Numerical Methods For Miscible And Immiscible Displacement In Deformed And Porous Reservoir

The solid medium consisted of a great deal of holes connected each other and located randomly with different shape and size is called porous media [1].According to different standard, there are many classified methods for porous media. The normal porous media observed in mostly literature is in the range of common oil reservoir which characterized by non-exceptional high pressure and low-seated.With the evolution of oil development technology, more and more deep-seated oil reservoirs were developed. During the development, due to the very high pressure and temperature the deep-stratum oil reservoir endured, the deposited layer tends to change in shape completely and partially. So the porous media has the property of deformation. We name this kind of media deformable porous media.Usually, there are two kinds of models to describe the incompressible fluid displacement problems in porous media[2]: in one model the fluids are assumed to be completely miscible; one example is given by oil and a detergent solution. In the other model the fluid, such as water and oil, are considered to be completely immiscible. The differential systems for the two models can be put into quite similar forms. It is usually modeled by a nonlinear coupled system of two partial differential equations, one of elliptic form for the pressure and the other of parabolic form for the concentration of one of the fluids.If the deformation happened, this will affect the dynamic character of reservoir seriously. Adding inappropriate development, very bad result will be obtained and the production of oil reduced sharply and beyond recovery. For the singularly high-pressure and deep-seated reservoir which media has the property of deformation, the methods of design are different from normal reservoirs. Especially for its non-linear rule of permeation, the common theory of permeation and reservoir development fail.Experiments and researches [3,63] show that, the changing ratio of permeation to pressure of layer in deformation reservoir is five to fifteen fold than the porosity. So, under high pressure, the change of permeation is huge. It is error to neglect the change of porosity in mine field computation. But, nowadays, it is common to assume the permeation is a constant in most computation of mine field. That is an evident departure from actual situation, always making huge error, even leading to whole failure. We should revise the common model of miscible displacement in porous media to describe the displacement problem in deformable porous media. [3] proposed some methods to describe permeation equation and development methods in deformable porous media. In[3,4,5], author has made some research about non-linear elastic permeation theory in deformable porous media, and discussed the characteristic of deformation reservoir development. In this paper, we reckon on the layer porosity and permeation of displacement model in deformation reservoir as function of pressure. In front part of this paper, we make the use of the finite difference methods to research One-dimensional and two-dimensional miscible and immiscible displacement problems in deformable porous media.The convection-dominated diffusion problem is often involved in fluid displacement problem in porous media. For treating this problem, the method of characteristics [6-7, 2, 9-19] has proved effective.Error estimates and numerical experiments have shown that this method permits the use of large time step, and avoids or sharply reduced the usual numerical difficulties. The method of characteristics has been combined with finite-element procedure in [6,9].Since convection-dominated diffusion problem involve time-changing localized phenomenon such as fronts. It is advantageous to apply the dynamic finite method [20]. This method has the capability for dynamically making local grid refinements or unrefinements and basic function adjustments. Numerical results have proved the validity of this method [21-31].Mixed finite-element method has been reported in many literatures. Via this method we can obtain approximation of the solution and its gradient simultaneously.Considering the advantages of the mixed finite-element method, and the advantages of dynamic finite-method and characteristics finite-element method in treating convection-dominated diffusion problem, We combine these three methods treat convection-dominated diffusion miscible displacement in porous media.Over the past few years, a series of research papers have been publish concerning the least-squares finite element methods for incompressible flow [32-34]. This method is same to mixed finite elements method, which can obtain approximation for function itself and flux with the same condition number at the same time, rather than by post processing in the standard formulation. But there are two problems: firstly, algebraic question derived from this method is non symmetry positive-definite usually. Secondly, the mixed finite element space have to satisfy Ladyzhenskaya Babuska Brezzi(LBB) [35] requirements. This has restricted the selection of spaces. In [36], author had pointed out the LSM can overcome these obstacle. Ladyzhenskaya Babuska Brezzi(LBB) is not necessary for this method, so, it is easy to select approximate spaces, even polynomial of different rank for function and its flux. The algebraic equation obtained is symmetry positive-definite easy to deal with.In the behind part of this paper, we use dynamic finite-method, characteristics finite-element method and least-squares finite element method separately to discuss one kind of displacement problem in porous media.Author has some interest in financial math and done a little researching work during the PhD period in Shandong University. In the last chapter, we study one kind of stochastic optimal control problem rising from one optimal portfolio choice problem.The whole dissertation consists of five chapters.In the chapter 1, we discussed the displacement of one incompressible fluid by another in deformable porous media. The section 1 discussed the finite difference methods for One-dimensional miscible displacement in deformable porous media. A convergence analysis is given for the method. In section 2, we extend the One-dimensional to two-dimensional problem, for the similarity, we give a brief analysis.In chapter 2, we discussed two-phase incompressible flow immiscible displacement in deformable porous media. We give the approximation scheme and convergence analysis. These results have been submitted.In the chapter 3, we combine dynamic finite-method, characteristics finite-element method etc. to treat convection-dominated diffusion miscible displacement in porous media. We give the approximation scheme and convergence analysis. These results have been published in the journal of Shandong University (natural science).In chapter 4, we discussed fully discrete least-squares mixed finite elements method for the miscible displacement of one incompressible fluid by another in a porous media. we use LSM to approximate pressure, fully discrete standard Galerkin method for concentration, by introducing the elliptic projection , error estimates with optimal rate of convergence are derived. These results have been submitted.In Chapter 5, we study one kind of stochastic optimal control problem rising from one optimal portfolio choice problem. Using the classical convex variational technique we derive the local necessary condition. Combining the necessary condition with a direct construction method we determine the portfolio problem. These result have been published in the journal of Shandong university (natural science).