Compact 99-Point Finite Difference Methods with High Accuracy Order and/or MM-Matrix Property for Elliptic Cross-Interface Problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order $9$-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an $M$-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the fourth (for the general case) and sixth (for the special case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an $M$-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.Comment: 25 pages, 13 figure

Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media

This paper introduces a new discrete fracture model accounting for non-isothermal compositional multiphase Darcy flows and complex networks of fractures with intersecting, immersed and non immersed fractures. The so called hybrid-dimensional model using a 2D model in the fractures coupled with a 3D model in the matrix is first derived rigorously starting from the equi-dimensional matrix fracture model. Then, it is dis-cretized using a fully implicit time integration combined with the Vertex Approximate Gradient (VAG) finite volume scheme which is adapted to polyhedral meshes and anisotropic heterogeneous media. The fully coupled systems are assembled and solved in parallel using the Single Program Multiple Data (SPMD) paradigm with one layer of ghost cells. This strategy allows for a local assembly of the discrete systems. An efficient preconditioner is implemented to solve the linear systems at each time step and each Newton type iteration of the simulation. The numerical efficiency of our approach is assessed on different meshes, fracture networks, and physical settings in terms of parallel scalability, nonlinear convergence and linear convergence

Modelling and quantification of structural uncertainties in petroleum reservoirs assisted by a hybrid cartesian cut cell/enriched multipoint flux approximation approach

Efficient and profitable oil production is subject to make reliable predictions about reservoir performance. However, restricted knowledge about reservoir distributed properties and reservoir structure calls for History Matching in which the reservoir model is calibrated to emulate the field observed history. Such an inverse problem yields multiple history-matched models which might result in different predictions of reservoir performance. Uncertainty Quantification restricts the raised model uncertainties and boosts the model reliability for the forecasts of future reservoir behaviour. Conventional approaches of Uncertainty Quantification ignore large scale uncertainties related to reservoir structure, while structural uncertainties can influence the reservoir forecasts more intensely compared with petrophysical uncertainty. What makes the quantification of structural uncertainty impracticable is the need for global regridding at each step of History Matching process. To resolve this obstacle, we develop an efficient methodology based on Cartesian Cut Cell Method which decouples the model from its representation onto the grid and allows uncertain structures to be varied as a part of History Matching process. Reduced numerical accuracy due to cell degeneracies in the vicinity of geological structures is adequately compensated with an enhanced scheme of class Locally Conservative Flux Continuous Methods (Extended Enriched Multipoint Flux Approximation Method abbreviated to extended EMPFA). The robustness and consistency of proposed Hybrid Cartesian Cut Cell/extended EMPFA approach are demonstrated in terms of true representation of geological structures influence on flow behaviour. In this research, the general framework of Uncertainty Quantification is extended and well-equipped by proposed approach to tackle uncertainties of different structures such as reservoir horizons, bedding layers, faults and pinchouts. Significant improvements in the quality of reservoir recovery forecasts and reservoir volume estimation are presented for synthetic models of uncertain structures. Also this thesis provides a comparative study of structural uncertainty influence on reservoir forecasts among various geological structures