One-dimensional Porous Medium Research Articles

Convergence of solutions of the porous medium equation with reactions

Consider the Cauchy problem of one-dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as t\to \infty to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behavior of the solutions for PME with monostable, bistable and combustion types of nonlinearities.

Simulations of flaming combustion and flaming-to-smoldering transition in wildland fire spread at flame scale

Our objective in the present study is to provide basic insights into the coupling between external-gas and solid biomass vegetation processes that control the dynamics of flame spread in wildland fire problems. We focus on a modeling approach that resolves processes occurring at vegetation and flame scales, i.e., the formation of flammable vapors due to the thermal degradation of the solid biomass, the subsequent combustion in ambient air, the thermal feedback to the biomass through radiative and convective heat transfer, and the possible transition from flaming combustion (taking place outside of the solid biomass) to smoldering combustion (taking place inside the solid biomass). The capability uses a multiphase combustion framework and treats external-gas processes through a Large Eddy Simulation solver and solid biomass processes through a discrete particle model. The discrete particle model adopts a one-dimensional porous medium formulation, includes descriptions of drying, thermal pyrolysis, oxidative pyrolysis, and char oxidation, as well as a description of the external-gas-to-solid-biomass diffusion of oxygen mass; the discrete particle model thereby provides a treatment of in-depth oxidative processes and allows the simulation of smoldering combustion. The modeling capability is applied to the simulation of fire spread across a surrogate biomass vegetation bed corresponding to a discrete array of cylindrical-shaped, vertically-oriented, pine wood sticks, characterized by a monomodal size distribution, in horizontal flat terrain and under wind-aided conditions. The numerical results demonstrate that the model can simulate successful flaming-to-smoldering transition followed by complete biomass consumption.Novelty and Significance statement:• A new computational model is proposed to simulate wildland fire behavior at high levels of resolution capable of capturing vegetation- and flame-scale phenomena.• Solid biomass vegetation processes are treated using a discrete particle model that features a porous medium formulation and includes a description of in-depth oxygen mass diffusion, and exothermic oxidative pyrolysis and char oxidation reactions.• The computational model is shown to be capable of simulating the transition from flaming to smoldering combustion, often observed in wildland fire spread problems.

Reaction dynamics and early-time behaviour of chemical decontamination

Abstract When a hazardous chemical soaks into a porous material such as a concrete floor, it can be difficult to remove. One approach is chemical decontamination, where a cleanser is added to react with and neutralise the contaminating agent. The goal of this paper is to investigate the reaction dynamics and the factors that affect the efficacy of the decontamination procedure. We consider a one-dimensional porous medium initially saturated with an oil-based agent. An aqueous cleanser is applied at the surface, so the two chemicals are immiscible and a boundary forms between them. A neutralising reaction takes place at this boundary in which cleanser and agent are consumed and reaction products are created. This is a Stefan problem, and the boundary between the cleanser and agent moves as the reaction proceeds. Reaction products formed at the interface may dissolve in one or both liquids. This may temporarily prevent cleanser and/or agent from reaching the reaction site, so diffusion of the chemical species, in particular the diffusion of product from the interface, plays a key role. The scenario described above was considered previously by Dalwadi et al. (2017, Mathematical modelling of chemical agent removal by reaction with an immiscible cleanser. SIAM J. Appl. Math., 77, 1937–1961) in the limit where the depth of the porous medium is large compared to the length scale over which concentrations vary inside the medium. Here, we present results that are valid for any ratio between these length scales and an analysis of agent removal times for various dimensionless parameter regimes. We also highlight the emergence of a boundary layer associated with diffusion in the oil phase for early times, where the thickness of the boundary layer is directly proportional to the square root of the time variable.

Liouville-type theorem for one-dimensional porous medium systems with sources

Liouville-type theorem for one-dimensional porous medium systems with sources

On Kjartansson model of thermoelastic attenuation

There is a mathematical analogy between the poroelastic and thermoelastic behavior of the compressional P wave, whose dissipation is due to the coupling between the elastic deformation and the phenomenon of pressure and heat diffusion, represented by Darcy’s law and heat conduction, respectively. This attenuation effect is more pronounced in heterogeneous media, where conversion of the fast P wave to the slow (Biot) and thermal diffusive modes occurs at material interfaces. Specifically, the problem is to obtain the P-wave properties of a porous medium due to temperature gradients between the solid and fluid phases. Then, we consider a simplified one-dimensional porous medium and obtain the quality factor, Q, and phase velocity as a a function of frequency, based on an isostress condition and using Gassmann equation and the Kramers Kronig relations.The model allows for the incorporation of a relaxation time to simulate a proper wave-like behavior at high frequencies, avoiding in this way infinite velocities. Moreover, we perform a complete analysis varying the different parameters, namely, the heat conduction, the specific heat, the thermal expansion, the types of solid and fluid and the pore size.

Counter-current imbibition and non-linear diffusion in fractured porous media: Analysis of early- and late-time regimes and application to inter-porosity flux

The displacement of non-aqueous phase liquid (NAPL) in permeable porous rock by water saturating the surrounding fractures is studied. In many situations of practical interest, capillarity is the dominant driving force. Using the global pressure concept, it is shown that the water saturation is driven by a generic non-linear diffusion equation governing the matrix block counter-current spontaneous imbibition. In addition, in most cases, the saturation-dependent diffusion coefficient vanishes at the saturation end points, that renders the driving equation highly singular.In this paper, two exact asymptotic solutions valid for short and long times are presented, under the assumption that the conductivity vanishes as a power-law of both phase saturations at the extreme values of the fluid saturation.Focusing on the late-time domain, the asymptotic solution is derived using an Ansatz that is written under the form of a power-law time decay of the NAPL saturation. In the spatial domain, this solution is an eigenvector of the non-linear diffusion operator driving the saturation, for a problem with Dirichlet boundary conditions. If the diffusion coefficient varies as a power law of the NAPL saturation, the spatial variations of the solution are given analytically for a one-dimensional porous medium corresponding to parallel fracture planes. The analytical solution is in very good agreement with results of numerical simulations involving various realistic sets of input transport parameters.Generalization to the case of two- or three-dimensional matrix blocks of arbitrary shape is proposed using a similar Ansatz, solution of a non-linear eigenvalue problem. A fast converging algorithm based on a fixed-point sequence starting from a suitable first guess was developed. Comparisons with full-time simulations for several typical block geometries show an excellent agreement.These results permit to set-up an analytical formulation generalizing linear single-phase representation of matrix-to-fracture exchange term. It accounts for the non-linearity of the local flow equations using the power-law dependence of the conductivity for low NAPL saturation. The corresponding exponent can be predicted from the input conductivity parameters. Similar findings are also presented and validated numerically for two- or three-dimensional matrix blocks. That original approach paves the way to research leading to a more faithful description of matrix-to-fracture exchanges when considering a realistic fractured medium composed of a population of matrix blocks of various size and shapes.

A second-order accurate, energy stable numerical scheme for the one-dimensional porous medium equation by an energetic variational approach

A second-order accurate, energy stable numerical scheme for the one-dimensional porous medium equation by an energetic variational approach

Asymptotic behavior of solutions to porous medium equations with boundary degeneracy

This article concerns the asymptotic behavior of solutions to a class of one-dimensional porous medium equations with boundary degeneracy on bounded and unbounded intervals. It is proved that the degree of degeneracy, the exponents of the nonlinear diffusion, and the nonlinear source affect the asymptotic behavior of solutions. It is shown that on a bounded interval, the problem admits both nontrivial global and blowing-up solutions if the degeneracy is not strong; while any nontrivial solution must blow up if the degeneracy is strong enough. For the problem on an unbounded interval, the blowing-up theorems of Fujita type are established. The critical Fujita exponent is finite if the degeneracy is not strong, while infinite if the degeneracy is strong enough. Furthermore, the critical case is proved to be the blowing-up case if it is finite. For more information see https://ejde.math.txstate.edu/Volumes/2021/96/abstr.html

Solving One-Dimensional Porous Medium Equation Using Unconditionally Stable Half-Sweep Finite Difference and SOR Method

A porous medium equation is a nonlinear parabolic partial differential equation that presents many physical occurrences. The solutions of the porous medium equation are important to facilitate the investigation on nonlinear processes involving fluid flow, heat transfer, diffusion of gas-particles or population dynamics. As part of the development of a family of efficient iterative methods to solve the porous medium equation, the Half-Sweep technique has been adopted. Prior works in the existing literature on the application of Half-Sweep to successfully approximate the solutions of several types of mathematical problems are the underlying motivation of this research. This work aims to solve the one-dimensional porous medium equation efficiently by incorporating the Half-Sweep technique in the formulation of an unconditionally-stable implicit finite difference scheme. The noticeable unique property of Half-Sweep is its ability to secure a low computational complexity in computing numerical solutions. This work involves the application of the Half-Sweep finite difference scheme on the general porous medium equation, until the formulation of a nonlinear approximation function. The Newton method is used to linearize the formulated Half-Sweep finite difference approximation, so that the linear system in the form of a matrix can be constructed. Next, the Successive Over Relaxation method with a single parameter was applied to efficiently solve the generated linear system per time step. Next, to evaluate the efficiency of the developed method, deemed as the Half-Sweep Newton Successive Over Relaxation (HSNSOR) method, the criteria such as the number of iterations, the program execution time and the magnitude of absolute errors were investigated. According to the numerical results, the numerical solutions obtained by the HSNSOR are as accurate as those of the Half-Sweep Newton Gauss-Seidel (HSNGS), which is under the same family of Half-Sweep iterations, and the benchmark, Newton-Gauss-Seidel (NGS) method. The improvement in the numerical results produced by the HSNSOR is significant, and requires a lesser number of iterations and a shorter program execution time, as compared to the HSNGS and NGS methods.

Mathematical modeling and numerical simulation of porous media single-phase fluid flow problem: a scientific review

The complexity of porous media makes the classical methods used to study hydrocarbon reservoirs inaccurate and insufficient to predict the performance and behavior of the reservoir. Recently, fluid flow simulation and modeling used to decrease the risks in the decision of the evaluation of the reservoir and achieve the best possible economic feasibility. This study deals with a brief review of the fundamental equations required to simulate fluid flow through porous media. In this study, we review the derivative of partial differential equations governing the fluid flow through pores media. The physical interpretation of partial differential equations (especially the pressures diffusive nature) and discretization with finite differences are studied. We restricted theoretic research to slightly compressible fluids, single-phase flow through porous media, and these are sufficient to show various typical aspects of subsurface flow numerical simulation. Moreover, only spatial and time discretization with finite differences will be considered. In this study, a mathematical model is formulated to express single-phase fluid flow in a one-dimensional porous medium. The formulated mathematical model is a partial differential equation of pressure change concerning distance and time. Then this mathematical model converted into a numerical model using the finite differences method.

Modeling and Simulation of Polymer Flooding with Time-Varying Injection Pressure.

Polymer flooding is one of the most incipient chemical-based enhanced oil recovery process that utilizes the injection of polymer solutions into oil reservoirs. The presence of a polymer in water increases the viscosity of the injected fluid, which upon injection reduces the water-to-oil mobility ratio and the permeability of the porous media, thereby improving oil recovery. The objective of this work is to investigate strategies that would help increase oil recovery. For that purpose, we have studied the effect of injection pressure and increasing polymer concentration on flooding performance. This work emphasizes on the development of a detailed mathematical model describing fluid saturations, pressure, and polymer concentration during the injection experiments and predicts oil recovery. The mathematical model developed for simulations is a black oil model consisting of a two-phase flow (aqueous and oleic) of polymeric solutions in one-dimensional porous media as a function of time and z-coordinate. The mathematical model consisting of heterogeneous, nonlinear, and simultaneous partial differential equations efficiently describes the physical process and consists of various parameters and variables that are involved in our lab-scale process to quantify and analyze them. A dimensionless numerical solution is achieved using the finite difference method. We implement the second-order high-accuracy central and backward finite-divided-difference formula along the z-direction that results in the discretization of the partial differential equations into ordinary differential equations with time as an independent variable. The input parameters such as porosity, permeability, saturation, and pore volume obtained from experimental data by polymer flooding are used in the simulation of the developed mathematical model. The model-predicted and commercial reservoir (CMG)-simulated oil production is in good agreement with experimental oil recoveries with a root-mean-square error (RMSE) in the range of 1.5–2.5 at a maximum constant pressure of 3.44 MPa as well as with temporal variation of the injection pressure between 2.41 and 3.44 MPa.

Thermal Impulse Response in Porous Media for Transpiration-Cooling Systems

A solution of the coupled differential equations for fluid and solid phases in a one-dimensional porous medium in thermal nonequilibrium is presented using the concept of analyzing the impulse response. The impulse response is shown to be sensitive to the volumetric heat transfer coefficient and the coolant mass flux. Experimental data obtained from surface heating of transpiration-cooled porous zirconium di-boride () samples are compared to a newly developed theoretical model. The surface and backside temperatures of the solid are measured using thermographic imaging and thermocouple instrumentation. The noninteger system identification approach is used to experimentally obtain the thermal impulse response, which is then compared to the model prediction. Good agreement is found between the simulated and experimental data with average deviations below 10%. The developed model provides the basis for inverse heat transfer measurements and further analysis of transpiration-cooled materials.

Long-time asymptotics for a 1D nonlocal porous medium equation with absorption or convection

In this paper, the long-time asymptotic behaviors of one-dimensional porous medium equations with a fractional pressure and absorption or convection are studied. In the parameter regimes when the nonlocal diffusion is dominant, the entropy method is adapted to derive the exponential convergence of relative entropy of solutions in similarity variables.

A comparative study of explicit high-resolution schemes for compositional simulations

PurposeThis paper aims to numerically study the compositional flow of two- and three-phase fluids in one-dimensional porous media and to make a comparison between several upwind and central numerical schemes.Design/methodology/approachImplicit pressure explicit composition (IMPEC) procedure is used for discretization of governing equations. The pressure equation is solved implicitly, whereas the mass conservation equations are solved explicitly using different upwind (UPW) and central (CEN) numerical schemes. These include classical upwind (UPW-CLS), flux-based decomposition upwind (UPW-FLX), variable-based decomposition upwind (UPW-VAR), Roe’s upwind (UPW-ROE), local Lax–Friedrichs (CEN-LLF), dominant wave (CEN-DW), Harten–Lax–van Leer (HLL) and newly proposed modified dominant wave (CEN-MDW) schemes. To achieve higher resolution, high-order data generated by either monotone upstream-centered schemes for conservation laws (MUSCL) or weighted essentially non-oscillatory (WENO) reconstructions are used.FindingsIt was found that the new CEN-MDW scheme can accurately solve multiphase compositional flow equations. This scheme uses most of the information in flux function while it has a moderate computational cost as a consequence of using simple algebraic formula for the wave speed approximation. Moreover, numerically calculated wave structure is shown to be used as a tool for a priori estimation of problematic regions, i.e. degenerate, umbilic and elliptic points, which require applying correction procedures to produce physically acceptable (entropy) solutions.Research limitations/implicationsThis paper is concerned with one-dimensional study of compositional two- and three-phase flows in porous media. Temperature is assumed constant and the physical model accounts for miscibility and compressibility of fluids, whereas gravity and capillary effects are neglected.Practical implicationsThe proposed numerical scheme can be efficiently used for solving two- and three-phase compositional flows in porous media with a low computational cost which is especially useful when the number of chemical species increases.Originality/valueA new central scheme is proposed that leads to improved accuracy and computational efficiency. Moreover, to the best of authors knowledge, this is the first time that the wave structure of compositional model is investigated numerically to determine the problematic situations during numerical solution and adopt appropriate correction techniques.

On exact solutions of a heat-wave type with logarithmic front for the porous medium equation

The paper deals with a nonlinear second-order parabolic equation with partial derivatives, which is usually called “the porous medium equation”. It describes the processes of heat and mass transfer as well as filtration of liquids and gases in porous media. In addition, it is used for mathematical modeling of growth and migration of population. Usually this equation is studied numerically like most other nonlinear equations of mathematical physics. So, the construction of exact solution in an explicit form is important to verify the numerical algorithms. The authors deal with a special solutions which are usually called “heat waves”. A new class of heat-wave type solutions of one-dimensional (plane-symmetric) porous medium equation is proposed and analyzed. A logarithmic heat wave front is studied in details. Considered equation has a singularity at the heat wave front, because the factor of the highest (second) derivative vanishes. The construction of these exact solutions reduces to the integration of a nonlinear second-order ordinary differential equation (ODE). Moreover, the Cauchy conditions lead us to the fact that this equation has a singularity at the initial point. In other words, the ODE inherits the singularity of the original problem. The qualitative analysis of the solutions of the ODE is carried out. The obtained results are interpreted from the point of view of the corresponding heat waves’ behavior. The most interesting is a damped solitary wave, the length of which is constant, and the amplitude decreases.

Excess Diffuse Light Absorption in Upper Mesophyll Limits CO2 Drawdown and Depresses Photosynthesis.

In agricultural and natural systems, diffuse light can enhance plant primary productivity due to deeper penetration into and greater irradiance of the entire canopy. However, for individual sun-grown leaves from three species, photosynthesis is actually less efficient under diffuse compared with direct light. Despite its potential impact on canopy-level productivity, the mechanism for this leaf-level diffuse light photosynthetic depression effect is unknown. Here, we investigate if the spatial distribution of light absorption relative to electron transport capacity in sun- and shade-grown sunflower (Helianthus annuus) leaves underlies its previously observed diffuse light photosynthetic depression. Using a new one-dimensional porous medium finite element gas-exchange model parameterized with light absorption profiles, we found that weaker penetration of diffuse versus direct light into the mesophyll of sun-grown sunflower leaves led to a more heterogenous saturation of electron transport capacity and lowered its CO2 concentration drawdown capacity in the intercellular airspace and chloroplast stroma. This decoupling of light availability from photosynthetic capacity under diffuse light is sufficient to generate an 11% decline in photosynthesis in sun-grown but not shade-grown leaves, primarily because thin shade-grown leaves similarly distribute diffuse and direct light throughout the mesophyll. Finally, we illustrate how diffuse light photosynthetic depression could overcome enhancement in canopies with low light extinction coefficients and/or leaf area, pointing toward a novel direction for future research.

Numerical Simulation that Provides Non-oscillatory Solutions for Porous Medium Equation and Error Approximation of Boussinesq's Equation

So far, so many works have been done in a different way to find the exact track that can grasp the true solution for the one-dimensional porous media equation (PME). For instance, Monika studied using relaxation [1], Q. Zhang and Z. Wu. have already done the similar work by local degenerate Galarkin (LDG) method [12] and so on. Still, this is a challenge to find an appropriate scheme that can track the true solution when adiabatic exponent increases monotonically. In this paper, we have studied numerical result for where we have used Explicit-Implicit Finite Difference Method (EIFDM). Since so far PME is a degenerate parabolic equation and analytically the existence and uniqueness occur weakly only in the Sobolev sense, it is very hard to track the true solution numerically. Our main objective is to study numerically the PME with mixed boundary conditions and shown the result is helpful to track the Barenblatt's self similar solution and its interface when adiabatic exponent larger than 3 that provides much less error. This paper will show a possibility that Finite Difference Method (FDM) is also helpful rather the Finite Elements Method to track the interface in the simulation with an appropriate initial guess. Also checked L1, L2 and L∞-error for Boussinesq's equation which is a fundamental equation of ground- water flow, hopefully, the simulated results can help when this equation is useful in the practical world. Finally, all studied results are given to show the advantage of the θ-scheme method in the simulation of the PME and its capability to capture accurately sharp interfaces without oscillation.

Entropy-dissipating semi-discrete Runge–Kutta schemes for nonlinear diffusion equations

Abstract. Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porousmedium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.

Band Fluctuation for the Average Temperature Describing a Front Wave Propagating in a Random Porous Medium

In this work, we have formulated a stochastic model to describe the dynamics of a wave of heat, which propagates through an inhomogeneous one-dimensional porous medium. Using a previously studied analytic but deterministic model for describing a combustion tube, we have stochastically extended it. To represent the undetermined heterogeneity of the transport properties of the medium, we have allowed simultaneously the diffusive and convective terms to be randomly functions of the position. We have calculated analytically the two first momenta of the spatially distributed temperature profile, which have permitted us to discern the effects of randomness on both the expected value of temperature profiles and its corresponding standard deviation. Our results have shown that it is possible to estimate characteristic periodic lengths associated to the spatial variations of both the convective and diffusive properties. Finally, in the limit case when the noise intensity is null, we have consistently recovered the deterministic model from which we have departed.

Dynamics of water imbibition through paper channels with wax boundaries

Although paper-based microfluidic devices have been of great interests because of their advantages in terms of manufacturing costs and simplicity, the dynamics of liquid imbibition through a porous medium has not entirely been understood. Washburn equation describes liquid imbibition with respect to time through a one-dimensional porous medium, but it has recently been reported that Washburn equation is not valid for paper channels with wax boundaries. Here, we present the results of a combined experimental and theoretical investigation of water imbibition through paper channels with wax boundaries. We propose a model that explains how hydrophobic channel boundaries affect the imbibition dynamics in the channels. The model was tested by comparing experimental results for paper channels with various forms of boundaries. Our results suggest simple ways to control the flow speeds in paper-based devices through the design of the hydrophobic boundaries.