of resident and solvent fluids) and Péclet numbers are considered. The numerical discretization is carried out by two stabilized finite-element formulations, with the concentration calculated via a fully Galerkin/least-squares space-time (GLS/ST) method and a streamline upwind Petrov–Galerkin semi-discrete approach. Darcys equation (velocity approximation) is treated via a precise post-processing technique. Some numerical test cases are exhibited demonstrating good physical behaviours in the presence of finger instabilities. Besides, the influence of the two parameters: mobility ratio and Péclet number on the reservoir recovery are also addressed showing that the GLS/ST approach is a good alternative to deal with miscible fingering problems.

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

Abstract In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.

SIMULATION OF TRACER INJECTION IN OILRESERVOIRS USING ADAPTIVE UNSTRUCTUREDGRIDS

Numerical simulation of tracer injection is used in many situations for reservoir characterization. Traditionally, finite differences techniques have been used in this area. In this work, we present a new approach, based on the finite element method. It has a great capacity of treating complex domains and the ability to handle unstructured meshes. Dynamic grid refinement techniques are used in the traces simulation minimizing grid orientation and dispersion. INTRODUCTION This work presents a computational strategy for the development of a numerical simulator for two-dimensional inactive tracer injection in oil reservoirs. An h-adaptive semi-discrete finite element formulation is employed. Local refmement/derefmement in complex domains can be achieved with this procedure. The transport equation for the concentration is discretized by the streamline upwind Petrov-Galerkin formulation^, considering a given velocity field. The water velocity field, that carries the tracer, is obtained by the finite element solution of the two-phase (water-oil) immiscible flow. A blockiterative scheme is used to solve the resulting finite element equations in time, combined with: element-by-element iterative techniques, a dynamic mesh partition algorithm, where the pressure equation is always implicit and the Transactions on the Built Environment vol 29,