Numerical Reservoir Simulation Method And Algorithm Design

Reservoir engineering is a comprehensive petroleum technology science , whichis based on the reservoir physics and reservoir flow mechanics and concerns aboutdecision of oil field development and engineering analysis. Its task is to study thelaw of motion and displacement mechanism of oil, gas and water in reservoir devel-opment process, and design appropriate engineering measures to improve the speedof production and recovery ratio.In the field of petroleum engineering, we generally use the mass and momentumconservation equations to describe the transport process of chemical materials suchas underground oil, water, gas and polymers. The main task of reservoir simulationis to use the oil and gas models as well as the previous data to simulate, analyze or fitthe movements of underground reservoir and the mining process, and to predict thefuture mining conditions. The main methods used in reservoir simulation are phys-ical simulation and numerical simulation. In the physical simulation the reservoir iswholly or locally reduced in proportion and made up of entitative model accordingto similar principles and similar standard number. Besides the model forms andparameters need to be similar to the reservoir, the similarity in the dynamics offluids is also required. This method is mostly used for the research of percolationphysical mechanism. In the numerical simulation, people study the physical pro-cess and variation of reservoir development by using numerical methods to solve thepartial diferential equations describing the reservoir development. The basic theoryof numerical simulation for reservoir is Darcy law. Its basic principle is to set theamount of production or injection as a determined value, and make the determinedvalue match the actual situation by adjusting the uncertain factors. The appearanceof numerical simulation methods make the study for reservoir simulation changed from qualitative research to quantitative research. Due to the complexity of theinterior structure and the non-repetitive of reservoir mining, numerical simulationmethods become more and more popular in reservoir research.The main contents of this paper is to establish some efcient numerical methodsfor concrete reservoir mathematical models, which can reflect the physical propertyof practical problems, prove the convergence analysis of these numerical methodsand design efcient algorithms. The outline is as follows:1. An optimal-order error estimate for a Galerkin-mixed finite element timestepping procedure for porous media flowsWhen Galerkin method and mixed finite element method are used for reservoirsimulation, there are some constraints about mesh parameters, which limits the useof methods. In order to solve these problems, we use a series of new techniques innumerical methods for partial diferential equations, and obtain some new theoreticalresults, which reduces the constraints on the mesh parameters and makes thesemethods more reasonable.2. A combined mixed and characteristic mixed finite element method for in-compressible miscible displacement problem in porous mediaIn this chapter we adopted mixed finite element methods to approximate thepressure and the Darcy velocity, while a characteristics mixed finite element methodwas used to approximate the concentration equation. In the characteristic mixed for-mulation characteristics method is used to deal with the convection term, which caneliminate the numerical dispersion and make the scheme more stable. Mixed finiteelement method is utilized to handle the difusion term so that we can approximateconcentration and its difusion flux simultaneously.3. Least-squares expanded mixed finite element method for incompressiblemiscible displacement problem in porous media Mixed finite element methods have been successfully used in reservoir simula-tion, but these methods have to satisfy LBB condition, which restricts the choosingof finite element spaces. In this paper, we apply least-squares expanded mixed finiteelement methods to approximate the pressure and the Darcy velocity, while a finiteelement method is used to approximate the concentration equation. This methodinherits all the advantages of least-squares and expanded mixed finite element meth-ods. i.e., it can explicitly treat three variables, the scalar unknown, its gradient andits flux, and the finite element spaces are not subject to the LBB condition.4. A broken P1-nonconforming finite element method for incompressible misci-ble displacement problem in porous mediaIn oil reservoirs, the properties of the porous medium often change abruptlywith sharp changes in lithology. For the pressure equation, the coefcientα(c) oftenchanges rapidly across fluid interfaces and this sharp change is accompanied by largechanges in the pressure gradient. As a compensatory, this yields a fairly smoothDarcy velocity u, so we can trade the pressure equation as an interface problem.When the interface is smooth enough, the solution of the interface problem is alsovery smooth in individual regions where the coefcient is smooth, but due to thejump of the coefcient across the interface, the global regularity is usually very lowand has order of H1+γ(0 <γ< 1). Due to the low global regularity and the irregulargeometry of the interface, it seems to be difcult for the standard finite elementor finite diference method to achieve high accuracy. For better approximation ofthe interface problem, we introduce an P1nonconforming finite element methodto pressure equation, which is based on the broken P1nonconforming piecewiselinear polynomials on interface elements having edge averages as degrees of freedom.Meanwhile, the basis function here are Crouzeix-Raviart-type, and the finite elementspace has good approximation property similar to that of the conforming linear finite element space. Moreover, theoretical analysis shows the efciency of the method.5. Algorithm DesignAccording to the mixed finite element method and Least-squares expandedmixed finite element method for incompressible miscible displacement problem inporous media, we design some efcient and stable algorithms. and then, numericalanalysis shows the efciency of the method.