The Study On The Multi-scale Methods For Permeability Inversion Of Nonlinear Diffusion Equations

Reservoir simulation based on the permeability inversion of nonlinear diffusion e-quation can help reservoir engineers make important decisions regarding the management of petroleum reservoirs, including selection of the type of recovery method, fluid produc-tion and injection rates, and well locations. The real reservoir model is of high uncertainty, the nonlinear diffusion equation model is of high complexity, the permeability inversion problem itself is of ill-posedness and high nonlinearity, and due to these features, there are several shortcomings in the permeability inversion process, such as ill-posedness, low computational efficiency, and local convergence. Hence, for scientists engaged in reser-voir simulation, there will be an increasing need for distribution parameter estimation techniques able to overcome these issues.As a recently developed inversion strategy, multiscale inversion has a world-wide reputation for speeding up convergence, enhancing stability of inversion, and avoiding impact of local minimum. So far, multiscale inversion can be roughly divided into two categories:one is the wavelet multiscale inversion method based on frequency-domain scale decomposition theory, the other is the multigrid multiscale inversion method based on grid scale decomposition theory. This paper designs and studies the above-metioned multiscale inversion methods for the permeability inversion problem of nonlinear diffu-sion equation, and popularizes the multigrid multiscale inversion method to the velocity inversion problem of wave equation, which plays an important role in geological survey.Firstly, an implicit finite-difference method is put forward for the forward math-ematical model of nonlinear diffusion equation, which can describe flow processes in multiphase porous media. In this method, due to the implicit treatment of the nonlinear diffusion term, the nonlinear system that needs to be solved for each time level can be solved by the Picard iteration. The results of numerical examples show that the implicit finite-difference method is an effective and stable numerical algorithm.Secondly, to overcome the shortcomings in the permeability inversion process, a wavelet multiscale method is designed to solve the permeability inversion problem of nonlinear diffusion equation by introducing the idea of wavelet multiresolution analysis. This method works by decomposing the inverse problem into multiple frequency-domain scales using wavelet transform so that the original inverse problem is reformulated to be a sequence of sub-inverse problems relying on scale variables, and successively solving these sub-inverse problems according to the size of scale from the longest to the shortest. Numerical examples verify the wide convergence, computational efficiency, anti-noise and de-noising abilities of the proposed algorithm.Thirdly, in order to further enlarge the convergence domain of wavelet multiscale inversion method, this thesis first designs an adaptive homotopy inversion method, which has global convergence properties, and then constructs a joint inversion method called the wavelet multiscale-adaptive homotopy method by combining it with the wavelet mul-tiscale inversion method. Numerical examples verify the effectiveness of the adaptive homotopy inversion method, and show that the wavelet multiscale-adaptive homotopy in-version method indeed combines the advantages of wavelet multiscale inversion method and adaptive homotopy inversion method.Fourthly, a multigrid multiscale method is applied to the permeability inversion problem of nonlinear diffusion equation. In the multigrid inversion process, in order to make the objective functionals at different grids compatible, they are dynamically adjust-ed. In this way, the necessary condition of "the optimal solution should be the fixed point of multigrid inversion method" can be met. By proving the theorem of multigrid mono-tone convergence, a set of sufficient conditions for monotone convergence of the multigrid multiscale inversion method is given. Through numerical examples, the superiority of the proposed method is validated.Finally, to overcome the difficulties within the velocity inversion problem of wave equation, such as large computational complexity, existence of numerous local mini-mum, ill-posedness, and to validate the universality of the multigrid multiscale inversion method, the multigrid multiscale inversion method is applied to solve the velocity inver-sion problem of wave equation. Some numerical examples are given to show that the multigrid multiscale inversion method has high precision, high speed, good stability and a broad scope of application.