Types Of Porous Media Research Articles

H\xf6lder regularity for porous medium systems

We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. ∂tu-div(m|u|m-1Du)=0,m>0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t \\mathbf{u}-\\mathrm{div}(m|\\mathbf{u}|^{m-1}D\\mathbf{u})=0,\\quad m>0. \\end{aligned}$$\\end{document}As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.

Large Deviation Principle for McKean–Vlasov Quasilinear Stochastic Evolution Equations

This paper is devoted to investigating the Freidlin–Wentzell’s large deviation principle for a class of McKean–Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean–Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean–Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations.

Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem

In this paper, we investigate a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati transformation method. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach’s fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space in an abstract setting. Some financial applications of the proposed result are presented in one-dimensional space.

Smoothing effects and infinite time blowup for reaction-diffusion equations: An approach via Sobolev and Poincaré inequalities

We consider reaction-diffusion equations either posed on Riemannian manifolds or in the Euclidean weighted setting, with power-type nonlinearity and slow diffusion of porous medium type. We consider the particularly delicate case p<m in problem (1.1), a case presently largely open even when the initial datum is smooth and compactly supported. We prove global existence for Lm data, and that solutions corresponding to such data are bounded at all positive times with a quantitative bound on their L∞ norm. We also show that on Cartan-Hadamard manifolds with curvature pinched between two strictly negative constants, solutions corresponding to sufficiently large Lm data give rise to solutions that blow up pointwise everywhere in infinite time, a fact that has no Euclidean analogue. The methods of proof are functional analytic in character, as they depend solely on the validity of the Sobolev and of the Poincaré inequalities. As such, they are applicable to different situations, among which we single out the case of (mass) weighted reaction-diffusion equation in the Euclidean setting. In this latter setting we also consider, with stronger results for large times, the case of globally integrable weights.

Traveling waves to one-dimensional Cauchy problems for scalar parabolic-hyperbolic conservation laws

In this paper, we introduce traveling waves to one-dimensional Cauchy problems (CP) for scalar parabolic-hyperbolic conservation laws. The equation is regarded as a linear combination of the scalar hyperbolic conservation laws and the porous medium type equations. Thus, this equation has both properties of hyperbolic equations and those of parabolic equations. Accordingly, it is difficult to investigate the behavior of solutions to (CP). Therefore, it is necessary to construct particular solutions and investigate their properties. In pure hyperbolic case, Riemann solutions are well studied because they are self-similar solutions. However, it cannot be expected in this paper. Hence, we focus on the traveling wave structure instead of the self-similar structure.At first, we construct traveling waves to (CP) and investigate their properties. Next, we discuss the asymptotic behavior of entropy solutions to (CP) using the constructed traveling waves. Finally, we estimate the propagation speed of support for entropy solutions to (CP) using the modified traveling waves.

A hysteresis model for the unfrozen liquid content in freezing porous media

The description of the freezing characteristics of porous media is one of the most conspicuous ingredients in flow and heat transport models that involve freezing and thawing processes. Unfrozen liquid content (ULC) shows strong hysteresis during freezing and thawing cycles in different types of soils and other porous media. We discuss the possible mechanisms of hysteresis in porous media and develop a numerical model for the unfrozen liquid content that is capable of describing the hysteresis phenomenon in freezing and thawing cycles. We present a coupled finite element model as the framework for the numerical simulation of fluid flow and heat transport in partially frozen porous media. The implementation aspects of the ULC model as well as its integration into numerical codes are discussed in detail. We investigate the potential impact of the hysteresis phenomenon on the numerical simulation of transport processes in porous media through benchmark examples and validate the behavior of the model against available laboratory measurement data.

Pore Space Reconstruction of Shale Using Improved Variational Autoencoders

Pore space reconstruction is of great significance to some fields such as the study of seepage mechanisms in porous media and reservoir engineering. Shale oil and shale gas, as unconventional petroleum resources with abundant reserves in the whole world, attract extensive attention and have a rapid increase in production. Shale is a type of complex porous medium with evident fluctuations in various mineral compositions, dense structure, and low hardness, leading to a big challenge for the characterization and acquisition of the internal shale structure. Numerical reconstruction technology can achieve the purpose of studying the engineering problems and physical problems through numerical calculation and image display methods, which also can be used to reconstruct a pore structure similar to the real pore spaces through numerical simulation and have the advantages of low cost and good reusability, casting light on the characterization of the internal structure of shale. The recent branch of deep learning, variational auto-encoders (VAEs), has good capabilities of extracting characteristics for reconstructing similar images with the training image (TI). The theory of Fisher information can help to balance the encoder and decoder of VAE in information control. Therefore, this paper proposes an improved VAE to reconstruct shale based on VAE and Fisher information, using a real 3D shale image as a TI, and saves the parameters of neural networks to describe the probability distribution. Compared to some traditional methods, although this proposed method is slower in the first reconstruction, it is much faster in the subsequent reconstructions due to the reuse of the parameters. The proposed method also has advantages in terms of reconstruction quality over the original VAE. The findings of this study can help for better understanding of the seepage mechanisms in shale and the exploration of the shale gas industry.

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge–Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply directly in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out.

Two-Dimensional Numerical Analysis of Non-Darcy Flow Using the Lattice Boltzmann Method: Pore-Scale Heterogeneous Effects

Abstract This study simulates pore-scale two-dimensional flows through porous media composed of circular grains with varied pore-scale heterogeneity to analyze non-Darcy flow effects on different types of porous media using the lattice Boltzmann method. The magnitude of non-Darcy coefficients and the critical Reynolds number of non-Darcy flow were computed from the simulation results using the Forchheimer equation. Although the simulated porous materials have similar porosity and representative grain diameters, larger non-Darcy coefficients and an earlier onset of non-Darcy flow were observed for more heterogeneous porous media. The simulation results were compared with existing correlations to predict non-Darcy coefficients, and the large sensitivity of non-Darcy coefficients to pore-scale heterogeneity was identified. The pore-scale heterogeneity and resulting flow fields were evaluated using the participation number. From the computed participation numbers and visualized flow fields, a significant channeling effect for heterogeneous media in the Darcy flow regime was confirmed compared with that for homogeneous media. However, when non-Darcy flow occurs, this channeling effect was alleviated. This study characterizes non-Darcy effect with alleviation of the channeling effect quantified with an increase in participation number. Our findings indicate a strong sensitivity of magnitude and onset of non-Darcy effect to pore-scale heterogeneity and imply the possibility of evaluating non-Darcy effect through numerical analysis of the channeling effect.

Effect of Nanopore Geometry in the Conformation and Vibrational Dynamics of a Highly Confined Molecular Glass.

The effect of nanoporous confinement on the glass transition temperature (Tg) strongly depends on the type of porous media. Here, we study the molecular origins of this effect in a molecular glass, N,N'-bis(3-methylphenyl)-N,N'-diphenylbenzidine (TPD), highly confined in concave and convex geometries. When confined in controlled pore glass (CPG) with convex pores, TPD's vibrational spectra remained unchanged and two Tg's were observed, consistent with previous studies. In contrast, when confined in silica nanoparticle packings with concave pores, the vibrational peaks were shifted due to more planar conformations and Tg increased, as the pore size was decreased. The strong Tg increases in concave pores indicate significantly slower relaxation dynamics compared to CPG. Given TPD's weak interaction with silica, these effects are entropic in nature and are due to conformational changes at molecular level. The results highlight the role of intramolecular degrees of freedom in the glass transition, which have not been extensively explored.

In Situ Investigation of Fluid‐Rock Interactions at Ångstrom Resolution

AbstractThe flow of aqueous phases through porous media shapes the world. It changes the courses of rivers, erodes mountainsides, dissolves historical monuments, builds caves, regulates the carbon cycle, and changes porosity in aquifers and hydrocarbon reservoirs. Despite the significance of this process both to natural phenomena and human endeavors, fundamental insight into the mechanisms that drive it at the Ångstrom scale is scarce. Here, we employ environmental transmission electron microscopy (ETEM) to visualize interactions between water and various types of porous media, including sandstone, limestone, and dolostone samples, at Ångstrom resolution in real time. Unlike other techniques presented in the literature, ETEM allows for direct visualization of both the aqueous phase and the porous media. As expected, dissolution and recrystallization were observed to significantly impact the shapes, sizes, and interconnectivities of the pores in the rocks. However, for the first time, we provide direct in situ experimental evidence that adsorption‐induced strain can also significantly impact the above‐mentioned parameters, especially in micropores. In some cases, adsorption was observed to reduce effective pore size by more than 60%. The magnitude of the change in pore size was found to vary with pore wall mineralogy and pore geometry. This work presents the first direct Ångstrom‐resolution observations of liquid adsorption films and adsorption‐induced strain.

Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations

We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs to the large class of PC kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the De Giorgi iteration technique. Sufficiently regular solutions are shown to be unique by means of an L1-contraction estimate.

Thermal Investigations of Double Pass Solar Air Heater with Two Types of Porous Media of Different Thermal Conductivity

This study describes an experimental investigation of the thermal efficiency of stainless steel mesh and steel wool as a porous medium in the lower channel of a double pass solar air heater. An experimental setup was planned and developed. Various types of porous media with high thermal conductivity and with different porosities have been tested. The effects of the porosity of wire mesh, the thermal conductivity of porous media, mass flow rate, and the intensity of radiation have been studied. Experimental results show that thermal efficiency with using porous media is greater than without using porous media. When used steel wool as a porous medium, the thermal efficiency reached 79.82 percent while it can be achieved 76. The percent by using stainless mesh as porous material. The reduction in porosity increasing thermal efficiency. The thermal efficiency of multi-pass solar air collector when used steel wool as porous media is 6, 12.6 and31.7percent higher than without porous media at porosity 98.75, 97.5, and 96.25percent. While it can increase 8.1 and 28.5 percent at porosity 97.875 and 95.75 percent when using stainless steel as porous media.

3TM: Software for the 3-Tau Model

The analysis of the Fast Field Cycling - Nuclear Magnetic Resonance dispersion curves (FFC-NMRD) of 1H nuclei provides fundamental information about the dynamics of fluid protons within a confined environment as well as information on the structure of the porous media. Extracting this information from FFC-NMRD curves requires the use of models which describe the dynamical behaviour of pair of spins through a set of physical parameters. Despite the large number of models available, a very few are accessible to those unfamiliar with programming languages. Those models that are accessible through simplified equations tend to yield limited information. This work presents a simple-to-use software, called 3TM, based on the 3-Tau Model, to stimulate the comparison between different models, and to encourage its application and extension to different types of porous media. The 3TM software provides best-fit solutions in terms of physical parameters using a least-squares minimization algorithm of an objective function. It yields a range of physical parameters which are presented and discussed.

Uniqueness of stationary states for singular Keller–Segel type models

We consider a generalised Keller–Segel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised Hardy–Littlewood–Sobolev type functional inequality in the radial setting.

Experimental study on methane explosion characteristics with different types of porous media

Porous media has a significant effect on flame and overpressure of methane explosion. In this paper, the pore diameter and thickness of porous media are studied. Nine experimental combinations of different pore diameter and thickness on the propagation of flame and overpressure of methane explosion in a tube are analyzed. The results show that the porous media not only can suppress the explosive flame propagation, but the porous media with large pore diameter can cause deflagration and accelerate the transition of flame from laminar to turbulent. The pore diameter of the porous media mainly determines the quenching of the flame. Simply increasing the thickness of porous media may cause the flame to temporarily stop propagating, but the flame is not completely extinguished for larger pore diameter. However, the deflagration propagation speed of flame is affected by the thickness. The attenuation of overpressure by porous media is mainly reflected in reducing the duration of overpressure and the peak value of overpressure. The smaller the pore diameter, the greater the thickness, and the more remarkable the reduction in overpressure duration and peak value. Suitable pore diameter and thickness of porous media can effectively suppress flame propagation and reduce the maximum value and duration of overpressure.

Semi-analytical model for pumping tests in discretely fractured aquifers

In this paper, a novel semi-analytical model for pumping tests in discretely fractured aquifers with randomly distributed and finitely conductive fractures is presented to investigate the wellbore drawdown transient behaviour. An aquifer model and discrete fracture model are established. The flux and drawdown equivalent conditions in the Laplace space are applied in the fracture wall to couple the fluid flow in both systems. The advantage of the proposed semi-analytical model is that only fractures must be separated into segments, not the entire domain. Then, a matrix of the fracture segment in Laplace space can be built and solved by using Gauss's elimination method. A final solution for wellbore drawdown can be easily evaluated using the numerical Laplace inversion algorithm of Stehfest. The model is compared with a previous result of an extended well. The proposed model allows us to consider the problems of pumping tests in complex fracture networks, including parallel uncrossed fracture networks, arbitrary distribution non-intersected fracture networks, and intersected fracture networks with isolated fractures. In the past studies, some scholars have found that the drawdown or pressure derivative curve will present only one dip valley which looks like a V-shaped curve for the naturally fractured confined aquifer or reservoirs, and number of dip valleys are only related to porous media type. However, this study illustrates that the drawdown transient behaviour and the number of dip valleys are closely related to fracture density, fracture conductivity, fracture length, fracture orientation, fracture distribution, position of a pumping well, and the distance between the pumping well and fracture.

The Effects of Hydrophobicity and Drainage Velocity on Water Retention Behaviour in Porous Media: A Computational Study

The present study investigates the water retention behavior in two different types of porous media, i.e., porous metal — a type of metallic foam and ideal geometry. The present study uses computational fluid dynamics (CFD) to model a decreasing water level in a reservoir consisting of a stationary porous medium beneath the water surface at initial stage. It mimics the setup of dynamics dip-testing which measures the amount of retained water for different types of fins-tubes heat exchangers. The study varies parameters like static contact angle ([Formula: see text]) and drainage velocity ([Formula: see text]). The literature review summarizes the unique water retention behaviors for different types of heat exchangers and the findings of the present study. Furthermore, the present study proposed new parameters for evaluating the structural variations in porous metal that explains the water saturation distribution in detail. The evaluation method could provide an insightful idea for performing the quality control check on metallic foam.

The obstacle problem for singular doubly nonlinear equations of porous medium type

In this paper we prove the existence of variational solutions to the obstacle problem associated with doubly nonlinear equations \partial_t (|u|^{m-1}u) - \mathrm {div}(D_\xi f(Du)) = 0 with m &gt; 1 and a convex function f satisfying a standard p -growth condition for an exponent p \in (1,\infty) in a bounded space-time cylinder \Omega_T := \Omega \times (0,T) . The obstacle function \psi and the boundary values g are time dependent. The proof relies on a nonlinear version of the method of minimizing movements.

Experimental study for laminar forced convection heat transfer enhancement from horizontal tube heated with constant heat flux, by using different types of porous media

Forced convective heat transfer for water flowing through a circular pipe oriented horizontally has been investigated experimentally. the pipe was heated at constant heat flux and also packed with two different types of particles (steel and mixture of steel & plastic) as a porous media.The flow features are an internal, laminar, incompressible and steady flow for a wide range of Reynolds numbers (500,1000 and 1500) with variable heat flux (200, 400 and 600 W) for each case and three times for each number.The experimental results comprehended temperature recording on different positions of test sections at steady state depending on the effect of the porous media type and Reynolds number employed (which indicate to the mass flux) in addition to the value of heat flux. The results of the study are represented in terms of temperature distribution to show temperature behaviors in the pipe, besides other diagrams uptake relation of the local and average Nusselt number with data aforementioned above. For each porous type, the results show an increase in local and average Nusselt number with increasing of the Reynold number for steel and mixture porous media respectively. It has obverse from this study that in a porous steel media the heat transfer will occur mainly by conduction, while in the other types of porous media that the heat transfer will occur mainly by forced convection.