Heterogeneous Media Research Articles

Intrinsic permeability of heterogeneous porous media

Intrinsic permeability of heterogeneous porous media

On the viscoelastic-electromagnetic-gravitational analogy

The analogy between electromagnetism and gravitation was achieved by linearizing the tensorial gravitational equations of general relativity and converting them into a vector form corresponding to Maxwell’s electromagnetic equations. On this basis, we use the equivalence with viscoelasticity and propose a theory of gravitational waves. We add a damping term to the differential equations, which is equivalent to Ohm’s law in electromagnetism and Maxwell’s viscosity in viscoelasticity, to describe the attenuation of the waves. The differential equations in viscoelasticity are those of cross-plane shear waves, commonly referred to as SH waves. A plane-wave analysis gives the phase velocity, the energy velocity, the quality factor and the attenuation factor of the field as well as the energy balance. To obtain these properties, we use the analogy with viscoelasticity; the properties of electromagnetic and gravitational waves are similar to those of shear waves. The presence of attenuation means that the transient field is generally a composition of inhomogeneous (non-uniform) plane waves, where the propagation and attenuation vectors do not point in the same direction and the phase velocity vector and the energy flux (energy velocity) are not collinear. The polarization of cross-plane field is linear and perpendicular to the propagation-attenuation plane, while the polarization of the field within the plane is elliptical. Transient wave fields in the space–time domain are analyzed with the Green function (in homogeneous media) and with a grid method (in heterogeneous media) based on the Fourier pseudospectral method for calculating the spatial derivatives and a Runge–Kutta scheme of order 4 for the time stepping. In the examples, wave propagation at the Sun–Earth and Earth–Moon distances using quadrupole sources is considered in comparison to viscoelastic waves. The Green and grid solutions are compared to test the latter algorithm. Finally, an example of propagation in heterogeneous media is presented.

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.

Deep neural helmholtz operators for 3D elastic wave propagation and inversion

Summary Numerical simulations of seismic wave propagation in heterogeneous 3D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full waveform inversion, which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent works in seismology have shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We employ a Helmholtz neural operator for 2D and 3D elastic wave modeling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly on the surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3D full-waveform inversion, reducing the computation time by a factor of 350.

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.

Pore-Scale Simulation of the Effect of Wettability Alteration on Flow Transport Kinetics in 3D Natural Porous Media

Summary Wettability alteration commonly occurs in subsurface two-phase displacements, such as enhanced hydrocarbon recovery, hydrogen storage, and carbon dioxide sequestration. A comprehensive understanding of two-phase flow transport kinetics during wettability alteration in natural rocks is essential for optimizing these processes. To address this, a wettability alteration model induced by low-salinity waterflooding (LSWF) was implemented based on the volume of fluid (VOF) method and the compressive continuous species transfer (C-CST) method in the OpenFOAM platform, which integrates the pore-scale two-phase fluid flow and the advection-diffusion of species. Following validation against experimental data from existing literature, extensive direct numerical simulations (DNSs) were conducted in an actual 3D sandstone sample obtained by microcomputed tomography (micro-CT) images. The effects of the wettability alteration degree, wettability alteration model, and capillary number on dynamic salt dispersion and fluid redistribution are considered in simulation works. The findings indicate that a higher wettability alteration degree facilitates the release of more oil trapped in smaller pores toward the outlet, while the mobilized oil might become trapped again due to snap-off in larger downstream pores. Moreover, due to the presence of alternative flow pathways in the system, the backflowed oil induced by heterogeneous salinity distribution might not be effectively recovered. A faster wettability alteration rate enhances the performance of LSWF because of the rapid reduction of entry capillary pressure and the delayed negative effect of salt dispersion. In terms of the capillary number, a higher capillary number accelerates the diffusion of species to the three-phase contact line and reduces the occurrence of snap-off retrapping, thereby increasing ultimate oil recovery. This study contributes to a deeper understanding of the microscopic displacement mechanism during the wettability alteration processes, especially for LSWF, in 3D heterogeneous porous media.

Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.

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

Nature's Load-Bearing Design Principles and Their Application in Engineering: A Review.

Biological structures optimized through natural selection provide valuable insights for engineering load-bearing components. This paper reviews six key strategies evolved in nature for efficient mechanical load handling: hierarchically structured composites, cellular structures, functional gradients, hard shell-soft core architectures, form follows function, and robust geometric shapes. The paper also discusses recent research that applies these strategies to engineering design, demonstrating their effectiveness in advancing technical solutions. The challenges of translating nature's designs into engineering applications are addressed, with a focus on how advancements in computational methods, particularly artificial intelligence, are accelerating this process. The need for further development in innovative material characterization techniques, efficient modeling approaches for heterogeneous media, multi-criteria structural optimization methods, and advanced manufacturing techniques capable of achieving enhanced control across multiple scales is underscored. By highlighting nature's holistic approach to designing functional components, this paper advocates for adopting a similarly comprehensive methodology in engineering practices to shape the next generation of load-bearing technical components.

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.

Chemical structure, morphology, and physicochemical properties of highly methylated pectin functionalized by amidation in homogeneous and heterogeneous medium

Chemical structure, morphology, and physicochemical properties of highly methylated pectin functionalized by amidation in homogeneous and heterogeneous medium

Evaluation of Grafting Parameters of Styrene onto Alkaline Modified Sugarcane Bagasse using Fe2+/H2O2 and Fe2+/H2O2-N2H4 initiator systems and Characterization of the Hydrolyzed Grafted Polystyrene Chains

Cellulose the most abundant natural fibre can be modified, via grafting with vinyl monomers, in homogenous or heterogeneous reaction medium. Therefore, the objective of this paper was to study the grafting parameters of styrene onto alkaline modified sugarcane bagasse using Fe2+/H2O2 and Fe2+/H2O2- N2H4 initiator systems and characterized the hydrolyzed grafted polystyrene chains for their viscometry and average molecular weight distributions using standard appropriate methods. The grafting parameters increases within 0.10-0.25 M styrene concentration in both initiator systems and initiator concentration within the range 1.0x10-3 to 6.0x10-3 for Fe2+/H2O2- N2H4 and 1.0x10-3 to 9.0x10-3 for Fe2+/H2O2 systems, albeit with higher values in the former system. These parameters decrease at higher monomer and initiator concentrations in both systems. The viscometry of hydrolyzed polystyrene chain prepared at optimum initiator concentrations and 0.10-0.30 M styrene concentration in both initiator systems are in the range (1.6416-4.5327 cP), (0.6416-3.5327 cP) and (73.2277-202.1927 cP) for relative, specific and inherent viscosities respectively, consistent with increases in activation energy of flow (1.24-5.30 KJmol-1) and average viscosity molecular weights (307.92-565.79 gmol-1) , these parameters decreases at higher monomer concentration in both initiator systems. The number and weight-average molecular weight distributions and polydispersity indices indicates that grafted polystyrene chains in Fe2+/H2O2 system are broader and more polydispersed than those for Fe2+/H2O2- N2H4 initiator system. The results showed that the hydrazine enhanced initiator system was more effective for the copolymerization reaction with polystyrene polydispersity indices comparable to those of synthetic chain-growth polymers.

Partially embedded metabarrier to suppress surface waves in granular mediaa).

The gravity-induced depth-dependent elastic properties of a granular half-space result in multiple dispersive surface modes and demand the consideration of material heterogeneity in metabarrier designs to suppress surface waves. Numerous locally resonant metabarrier configurations have been proposed in the literature to suppress Rayleigh surface waves in homogeneous media, with little focus on extending the designs to a heterogeneous half-space. In this work, a metabarrier comprising partially embedded rod-like resonators to suppress the fundamental dispersive surface wave modes in heterogeneous granular media known as first order PSV (PSV1; where P is the longitudinal mode and SV is the shear-vertical mode) and second order PSV (PSV2) is proposed. The unit-cell dispersion analysis, together with an extensive frequency-domain finite element analysis, reveals preferential hybridization of the PSV1 and PSV2 modes with the longitudinal and flexural resonances of the resonators, respectively. The presence of the cutoff frequency for the longitudinal-resonance hybridized mode facilitates straightforward suppression of the PSV1 mode, while PSV2 mode suppression is possible by tailoring the hybridized flexural resonance modes. These PSV1 and PSV2 bandgaps are realized experimentally in a granular testbed comprising glass beads by embedding 3D-printed resonator rods. Also explored are novel graded metabarriers capable of suppressing both PSV1 and PSV2 modes over a broad frequency range for potential applications in vibration control and seismic isolation.

Density-driven free convection in heterogeneous aquifers with connectivity features

Free convection usually happens in variable-density groundwater flow systems, and it favors contaminant transport by enlarging length scales and shortening timescales compared to advection and diffusion/dispersion alone. Previous studies have demonstrated that heterogeneity with multi-variate Gaussian distribution for logarithmic permeabilities (log10k) plays an important role in the onset, growth, and/or decay of instability during the density-driven convective process. Nevertheless, the connectivity features (i.e., connected structures of extremely high/low-k values), which are common in natural aquifers, have received little attention. In this study, the classical problem of transient free convection has been modified and numerically simulated by Monte Carlo approach to investigate the effects of connectivity features in heterogeneity on the unstable convective processes. Results show that free convection is promoted by the connected high-k structures and retarded by the connected low-k structures during mainly the early-stage mass loading. The impacts of connectivity features tend to be amplified by higher variation in log10k distributions, and can be secondarily influenced by correlation lengths and anisotropy. Under the multi-variate Gaussian assumption, the existence of connected high-k structures leads to underestimation of density-driven instability, in which the risk differs based on the statistics of permeability fields, metrics of interest and timeframe. This study highlights the importance of understanding connectivity features in heterogeneous geological media when assessing density-dependent solute transport in groundwater systems.

Signal de-noising based on ICEEMDAN for delamination detection by phased array ultrasonic testing

Abstract Aiming at the problem of clutter noise of inter-layer reflections that occurs in multilayer heterogeneous medium such as hybrid fibre composites during ultrasonic inspection, a signal de-noising method based on the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) is proposed in this paper. In the method, the cross-correlation coefficient is utilized to identify the intrinsic mode function (IMF) that is related to raw ultrasonic signal. Particularly, the root mean square error (RMSE), as selection criterions, are employed conjunctively to sift the optimal IMF among selected decomposition signals. Hilbert transform is used to extract the envelope of the signal after removing the interlayer reflection noise. Experimental results show that, the proposed method can adaptively decompose the non-stationary ultrasonic signal into several IMFs with different spectrum so as to dealing with clutter noises. For delaminate detection by phased array ultrasonic testing for hybrid fibre composites, structural noises are significantly suppressed in the reconstructed signal that retains the distinct echo characteristics of a delaminated defect.

Circular phase shift migration based photoacoustic computational tomography

Abstract In the recent several decades, photoacoustic technique has proved to be capable of noninvasive biomedical imaging with scalable spatial resolution and high penetration depth. To achieve faithful image reconstruction, various method has been proposed, such as back projection, phase shift migration, model-based method, etc. These methods have simple implementations in homogeneous media, whereas in the cases of heterogeneous media (e.g., layered media), complicated modifications with respect to wave propagation at layer interface are always necessary. Here, we propose a frequency domain algorithm extension of phase shift migration in polar coordinates, which is obtained by making second order approximation to the optoacoustic point spread function in cylindrical coordinates and thereafter step by step wavenumber-frequency domain wave field extrapolation along the axial direction. Theoretical simulation and phantom experimental results demonstrate that the proposed circular phase shift migration method can well solve the layered heterogeneous reconstructive problem without modifying the wave propagation model. The acoustic diffraction limited acoustic resolution (up to 225 μm) and high imaging SNR (at least 16 dB) are obtained.

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.

NEUTRON POINT KINETICS MODEL WITH A DISTRIBUTED-ORDER FRACTIONAL DERIVATIVE

In this paper, the solutions of an extended form of the Fractional-order Neutron Point Kinetics (FNPK) equation in terms of Caputo-time derivatives of the same order are investigated. Instead of using a Caputo derivative, a distributed-order fractional derivative in the Caputo sense was employed in the term of the FNPK equation which is multiplied by the reactivity. This term plays an important role in the description of neutron kinetics during the start-up, shutdown, and steady-state processes in nuclear reactors. The extended (DFNPK) model was solved using the beta, normal, bimodal and Dirac delta distributions to investigate their effect on the transient state solutions of the neutron density. Regardless of the distribution used, the most significant finding is that a destabilizing effect on the neutron density is induced when the mode (or the instant of application of the Dirac delta) of the distribution tends to one while maintaining the orders of the Caputo-time derivatives constant. What defines the destabilizing effect are large magnitude oscillations, a rapid decay, and an oscillation-free steady state with a monotonic increase that is parallel to but somewhat above the trend determined by the FNPK equation. The extended model is anticipated to be effective for modeling neutron density dispersion in a highly heterogeneous medium that may be described using distributed derivatives.

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.