arXiv subjects

Hamdi A. Tchelepi

Publications and source records attributed to Hamdi A. Tchelepi.

At least 19 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.

math.NA↗

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.

math.NA↗

arXiv2025

Sensitivity of Trapping Efficiency and Relative Permeability to Experimental Methodology in Laboratory Core Flooding

Understanding the migration and trapping of CO$_2$ in the subsurface is vital to geologic carbon storage projects. Traditional characterization methods employ steady-state co-injection experiments to determine relative permeability and trapping efficiency. Although laboratory studies aim to replicate reservoir conditions, co-injection experiments are often selected because they facilitate steady-state flow and reduce capillary end effects. The fundamental influence of this experimental design choice on measured petrophysical parameters remains inadequately characterized. This study presents a comparative analysis between co-injection and single-phase injection experiments, specifically investigating how experimental methodology influences both trapping efficiency and pressure differential across the core sample (which is used to calculate relative permeability). Our results demonstrate significant variations in trapping behavior between these injection techniques, suggesting that the traditional focus on co-injection experiments may overlook important physical flow phenomena. Differences between injection techniques could be strategically exploited in field applications to enhance residual trapping capacity in subsurface CO$_2$ storage operations. This work highlights the importance of understanding experimental artifacts in core flooding studies and their potential applications for improving carbon storage efficiency.

physics.geo-ph↗

arXiv2024

Python workflow for segmenting multiphase flow in porous rocks

X-ray micro-computed tomography (X-ray micro-CT) is widely employed to investigate flow phenomena in porous media, providing a powerful alternative to core-scale experiments for estimating traditional petrophysical properties such as porosity, single-phase permeability or fluid connectivity. However, the segmentation process, critical for deriving these properties from greyscale images, varies significantly between studies due to the absence of a standardized workflow or any ground truth data. This introduces challenges in comparing results across different studies, especially for properties sensitive to segmentation. To address this, we present a fully open-source, automated workflow for the segmentation of a Bentheimer sandstone filled with nitrogen and brine. The workflow incorporates a traditional image processing pipeline, including non-local means filtering, image registration, watershed segmentation of grains, and a combination of differential imaging and thresholding for segmentation of the fluid phases. Our workflow enhances reproducibility by enabling other research groups to easily replicate and validate findings, fostering consistency in petrophysical property estimation. Moreover, its modular structure facilitates integration into modeling frameworks, allowing for forward-backward communication and parameter sensitivity analyses. We apply the workflow to exploring the sensitivity of the non-wetting phase volume, surface area, and connectivity to image processing. This adaptable tool paves the way for future advancements in X-ray micro-CT analysis of porous media.

physics.geo-ph↗

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.

math.NA↗

arXiv2024

Dynamic Mode Decomposition of real-time 4D imaging data to explore intermittent fluid connectivity in subsurface flows

The interaction of multiple fluids through a heterogeneous pore space leads to complex pore-scale flow dynamics, such as intermittent pathway flow. The non-local nature of these dynamics, and the size of the 4D datasets acquired to capture them, presents challenges in identifying key fluctuations controlling fluid connectivity. To address these challenges, this work employs Dynamic Mode Decomposition (DMD), a data-driven algorithm that decomposes complex nonlinear systems into dominant spatio-temporal structures without relying on prior system assumptions. We present a workflow that identifies critical spatio-temporal regions exhibiting intermittent flow dynamics. This workflow is validated through three test cases, each exploring the impact of viscosity ratio on flow dynamics while maintaining a constant capillary number. Our findings demonstrate DMD's potential in analyzing extensive experimental datasets and identifying crucial intermittent flow structures, offering a powerful tool for understanding complex fluid behaviors in heterogeneous pore spaces. Using our method, we can quickly identify the timescales and locations of interest in an objective manner, providing a valuable diagnostic tool for analysing large synchrotron datasets.

physics.geo-ph↗

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.

math.NA↗

arXiv2023

Learning CO$_2$ plume migration in faulted reservoirs with Graph Neural Networks

Deep-learning-based surrogate models provide an efficient complement to numerical simulations for subsurface flow problems such as CO$_2$ geological storage. Accurately capturing the impact of faults on CO$_2$ plume migration remains a challenge for many existing deep learning surrogate models based on Convolutional Neural Networks (CNNs) or Neural Operators. We address this challenge with a graph-based neural model leveraging recent developments in the field of Graph Neural Networks (GNNs). Our model combines graph-based convolution Long-Short-Term-Memory (GConvLSTM) with a one-step GNN model, MeshGraphNet (MGN), to operate on complex unstructured meshes and limit temporal error accumulation. We demonstrate that our approach can accurately predict the temporal evolution of gas saturation and pore pressure in a synthetic reservoir with impermeable faults. Our results exhibit a better accuracy and a reduced temporal error accumulation compared to the standard MGN model. We also show the excellent generalizability of our algorithm to mesh configurations, boundary conditions, and heterogeneous permeability fields not included in the training set. This work highlights the potential of GNN-based methods to accurately and rapidly model subsurface flow with complex faults and fractures.

cs.LG↗

arXiv2023

Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media

This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approach-and in particular, FSMSN-results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance-i.e., measured by wall-clock time-will be studied in a subsequent publication.

math.NA↗

arXiv2021

Smooth Implicit Hybrid Upwinding for Compositional Multiphase Flow in Porous Media

In subsurface multiphase flow simulations, poor nonlinear solver performance is a significant runtime sink. The system of fully implicit mass balance equations is highly nonlinear and often difficult to solve for the nonlinear solver, generally Newton(-Raphson). Strong nonlinearities can cause Newton iterations to converge very slowly. This frequently results in time step cuts, leading to computationally expensive simulations. Much literature has looked into how to improve the nonlinear solver through enhancements or safeguarding updates. In this work, we take a different approach; we aim to improve convergence with a smoother finite volume discretization scheme which is more suitable for the Newton solver. Building on recent work, we propose a novel total velocity hybrid upwinding scheme with weighted average flow mobilities (WA-HU TV) that is unconditionally monotone and extends to compositional multiphase simulations. Analyzing the solution space of a one-cell problem, we demonstrate the improved properties of the scheme and explain how it leverages the advantages of both phase potential upwinding and arithmetic averaging. This results in a flow subproblem that is smooth with respect to changes in the sign of phase fluxes, and is well-behaved when phase velocities are large or when co-current viscous forces dominate. Additionally, we propose a WA-HU scheme with a total mass (WA-HU TM) formulation that includes phase densities in the weighted averaging. The proposed WA-HU TV consistently outperforms existing schemes, yielding benefits from 5\% to over 50\% reduction in nonlinear iterations. The WA-HU TM scheme also shows promising results; in some cases leading to even more efficiency. However, WA-HU TM can occasionally also lead to convergence issues. Overall, based on the current results, we recommend the adoption of the WA-HU TV scheme as it is highly efficient and robust.

math.NA↗

arXiv2021

Probabilistic forecast of multiphase transport under viscous and buoyancy forces in heterogeneous porous media

In this study, we develop a probabilistic approach to map the parametric uncertainty to the output state uncertainty in first-order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two-phase transport in heterogeneous porous media in the presence of a stochastic velocity field. The uncertainty in the velocity field can arise from the incomplete description of either porosity field, injection flux, or both. The uncertainty in the total-velocity field leads to the spatiotemporal uncertainty in the saturation field. Given information about the spatial/temporal statistics of the correlated heterogeneity, we leverage method of distributions to derive deterministic equations that govern the evolution of single-point CDF of saturation. Unlike Buckley Leverett equation, the equation for the raw CDF function is linear in space and time. Hereby, we give routes to circumventing the computational cost of Monte Carlo scheme while obtaining the full statistical description of saturation. We conduct a set of numerical experiments and compare statistics of saturation computed with the method of distributions, against those obtained using the statistical moment equations approach and kernel density estimation post-processing of high-resolution Monte Carlo simulations. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties, i.e. standard deviation and correlation length of the underlying random fields, while the corresponding low-order statistical moment equations significantly deviate from Monte Carlo results, unless for very small values of standard deviation and correlation length.

physics.comp-ph↗

arXiv2020

Nonlinear Acceleration of Sequential Fully Implicit (SFI) Method for Coupled Flow and Transport in Porous Media

The sequential fully implicit (SFI) method was introduced along with the development of the multiscale finite volume (MSFV) framework, and has received considerable attention in recent years. Each time step for SFI consists of an outer loop to solve the coupled system, in which there is one inner Newton loop to implicitly solve the pressure equation and another loop to implicitly solve the transport equations. Limited research has been conducted that deals with the outer coupling level to investigate the convergence performance. In this paper we extend the basic SFI method with several nonlinear acceleration techniques for improving the outer-loop convergence. Specifically, we consider numerical relaxation, quasi-Newton (QN) and Anderson acceleration (AA) methods. The acceleration techniques are adapted and studied for the first time within the context of SFI for coupled flow and transport in porous media. We reveal that the iterative form of SFI is equivalent to a nonlinear block Gauss-Seidel (BGS) process. The effectiveness of the acceleration techniques is demonstrated using several challenging examples. The results show that the basic SFI method is quite inefficient, suffering from slow convergence or even convergence failure. In order to better understand the behaviors of SFI, we carry out detailed analysis on the coupling mechanisms between the sub-problems. Compared with the basic SFI method, superior convergence performance is achieved by the acceleration techniques, which can resolve the convergence difficulties associated with various types of coupling effects. We show across a wide range of flow conditions that the acceleration techniques can stabilize the iterative process, and largely reduce the outer iteration count.

physics.comp-ph↗

arXiv2020

Dissipation-Based Continuation Method for Multiphase Flow in Heterogeneous Porous Media

In reservoir simulation, solution of the coupled systems of nonlinear algebraic equations that are associated with fully-implicit (backward Euler) discretization is challenging. Having a robust and efficient nonlinear solver is necessary in order for reservoir simulation to serve as the primary tool for managing the recovery processes of large-scale reservoirs. Here, we develop a continuation method based on the use of a dissipation operator. We focus on nonlinear two-phase flow and transport in heterogeneous formations in the presence of viscous, gravitational, and capillary forces. The homotopy is constructed by adding numerical dissipation to the coupled discrete conservation equations. A continuation parameter is introduced to control the amount of dissipation. Numerical evidence of multi-dimensional models and detailed analysis of single-cell problems are used to explain how the dissipation operator improves the nonlinear convergence of the coupled system of equations. An adaptive strategy to determine the dissipation coefficient is proposed. The dissipation level is computed locally for each cell interface. We demonstrate the efficiency of the dissipation-based continuation (DBC) nonlinear solver using several examples, including 1D scalar transport and 2D heterogeneous problems with fully-coupled flow and transport. The DBC solver has better convergence properties compared with the standard damped-Newton solvers used in reservoir simulation.

physics.comp-ph↗

arXiv2020

MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework

We propose MeshfreeFlowNet, a novel deep learning-based super-resolution framework to generate continuous (grid-free) spatio-temporal solutions from the low-resolution inputs. While being computationally efficient, MeshfreeFlowNet accurately recovers the fine-scale quantities of interest. MeshfreeFlowNet allows for: (i) the output to be sampled at all spatio-temporal resolutions, (ii) a set of Partial Differential Equation (PDE) constraints to be imposed, and (iii) training on fixed-size inputs on arbitrarily sized spatio-temporal domains owing to its fully convolutional encoder. We empirically study the performance of MeshfreeFlowNet on the task of super-resolution of turbulent flows in the Rayleigh-Benard convection problem. Across a diverse set of evaluation metrics, we show that MeshfreeFlowNet significantly outperforms existing baselines. Furthermore, we provide a large scale implementation of MeshfreeFlowNet and show that it efficiently scales across large clusters, achieving 96.80% scaling efficiency on up to 128 GPUs and a training time of less than 4 minutes.

cs.LG↗

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.

math.NA↗

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.

math.NA↗

arXiv2018

The stochastic counterpart of conservation laws with heterogeneous conductivity fields: application to deterministic problems and uncertainty quantification

Conservation laws in the form of elliptic and parabolic partial differential equations (PDEs) are fundamental to the modeling of many problems such as heat transfer and flow in porous media. Many of such PDEs are stochastic due to the presence of uncertainty in the conductivity field. Based on the relation between stochastic diffusion processes and PDEs, Monte Carlo (MC) methods are available to solve these PDEs. These methods are especially relevant for cases where we are interested in the solution in a small subset of the domain. The existing MC methods based on the stochastic formulation require restrictively small time steps for high variance conductivity fields. Moreover, in many applications the conductivity is piecewise constant and the existing methods are not readily applicable in these cases. Here we provide an algorithm to solve one-dimensional elliptic problems that bypasses these two limitations. The methodology is demonstrated using problems governed by deterministic and stochastic PDEs. It is shown that the method provides an efficient alternative to compute the statistical moments of the solution to a stochastic PDE at any point in the domain. A variance reduction scheme is proposed for applying the method for efficient mean calculations.

physics.comp-ph↗

arXiv2018

Temporal Markov Processes for Transport in Porous Media: Random Lattice Networks

Monte Carlo (MC) simulations of transport in random porous networks indicate that for high variances of the log-normal permeability distribution, the transport of a passive tracer is non-Fickian. Here we model this non-Fickian dispersion in random porous networks using discrete temporal Markov models. We show that such temporal models capture the spreading behavior accurately. This is true despite the fact that the slow velocities are strongly correlated in time, and some studies have suggested that the persistence of low velocities would render the temporal Markovian model inapplicable. Compared to previously proposed temporal stochastic differential equations with case specific drift and diffusion terms, the models presented here require fewer modeling assumptions. Moreover, we show that discrete temporal Markov models can be used to represent dispersion in unstructured networks, which are widely used to model porous media. A new method is proposed to extend the state space of temporal Markov models to improve the model predictions in the presence of extremely low velocities in particle trajectories and extend the applicability of the model to higher temporal resolutions. Finally, it is shown that by combining multiple transitions, temporal models are more efficient for computing particle evolution compared to correlated CTRW with spatial increments that are equal to the lengths of the links in the network.

physics.comp-ph↗