Recovering the Diffusion Coefficient in the Porous Medium Equation from a Large-Time Measurement

An efficient numerical parameter estimation scheme for the space-dependent dispersion coefficient of a solute transport equation in porous media

The porous medium equation (PME) $$ {{\text{u}}_{\text{t}}} = \Delta ({{\text{u}}^{\text{m}}}),\,{\text{m>1}} $$ is one of the simplest models of nonlinear diffusion equations. It arises naturally in the study of a number of problems describing the evolution of a continuous quantity subject to a nonlinear diffusion mechanism, which we can instance explain as caused by a diffusion coefficient of the form $$ {\text{c(u) = m}}{{\text{u}}^{{{\text{m - 1}}}}} $$ (1.1) if we write the PME as ut = div(c(u) ∇u). Among the applications of the PME have (i) Percolation of gas through porous media, where m ≥ 2 [M], (ii) Radiative heat transfer in ionized plasmas, where m ≃ 6 [ZR], (iii) Thin liquid films spreading under gravity, where m = 4 [Bu], (iv) Crowd-avoiding population spreading, where m>1 [GM].

Global warming and reducing fossil fuel resources have increased the interest in using renewable resources such as geothermal energy. In this paper, in the first step, heat transfer equations have been presented for reservoir during water (steam) injection by considering heat loss to adjacent formations. According to radius of thermal front, the reservoir is partitioned into two regions with different fluid physical properties. The heat transfer model is coupled with a fluid flow model which is used to calculate the reservoir pressure or fluid flow rates. Then by calculating outer radius of heated region and using radial composite reservoir model, the fluid flow equations in porous media are solved. Using pressure derivative plot in regions with different thermal conductivity coefficients, a type curve plot is presented. The reservoir and adjacent formation thermal conductivity coefficients can be calculated by matching the observed pressure data on the thermal composite type curve. Additionally, the interference test in composite geothermal reservoir is discussed. In the composite reservoir model, parameters such as diffusivity coefficient, conductivity ratio and the distance to the radial discontinuity are considered. New type curves are provided to introduce the effect of diffusivity/conductivity contrast ratios on temperature behavior. Improving interpretations, and performing fast computations and fast sensitivity analysis are the benefits of the presented solutions.

SUMMARYThe paper examines ion (chloride) transport equations in porous media (concrete) integrated over a representative elementary volume, that is to say, averaging over the macroscopic level the phenomena that occur really at the pore scale. There are three basic variables to be used: chloride concentration, moisture and temperature. The diffusion process is examined, in addition to other phenomena such as convection (the motion of dissolved substances caused by flow of water in a pore solution of partially saturated media) or chloride binding (the capacity of free chloride of being chemically bound, particularly with C3A to form Friedel salts). Contrary to other approaches, such effects are not considered by means of apparent diffusion coefficients but by developing the complete set of time‐dependent equations for both the chloride concentration within the pore solution and the moisture content within the pore space.Once the general model is described, the system of equations can be solved numerically by means of a two‐dimensional finite element formulation. The main objective is to reproduce results of experimental tests by means of a priori parameter estimation, according to the characteristics of materials and external environment conditions, thereby superseding the well‐known best fit a posteriori through Fick's second equation.While the introduction of hygrometric conditions and convection phenomena appears to be of high significance, other factors like temperature, surface concentration, chloride binding or equivalent hydration time are analysed too. The proposed model can reproduce bidimensional complex geometries, for example, cracked concrete cover, as well as variable surface condition. An application case is developed through a realistic model of the geometry of a crack. Copyright © 2014 John Wiley & Sons, Ltd.

The article assesses the fire barrier capability of a glass-reinforced intumescent flame retardant system under burn-through conditions. The material is a glass fiber-reinforced intumescent flame retardant polypropylene. The experimental evaluation involved a burn-through test close to the UL2596 standardized test (without grit) for assessing Electrical Vehicle battery protective materials under thermal runaway. The backside time/temperature curves demonstrated the ability of the intumescent material to withstand for 30 min without burn-through. Numerically, a three-dimensional pyrolysis model, including mass and energy conservation principles, heat transfer equations in porous media, and a moving boundary condition (expansion rate) to simulate the thermal expansion of an intumescent material, was implemented. The comparison between the model and experiments showed a good prediction. A parametric study on critical parameters, including emissivity, convective heat coefficient, gas mass diffusion coefficients, and flame heat flux, provided a coherent description of the barrier properties of the material as analyzed by the numerical model.

ABSTRACTSulfur dioxide uptake by pre‐peeled potatoes from dipping solutions was mathematically modeled. Diffusive mass transfer equations in porous medium were experimentally verified; residual levels of sulfur dioxide were measured in the range of industrial operating conditions. Effects of sodium bisulfite solution concentration, immersion time, size, shape, dry matter, density and velocity of the product were analyzed. Three geometric approximations to pre‐peeled potato cuts were examined (spheres, cubes and parallelepipeds). The fitting of equations to experimental data determined the effective diffusion coefficient of sulfur dioxide in potato tissue, which was compared to theoretical predictions in terms of molecular diffusivity, total solids content and tortuosity factor.

We present a theoretical description of the generation of ultra‐short, high‐energy pulses in two laser cavities driven by periodic spectral filtering or dispersion management. Critical in driving the intra‐cavity dynamics is the nontrivial phase profiles generated and their periodic modification from either spectral filtering or dispersion management. For laser cavities with a spectral filter, the theory gives a simple geometrical description of the intra‐cavity dynamics and provides a simple and efficient method for optimizing the laser cavity performance. In the dispersion managed cavity, analysis shows the generated self‐similar behavior to be governed by the porous media equation with a rapidly‐varying, mean‐zero diffusion coefficient whose solution is the well‐known Barenblatt similarity solution with parabolic profile.

We analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our investigation by considering the linear case and the nonlinear case afterward. The linear case is discussed taking fractional time and spatial derivatives into account in a unified approach. We also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by D(x,t)=D(t)|x|−θ. For the nonlinear case, we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation.

In recent years, research and development in nanoscale science and technology have grown significantly, with electrical transport playing a key role. A natural challenge for its description is to shed light on anomalous behaviours observed in a variety of low-dimensional systems. We use a synergistic combination of experimental and mathematical modelling to explore the transport properties of the electrical discharge observed within a micro-gap based sensor immersed in fluids with different insulating properties. Data from laboratory experiments are collected and used to inform and calibrate four mathematical models that comprise partial differential equations describing different kinds of transport, including anomalous diffusion: the Gaussian Model with Time Dependent Diffusion Coefficient, the Porous Medium Equation, the Kardar-Parisi-Zhang Equation and the Telegrapher Equation. Performance analysis of the models through data fitting reveals that the Gaussian Model with a Time-Dependent Diffusion Coefficient most effectively describes the observed phenomena. This model proves particularly valuable in characterizing the transport properties of electrical discharges when the micro-electrodes are immersed in a wide range of insulating as well as conductive fluids. Indeed, it can suitably reproduce a range of behaviours spanning from clogging to bursts, allowing accurate and quite general fluid classification. Finally, we apply the data-driven mathematical modeling approach to ethanol-water mixtures. The results show the model's potential for accurate prediction, making it a promising method for analyzing and classifying fluids with unknown insulating properties.

Fractional and nonlinear diffusion equation: additional results

We analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions. In particular, the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation. We also discuss some implications by considering a diffusion coefficient D(x,t)=D(t)/x/(-theta) (theta in R) and a drift force F=-k(1)(t)x+k(alpha)x/x/(alpha-1). In both situations, we relate our solutions to those obtained within the maximum entropy principle by using the Tsallis entropy.

In recent years, research and development in nanoscale science and technology have grown significantly, with electrical transport playing a key role. A natural challenge for its description is to shed light on anomalous behaviours observed in a variety of low-dimensional systems. We use a synergistic combination of experimental and mathematical modelling to explore the transport properties of the electrical discharge observed within a micro-gap based sensor immersed in fluids with different insulating properties. Data from laboratory experiments are collected and used to inform and calibrate four mathematical models that comprise partial differential equations describing different kinds of transport, including anomalous diffusion: the Gaussian Model with Time Dependent Diffusion Coefficient, the Porous Medium Equation, the Kardar-Parisi-Zhang Equation and the Telegrapher Equation. Performance analysis of the models through data fitting reveals that the Gaussian Model with a Time-Dependent Diffusion Coefficient most effectively describes the observed phenomena. This model proves particularly valuable in characterizing the transport properties of electrical discharges when the micro-electrodes are immersed in a wide range of insulating as well as conductive fluids. Indeed, it can suitably reproduce a range of behaviours spanning from clogging to bursts, allowing accurate and quite general fluid classification. Finally, we apply the data-driven mathematical modeling approach to ethanol-water mixtures. The results show the model's potential for accurate prediction, making it a promising method for analyzing and classifying fluids with unknown insulating properties.

We study nonlinear time-inhomogeneous Markov processes in the sense of McKean’s (Proc Natl Acad Sci USA 56(6):1907–1911, 1966) seminal work. These are given as families of laws Ps,ζ, s≥0, on path space, where ζ runs through a set of admissible initial probability measures on Rd. In this paper, we concentrate on the case where every Ps,ζ is given as the path law of a solution to a McKean–Vlasov stochastic differential equation (SDE), where the latter is allowed to have merely measurable coefficients, which in particular are not necessarily weakly continuous in the measure variable. Our main result is the identification of general and checkable conditions on such general McKean–Vlasov SDEs, which imply that the path laws of their solutions form a nonlinear Markov process. Our notion of nonlinear Markov property is in McKean’s spirit, but more general in order to include processes whose one-dimensional time marginal densities solve a nonlinear parabolic partial differential equation, more precisely, a nonlinear Fokker–Planck–Kolmogorov equation, such as Burgers’ equation, the porous media equation and variants thereof with transport-type drift, and also the very recently studied two-dimensional vorticity Navier–Stokes equation and the p-Laplace equation. In all these cases, the associated McKean–Vlasov SDEs are such that both their diffusion and drift coefficients singularly depend (i.e., Nemytskii type) on the one-dimensional time marginals of their solutions. We stress that for our main result the nonlinear Fokker–Planck–Kolmogorov equations do not have to be well posed. Thus, we establish a one-to-one correspondence between solution flows of a large class of nonlinear parabolic PDEs and nonlinear Markov processes.

We present a theoretical description of the generation of ultra-short, high-energy pulses in two laser cavities driven by periodic spectral filtering or dispersion management. Critical in driving the intra-cavity dynamics is the nontrivial phase profiles generated and their periodic modification from either spectral filtering or dispersion management. For laser cavities with a spectral filter, the theory gives a simple geometrical description of the intra-cavity dynamics and provides a simple and efficient method for optimizing the laser cavity performance. In the dispersion managed cavity, analysis shows the generated self-similar behavior to be governed by the porous media equation with a rapidly-varying, mean-zero diffusion coefficient whose solution is the well-known Barenblatt similarity solution with parabolic profile.

This work deals with a continuation method for computing solutions to a self-similar two-component Stefan system in which the diffusion coefficients depend on the concentrations. The procedure computes a one-parameter homotopy connecting the known solution of a simplified problem (when the parameter is zero) to the solution of the problem at hand (when the parameter is one). Local convergence of the method and local existence and uniqueness of solutions for the original system are proven. Also, several examples coming from precipitant-driven protein crystal growth are discussed. The most interesting of these is a Stefan problem containing a porous media equation that corresponds to the liquid phase being in a meta-stable state near the spinodal region. The bifurcation code AUTO is used in the computations.