Frederico Furtado

On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs

We present a new, naturally parallelizable, accurate numerical method for the solution of transport-dominated diffusion processes in heterogeneous porous media. For the discretization in time of one of the governing partial differential equations, we introduce a new characteristics-based procedure which is mass conservative, the modified method of characteristics with adjusted advection (MMOCAA). Hybridized mixed finite elements are used for the spatial discretization of the equations and a new strip-based domain decomposition procedure is applied towards the solution of the resulting algebraic problems.We consider as a model problem the two-phase immiscible displacement in petroleum reservoirs. A very detailed description of the numerical method is presented. Following that, numerical experiments are presented illustrating the important features of the new method and comparing computed results with ones derived from previous, related techniques.

Crossover from nonlinearity controlled to heterogeneity controlled mixing in two-phase porous media flows

We use high resolution Monte Carlo simulations to study the dispersive mixing in two-phase, immiscible, porous media flow that results from the interaction of the nonlinearities in the flow equations with geologic heterogeneity. Our numerical experiments show that distinct dispersive regimes occur depending on the relative strength of nonlinearity and heterogeneity. In particular, for a given degree of multiscale heterogeneity, controlled by the Hurst exponent which characterizes the underlying stochastic model for the heterogeneity, linear and nonlinear flows are essentially identical in their degree of dispersion, if the heterogeneity is strong enough. As the heterogeneity weakens, the dispersion rates cross over from those of linear heterogeneous flows to those typical of nonlinear homogeneous flows.

Approximate similarity solutions to the Boussinesq equation

We derive a new cubic polynomial approximation to the similarity solution of the Boussinesq equation for one-dimensional, unconfined groundwater flow in a horizontal, initially dry aquifer with a power law inlet condition. This derivation builds upon the work of Lockington et al. (Adv. Wat. Resour. 23 (7) (2000) 725), where a quadratic approximation is constructed, and uses an additional condition for the determination of the (approximate) scaling function. The approximate analytical solutions are compared with the direct numerical solution obtained by Shampines method (ZAMM 53 (1973) 421). In all the tests performed, the new cubic approximation gives better results than the quadratic one, while it is only slightly harder to use.

On the Numerical Simulation of Three‐Phase Reservoir Transport Problems

Abstract We introduce a new method for the numerical solution of three‐phase immiscible displacement in porous media. An operator splitting technique allows for the use of distinct time steps for the solution of advection and diffusion problems defined by the procedure. A nonlinear system of conservation laws is approximated by a nonoscillatory, second order, conservative central difference scheme in the advection step. Locally conservative mixed finite elements are used to discretize the nonlinear diffusion system. A simulator was developed to solve one‐dimensional problems and it was shown to be accurate and efficient through a set of numerical experiments. The numerical solutions are in very good agreement with semi‐analytic results available in the literature. In these numerical solutions transitional waves, which have a strong dependency upon the physical diffusion being modeled, have been captured.

Hysteretic enhancement of carbon dioxide trapping in deep aquifers

The sequestration of supercritical carbon dioxide in saline aquifers has been proposed to mitigate global climate change. An important issue is whether it escapes to the atmosphere: chemical retention, for instance, is permanent in well chosen aquifers. The effects of chemical reactions may take time, so one needs short-term containment techniques such as CO2 capillary retention enhanced by permeability hysteresis during water imbibition. This retention is predicted in a class of permeability models for the capillary-dominated regime. Here, we use high-quality laboratory-measured permeabilities as well as exact analytic solutions and accurate simulation techniques to quantify the amount of carbon dioxide that can be trapped. This study on one-dimensional, vertically upward flow demonstrates that under ideal conditions, all of the carbon dioxide is immobilized permanently. This is achieved by a trapping shock due to switching under hysteresis from CO2 to chase-brine injection. This shock is peculiar: it possesses a very small amplitude but a large speed, making it difficult to simulate and detect.

Uniqueness of the Riemann Solution for Three-Phase Flow in a Porous Medium

We solve injection problems for immiscible three-phase flow described by a system of two conservation laws with fluxes originating from Coreys model with quadratic permeabilities. A mixture of water, gas, and oil is injected into a porous medium containing oil, which is partially displaced. We prove that the resulting Riemann problems have solutions, which are unique under technical hypotheses that can be verified numerically. The wave curve method constructs the solutions straightforwardly, despite loss of strict hyperbolicity at an isolated point in state space. This umbilic point induces the separation of the solutions into two types, according to the water/gas proportion in the injection mixture.

On a universal structure for immiscible three-phase flow in virgin reservoirs

We discuss the solution for commonly used models of the flow resulting from the injection of any proportion of three immiscible fluids such as water, oil, and gas in a reservoir initially containing oil and residual water. The solutions supported in the universal structure generically belong to two classes, characterized by the location of the injection state in the saturation triangle. Each class of solutions occurs for injection states in one of the two regions, separated by a curve of states for most of which the interstitial speeds of water and gas are equal. This is a separatrix curve because on one side water appears at breakthrough, while gas appears for injection states on the other side. In other words, the behavior near breakthrough is flow of oil and of the dominant phase, either water or gas; the non-dominant phase is left behind. Our arguments are rigorous for the class of Corey models with convex relative permeability functions. They also hold for Stone’s interpolation I model [5]. This description of the universal structure of solutions for the injection problems is valid for any values of phase viscosities. The inevitable presence of an umbilic point (or of an elliptic region for the Stone model) seems to be the cause of this universal solution structure. This universal structure was perceived recently in the particular case of quadratic Corey relative permeability models and with the injected state consisting of a mixture of water and gas but no oil [5]. However, the results of the present paper are more general in two ways. First, they are valid for a set of permeability functions that is stable under perturbations, the set of convex permeabilities. Second, they are valid for the injection of any proportion of three rather than only two phases that were the scope of [5].

Renormalization Group Analysis of Nonlinear Diffusion Equations with Periodic Coefficients

In this paper we present an efficient numerical approach based on the renormalization group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear parabolic partial differential equations. We illustrate the approach with the verification of a conjecture about the long-time behavior of solutions to a certain class of nonlinear diffusion equations with periodic coefficients. This conjecture is based on a mixed argument involving ideas from homogenization theory and the renormalization group method. Our numerical approach provides a detailed picture of the asymptotics, including the determination of the effective or renormalized diffusion coefficient.

Fluid mixing in multiphase porous media flows

Publisher Summary This chapter discusses the fluid-mixing dynamics for two-phase immiscible flow in heterogeneous porous media. The discussion focuses on the interplay between nonlinearity and stochastically described heterogeneity in determining fluid mixing regimes. Fluid mixing and multiphase fluid flow are important in various scientific and technological contexts. The examples include—environmental remediation and the management of petroleum reservoirs. Monte Carlo simulations can be used for a quantitative analysis of the two-fluid mixing in heterogeneous porous media flows. The mixing regions in linear and nonlinear flows grow at essentially identical asymptotic rates when heterogeneity is strong enough. As the heterogeneity weakens, the growth rates cross over to those typical of nonlinear flows in homogeneous geologies. Both nonlinearity of the flow equations and permeability heterogeneity can cause dispersive mixing of the fluid transport.

Parallel methods for immiscible displacement in porous media

High resolution numerical simulation of instabilities (viscous fingering) in water-oil fronts in heterogeneous porous media requires efficient computations on fast, multi-processor computers. This paper is devoted to the description of an appropriate algorithm based on a domain-decomposition technique applied to a mixed finite element approximation of the nonlinear system of equations for two-phase, immiscible, incompressible flow expressed in a “modified method of characteristics” form and its implemented to be portable to several state-of-the-art parallel architectures.