Heterogeneous Porous Media Research Articles

Vertical displacement of a non-Newtonian Bingham plastic by a Newtonian phase in an axially composite reservoir

In general, oil reservoirs may consist of composite sedimentary structures composed of materials such as sand, clay, or limestone, which exhibit varying lithology due to sedimentary processes. A comprehensive knowledge of this lithology is essential for accurately assessing their hydrocarbon storage and production capacity. Additionally, this information is indispensable for the implementation of various recovery techniques such as waterflooding, gas injection, surfactant injection, polymer injection, alkaline water solution injection, and others. Highly viscous oil can exhibit non-Newtonian behavior during water injection in certain cases. The use of surfactants, alkaline, or polymer solutions in enhanced oil recovery also introduces non-Newtonian behavior. Recovery methods face challenges when non-Newtonian phases, gravity, and reservoir heterogeneity are combined. Against this backdrop, this work presents a mathematical model for immiscible two-phase flow in an axially composite reservoir with a periodic-layered structure. The model considers a non-Newtonian plastic Bingham-type phase extension of the Buckley-Leverett model. To address the porous medium’s heterogeneity with discontinuous flux functions, the numerical solutions were obtained using the Lax-Friedrichs and Lagrangian-Eulerian schemes. The numerical solutions were compared to analytical solutions obtained using an extended version of Oleinik’s geometric construction for discontinuous flux functions. The outcomes display shock and rarefaction waves, as well as a fixed shock due to the porous medium heterogeneity. The numerical results closely correspond with the analytical solutions, seen particularly in the greater accuracy of the Lagrangian-Eulerian method compared to the Lax-Friedrichs method.

A lattice Boltzmann approach to mathematical modeling of myocardial perfusion.

A mathematical model of myocardial perfusion based on the lattice Boltzmann method (LBM) is proposed and its applicability is investigated in both healthy and diseased cases. The myocardium is conceptualized as a porous material in which the transport and mass transfer of a contrast agent in blood flow is studied. The results of myocardial perfusion obtained using LBM in 1D and 2D are confronted with previously reported results in the literature and the results obtained using the mixed-hybrid finite element method. Since LBM is not suitable for simulating flow in heterogeneous porous media, a simplified and computationally efficient 1D-analog approach to 2D diseased case is proposed and its applicability discussed.

Time-dependent dispersion coefficients for the evolution of displacement fronts in heterogeneous porous media

We present an approach for the quantification of displacement fronts in heterogeneous porous media based on the concept of time-dependent apparent dispersion coefficients. The concept of constant asymptotic macrodispersion generally overestimates the area swept by a displacement front and leads to unrealistic upstream dispersion. We show that the large-scale front spreading can be captured by a one-dimensional advection–dispersion equation that is parameterized by a suitably chosen, temporally evolving dispersion coefficient. For purely advective front spreading, we derive an analytical expression based on a predictive continuous time random walk approach, which applies to highly heterogeneous porous media. This analysis elucidates the variability of solute travel times as the key longitudinal spreading mechanisms. It shows that the evolution of dispersion can be captured the sum of exponentials that decay on two dominant time scales. In a particle-based picture, these scales mark the (short) time at which transported particles start exploring the flow variability, and the (large) time at which the slowest particles start decorrelating their transport velocity. Based on these insights, we propose a heuristic formula that accounts for the impact of local scale dispersion as an additional decorrelation mechanisms. The heuristic expression for the longitudinal dispersion coefficient captures solute spreading for a broad range of Péclet numbers and heterogeneity variances. The proposed approach is tested against direct numerical simulations. It provides a robust and fast method for quantifying the evolution of displacement fronts in heterogeneous porous media, with possible applications, for example, in the modelling of groundwater contamination, underground gas storage, and geothermal energy production.

Modeling gas flow in low-permeability formations: an efficient combination of mixed finite elements and high order time integration schemes

Numerical simulation of gas flow in low permeability formations is a challenging task due to the high nonlinearity induced by (i) the compressibility of the gas, (ii) the Klinkenberg slippage effect and (iii) the Langmuir adsorption of the gas on the pore surface. Because of these nonlinearities, modeling gas flow in low permeability formations requires a great deal of computational effort. In this work, we develop an efficient numerical model using advanced spatial and temporal discretization methods for a simultaneous solution of the coupled equations of gas flow, cubic Peng-Robinson equation of state, slippage effect and Langmuir adsorption. The spatial discretization is performed with the lumped hybrid formulation of the mixed finite element method which is well adapted for fluid flow in heterogeneous porous media. The time integration is performed with high-order methods via the method of lines (MOL) which allows large time steps and efficient solution of the nonlinear system of equations. Numerical experiments, performed for gas extraction in a heterogeneous domain, point out the high efficiency of the new model since it can be until 10 times more efficient than the classical first-order time discretization method. Results of global sensitivity analysis show that the intrinsic gas-phase permeability and the Klinkenberg factor are the most influential parameters controlling the pumping rate in the case of gas extraction in a homogeneous domain.

Study on Plugging the Multiscale Water Channeling in Low-Permeability Heterogeneous Porous Media Based on the Growth of Bacteria

Summary To promote the effect of waterflooding of a heterogeneous low-permeability reservoir in the Ordos Basin, a microbial plugging agent is developed to plug the multiscale water channeling. Based on the characteristics of the growth of bacteria, the microbial plugging agent can plug both porous media and microfractures with different scales. The microbial plugging agent is prepared by activating the native bacteria present in low-permeability reservoirs by using the fermentation nutrients. After growing in the fermentation nutrient solution for 4 days in a beaker, the growth of microbial strains begins to stabilize. After that, the main particle size of the prepared microbial plugging agent is between 40 μm and 160 μm and the median particle size (D50) is near 90 μm. The microbial plugging agent has good shear resistance, salt resistance, and stability. At the initial state, due to good injectivity, the microbial plugging agent can smoothly enter into a low-permeability core, a heterogeneous core, and a fractured core, respectively. Thus, it can grow and reproduce in the cores. Based on the characteristics of growth, it can match with the spatial scale of pore or fracture in the cores, so that it cannot only plug the porous media water channeling with different scales but also plug the microfracture water channeling with different scales. This phenomenon has been confirmed by microscopic visualization flow experiments and core flow experiments. The developed microbial plugging agent can be applied to plug the multiscale water channeling to enhance oil recovery of low-permeability heterogeneous reservoirs.

Geo-guided deep learning for spatial downscaling of solute transport in heterogeneous porous media

Resolving solute transport in heterogeneous porous media is a complex task, because of the sparse experimental data and the high computational cost of numerical simulations. This work proposes a unique two-stage deep learning architecture comprising a dual-branch autoencoder and a geo-guided super-resolution generative adversarial network (Gg-SRGAN) to address this dual challenge. The dual-branch autoencoder addresses the issue of sparsity by constructing a continuous, but coarse representation of concentration and pressure profiles from a sparse, discontinuous profile with up to 85% missing data points. The Gg-SRGAN is then employed to generate a finer representation of field variables from the outputs generated by the dual-branch autoencoder (i.e., downscaling). We train and test our framework using six solute transport cases with varying levels of heterogeneity and compare the results with standalone methods, namely the vanilla autoencoder and vanilla SRGAN, in addition to ground truth profiles generated by the finite element method (FEM). The comparisons are performed based on several statistical metrics, such as absolute point error (APE), mean squared error (MSE), peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and learned perceptual image patch similarity (LPIPS). The first four cases are used for training, evaluation, and testing. The last two cases are utilized for blind testing to determine the generalizability of the framework. Our results show that the dual-branch autoencoder outperforms the vanilla autoencoder, and the Gg-SRGAN outperforms the SRGAN during both the training and evaluation phases. Moreover, the proposed framework can successfully construct the fine representation of concentration profiles, compared to FEM, using the coarse representation of the pressure, concentration, and domain permeability fields. When tested using the two blind test cases, the proposed dual-branch autoencoder and Gg-SRGAN exhibit superior performance compared to their counterparts in terms of all evaluation metrics.

A higher mass flux from the effected DNAPL migration and distribution patterns by brine flow in saturated porous media

This study investigated the influence of groundwater salinity on DNAPL migration and distribution in saturated physically homogeneous and heterogeneous porous media. Interfacial tension (IFT) of PCE/aqueous phase containing NaCl and contact angles through the aqueous phase measured results showed that the contact angle decreased and IFT increased with NaCl concentration, leading to the fact that the wettability of quartz sand can be changed from water-wet to strong water-wet. Three sets of two-dimensional (2-D) sandbox experiments conducted with different salinity showed DNAPL migrated faster in the vertical direction in salinity, yielding decreased residual DNAPL entrapped in the migration path. In addition, the salinity-induced strong water-wet media promoted DNAPL trapping at relatively lower saturation both in the coarse or medium sand, and the volume of PCE entrapped as ganglia increased in the salinity cells. The salinity enhanced the dissolution of PCE from the source zone due to the PCE zone architecture change.

Probabilistic assessment of scalar transport under hydrodynamically unstable flows in heterogeneous porous media

Quantitative predictions of scalar transport in natural porous media is a nontrivial task given the presence of multi-scale spatial heterogeneity in the permeability field. Due to data scarcity, the structural map of the permeability field is subject to uncertainty and therefore, model predictions are uncertain. For such reasons, probabilistic models of flow and transport in natural porous media are required in risk assessment and to provide reliable decision making under uncertainty. Further complexities arise when the viscosity of the injected solute differs from that of the ambient fluid. Under the presence of viscosity contrast, hydrodynamic instabilities give rise to viscous fingering, which induces additional disorder in both velocity and solute concentration fields. This work examines the combined role of viscous fingering and permeability heterogeneity in the probabilistic description of transport predictions. In particular, we focus on metrics that are important for risk analysis, such as the solute plume’s early arrival times and the maximum concentration observed at a given location. We propose to use the Projection Pursuit Adaptation (PPA) method in the Polynomial Chaos Expansion (PCE) framework to quantify uncertainty in transport model predictions. The PPA method is a data-driven approach that optimally represents a given quantity of interest in a low-dimensional manifold. Unlike other dimension reduction techniques in uncertainty quantification, the PPA method utilizes non-linear information of the quantity of interest to identify the low-dimensional manifold, thereby increasing the likelihood of finding a more accurate lower-dimensional space. Moreover, the PPA model converges to the physical solution in a mean squared sense with respect to the polynomial order, enabling the construction of a converged model even with limited available data. Then, the PPA results are compared to Monte Carlo simulations using the same amount of data. This comparison illustrates that while Monte Carlo simulations are able to capture low-order statistics, they struggle to represent more detailed probability density functions. Our results show how the combined effect of permeability heterogeneity and viscosity contrast can enhance the mobility of the solute plume.

Existence of solution for an optimal control problem in a heterogeneous porous medium

Abstract The paper explores an optimal control problem concerning transport processes in a porous medium. The transport phenomena is governed by diffusion-reaction equations, which is basically a semi-linear parabolic system. An $L^{2}$-cost (energy) functional is introduced, and control (distributed) is applied in the porous part of the medium. The primary objective is to characterize a given control to be an optimal control and analyse the relationship between optimal control and the adjoint state. The analysis commences with the microscopic description of the controlled system followed by upscaling the system via periodic homogenization.

A Data Assimilation Methodology to Analyze the Unsaturated Seepage of an Earth–Rockfill Dam Using Physics-Informed Neural Networks Based on Hybrid Constraints

Data assimilation for unconfined seepage analysis has faced significant challenges due to hybrid causes, such as sparse measurements, heterogeneity of porous media, and computationally expensive forward models. To address these bottlenecks, this paper introduces a physics-informed neural network (PINN) model to resolve the data assimilation problem for seepage analysis of unsaturated earth–rockfill dams. This strategy offers a solution that decreases the reliance on numerical models and enables an accurate and efficient prediction of seepage parameters for complex models in the case of sparse observational data. For the first attempt in this study, the observed values are obtained by random sampling of numerical solutions, which are then contributed to the synchronous constraints in the loss function by informing both the seepage control equations and boundary conditions. To minimize the effects of sharp gradient shifts in seepage parameters within the research domain, a residual adaptive refinement (RAR) constraint is introduced to strategically allocate training points around positions with significant residuals in partial differential equations (PDEs), which could facilitate enhancing the prediction accuracy. The model’s effectiveness and precision are evaluated by analyzing the proposed strategy against the numerical solutions. The results indicate that even with limited sparse data, the PINN model has great potential to predict seepage data and identify complex structures and anomalies inside the dam. By incorporating coupling constraints, the validity of our PINN model could lead to theoretically viable applications of hydrogeophysical inversion or multi-parameter seepage inversion. The results show that the proposed framework can predict the seepage parameters for the entire research domain with only a small amount of observation data. Furthermore, with a small amount of observation data, PINNs are able to obtain more accurate results than purely data-driven DNNs.

The Method of Finite Averages: A rigorous upscaling methodology for heterogeneous porous media

Rigorous upscaling techniques offer accurate and computationally-efficient strategies for modeling the average behaviors of multi-physical, multiscale phenomena in geological porous media. However, such techniques often rely on a variety of methodological assumptions that prohibit their rigorous application to practical systems (e.g., systems involving heterogeneous porous media, system-scale boundary conditions, and fine-scale dynamics that are not diffusion-dominant). In this work, we aim to formulate an upscaling methodology with few methodological assumptions to provide high levels of model generality and foster the utilization of rigorously-derived upscaled models in practice. In particular, we introduce the Method of Finite Averages (MoFA), a novel upscaling methodology for rigorously modeling heterogeneous porous media and system-scale boundary conditions. We detail MoFA’s implementation for the advective-diffusive transport of a single species and compare the methodology with classic numerical techniques, as well as other rigorous upscaling techniques, to highlight MoFA’s unique combination of rigor and generality. We then validate the derived model while demonstrating its benefits in three numerical experiments. The results suggest that (1.) the applicability and a priori error guarantees of MoFA models do not directly depend on system geometry, (2.) a model’s applicability and error guarantees can be can arbitrarily expanded and reduced, respectively, with further computational expense, and (3.) downscaling with MoFA provides an efficient strategy for generating accurate pore-scale solutions from upscaled results. Ultimately, the results evidence that upscaled models can be rigorously derived for heterogeneous porous media systems and resolved in a fraction of the time it takes to perform the equivalent pore-scale simulations.

Spatiotemporal dynamic temperature variation dominated by ion behaviors during groundwater remediation using direct current

Direct current (DC) electric field has shown promising performance in contaminated site remediation, in which the Joule heating effect plays an important role but has been previously underappreciated. This study focuses on the spatiotemporal characteristics and mechanism of temperature change in heterogeneous porous media with applied DC. The heating process can be divided into four phases: preferential heating of the low permeability zone (LPZ), rapid heating in the middle region, temperature drop and hot zone shift, and reheating. The dynamic ion behaviors with complex interplays among reactions, electrokinetic-driven migration, and mixed convection induced an uneven redistribution of ions and dominated the heating rate and temperature distribution. The concentration of major ions near the pH jump decreased to 1% of the initial value, even though ions were continuously pumped into the heating zone. This ion depletion caused a drop in current, heating rate, and temperature. Here ions cannot be delivered rapidly into the ion-depleted zone by electromigration due to the potential flattening in the surrounding region. The presence of LPZ intensified the nonuniformity of ion redistribution, where a regional focusing of water-soluble ions was observed, and weakened the temperature rebound compared with that using homogeneous sand. These results provide a new perspective on the regulation of DC heating in site remediation.

Ensemble variational Bayesian approximation for the inversion and uncertainty quantification of Darcy flows in heterogeneous porous media with random parameters

The inversion and uncertainty quantification of physical parameters are important for many science and engineering problems. For nonlinear parameter inversion problems of flows in hydrocarbon reservoirs, estimating the true posterior probability distribution function for the random parameter is difficult when the dimension of the parameter is relatively high. Inversion methods such as gradient-based or heuristic optimisation methods searching for the maximum a posterior are generally point-estimate, while uncertainty quantification using MCMC is computationally expensive. In the current study, a new ensemble variational Bayesian approximation method is developed to estimate the posterior probability density using an ensemble of realisations regarded as a trial sample distribution. The objective is to optimize the trial sample distribution such that the evidence lower bound is maximized indicating that the distance between the trial and true posterior distribution is minimized. A subspace is built using principle component analysis based on finite samples to obtain a diagonal positive-definite covariance matrix, and ensemble-based data assimilation is used to optimize the trial distribution efficiently. Heterogeneous 2D and 3D test cases of transient single- and two-phase Darcy flows are presented for validation.

Numerical simulation of foam displacement impacted by kinetic and equilibrium surfactant adsorption

This study explores the effectiveness of foam in reducing gas mobility in porous media impacted by surfactant adsorption in highly heterogeneous porous media. The research focuses on assessing the influence of kinetic and equilibrium adsorption during foam displacement on reservoir sweep efficiency, considering co-injection and SAG (Surfactant Alternating Gas) injection strategies. Including the surfactant on the liquid phase and considering the surfactant adsorption effects fitted by experimental data, the mathematical modeling combines Darcy’s law, fluid phase conservation, surfactant transport equations, and a non-Newtonian foam model in local equilibrium. The simulations utilize FOSSIL (FOam diSplacement SImuLator), an in-house software composed of a stable and conservative numerical algorithm with reduced numerical diffusion. These properties are achieved by combining a hybrid mixed finite element method to solve the Darcy system with a central-upwind finite volume scheme to approximate the transport equations and adopting an implicit adaptive method in time. The results reveal that while omitting adsorption phenomena enhances sweep efficiency due to reduced surfactant loss, differences between equilibrium and kinetic adsorption models are minor (up to 0.3% in production). Consequently, these results suggest using simple adsorption models, such as equilibrium isotherms, rather than more complex models, such as kinetics. Additionally, the SAG approach outperforms co-injection (up to 19% in production), despite the fact that the adsorption is detrimental to both of them, aligning with literature results and previous findings.

Flow prediction of heterogeneous nanoporous media based on physical information neural network

The simulation and prediction of fluid flow in porous media play a profoundly significant role in today's scientific and engineering domains, particularly in gaining a deeper understanding of phenomena such as the migration and fluid flow in underground rock formations and the enhancement of oil recovery rates. The flow of fluids in nanoscale porous media requires consideration of the effects of microscale phenomena, which are challenging to accurately describe using traditional physical models. Currently, research in deep learning for porous media predominantly focuses on conventional porous media, and there is an urgent need for investigations into heterogeneous nanoporous media. Simultaneously, there is a necessity to overcome the limitations of traditional data-driven models lacking physical prior knowledge. Therefore, the integration of physics-informed neural networks, which combine deep learning with physical principles, becomes essential for inferring relatively accurate results from sparse data. In this work, based on the heterogeneity of porous media in shale, we have introduced a deep learning model that couples physical information to predict the flow in heterogeneous nanoscale porous media. In the Physical Information Neural Network model, we utilize point clouds and couple them with deep residual networks. Discrete sampling points are used as inputs, and a multi-level residual connection, along with dimension concatenation, is employed to fuse feature information. The network, through backpropagation, takes into account the Navier-Stokes equations and wall conditions in heterogeneous nanoscale porous media. The results indicate that the apparent permeability and pressure field accuracy are over 90% and 95%, respectively. The Physical Information Neural Network demonstrates promising prospects for predicting flow in nanoscale porous media. Future work will extend to the multiphase complex flow in three-dimensional porous media.

Dual roles of goethite coating on the transport of plastic nanoparticles in heterogeneous porous media: The significance of collector surface roughness

This study systematically examines the roles of positive goethite on the retention and release of negative plastic nanoparticles (PSNPs) with different surface functional groups (Blank, -COOH, and -NH2). It provides the first evidence for the dual roles of goethite coatings on colloid transport; e.g., increased transport caused by surface morphology modification or decreased transport due to increased surface roughness and charge heterogeneity. Although previous work has shown that goethite-coated sand increases the retention of negative colloids, this work demonstrates that collector surface roughness can also reduce the retention of PSNPs due to increased interaction energy profiles. Nonmonotonic retention of all the different functionalized PSNPs was observed in goethite-coated rough sand, and the magnitude of variations was contingent on the PSNP functionalization, the solution ionic strength (IS), and the goethite coating. The release of PSNPs with IS decrease (phase I) and pH increase (phase II) varied significantly due to differences in energy barriers to detachment, e.g., release in phase I was inhibited in both goethite-coated sands, whereas release in phase II was enhanced in coated smooth sand but completely inhibited in rough sand. The findings of this study provide innovative insight into transport mechanisms for colloidal and colloid-associated contaminants.

Analysis of Love-type wave in a piezoelectric layer bonded between fiber-reinforced viscoelastic and dual porous media

Love-type wave propagation in a piezoelectric layer bonded by a layer of fiber-reinforced viscoelastic and a heterogeneous fluid-saturated cracked dual porous half-space is investigated in this paper. The heterogeneity of fracture double porous medium is assumed to be exponential distribution functions. With the help of the variable separation technique, the governing equations of each layer and half-space are solved to get the solutions of mechanical displacement and electric potential function. The results are confirmed by comparison with the classical case of Love wave, and special cases of the problem are also discussed. The phase velocity is calculated numerically and plotted graphically for various material characteristics of the layers and half-space. The analysis reveals that the phase velocity rises with an increase in viscoelastic parameters and the dielectric constant, as well as the reinforcement parameters, while it decreases with an increase in the inhomogeneity parameter, piezoelectric constant, and thickness ratio parameter. A comparative study has also been conducted to analyze the effect of these parameters on the phase velocity using two piezoelectric materials, PZT-4 and PZT-7. The importance of parameter effects increases for Love-type waves with shorter wavelengths and small guiding layer thicknesses.

Multiscale modeling of environment-dependent heterogeneous media by a Thermo-Hydro-mechanical damage model

Evaluating the combined environmental effects, such as external loads, water saturation, and thermal variations, on the mechanical performance of porous media is challenge, mainly due to its inherent heterogeneity and complex multi-physical field mechanisms. This study developed a multiscale thermo-hydro-mechanical damage modeling framework to investigate environment-dependent heterogeneous porous media, with a specific focus on pavement infrastructure. The notable features of this framework comprise: (1) introducing an upscaling homogenization method to connect material thermo-mechanical properties across various scales while considering seasonal variations, (2) creating a downscaling transfer procedure to translate structural responses and temperature fields from macroscopic to mesoscopic scales, providing insights into their multiphysics mechanisms at different scales, and (3) proposing a new sequential coupled algorithm in multiscale simulations for comprehensive multi-field cross-scale calculations. The key findings demonstrated that under hydro-thermo-mechanical loading, shear transverse damage occurs mainly at the macro media surface, while tension longitudinal cracks appear at the bottom of the meso-scale material layer. Damage patterns vary with seasons, with more shear damage prevalent during the summer and an increased occurrence of tension cracks during the winter. Additionally, water-saturated media are more susceptible to “down-top” cracking compared to dry ones. The proposed model can be easily extended to investigate the mechanical behaviors of various heterogeneous porous media under a range of environmental conditions.

Simulation of two-phase flow in 3D fractured reservoirs using a projection-based Embedded Discrete Fracture Model on Unstructured tetrahedral grids (pEDFM-U)

The modeling of fluid flow in heterogeneous, anisotropic and fractured porous media is relevant in many applications, including hydrocarbon and groundwater extraction, dispersion of contaminants, hydrogen or carbon dioxide (CO2) storage. Thus, accurate and scalable simulation of fluid flow through these formations continues to be a great challenge. The presence of fractures that are explicitly modeled, ranging from flow barriers to highly conductive channels, significantly increases the complexity of the numerical simulation. The Embedded Discrete Fracture Model (EDFM) produces results that can be as accurate as those obtained using equidimensional Discrete Fracture Models (DFM) at a much lower computational cost for highly conductive fractures, but it is not adequate for impermeable barriers. In contrast, the pEDFM (projection-based Embedded Discrete Fracture Model) produces accurate results for both, channels and barriers by using additional matrix-fracture connectivities. In its current versions, it is restricted to cartesian and corner-point grid geometries and to the classical Two Point Flux Approximation (TPFA) scheme. In this work, for the first time, the pEDFM is extended to handle unstructured tetrahedral meshes (pEDFM-U). In addition, interface fluxes are approximated by using the Multipoint Flux Approximation method with a Diamond stencil (MPFA-D). This allows for simulation of highly heterogeneous and anisotropic geo-models. The developed method is robust and can deal with full permeability tensors on arbitrary tetrahedral meshes. The advective terms are discretized considering an implicit First Order Upwind (FOU) method. Our method is implemented within the DARSim (Delft Advanced Reservoir Simulation) open-source simulator framework. Through several test cases, the proposed method showed to be robust and capable to accurately capture the effects of both high and low permeability fractures for general tetrahedral meshes, under arbitrary heterogeneous and anisotropic permeability tensors for the rock matrix.

Modeling groundwater flow with random hydraulic conductivity using radial basis function partition of unity method

Simulating groundwater flows in heterogeneous aquifers is one of the most widely studied problems. The heterogeneity is modeled through random hydraulic conductivity fields log-normally distributed. In this paper, we aim to generate the realization of the log-normal hydraulic conductivity by summing up a finite number of random periodic modes with the Kraichnan algorithm. To address Neumann conditions, we propose a change of variables that results in Dirichlet boundary conditions along all boundaries of the computational domain. Then, we suggest using the radial basis function partition of unity method (RBFPUM) and RBFPUM with QR factorization (RBFPUM-QR) to analyze groundwater flow in the heterogeneous porous medium. We evaluate the performance of the proposed methods by increasing the number of random modes and the variance of the log-hydraulic conductivity fields using Gaussian and exponential correlation. In addition, we verify the effectiveness of the presented approaches in one and two dimensions using the method of manufactured solutions. Furthermore, we provide statistical inferences through Monte Carlo simulations. Numerical results demonstrate that RBFPUM-QR performs well in handling higher heterogeneity in both single realizations and Monte Carlo simulations.