Heterogeneous Porous Media Research Articles

Biofilm Growth in Porous Media Well Approximated by Fractal Multirate Mass Transfer With Advective‐Diffusive Solute Exchange

AbstractBiofilm growth in porous media changes not only the hydrodynamic properties of the medium (reduction in porosity and permeability, and increase in dispersivity), but also the transport itself (breakthrough curves display increasingly fast first arrivals and long tails). These features are well reproduced by multicontinuum models (Multi‐Rate Mass Transfer, MRMT) which can be used to describe reactive transport in heterogeneous porous media and facilitate the simulation of reactions that are localized within biofilms. Here, we present a conceptual and numerical model of biochemical reactive transport with dynamic biofilm growth based on MRMT formulations. Mass exchange between mobile water and immobile biofilm aggregates is represented by a memory function, which simplifies definition of MRMT parameters. We successfully tested this model on two sets of laboratory data and found that (a) a basic model based on the growth of uniformly sized biofilm aggregates fails to reproduce laboratory tracer tests and rate of biofilm growth, while a fractal growth model, which we obtain by integrating the memory functions of biofilm aggregates with a power law distribution, does; (b) the biofilm memory function evolves as the biofilm grows; and (c) the early time portion of eluted volume tracer breakthrough curves are independent of flow rate, whereas the tail becomes heavier when the flow rate is decreased, which implies that both advection controlled and diffusion controlled mass exchange coexist in biofilms. These findings imply that porous media biofilms are essentially different from those developing in human tissues or open spaces.

Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.

Strong effect of often-overlooked initial spilled oil distribution on subsequent soil remediation: A pore-scale perspective

The removal of spilled oil within soil by water injection is essentially an immiscible fluid displacement process. Numerous studies have been conducted to investigate the way to achieve a higher displacement efficiency. However, these studies often assumed that the pore space was initially saturated by oil only, ignoring the effect of initial oil distribution on the remediation, which led to incorrectly computed flow field and biased analysis. This work aimed to reveal the effect of initial oil distribution on flow physics during the remediation. Based on a pore-scale model and the phase-field method, the spilled oil invasion and remediation in a heterogeneous porous media were investigated, and the dynamic evolution of the phase interface was examined in particular. Four different distribution patterns of spilled oil were obtained through various velocities during the oil invasion stage, which were considered as initial states for the subsequent remediation stage. The results showed that ignoring the oil invasion process led to a more than 10% reduction in the ultimate remediation efficiency. However, the maximum interfacial area of fluids was larger and the obtained relative permeability curve was much smoother for the scenario considering the oil invasion. Besides, a higher initial water saturation led to a lower starting pressure. This work provided new pore-scale insights into the often-overlooked roles of initial phase distribution in fluid dynamics behind the remediation of oil-contaminated soil by water injection, which was of fundamental importance to the development of effective response strategy for soil contamination.

Non-Intrusive Reduced Basis two-grid method for flow and transport problems in heterogeneous porous media

Due to its non-intrusive nature and ease of implementation, the Non-Intrusive Reduced Basis (NIRB) two-grid method has gained significant popularity in numerical computational fluid dynamics simulations. The efficiency of the NIRB method hinges on separating the procedure into offline and online stages. In the offline stage, a set of high-fidelity computations is performed to construct the reduced basis functions, which is time-consuming but is only executed once. In contrast, the online stage adapts a coarse-grid model to retrieve the expansion coefficients of the reduced basis functions. Thus it is much less costly than directly solving a high-fidelity model. However, coarse grids in heterogeneous porous media of flow models are often accompanied by upscaled hydraulic parameters (e.g. hydraulic conductivity), thus introducing upscaling errors. In this work, we introduce the two-scale idea to the existing NIRB two-grid method: when dealing with coarse-grid models, we also employ upscaled model parameters. Both the discretization and upscaling errors are compensated by the rectification post-processing. The numerical examples involve flow and heat transport problems in heterogeneous hydraulic conductivity fields, which are generated by self-affine random fields. Our research findings indicate that the modified NIRB method can effectively capture the large-scale features of numerical solutions, including pressure, velocity, and temperature. However, accurately retrieving velocity fields with small-scale features remains highly challenging.

Stochastic analysis of the interaction between excess fluid flow and soil deformation in heterogeneous deformable porous media

Stochastic analysis of the interaction between excess fluid flow and soil deformation in heterogeneous deformable porous media

Numerical Investigation of Heterogeneous Calcite Distributions in MICP Processes

Microbially induced calcite precipitation (MICP) is a sustainable and environmentally friendly technology with applications in soil stabilization, concrete crack repair, and wastewater treatment. This study presents an improved Darcy-scale numerical model to simulate the MICP processes in heterogeneous porous media. It focuses on the effects of porosity heterogeneity, characterized by average porosity and correlation length, as well as injection strategies. Both average porosity and correlation length are critical factors influencing mass transport and calcite distribution during MICP treatment. An increase in average porosity leads to significant reductions in transport distance and total calcite mass. Notably, in the case of low averaged porosity, a larger correlation length results in more heterogeneous calcite distributions. However, there exists an upper threshold value of the initial averaged porosity (ϕ0=0.45) above which the heterogeneity of the calcite does not present clear dependence on the correlation length. Additionally, injection strategies significantly impact the consolidation effects. Compared to continuous injection, using the phased injection strategy can greatly improve the precipitated calcite area and mass due to its high utility and the efficiency of reactants.

Two-Phase Incompressible Flow with Dynamic Capillary Pressure in a Heterogeneous Porous Media

We prove the existence of weak solutions of a two-incompressible immiscible wetting and non-wetting fluids phase flow model in porous media with dynamic capillary pressure. This model is a coupled system which includes a nonlinear parabolic saturation equation and an elliptic pressure–velocity equation. In the regularized case, the existence and uniqueness of the weak solution are obtained. We let the regularization parameter be η→0 to prove the existence of weak solutions.

CFD analysis of pore morphology, gravity, and fluid characteristics influences on water flooding process

In this research, computational fluid dynamics (CFD) was employed to examine the two-phase flow of water and oil in a porous medium. For this purpose, the Navier-Stockes equations, which describe fluid motion, and the Cahn-Hillard phase field, which defines the interface between two phases, are coupled. The numerical discretization system of equations was solved using the finite element method (FEM) with the COMSOL software. To validate the results and the phase-field model (PFM), the Lucas and Washburn equation was used together with the experimental data. Through this study, the effective parameters were evaluated in two parts. In the first part, seven models with different pore morphologies were designed, and the impact of petrophysical parameters of the reservoir, including the shape of pores, connectivity of pores with or without throat lines, porous media heterogeneity, and relative permeability, on oil recovery was investigated. The second part was devoted to performing sensitivity analysis on the effect of fluid properties, including interfacial tension, wettability, viscosity ratio, injected fluid flow, and gravity, upon enhanced oil recovery (EOR) in different morphologies. Owing to the uniform distribution of capillary pressure in the patterns with the throat lines, the sweep efficiency of the injected fluid was found to be better, and thereby oil production increased. The results of the present work proved the significant influence of gravity on EOR, so that by applying gravity to the solution domain, the breakthrough time and oil recovery factor increased by 20 minutes and 28.6 %, respectively. Moreover, the velocity of the injected fluid as a representative of the flow rate had the greatest effect on EOR, with a 35 % increase in production.

A combined Markov Chain Monte Carlo and Levenberg–Marquardt inversion method for heterogeneous subsurface reservoir modeling

In this study, we systematically investigate an inverse method for heterogeneous porous media to obtain porosity and permeability fields considering only production well data. The forward modeling of the input data, namely pressure and saturation fields, is based on the motion equations of a coupled Darcy flow system involving two phases of isothermal fluid flow. We discretize these equations using a multiscale finite volume simulation technique in the spatial domain, and the backward Euler method in time domain. In the inversion procedure, we combine a global optimization method, the Markov Chain Monte Carlo (MCMC) method, with a local optimization method, the Levenberg–Marquardt (LM) method. The MCMC was implemented as the Random Walk algorithm, and to generate the samples of the porosity and permeability fields, we employed Karhunen–Loève (KL) expansion of second-order stationary fields with Gaussian covariance. The coefficients of the KL expansion are estimated by minimizing the norm of the residual dependent on these fields. At the end of the MCMC iterations, we refine the KL coefficients associated with the porosity and permeability fields using the LM method. We verify in the numerical experiments that the accuracy of the porosity and permeability fields is improved by the LM refinement step.

Intrinsic permeability of heterogeneous porous media

Intrinsic permeability of heterogeneous porous media

Modelling of 2-D seismic wave propagation in heterogeneous porous media: a frequency-domain finite-element method formulated by variational principles

SUMMARY We propose a frequency-domain finite-element (FDFE) method to model the 2-D P–SV waves propagating in porous media. This specific finite-element method (FEM) is based on the framework of variational principles, which differs from previously widely used FEMs that rely on the weak formulations of the governing equations. By applying the calculus of variations, we establish the equivalence between solving the stress–strain relations, equations of motion and boundary conditions that govern the propagation of P–SV waves, and determining the extremum or stationarity of a properly defined functional. The structured rectangular element is utilized to partition the entire computational region. We validate the FDFE method by conducting a comparison with an analytically-based method for models of a horizontal planar contact of two poroelastic half-spaces, and a poroelastic half-space with a free surface. The excellent agreements between the analytically-based solutions and the FDFE solutions indicate the effectiveness of the FDFE method in modelling the poroelastic waves. Modelling results manifest that both propagative and diffusive natures of the Biot slow P wave can be effectively modelled. The proposed FDFE method simulates wavefields in the frequency domain, allowing for easy incorporation of frequency-dependent parameters and enabling parallel computational capabilities at each frequency point (sample). We further employ the developed FDFE method to model two simplified poroelastic reservoirs, one with gas-saturated sandstone and the other with oil-saturated sandstone. The results suggest that changing the fluid phase of the sandstone reservoir from gas to oil can substantially impact the recorded solid and relative fluid–solid displacements. The modelling suggests that the proposed FDFE algorithm is a useful tool for studying the propagation of poroelastic waves.

U-DeepONet: U-Net enhanced deep operator network for geologic carbon sequestration

Learning operators with deep neural networks is an emerging paradigm for scientific computing. Deep Operator Network (DeepONet) is a modular operator learning framework that allows for flexibility in choosing the kind of neural network to be used in the trunk and/or branch of the DeepONet. This is beneficial as it has been shown many times that different types of problems require different kinds of network architectures for effective learning. In this work, we design an efficient neural operator based on the DeepONet architecture. We introduce U-Net enhanced DeepONet (U-DeepONet) for learning the solution operator of highly complex CO2-water two-phase flow in heterogeneous porous media. The U-DeepONet is more accurate in predicting gas saturation and pressure buildup than the state-of-the-art U-Net based Fourier Neural Operator (U-FNO) and the Fourier-enhanced Multiple-Input Operator (Fourier-MIONet) trained on the same dataset. Moreover, our U-DeepONet is significantly more efficient in training times than both the U-FNO (more than 18 times faster) and the Fourier-MIONet (more than 5 times faster), while consuming less computational resources. We also show that the U-DeepONet is more data efficient and better at generalization than both the U-FNO and the Fourier-MIONet.

Effects of Varying Oil/Water Saturations on Miscible and Immiscible CO2 Flooding in Heterogeneous Porous Media: A Lattice Boltzmann Method Simulation Study

Effects of Varying Oil/Water Saturations on Miscible and Immiscible CO₂ Flooding in Heterogeneous Porous Media: A Lattice Boltzmann Method Simulation Study

Experimental and modeling insights into mixing-limited reactive transport in heterogeneous porous media: Role of stagnant zones

The understanding of mixing-controlled reactive dynamics in heterogeneous porous media remains limited, presenting significant challenges for modeling subsurface contaminant transport processes and for designing cost-effective environmental remedial efforts. The complexity of accurately observing, measuring, and modeling mixing-limited reactive transport has led to inadequate exploration of these critical processes. This study investigates the mixing and reaction kinetics affected by stagnant zones, which are commonly found in alluvial aquifers-aquitards and fracture-matrix systems. By conducting experiments involving conservative and bimolecular reactive transport through porous media within translucent chambers filled with two sizes of glass beads and under varying flow rates, we explored the effects of grain size and hydrodynamic conditions. Using a high-resolution camera, we monitored the concentration changes of conservative and reactive tracers, with subsequent interpretation through three-dimensional numerical simulations. The outcomes revealed the emergence of distinct mixing interfaces within both mobile and stagnant zones, culminating in a bi-peaked plume formation. Notably, the mixing and reaction times in media containing stagnant zones were found to be approximately 10 times longer than in homogeneous media. These findings, through experimental and modeling efforts, advance our understanding of mixing-limited reactive transport phenomena within heterogeneous media, underscoring the significant role of stagnant zones—a topic previously underexplored.

Hydrodynamic dipole-driven theory for active flow control in heterogeneous porous media

Although significant efforts have been directed toward refining active control methods for porous media flows, limited explorations have been devoted to the effects of heterogeneous permeability on fluid flow in such environments. These gaps in understanding pose a challenge in developing effective strategies for regulating flow states in porous media with varying permeability. To address these issues, we propose a hydrodynamic dipole-driven theory, solely leveraging a pair of hydrodynamic point source and sink, to rectify flow in heterogeneous porous media systems, thus enabling precise manipulation of the flow field. By carefully tuning the moment of the hydrodynamic dipole, we demonstrate the complete elimination of flow disturbances arising from permeability heterogeneity, and this restoration of the original uniform flow state effectively homogenizes overall permeability. Furthermore, our theory transcends limitations associated with electroosmotic and magnetic methods that require fluids respond to such physical fields, offering broader applicability and minimizing potential contamination risks. Finally, the inherent relation between potential function and pressure distributions in Dracy's law is established with rigorous theoretical analysis, which lays the foundation for active hydrodynamic metamaterials assisted with hydrodynamic dipole strategy. We anticipate that our findings will significantly advance the field of active flow control, particularly in addressing heterogeneous permeability in complex porous media flows, and provide valuable insights for the development of hydrodynamic metamaterial without reliance on heterogeneous or anisotropic materials.

Modeling fluid flow in heterogeneous porous media with physics-informed neural networks: Weighting strategies for the mixed pressure head-velocity formulation

Physics-informed neural networks (PINNs) are receiving increased attention in modeling flow in porous media because they can surpass purely data-driven approaches. However, in heterogeneous domains, PINNs often face convergence challenges due to discontinuities in rock properties. A promising alternative is the mixed formulation of PINNs, which utilizes pressure head and velocity fields as primary variables. This formulation introduces a multi-term loss function whose terms must be carefully balanced to ensure effective convergence during training. The main goal of this work is to identify the most suitable weighting technique to overcome convergence issues and enhance the applicability of the mixed formulation of PINNs for modeling flow in heterogeneous porous media. Thus, we implement and adapt different global and local weighting techniques and evaluate their performance through multiple test scenarios, involving stochastic and block heterogeneity. The results reveal that the most appropriate weighting strategy is the max-average technique. In the case of stochastic heterogeneity, this technique allows for improving the convergence of the training algorithm. In the case of discontinuous heterogeneity, the max-average method is the only strategy that achieved convergence, highlighting its robustness. The results also show that under high heterogeneity, using an appropriate weighting technique becomes imperative because baseline PINN failed to converge. Implementing an optimal weighting strategy can improve convergence and yield accurate solutions with fewer learnable parameters, thereby enhancing overall model performance.

Comparative analysis of double and single porosity effects on SH-wave induced vibrations in periodic porous lattices

The manuscript presents a novel study comparing the impact of single and double porosity on horizontally polarized shear wave (SH-wave) propagation in corrugated elastic void materials. The considered mathematical model comprises two cases; the first one depicts SH-wave propagation through a void porous layer having creased boundaries and resting over a heterogeneous anisotropic fluid-saturated fractured porous half-space, whereas according to the second case, heterogeneous anisotropic fluid-saturated porous semi-infinite medium without fractures has been considered. In both cases, rigidity and density of the half-space vary quadratically with depth. The separable variable method is used to attain the complex frequency equation for each case that leads to two different dispersion relations associated with two distinct wave fronts. The first wave front depends on the void parameters, whereas the second wave front defines the propagation of SH-waves in an elastic layer without void pores. In each case, the complex dispersion relation has been separated into two equations that illustrate the dispersion and attenuation properties of SH-waves. Using the variations in the inhomogeneity, position, fluctuation, flatness, total porosity, and anisotropy parameters, case I and case II have been compared in each graphical execution. In addition, the surface response of shear stress and displacement have been implemented graphically.

Convection heat and mass transfer of non-Newtonian fluids in porous media with Soret and Dufour effects using a two-sided space fractional derivative model

Non-Newtonian fluids within heterogeneous porous media may give rise to complex spatial energy and mass distributions owing to non-local mechanisms, the modeling of which remains unclear. This study investigates the natural convection heat and mass transfer of non-Newtonian fluids in porous media, considering the Soret and Dufour effects. A strongly coupled model is developed to quantify the coupled transport of energy and reactive pollutants with the non-Newtonian fluid. The constitutive equation for the non-Newtonian fluid is described by a two-sided Caputo type space fractional velocity gradient. The governing equation, with a symmetric diffusion term, is effectively solved using a stable and convergent shifted Grünwald–Letnikov formula. The influences of three important parameters, which are the average skin friction coefficient, the average Nusselt number, and the Sherwood number, on fluid heat and mass transfer are calculated and analyzed. Numerical results reveal a significant interaction between the fractional derivative and the buoyancy ratio number, both of which affect the average skin friction coefficient. Furthermore, the average Nusselt number increases with the Dufour number while decreasing with the average Sherwood number. These findings enhance our understandings of the dynamics of energy and mass co-transport in non-Newtonian fluids, particularly in relation to their constitutive equation featuring spatial non-local properties.

Investigation of non-chemical CO2 microbubbles for enhanced oil recovery and carbon sequestration in heterogeneous porous media

Non-chemical CO2 microbubble (MB) is a promising environmentally friendly technology for enhanced oil recovery (EOR) and carbon sequestration. Understanding the interaction between non-chemical CO2 MB and oil phases in porous media is crucial for optimizing the design of CO2 MB to enhance both EOR and carbon sequestration efforts. This study explores the impact of oil on the conformance control of non-chemical MB and conducts a case study on their potential for EOR and carbon sequestration in a heterogeneous reservoir. First, experimental results reveal that MB's conformance control is impacted by the presence of oil, with varying degrees of impairment depending on oil composition. Then, reservoir simulations demonstrate that MB outperforms conventional CO2 flooding and water-alternating-gas (WAG), achieving a final recovery of 77.48% of original oil in place compared to 66.79% of original oil in place for CO2 flooding and 68.19% of original oil in place for WAG. Sensitivity analyses highlight the importance of parameters such as gas-liquid ratio of MB, reservoir heterogeneity, and reservoir thickness in MB's EOR performance. Finally, CO2 storage efficiency analyses show that MB exhibits superior CO2 storage efficiency, with higher structural and solubility trapping compared to WAG. The study concludes that MB technology holds promise for EOR and carbon sequestration, contributing to efforts toward carbon neutrality and sustainable oil recovery practices. These findings provide valuable insights for optimizing MB applications in field-scale operations and advancing toward a more environmentally sustainable energy industry.

Peridynamics modeling of coupled gas convection transport and thermal diffusion in heterogeneous porous media

Peridynamics modeling of coupled gas convection transport and thermal diffusion in heterogeneous porous media