Zhangxin Chen

Reservoir simulation : mathematical techniques in oil recovery

List of figures List of tables List of notation Preface 1. Introduction 2. A glossary of petroleum terms 3. Single-phase flow and numerical solution 4. Well modeling 5. Two-phase flow and numerical solution 6. The black oil model and numerical solution 7. Transport of multicomponents in a fluid and numerical solution 8. Compositional flow and numerical solution 9. Nonisothermal flow and numerical solution 10. Practical topics in reservoir simulation Bibliography Index.

On the implementation of mixed methods as nonconforming methods for second-order elliptic problems

In this paper we show that mixed finite element methods for a fairly general second-order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower-order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection methods finite element space M h satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces M h for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in R 2 and R 3 , on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal-order multigrid methods for the solution of the linear system.

Prismatic mixed finite elements for second order elliptic problems

In this paper, three families of mixed finite elements based on prisms are introduced. These spaces are analogues to those based on simplices and cubes in three space variables. Error estimates in L2 and H−5 are given.

Derivation of the Forchheimer Law via Homogenization

In this paper we derive the Forchheimer law via the theory of homogenization. In particular, we study the nonlinear correction to Darcys law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived directly from the Navier–Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in velocity for isotropic media, the present work shows that the nonlinear correction is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. Both incompressible and compressible fluids are considered.

A new stabilized finite volume method for the stationary Stokes equations

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P1–P1 pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H1-norm for velocity and an estimate in the L2-norm for pressure are obtained. An optimal error estimate in the L2-norm for the velocity is derived under an additional assumption on the body force.

An improved IMPES method for two-phase flow in porous media

In this paper we develop and numerically study an improved IMPES method for solving a partial differential coupled system for two-phase flow in a three-dimensional porous medium. This improved method utilizes an adaptive control strategy on the choice of a time step for saturation and takes a much larger time step for pressure than for the saturation. Through a stability analysis and a comparison with a simultaneous solution method, we show that this improved IMPES method is effective and efficient for the numerical simulation of two-phase flow and it is capable of solving two-phase coning problems.

Wettability effect on nanoconfined water flow

Significance The flow of water confined in nanopores is significantly different from that of bulk water. Moreover, understanding and controlling the flow of the confined water remains an open question, especially concerning whether the flow capacity of the confined water increases or not compared with that of bulk water. Here, combining a theoretical analysis and data from molecular dynamics simulations and experiments in the literature we develop a simple model for the flow of water confined in nanopores. We find that a contact angle and a nanopore dimension may substantially affect the confined water flow. We also quantitatively explain a controversy over an increase or decrease in flow capacity. Understanding and controlling the flow of water confined in nanopores has tremendous implications in theoretical studies and industrial applications. Here, we propose a simple model for the confined water flow based on the concept of effective slip, which is a linear sum of true slip, depending on a contact angle, and apparent slip, caused by a spatial variation of the confined water viscosity as a function of wettability as well as the nanopore dimension. Results from this model show that the flow capacity of confined water is 10−1∼107 times that calculated by the no-slip Hagen–Poiseuille equation for nanopores with various contact angles and dimensions, in agreement with the majority of 53 different study cases from the literature. This work further sheds light on a controversy over an increase or decrease in flow capacity from molecular dynamics simulations and experiments.

Mathematical analysis for reservoir models

In the first part of this paper, the mathematical analysis is presented in detail for the single-phase, miscible displacement of one fluid by another in a porous medium. It is shown that initial boundary value problems with various boundary conditions for this miscible displacement possess a weak solution under physically reasonable hypotheses on the data. In the second part of this paper, it is proven how the analysis can be extended to two-phase fluid flow and transport equations in a porous medium. The flow equations are written in a fractional flow formulation so that a degenerate elliptic-parabolic partial differential system is produced for a global pressure and a saturation. This degenerate system is coupled to a parabolic transport equation which models the concentration of one of the fluids. The analysis here does not utilize any regularized problem; a weak solution is obtained as a limit of solutions to discrete time problems.

Mixed-RKDG finite element methods for the 2-D hydrodynamic model for semiconductor device simulation

In this paper we introduce a new method for numerically solving the equations of the hydrodynamic model for semiconductor devices in two space dimensions. The method combines a standard mixed finite element method, used to obtain directly an approximation to the electric field, with the so-called Runge-Kutta Discontinuous Galerkin (RKDG) method, originally devised for numerically solving multi-dimensional hyperbolic systems of conservation laws, which is applied here to the convective part of the equations. Numerical simulations showing the performance of the new method are displayed, and the results compared with those obtained by using Essentially Nonoscillatory (ENO) finite difference schemes. From the perspective of device modeling, these methods are robust, since they are capable of encompassing broad parameter ranges, including those for which shock formation is possible. The simulations presented here are for Gallium Arsenide at room temperature, but we have tested them much more generally with considerable success.

A new local stabilized nonconforming finite element method for the Stokes equations

In this paper, we propose and study a new local stabilized nonconforming finite method based on two local Gauss integrations for the two-dimensional Stokes equations. The nonconforming method uses the lowest equal-order pair of mixed finite elements (i.e., NCP1–P1). After a stability condition is shown for this stabilized method, its optimal-order error estimates are obtained. In addition, numerical experiments to confirm the theoretical results are presented. Compared with some classical, closely related mixed finite element pairs, the results of the present NCP1–P1 mixed finite element pair show its better performance than others.