arXiv subjects

Nicola Castelletto

Publications and source records attributed to Nicola Castelletto.

18 recordsLinked to original sources

arXiv2026

Adaptive Consistency for Mimetic Finite Differences

The mimetic finite difference (MFD) method provides a robust discretization for flow simulation on general polyhedral meshes, but its computational cost can become significant due to dense local operators and reduced global sparsity. While the two-point flux approximation (TPFA) offers a substantially cheaper alternative, its accuracy is generally restricted to $K$-orthogonal grids. To balance these competing considerations, we present an adaptive MFD framework based on a residual-based consistency indicator derived from the discrete constitutive equations. The indicator measures local inconsistency and enables adaptive TPFA/MFD stencil selection through a user-prescribed tolerance $τ$. Because the adaptation is performed within a mimetic framework, arbitrary TPFA/MFD partitions remain stable and structure preserving. Theoretical analysis establishes uniform coercivity and proves explicit tolerance-controlled convergence of the relative flux error. Numerical experiments on challenging polyhedral reservoir benchmarks demonstrate accuracy comparable to full MFD discretizations while substantially reducing matrix density and computational cost.

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arXiv2025

A robust framework for frictional fault contact in geological formations using a stabilized augmented Lagrangian approach

Numerical simulations are essential for evaluating the performance and safety of geological engineered systems such as geologic carbon storage sites, enhanced geothermal fields, and oil and gas reservoirs. A key challenge lies in accurately modeling the frictional contact behavior along fault surfaces. This problem involves inequality constraints that arise from the physics of frictional slip, requiring specialized numerical methods to handle the resulting highly nonlinear and path-dependent behavior. In this work, we address this challenge using an augmented Lagrangian method implemented via the Uzawa algorithm. The formulation employs mixed finite element spaces, combining low-order piecewise linear displacements within the 3D domain cells with piecewise constant tractions defined on the fault surfaces. Furthermore, to ensure stability and satisfy the inf-sup condition, the discrete displacement space is enriched with face bubble functions on both sides of the contact interfaces. This approach offers several advantages over other stabilization techniques that rely on additional terms, as it does not require extra parameters for implementation, and it integrates naturally in the Uzawa framework.

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arXiv2024

Pressure stability in explicitly coupled simulations of poromechanics with application to CO$_2$ sequestration

We study in detail the pressure stabilizing effects of the non-iterated fixed-stress splitting in poromechanical problems which are nearly undrained and incompressible. When applied in conjunction with a spatial discretization which does not satisfy the discrete inf-sup condition, namely a mixed piecewise linear - piecewise constant spatial discretization, the explicit fixed-stress scheme can have a pressure stabilizing effect in transient problems. This effect disappears, however, upon time step refinement or the attainment of steady state. The interpretation of the scheme as an Augmented Lagrangian method similar to Uzawa iteration for incompressible flow helps explain these results. Moreover, due to the slowly evolving solution within undrained seal regions, we show that the explicit fixed-stress scheme requires very large time steps to reveal its pressure stabilizing effect in examples of geologic CO$_2$ sequestration. We note that large time steps can result in large errors in drained regions, such as the aquifer or reservoir regions of these examples, and can prevent convergence of nonlinear solvers in the case of multiphase flows, which can make the explicit scheme an unreliable source of pressure stabilization. We conclude by demonstrating that pressure jump stabilization is as effective in the explicit fixed-stress setting as in the fully implicit setting for undrained problems, while maintaining the stability and convergence of the fixed-stress split for drained problems.

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arXiv2024

Pressure-stabilized fixed-stress iterative solutions of compositional poromechanics

We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.

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arXiv2023

Performant low-order matrix-free finite element kernels on GPU architectures

Numerical methods such as the Finite Element Method (FEM) have been successfully adapted to utilize the computational power of GPU accelerators. However, much of the effort around applying FEM to GPU's has been focused on high-order FEM due to higher arithmetic intensity and order of accuracy. For applications such as the simulation of subsurface processes, high levels of heterogeneity results in high-resolution grids characterized by highly discontinuous (cell-wise) material property fields. Moreover, due to the significant uncertainties in the characterization of the domain of interest, e.g. geologic reservoirs, the benefits of high order accuracy are reduced, and low-order methods are typically employed. In this study, we present a strategy for implementing highly performant low-order matrix-free FEM operator kernels in the context of the conjugate gradient (CG) method. Performance results of matrix-free Laplace and isotropic elasticity operator kernels are presented and are shown to compare favorably to matrix-based SpMV operators on V100, A100, and MI250X GPUs.

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arXiv2023

Multilevel well modeling in aggregation-based nonlinear multigrid for multiphase flow in porous media

A full approximation scheme (FAS) nonlinear multigrid solver for two-phase flow and transport problems driven by wells with multiple perforations is developed. It is an extension to our previous work on FAS solvers for diffusion and transport problems. The solver is applicable to discrete problems defined on unstructured grids as the coarsening algorithm is aggregation-based and algebraic. To construct coarse basis that can better capture the radial flow near wells, coarse grids in which perforated well cells are not near the coarse-element interface are desired. This is achieved by an aggregation algorithm proposed in this paper that makes use of the location of well cells in the cell-connectivity graph. Numerical examples in which the FAS solver is compared against Newton's method on benchmark problems are given. In particular, for a refined version of the SAIGUP model, the FAS solver is at least 35% faster than Newton's method for time steps with a CFL number greater than 10.

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arXiv2021

An Aggregation-based Nonlinear Multigrid Solver for Two-phase Flow and Transport in Porous Media

A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. The resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.

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arXiv2021

Enhanced Relaxed Physical Factorization preconditioner for coupled poromechanics

The relaxed physical factorization (RPF) preconditioner is a recent algorithm allowing for the efficient and robust solution to the block linear systems arising from the three-field displacement-velocity-pressure formulation of coupled poromechanics. For its application, however, it is necessary to invert blocks with the algebraic form $\hat{C} = ( C + βF F^T)$, where $C$ is a symmetric positive definite matrix, $FF^T$ a rank-deficient term, and $β$ a real non-negative coefficient. The inversion of $\hat{C}$, performed in an inexact way, can become unstable for large values of $β$, as it usually occurs at some stages of a full poromechanical simulation. In this work, we propose a family of algebraic techniques to stabilize the inexact solve with $\hat{C}$. This strategy can prove useful in other problems as well where such an issue might arise, such as augmented Lagrangian preconditioning techniques for Navier-Stokes or incompressible elasticity. First, we introduce an iterative scheme obtained by a natural splitting of matrix $\hat{C}$. Second, we develop a technique based on the use of a proper projection operator annihilating the near-kernel modes of $\hat{C}$. Both approaches give rise to a novel class of preconditioners denoted as Enhanced RPF (ERPF). Effectiveness and robustness of the proposed algorithms are demonstrated in both theoretical benchmarks and real-world large-size applications, outperforming the native RPF preconditioner.

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arXiv2021

Multigrid reduction preconditioning framework for coupled processes in porous and fractured media

Many subsurface engineering applications involve tight-coupling between fluid flow, solid deformation, fracturing, and similar processes. To better understand the complex interplay of different governing equations, and therefore design efficient and safe operations, numerical simulations are widely used. Given the relatively long time-scales of interest, fully-implicit time-stepping schemes are often necessary to avoid time-step stability restrictions. A major computational bottleneck for these methods, however, is the linear solver. These systems are extremely large and ill-conditioned. Because of the wide range of processes and couplings that may be involved--e.g. formation and propagation of fractures, deformation of the solid porous medium, viscous flow of one or more fluids in the pores and fractures, complicated well sources and sinks, etc.--it is difficult to develop general-purpose but scalable linear solver frameworks. This challenge is further aggravated by the range of different discretization schemes that may be adopted, which have a direct impact on the linear system structure. To address this obstacle, we describe a flexible framework based on multigrid reduction that can produce purely algebraic preconditioners for a wide spectrum of relevant physics and discretizations. We demonstrate its broad applicability by constructing scalable preconditioners for several problems, notably: a hybrid discretization of single-phase flow, compositional multiphase flow with complex wells, and hydraulic fracturing simulations. Extension to other systems can be handled quite naturally. We demonstrate the efficiency and scalability of the resulting solvers through numerical examples of difficult, field-scale problems.

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arXiv2020

Hybrid mimetic finite-difference and virtual element formulation for coupled poromechanics

We present a hybrid mimetic finite-difference and virtual element formulation for coupled single-phase poromechanics on unstructured meshes. The key advantage of the scheme is that it is convergent on complex meshes containing highly distorted cells with arbitrary shapes. We use a local pressure-jump stabilization method based on unstructured macro-elements to prevent the development of spurious pressure modes in incompressible problems approaching undrained conditions. A scalable linear solution strategy is obtained using a block-triangular preconditioner designed specifically for the saddle-point systems arising from the proposed discretization. The accuracy and efficiency of our approach are demonstrated numerically on two-dimensional benchmark problems.

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arXiv2020

Simulation of coupled multiphase flow and geomechanics in porous media with embedded discrete fractures

In fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using uncoupled methods. In recent years, significant research has focused on discretization strategies for these coupled systems, particularly in the presence of complicated fracture network geometries. In this work, we explore a finite-volume discretization for the multiphase flow equations coupled with a finite-element scheme for the mechanical equations. Fractures are treated as lower dimensional surfaces embedded in a background grid. Interactions are captured using the Embedded Discrete Fracture Model (EDFM) and the Embedded Finite Element Method (EFEM) for the flow and the mechanics, respectively. This non-conforming approach significantly alleviates meshing challenges. EDFM considers fractures as lower dimension finiten volumes which exchange fluxes with the rock matrix cells. The EFEM method provides, instead, a local enrichment of the finite-element space inside each matrix cell cut by a fracture element. Both the use of piecewise constant and piecewise linear enrichments are investigated. They are also compared to an Extended Finite Element (XFEM) approach. One key advantage of EFEM is the element-based nature of the enrichment, which reduces the geometric complexity of the implementation and leads to linear systems with advantageous properties. Synthetic numerical tests are presented to study the convergence and accuracy of the proposed method. It is also applied to a realistic scenario, involving a heterogeneous reservoir with a complex fracture distribution, to demonstrate its relevance for field applications.

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arXiv2020

Efficient solvers for hybridized three-field mixed finite element coupled poromechanics

We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures.

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arXiv2020

Enhanced multiscale restriction-smoothed basis (MsRSB) preconditioning with applications to porous media flow and geomechanics

A novel method to enable application of the Multiscale Restricted Smoothed Basis (MsRSB) method to non M-matrices is presented. The original MsRSB method is enhanced with a filtering strategy enforcing M-matrix properties to enable the robust application of MsRSB as a preconditioner. Through applications to porous media flow and linear elastic geomechanics, the method is proven to be effective for scalar and vector problems with multipoint finite volume (FV) and finite element (FE) discretization schemes, respectively. Realistic complex (un)structured two- and three-dimensional test cases are considered to illustrate the method's performance.

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arXiv2020

Multi-Stage Preconditioners for Thermal-Compositional-Reactive Flow in Porous Media

We present a family of multi-stage preconditioners for coupled thermal-compositional-reactive reservoir simulation problems. The most common preconditioner used in industrial practice, the Constrained Pressure Residual (CPR) method, was designed for isothermal models and does not offer a specific strategy for the energy equation. For thermal simulations, inadequate treatment of the temperature unknown can cause severe convergence degradation. When strong thermal diffusion is present, the energy equation exhibits significant elliptic behavior that cannot be accurately corrected by CPR's second stage. In this work, we use Schur-complement decompositions to extract a temperature subsystem and apply an Algebraic MultiGrid (AMG) approximation as an additional preconditioning stage to improve the treatment of the energy equation. We present results for several two-dimensional hot air injection problems using an extra heavy oil, including challenging reactive In-Situ Combustion (ISC) cases. We show improved performance and robustness across different thermal regimes, from advection dominated (high Peclet number) to diffusion dominated (low Peclet number). The number of linear iterations is reduced by 40-85% compared to standard CPR for both homogeneous and heterogeneous media, and the new methods exhibit almost no sensitivity to the thermal regime.

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arXiv2020

Algebraically stabilized Lagrange multiplier method for frictional contact mechanics with hydraulically active fractures

Accurate numerical simulation of coupled fracture/fault deformation and fluid flow is crucial to the performance and safety assessment of many subsurface systems. In this work, we consider the discretization and enforcement of contact conditions at such surfaces. The bulk rock deformation is simulated using low-order continuous finite elements, while frictional contact conditions are imposed by means of a Lagrange multiplier method. We employ a cell-centered finite-volume scheme to solve the fracture fluid mass balance equation. From a modeling perspective, a convenient choice is to use a single grid for both mechanical and flow processes, with piecewise-constant interpolation of Lagrange multipliers, i.e., contact tractions and fluid pressure. Unfortunately, this combination of displacement and multiplier variables is not uniformly inf-sup stable, and therefore requires a stabilization technique. Starting from a macroelement analysis, we develop two algebraic stabilization approaches and compare them in terms of robustness and convergence rate. The proposed approaches are validated against challenging analytical two- and three-dimensional benchmarks to demonstrate accuracy and robustness. These benchmarks include both pure contact mechanics problems and well as problems with tightly-coupled fracture flow.

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arXiv2020

Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems

This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches -- Picard's and Newton's methods -- are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness.

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arXiv2019

A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

Simulation of multiphase poromechanics involves solving a multi-physics problem in which multiphase flow and transport are tightly coupled with the porous medium deformation. To capture this dynamic interplay, fully implicit methods, also known as monolithic approaches, are usually preferred. The main bottleneck of a monolithic approach is that it requires solution of large linear systems that result from the discretization and linearization of the governing balance equations. Because such systems are non-symmetric, indefinite, and highly ill-conditioned, preconditioning is critical for fast convergence. Recently, most efforts in designing efficient preconditioners for multiphase poromechanics have been dominated by physics-based strategies. Current state-of-the-art "black-box" solvers such as algebraic multigrid (AMG) are ineffective because they cannot effectively capture the strong coupling between the mechanics and the flow sub-problems, as well as the coupling inherent in the multiphase flow and transport process. In this work, we develop an algebraic framework based on multigrid reduction (MGR) that is suited for tightly coupled systems of PDEs. Using this framework, the decoupling between the equations is done algebraically through defining appropriate interpolation and restriction operators. One can then employ existing solvers for each of the decoupled blocks or design a new solver based on knowledge of the physics. We demonstrate the applicability of our framework when used as a "black-box" solver for multiphase poromechanics. We show that the framework is flexible to accommodate a wide range of scenarios, as well as efficient and scalable for large problems.

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arXiv2018

A Two-Stage Preconditioner for Multiphase Poromechanics in Reservoir Simulation

Many applications involving porous media--notably reservoir engineering and geologic applications--involve tight coupling between multiphase fluid flow, transport, and poromechanical deformation. While numerical models for these processes have become commonplace in research and industry, the poor scalability of existing solution algorithms has limited the size and resolution of models that may be practically solved. In this work, we propose a two-stage Newton-Krylov solution algorithm to address this shortfall. The proposed solver exhibits rapid convergence, good parallel scalability, and is robust in the presence of highly heterogeneous material properties. The key to success of the solver is a block-preconditioning strategy that breaks the fully-coupled system of mass and momentum balance equations into simpler sub-problems that may be readily addressed using targeted algebraic methods. Numerical results are presented to illustrate the performance of the solver on challenging benchmark problems.

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