Medium Equation Research Articles

Scaled asymptotic solution nets for unlabeled seepage equation solutions with variable well flow

The seepage equation is essential for understanding fluid flow in porous media, crucial for analyzing fluid behavior in various pore structures and supporting reservoir engineering. However, solving this equation under complex conditions, such as variable well flow rates, poses significant challenges. Although physics-informed neural networks have been effective in addressing partial differential equations, they often struggle with the complexities of such physical phenomena. This paper presents an improved method using physical asymptotic solution nets combined with scaling before activation (SBA) and gradient constraints to solve the seepage equation in porous media under varying well flow rates without labeled data. The model consists of two neural networks: one that approximates the asymptotic solution of the seepage equation and another that corrects approximation errors to ensure both mathematical and physical accuracy. When the well flow rate changes, the network may fail to fully satisfy the asymptotic solution due to pressure distribution variations, resulting in sub-optimal outcomes. To address this, we incorporate gradient information into the loss function to reinforce physical constraints and utilize the SBA method to enhance the approximation. This gradient information is derived from the pressure distribution at the previous flow rate, and the SBA method regulates weight adjustments through an adjustment coefficient constrained by the loss function, preventing sub-optimal local minima during optimization. Experimental results show that our method achieves an accuracy range of 10−4 to 10−2.

A Pressure- and Frequency-Dependent Multiscale Model for Describing the Wave Propagation Characteristics of Fluid-Saturated Porous Media

Abstract The mechanism of wave propagation in fluid-saturated porous media is influenced by pressure and frequency. Pressure dependence is mainly dominated by the opening and closing of compliant and stiff pores in rocks, as well as nonlinear deformation respect to high-order elastic constants. Frequency dependence is mainly reflected in the dispersion and attenuation caused by wave-induced fluid flow (WIFF). Therefore, the propagation characteristics of seismic waves in subsurface rocks when pressure and frequency are coupled have broad practical significance, such as geofluid discrimination and in situ stress detection. A new equivalent elastic modulus applicable to fluid-saturated porous media has been established, which simultaneously considers the effects of pressure and WIFF. First, the dual-porosity model is incorporated to account for the changes in rock porosity under pressure and corresponding linear and nonlinear deformations. Then, based on the heterogeneity of rock at the mesocale and microscale, a unified pressure- and frequency-dependent elastic modulus over a wide frequency band is established using the Zener model. The wave equation of fluid-saturated porous media is constructed using the new model, and the pressure- and frequency-dependent phase velocities are derived. Rock physics and digital simulation experiments are applied to analyze the variation of elastic parameters and velocity with pressure and frequency. Comparison with experimental measurement data shows that the new model has higher accuracy than traditional models, especially in the low effective pressure region and the frequency band respect to seismic exploration.

Time-lapse amplitude variation with offset difference inversion based on the reformulated Biot-Squirt model

Time-lapse AVO (Amplitude variation with offset) inversion is a significant technique for the estimation of dynamic reservoir changes. The main theoretical foundation is Zoeppritz equations in single-phase media. At present, there are studies on the inversion method of using Biot's theory. Although it has some improvements in accuracy compared with the method of using Zoeppritz equations, the squirt-flow mechanism is not considered. Furthermore, the precision of difference data equations still needs to be improved in difference inversion. In view of the above problems, the reformulated BISQ (Biot-Squirt) model is introduced into the inversion, and the iterative optimization is used to increase the precision of difference data equations continually in this paper. First, we derive the difference data equations about the reservoir-parameter changes. The reformulated BISQ model is applied, and can describe the characteristics of P-wave reflection accurately and expediently. Secondly, the objective function about the reservoir-parameter changes is constructed. The Bayesian theory and the model smoothing constraint are used, and are helpful to obtain accurate and stable solutions. Thirdly, we derive the equation for estimating the reservoir-parameter changes, and the inversion is iterative for the continuous improvement of accuracy. The difference inversion method is applied, and is helpful to enhance the data repeatability and reduce the calculation amount. Finally, synthetic and real data are applied for the inversion and testing to prove the effectiveness and feasibility of the method.

Approximate P3 equation analysis in multi-layer slab media: Steady-state and time-domain based on the diffusion model

This study aimed to establish a steady-state and time-domain solutions for the approximate P3 equation in arbitrary multi-layer slab media for light propagation, based on a diffusion model with an isotropic point source in the first layer and extrapolated boundary conditions. Spatially resolved diffuse reflectance and transmittance were calculated using the steady-state approximation of the P3 equation, whereas temporally resolved diffuse reflectance and transmittance were derived from the time-domain approximation. The validity of the approximate P3 solutions was confirmed by comparing their results with Monte Carlo simulations. In steady-state analysis, the approximate P3 equation demonstrated superior accuracy to the diffusion equation for reflectance, particularly at smaller thicknesses. Accuracy further improved as the absorption coefficient and detection distance increased. For transmittance, the approximate P3 equation closely matched the diffusion equation at low thickness, but divergence occurred with higher absorption. In time-domain analysis, the approximate P3 equation aligned closely with Monte Carlo simulations at peak values, while its numerical values were close approximations of the diffusion equation away from the peak. The potential application of the approximate P3 equation in multi-layer media offers significant advancements for optical non-invasive detection and treatment techniques, enabling the extraction of optical parameters from such media.

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.

Quantitative photothermal investigation of nonradiative recombination parameters in GaAs/InAs(QD)/GaAs quantum dot structures using a three-layer laser beam deflection model

In this paper, we developed a theoretical model for the photothermal deflection technique in order to investigate the electronic parameters of three-layer semiconductor structures. This model is based on the resolution of thermal and photogenerated carrier diffusion-wave equations in different media. Theoretical results show that the amplitude and phase of the photothermal deflection signal is very sensitive to the nonradiative recombination parameters. The theoretical model is applied to one layer of InAs quantum dots (QDs) inserted in GaAs matrix InAs/GaAs QDs in order to investigate the QD density effects on nonradiative recombination parameters in InAs through fitting the theoretical photothermal beam deflection signal to the experimental data. It was found that the minority carrier lifetime and the electronic diffusivity decrease as functions of increasing InAs QD density. This result is also related to the decrease in the mobility from 21.58 to 4.17 (±12.9%) cm2/V s and the minority carrier diffusion length from 0.62 (±5.8%) to 0.14 (±10%) μm, respectively. Furthermore, both interface recombination velocities S2/3 of GaAs/InAs (QDs) and S1/2 of InAs (QDs)/GaAs increase from 477.7 (±6.2%) to 806.5 (±4%) cm/s and from 75 (±7.8%) to 148.1 (±5.5%) cm/s, respectively.

Simple unidirectional few-cycle electromagnetic pulses

The paper is aimed at constructing exact solutions of Maxwell’s equations for homogeneous media, convenient for modeling ultrashort pulses of various shapes. An analytical description of a family of simple closed-form few-cycle electromagnetic pulses that are free of backward propagating components and have finite energy is presented. The mathematical framework rests on using, as a component of Hertz’s potential, a certain axisymmetric exact solution of the linear wave equation, which is studied here in detail. Depending on the choice of free parameters in this solution and on polarization of the potential, the resulting electromagnetic pulses can be pancake-like, ball-like, needle-like, and doughnut-like. Expressions for spectra of the electric field components of the pulses are obtained. Based on the derived formulas, typical examples of pulses with different types of localization and their spectra are calculated and plotted.

Inverse problems of the wave equation for media with mixed but separated heterogeneous parts

In this article, the inverse problems for the wave equation in a medium in which multiple types of cavities and inclusion exist in a mixture are considered. From the point of view of the indicator function of the enclosure method, there are two types of heterogeneous parts: “minus group” and “plus group.” For example, cavities with the Dirichlet boundary condition belong to the minus group, while inclusions with smaller propagation velocity belong to the plus group. The heterogeneous part of the minus group gives a negative sign to the indicator function, and the heterogeneous part of the plus group gives a positive sign. In general, the presence of many types of heterogeneous parts causes cancelation of the sign of the indicator function. Such cases are referred to as “mixed cases.” Here, we consider the case that the shortest length obtained from the indicator function is attained only by heterogeneous parts of the same group. This case is called the “mixed but separated case,” and it is shown that the method of elliptic estimates developed by Ikehata works well. We also show that the case of a two‐layered background medium with a flat layer can be considered in the same way as the case of a homogeneous background medium.

Asymptotic behavior of the indicator function in the inverse problem of the wave equation for media with multiple types of cavities

This paper discusses an inverse problem of recovering information about cavities by measuring the solutions of a wave equation. When the medium has multiple types of cavities and the distances from the observation site to the ‘positive cavity’ and to the ‘negative cavity’ coincide, the sign of the indicator function of the enclosure method cannot be determined. Thus, sign cancellation may occur, and information may be lost. In this study, by using an asymptotic solution and examining the top term of the indicator function in detail, it is shown that the shortest distance to the cavity can be obtained, even in the above case.

Generalized Transition Waves for Neural Field Equations in Time Heterogeneous Media

Generalized Transition Waves for Neural Field Equations in Time Heterogeneous Media

Seismic scattering inversion for multiple parameters of overburden-stressed isotropic media

Subsurface reservoirs are compressed by overburden stress resulting from the gravity of overlying masses, and the resulting stress changes significantly affect the seismic reflection responses generated at the reservoir interfaces. Several exact reflection coefficient equations have been well established to delineate the role that in situ stress plays in altering the energy transition and amplitude of seismic reflection responses. These exact equations, however, cannot be effectively used in practice due to their intricate formulations and the difficult geophysical estimation for the third-order elastic constants (3oECs) embedded in reflection coefficients. Based on the theories of nonlinear elasticity and elastic wave inverse scattering, we derive an approximate seismic reflection coefficient equation for overburden-stressed isotropic media in terms of the P-wave modulus, shear modulus, density, and a defined stress-related parameter (SRP). The SRP is a combined quantity of elastic moduli, 3oECs, and overburden stress, which can be naturally treated as a dimensionless stress-induced anisotropy parameter. Its inclusion effectively eliminates the need for 3oECs information when using our equation to estimate the desired reservoir properties from seismic observations. By comparing our equation to the exact one, we confirm its validity within the moderate-stress range. Then, we introduce a Bayesian inversion approach incorporating the new reflection coefficient equation to estimate four model parameters. In our approach, the Cauchy and Gaussian distribution functions are used for the a priori probability and the likelihood distributions, respectively. The synthetic tests from two well-log data sets and a field example demonstrate that four parameters can be reasonably inverted using our approach with rather smooth initial models, which illustrates the feasibility of our inversion approach.

Stabilized equal lower-order finite element methods for simulating Brinkman equations in porous media

This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the L 2 norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations.

A Domain-Adaptive Physics-Informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Media

A Domain-Adaptive Physics-Informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Media

MultIHeaTS: A Fast and Stable Thermal Solver for Multilayered Planetary Surfaces

A fully implicit scheme is proposed for solving the heat equation in 1D heterogeneous media, available as a computationally efficient open-source Python code. The algorithm uses finite differences on an irregular grid and is unconditionally stable due to the implicit formulation. The thermal solver is validated against a stiff analytical solution, demonstrating its robustness in handling stiff initial conditions. Its general applicability for heterogeneous cases is demonstrated through its use in a planetary surface scenario with nonlinear boundary conditions induced by blackbody thermal emission. MultIHeaTS's advantageous stability allows for computation times up to 100 times faster than Spencer’s explicit solver, making it ideal for simulating processes on large timescales. This solver is used to compare the thermal signatures of homogeneous and bilayer profiles on Europa. Results show that homogeneous materials cannot reproduce the thermal signature observed in bilayer profiles, emphasizing the need for multilayer solvers. In order to optimize the scientific return of a space mission, we propose a strategy made of three local time observations that is enough to identify bilayer media, for instance, for the next missions to the Jovian system. A second application of the solver is the estimation of the temperature profile of Europa’s near surface (first 10s m) throughout a 1 million yr simulation with varying orbital parameters. The probability distribution of temperature through depth is obtained. Among its various applications, MultIHeaTS serves as the core thermal solver in a multiphysics simulation model detailed in the companion article by C. Mergny & F. Schmidt.

Stochastic analysis and soliton solutions of the Chaffee–Infante equation in nonlinear optical media

The Chaffee–Infante (CI) equation is a nonlinear equation that may be used to predict the complex dynamics of soliton propagation in a nonlinear optical medium. In this study, we elucidate soliton solutions of the CI equation by stochastic differential equations (SDEs) with the Wiener process. In excess of the modified auxiliary equation (MAE) method, we obtain new exact soliton solutions. By combining stochastic differential equations with the Wiener process, we explain the stochastic processes directing the magnitude of solitons within the basis of the CI equation, providing dynamic views on their behavior. The acquired solitary waves are depicted via 3D and 2D graphs to show their dynamics with and without Brownian motion.

Granular media equation with double-well external landscape: limiting steady state

Granular media equation with double-well external landscape: limiting steady state

Deep learning method for finding eigenpairs in Sturm-Liouville eigenvalue problems

Solving the eigenvalue problem for differential equations in inhomogeneous media poses a significant challenge across diverse scientific fields. While classical finite difference methods and finite element methods have produced numerous outcomes, they heavily rely on discretizing the computational domain, which can introduce complexities and limitations. In this study, we present an unsupervised neural network approach tailored for finding eigenpairs in Sturm-Liouville eigenvalue problems within inhomogeneous media. Our method introduces eigenvalues as trainable parameters, crafts a novel cost function, incorporates an adaptive hyper-parameter tuning strategy, and sequentially trains the eigenpairs. The simplicity, accuracy, and interpretability of our approach significantly expand its applicability across various domains. The method we present in this paper can easily tackle boundary value conditions with derivatives, resulting in orthogonal eigenfunctions. This is a very important advantage of deep learning methods that has not yet been noticed. Quantitative estimation of eigenpairs is given for the Sturm-Liouville eigenvalue problems. Furthermore, we extend the proposed methodology to tackle two-dimensional cases, periodic scenarios, demonstrating its versatility and broad potential. For more information see https://ejde.math.txstate.edu/Volumes/2024/53/abstr.html

Nonlinear evolution equations in dispersive media: Analyzing the semilinear dispersive-fisher model

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.

An unconditional stable modified finite element methods for Maxwell’s equation in Kerr-type nonlinear media

The purpose of this paper is to construct a newly efficient finite element method which named as modified finite element method, for Maxwell’s equation in Kerr-type nonlinear media. Because the efficient unconditional stable alternative direction implicit method can’t extend to the Maxwell’s equations in Kerr-type nonlinear media (the difficulty will be discussed in section 3.1), we design the modified method to construct an efficient unconditional stable scheme for Maxwell’s equations in Kerr-type nonlinear media. Furthermore, combined with the direct method, we construct a more efficient direct modified scheme. Comparisons of the efficiency, stability, convergence rate for our modified method and classical explicit leapfrog method were done by theoretical analysis and numerical experiments. The modified method is unconditional stable and its efficiency is not less than leapfrog method.

Machine-learning aided calibration and analysis of porous media CFD models used for rotating packed beds

The proposed research is an attempt to advance the state-of-the-art of the numerical modelling of RPB by combining Computational Fluid Dynamics and Machine Learning approaches. The latter creates an accurate framework that should help quantify the potential of Rotating Packed Beds (RPB) technology to intensify conventional CO2 capture processes. To this end, a direct sensitivity analysis is detailed to supplement a machine-learning (ML) algorithm built for calibrating resistance coefficients needed for porous media modelling. The algorithm is used to improve CFD predictions of dry pressure drop in rotating packed beds (RPBs) for a wide range of operating conditions. The sensitivity derivatives with respect to the packing resistance coefficients are demonstrated for the first time in RPBs, which is not available in the current CFD open source and commercial codes. In this regard, sensitivity differential equations are derived from three-dimensional Navier-Stokes equations for porous media in a rotating reference frame. These sensitivity equations are discretized using a finite volume scheme and solved to obtain the sensitivity pressure drop differences at the packing edges. The results are validated against the predictions of the analytical sensitivity analysis and the finite difference approximation. After, the Newton – Gauss method that employs the sensitivity pressure drop derivatives, is used to minimize the error (cost function) between the pressure drop obtained from CFD simulations and the available experimental data. This is achieved by tuning the packing resistance coefficients to the RBPs' operating conditions (gas flowrate and rotating speed) and correlate them using an artificial neural network (ANN). The results of the proposed approach show a significant improvement in porous media-based CFD predictions of RPBs' pressure drop across a wide range of operating conditions and this over conventional porous media-based CFD models. This is necessary for CFD models to be reliably used as a tool that can efficiently improve existing RPBs' designs and/or participate in RPBs' design innovation.