This research presents a new method, the decoupled overlapping grids method, for the numerical modelling of transient pressure and rate properties of oil wells. The method is implemented in two stages: a global stage solved in the entire domain with a point or line source well approximation, and a local (post-process) stage solved in the near-well region with the well modelled explicitly and boundary data interpolated from the global stage results. We have carried out simulation studies in twoand threedimensions to investigate the accuracy of the method. For homogeneous case studies in 2D, we have demonstrated the convergence rate of the maximum error in the quantities of interest of the global and local stage computations by numerical and theoretical means. We also proposed a guideline for the selection of the relative mesh sizes of the local and global simulations based on error trends. Comparison to other methods in the literature showed better performance of the decoupled overlapping grids method in all cases. We carried out further investigations for heterogeneous case studies in 2D and partially-penetrating wells in 3D which show that the error trends observed for the 2D homogeneous case deteriorate only slightly, and that a high level of accuracy is achieved. Overall the results in this thesis demonstrate the potential of the method of decoupled overlapping grids to accurately model transient wellbore properties for arbitrary well configurations and reservoir heterogeneity, and the gain in computational efficiency achieved from the method. For Kaycee, Blues, Zeros.

/pdf/decoupled-overlapping-grids-for-modelling-transient-3mgj6739jv.pdf

Decoupled overlapping grids for modelling transient behaviour of oil wells

Numerical upscaling for the eddy-current model with stochastic magnetic materials

We propose a new reduced order modeling strategy for tackling parametrized Partial Differential Equations (PDEs) with linear constraints, in particular Darcy flow systems in which the constraint is given by mass conservation. Our approach employs classical neural network architectures and supervised learning, but it is constructed in such a way that the resulting Reduced Order Model (ROM) is guaranteed to satisfy the linear constraints exactly. The procedure is based on a splitting of the PDE solution into a particular solution satisfying the constraint and a homogenous solution. The homogeneous solution is approximated by mapping a suitable potential function, generated by a neural network model, onto the kernel of the constraint operator; for the particular solution, instead, we propose an efficient spanning tree algorithm. Starting from this paradigm, we present three approaches that follow this methodology, obtained by exploring different choices of the potential spaces: from empirical ones, derived via Proper Orthogonal Decomposition (POD), to more abstract ones based on differential complexes. All proposed approaches combine computational efficiency with rigorous mathematical interpretation, thus guaranteeing the explainability of the model outputs. To demonstrate the efficacy of the proposed strategies and to emphasize their advantages over vanilla black-box approaches, we present a series of numerical experiments on fluid flows in porous media, ranging from mixed-dimensional problems to nonlinear systems. This research lays the foundation for further exploration and development in the realm of model order reduction, potentially unlocking new capabilities and solutions in computational geosciences and beyond. 

https://export.arxiv.org/pdf/2311.14554v1.pdf

Deep learning based reduced order modeling of Darcy flow systems with local mass conservation

Decoupled overlapping grids for the numerical modeling of oil wells

Stochastic direct integration schemes for dynamic systems subjected to random excitations

The paper presents a new method for generating isotropic realizations of a random field with varying resolution scales and local block averages in one, two, or three dimensions. The random field has a lognormal distribution as found for the hydraulic conductivity in Darcy's flow in hydrology. We also present a fast method for generating anisotropic realizations of aGaussian random field using superposition of harmonic modes. Both methods are implemented in a software package which is presented. The realizations of the rescaled random field are made up ofupscaled local averages of the underlying random field which are consistentwith the level of resolution and which can be conditioned. The approach is motivated by the need to represent engineering properties aslocal averages and to be able to easily condition the realizations to incorporate known data or to change the resolution within sub-regions. The numerical results show that the method is very efficient computationally.

Simulation of lognormal random fields with varying resolution scale and local average for Darcy flow

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Simulation of lognormal random fields with varying resolution scale and local average for Darcy flow