Relaxation schemes for multiphase, multicomponent flow in gas injection processes

Abstract

Gas injection is a well established tertiary recovery process for increasing oil recovery above primary depletion and secondary recovery by water injection. Over the past thirty years significant advances in numerical and computational modeling of gas injection processes have been achieved. This thesis provides further advancement in the computational modeling of gas injection processes.

In this thesis we are interested in numerical methods that accurately resolve transport of fluids in two phase gas injection displacements. Though some physical diffusion is always present, field scale gas injection processes are mostly advective in nature. Under the assumption that the diffusion can be neglected, the nonlinear governing systems of equations is hyperbolic, with weak hyperbolicity at isolated points in phase space. The strong nonlinear coupling and weak hyperbolicity of the governing equations render numerical solutions highly sensitive to modeling and discretization errors. High resolution models are essential. The phase equilibrium calculations, needed at every time step to partition the components in the two phases, further increase computational complexity.

First order numerical methods introduce excessive numerical diffusion leading not only to severe smoothing of displacement fronts, but also to incorrect predictions of frontal speeds. To achieve reasonably accurate predictions, very fine grids are needed with first order methods. This increases the number of the equilibrium computations required, thus making the simulation prohibitively expensive. Higher order upwind schemes can help improve the accuracy and efficiency of the displacement predictions. But traditional applications of upwind schemes necessitate characteristic decompositions that involve solving local Riemann problems. Since Riemann solutions are not readily available for systems with complex phase behavior, such methods are impractical. The characteristic approach also requires a full set of eigenvectors, which is not available at weakly hyperbolic points.

Here we present a new class of higher order schemes, called variable relaxation schemes, based on relaxation formulations introduced by Jin and Xin [S. Jin and Z. P. Xin, Comm. Pure Appl. Math., 48(1995), pp. 235-277]. The relaxation framework enables the construction of schemes that are free of nonlinear Riemann solvers and are independent of the underlying eigenstructure of the problem. However the relaxation schemes of Jin & Xin schemes can render numerical solutions very diffusive in the presence of global maximum speeds that are high compared to the average speeds in the domain. This is especially true in multiple dimensions. By using an estimate of local maximum speeds, our relaxation schemes not only reduce diffusion considerably but also are independent of the range of speeds present. We use a modified equation analysis on a ID scalar equation and compare the coefficients of numerical diffusion of the Jin-Xin scheme and an upwind scheme, to understand the cause of this excessive diffusion. Then we construct a general relaxation formulation and derive various possible relaxation models from the general formulation. Using the observation from the above mentioned analysis, we construct variable relaxation schemes with reduced diffusion. This is achieved by including minimal characteristic information in the construction of these schemes. The first order version of this scheme with symmetric relaxation parameters (called subcharacteristic speeds) results in the local Lax-Friedrichs scheme and the first order scheme with optimal subcharacteristic speeds is similar to the Harten-Lax-van Leer (HLL) solver. The high resolution 1D variable relaxation schemes are very competitive to the high resolution upwind schemes. The first order scheme is proved to be monotone and the second order scheme is proved to be a total variation diminishing (TVD) scheme. The multidimensional scheme is developed by a dimension-by-dimension approach. The variable relaxation schemes show excellent resolution in multidimensions; they also cost less since they are independent of the sign of the velocities. Because of their accuracy, simplicity and efficiency, variable relaxation schemes have great potential in simulation of gas injection processes and other compositional applications. (Abstract shortened by UMI.)