arXiv subjects

Jhabriel Varela

Publications and source records attributed to Jhabriel Varela.

4 recordsLinked to original sources

arXiv2025

A posteriori error estimates for mixed-dimensional Darcy flow using non-matching grids

In this article, we extend the a posteriori error estimates for hierarchical mixed-dimensional elliptic equations developed in [Varela et al., J. Numer. Math., 48 (2023), pp. 247-280] to the setting of non-matching mixed-dimensional grids. The extension is achieved by introducing transfer grids between the planar subdomain and interface grids, together with stable discrete projection operators for primal (potential) and dual (flux) variables. The proposed non-matching estimators remain fully guaranteed and computable. Numerical experiments, including three-dimensional problems based on community benchmarks for incompressible Darcy flow in fractured porous media, demonstrate reliable performance of the estimators for the non-matching grids and effectivity that is comparable to the estimators for matching grids.

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arXiv2023

Flexible and rigorous numerical modelling of multiphysics processes in fractured porous media using PorePy

Multiphysics processes in fractured porous media is a research field of importance for several subsurface applications and has received considerable attention over the last decade. The dynamics are characterised by strong couplings between processes as well as interaction between the processes and the structure of the fractured medium itself. The rich range of behavior calls for explorative mathematical modelling, such as experimentation with constitutive laws and novel coupling concepts between physical processes. Moreover, efficient simulations of the strong couplings between multiphysics processes and geological structures require the development of tailored numerical methods. We present a modelling framework and its implementation in the open-source simulation toolbox PorePy, which is designed for rapid prototyping of multiphysics processes in fractured porous media. PorePy uses a mixed-dimensional representation of the fracture geometry and generally applies fully implicit couplings between processes. The code design follows the paradigms of modularity and differentiable programming, which together allow for extreme flexibility in experimentation with governing equations with minimal changes to the code base. The code integrity is supported by a multilevel testing framework ensuring the reliability of the code. We present our modelling framework within a context of thermo-poroelasticity in deformable fractured porous media, illustrating the close relation between the governing equations and the source code. We furthermore discuss the design of the testing framework and present simulations showcasing the extendibility of PorePy, as well as the type of results that can be produced by mixed-dimensional simulation tools.

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arXiv2022

A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution has conservation properties, i.e.: (1) no conservation, (2) subdomain conservation, (3) grid-level conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either at the grid level or exactly, with a comparable structure to those obtained from grid-based a posteriori techniques. We demonstrate the practical effectiveness of our theoretical results through numerical experiments using four different discretization methods for synthetic problems and applications based on benchmarks of flow in fractured porous media.

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arXiv2019

PorePy: An Open-Source Software for Simulation of Multiphysics Processes in Fractured Porous Media

Development of models and dedicated numerical methods for dynamics in fractured rocks is an active research field, with research moving towards increasingly advanced process couplings and complex fracture networks. The inclusion of coupled processes in simulation models is challenged by the high aspect ratio of the fractures, the complex geometry of fracture networks and the crucial impact of processes that completely change characteristics on the fracture-rock interface. This paper provides a general discussion of design principles for introducing fractures in simulators, and defines a framework for integrated modeling, discretization and computer implementation. The framework is implemented in the simulation software PorePy, which can serve as a flexible prototyping tool or multiphysics problems in fractured rocks. Based on a representation of the fractures and their intersections as lower-dimensional objects, we discuss data structures for mixed-dimensional meshes, formulation of multiphysics problems and discretizations that utilize existing software. We further present the implementation of these concepts in the PorePy open-source software tool, which is aimed at coupled simulation of flow and transport in three-dimensional fractured reservoirs as well as deformation of fractures and the reservoir in general. We present validation by benchmarks for flow, poroelasticity and fracture deformation in fractured porous media. The flexibility of the framework is then illustrated by simulations of fully coupled flow and transport and of injection driven deformation of fractures. All results reported herein can be reproduced by openly available simulation scripts.

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