Numericai Simulation And Uncertainty Quantification Of Reservoir And CO₂ Sequestration And Numerical Methods For Fractional Differential Equations

Porous medium flow and transport model occurs in many importan-t applications and disciplines([1][6][9][19][20][24][30][36][39][42][68][92]), such as petroleum reservoir simulation, basin modeling, groundwater contami-nant transport and remediation, geological storage of carbon dioxide, CO2 sequestration, seawater intrusion, coalbed methane, semiconductor devices, fuel cells, heat and moisture transfer in clothes and shoes, spread of ink in paper, plants, agriculture, etc. In this dissertation, we mainly study the numerical methods in petroleum reservoir simulation and CO2 sequestration.Petroleum, is an major source of energy. Petroleum reservoir simulation is an important and basic tool to design or optimize the product strategy[6]. So the international petroleum companies such as the Royal Dutch/Shell Group of Companies and Chevron Corporation develop their own reservoir simulation software. By comparison, the present situation of reservoir sim-ulation in China is, although the oil field companies have to spend a lot of money to pay the fee for the key of the the commercial reservoir simulation software every year, the softwares sometimes may not adapt to the geological condition of the oil field in China. For example, the domestic Lab might fail to provide the the parameters that the commercial software needs, whereas the important experimental data that the Lab detected to characterize the reservoir may lack the access to the commercial software. To get rid of this dilemma, the oil field companies, together with the universities in China, need to develop their own petroleum reservoir softwares. The computational mathematical team in Shandong University, has many years of experience in research and development of the reservoir simulation. When I pursued a doctorate in Shandong University, I fortunately participate in the project of reservoir simulation conducted by the oil company and Shandong Universi-ty. Some part of the content of the dissertation is about the mathematical problems arising from this project.The oilfield developing in the early stage has already reached the high water cut stage with water flooding. The chemical flooding[20], including polymer flooding, surfactant flooding and alkaline flooding, has a larger s-cale of oil recovery efficiency than water flooding. So the research and de-velopment of the software especially for chemical flooding is on the agenda. In this article, I would like to study two mathematical problems arising in the numerical simulation of the chemical flooding. The first one is about the interpolation of the relative permeability in surfactant flooding, and the second one is about the flow and transport of petroleum acid in oil phase and water phase system. The surfactant flooding will decrease the interfacial tension, increase the capillary number, then reduce the saturation of residual oil. The reduction of the saturation of the residual oil and water will change the relative permeability of oil phase and water phase, and finally increase the mobility of both oil and water. The interpolation of the relative perme-ability is to compute the new relative permeability according to the current saturation of the residual oil and water. In the famous chemical flooding simulation software UTCHEM[20][77], the relative permeability is given by empirical formula. The interpolation of the relative permeability is equal to the interpolation of the corresponding parameters in the empirical formula. But the engineers in oil field prefer using the discrete points of the relative permeability detected by their own Lab instead of the empirical formula. In this dissertation, we introduce a reasonable interpolation of the rela-tive permeability. In the mathematical model of the chemical flooding, the components transport with water and oil[20][77]. Most of the components, such as polymer, surfactant and ion, only exist in the water phase, but the petroleum acid exists in both oil phase and water phase. Moreover, there is mass transfer of petroleum acid between the oil phase and water phase, so the mass of the petroleum acid in a particular phase is not conserved. Only the total mass of petroleum acid among all the phases is conserved. There-fore, we solve the total mass balance equation of the petroleum acid to get the total concentration of the petroleum acid, then distribute the petroleum acid in each phase according to the PH.Anthropogenic emissions of carbon dioxide continue to cause increases in the atmospheric concentration of carbon dioxide([13] [15] [38] [41] [43] [56] [58] [66] [67] [75] [76]), which are leading to global warming of the earth with wide-ranging environmental implications. The major source of anthropogenic e-missions of CO2 is fossil fuel combustion such as petroleum hydrocarbon and coals, and the ideal solution to reduce the amount of fossil fuel burned is to replace fossil fuels with clean renewables. However, this is infeasible in the near and intermediate future, since carbon-free energy sources are not avail-able at the scale required for modern economies. This leaves carbon capture and storage of captured CO2 in geological materials as the only large-scale technology to allow both the continued use of coal and petroleum hydrocar-bon and a solution to carbon problem ([17] [23] [38] [60] [62] [63] [64]). The idea of CCS is to inject captured CO2 into geological formations. The geological formations could be exploited gas or oil reservoirs or saline aquifers. The CO2 is injected into the formation in a supercritical status. Supercritical CO2 is slightly miscible in brine, while most of the CO2 will form a separate fluid phase. A small fraction of the water can also evaporate into the CO2, pro-ducing "wet" CO2 from the injected dry CO2. The dissolved CO2 leads to reductions in PH in the aqueous phase, which can drive different geochemical reactions. In this dissertation, we just focus on the flow and transfer of brine and CO2 with the injection of CO2. We use ELL AM scheme to solve this pro-cess. Although the oil reservoirs or saline aquifers are the ideal formations for CO2, the drilled well for petroleum exploitation break the closure of the for-mations. The injected CO2 may leak to the atmosphere or leak and contam-inate the ground water through the drilled or abandoned well[17][23][61][65]. In this dissertation, we also study the numerical simulation of CO2 leak-age. Due to the high spatial variability of geological properties of porous media and the scarcity of available data, the hydrological parameters de-scribing continuous or macroscopic properties of porous media, such as the intrinsic permeability and porosity, cannot be accurately characterized in de-tail and are often modeled as spatially correlated random field. We assume the intrinsic permeability is spatially correlated random variable, then use the Karhunen-Loeve expansion [14] [31] [40] [51] [54] [59] [73] [74] [90] [91] [92] and sparse grid technology [14][73][74]to solve the system, to get the mean and standard deviation of CO2 leakage.Fractional diffusion equations provide an adequate and accurate descrip-tion of transport processes that exhibit anomalous diffusion([11][57][69][71]), which can not be modeled properly by second-order diffusion equations. How-ever, fractional differential equations introduce new mathematical difficul-ties [11][57][86]that have not been encountered in the context of traditional second-order diffusion equations. In this article, we present the finite ele-ment method for three different types of space-fractional diffusion equations coupled with different types of Neumann boundary conditions. We find the matrix of the finite element method could be factorized into the products of several simple matrix. We implement the numerical experiment on the uniform grid and geometric refined grid.Finite volume method is a common method to solve the concentration equation arising in chemical flooding([6][9][20][36][42][68]), which is an ad-vection domainated advection diffusion equation. In this article, we prove an optimal-order error estimate in a weighted energy norm for bilinear fi-nite volume method for two dimensional time-dependent advection-diffusion equations on a uniform space-time partition of the domain. The generic con-stants in the estimates depend only on certain norms of the true solution but not on the scaling parameter ε. These estimates, combined with a priori stability estimates of the governing partial differential equations with full reg-ularity, yield ε-uniform estimate of the bilinear finite volume method, in which the generic constants depend only on the Sobolev norms of the initial and right data but not on the scaling parameter e.We use the interpolation of spaces and stability estimates to derive an ε-uniform estimate for problem-s with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right-hand side data.The outline of the dissertation is as followsIn chapter 1, we introduce the mathematical model of the porous medi-um flow, including the mathematical models of single phase, compositional model, black oil model.In chapter 2, based on the mathematical models in chapter 1, we study two mathematical problems arising in the reservoir simulation. The first one is the interpolation of relative permeability in surfactant flooding, and the second one is to study numerical method to solve the transport of petroleum acid in oil phase and water phase.In chapter 3, we study the numerical method that simulate the storage and leakage of CO2.We implement ELLAM scheme to solve the saturation equation. We assume the intrinsic permeability is spatially correlated random variable. We implement the Karhunen-Loeve expansion and sparse grid skill to the system, to get the mean and standard deviation of CO2 leakage.In chapter 4, we study the fractional diffusion equations. We present the finite element method for three different types of space-fractional diffusion equations coupled with different types of Neumann boundary conditions. We implement the numerical experiment on the uniform grid and geometric refined grid.In chapter 5, we analysis the finite volume method advection diffusion equation with ε parameter. We prove an optimal-order error estimate in a weighted energy norm with or lack of full regularity. The general constants don’t depent on the parameter ε.