Numerical modelling of near wellbore flow

This thesis is an examination of numerical approximations of physical phenomena occurring in porous media. It focuses on the development of robust multiphysics solvers, which could mimic interactions among different physical processes. Their effects on fluid flow behavior and well productivity reduction resulting from various phenomena,e.g., solid deformation and mineral dissolution, are also illustrated. This thesis is divided into three parts summarized as follows: The first part of this thesis begins with the numerical approximations of nonlinear poroelasticity with permeability alteration using the enriched Galerkin method. This method is defined as the continuous Galerkin function space augmented by a piecewise-constant function at the center of each element. Poroelastic solvers, such as the continuous Galerkin, the discontinuous Galerkin,the enriched Galerkin, and the mixed methods, e.g., three-field formulation, are compared, and their advantages and disadvantages are discussed. The enriched Galerkin approximation is then extended to the coupled hydro-mechanical-chemical model and mixed-dimensional abstraction. The results suggest that the enriched Galerkin method has the same advantages as the discontinuous Galerkin and mixed methods; each of these methods conserves local and global fluid mass, capture pressure discontinuity, and provide optimal error convergence rates. The enriched Galerkin method, however, requires much fewer degrees of freedom than the other techniques, e.g., discontinuous Galerkin and mixed method, in their classical form. Note that this comparison can vary based on advanced developments of each method, e.g., a hybridized discontinuous Galerkin method,fastsolvers, use of hanging nodes, or variable approximation orders. The second part of this thesis presents the possibilities of utilizing deep learning approaches to enhance the interior penalty of discontinuous and enriched Galerkin methods. Subsequently, deep learning is combined with physical information to solve a set of partial differential equations, which is called a physics-informed neural network (PINN). It could be seen as a multiphysics solver for both forward and inverse nonlinear modeling. The impacts of stochastic variations between various training realizations (i.e., different initialization) are also investigated. The effects of noisy measurements are incorporated for inverse modeling. Besides, the challenge of selecting the hyperparameters of inverse models is addressed by identifying keyhyper parameters in forward models. Additionally,thebatchtrainingapplyinginthetrainingphaseoftheneuralnetworkscouldincrease boththeaccuracyandstabilityofthePINNmethodforinversemodeling. Specifically,thesizeof each batch should not be too small since very small batch sizes require very long training times withoutmeasurableimprovementinthemodelaccuracy. The third part of this thesis describes the impact of solid deformation on well productivity in fractured porous media. Combined analytical and numerical solutions are used to describe well productivity in single-fracture media. Further, an equivalent aperture model is developed for the single-fractureundervariablecontactstressesthatcapturestheflowbehaviorofthefracturewith variable apertures obtained by the deformable fracture model. Besides, the full factorial experimental design, combined with numerical modeling, is employed to quantify the effect of various physicalquantitiessuchasinitialreservoirpressure,matrixpermeability,andfracturedrockstiffness on the productivity index reduction in complex fractured media. The results suggest that six mainquantitiesaffectwellproductivitywithdifferentdrawdownpressures. Theyareinitialreservoirpressure,matrixpermeability,far-fieldstresses,fracturedrockstiffness,fracturedensity,and fractureconnectivity.