Boris S. Maryshev

Non Fickian flux for advection-dispersion with immobile periods

The fractal mobile–immobile model (MIM) is intermediate between advection–dispersion (ADE) and fractal Fokker–Planck (FFKPE) equations. It involves two time derivatives, whose orders are 1 and γ (between 0 and 1) on the left-hand side, whereas all mentioned equations have identical right-hand sides. The fractal MIM model accounts for non-Fickian effects that occur when tracers spread in media because of through-flow, and can get trapped by immobile sites. The solid matrix of a porous material may contain such sites, so that non-Fickian spread is actually observed. Within the context of the fractal MIM model, we present a mapping that allows the computation of fluxes on the basis of the density of spreading particles. The mapping behaves as Fickian flux at early times, and tends to a fractional derivative at late times. By means of this mapping, we recast the fractal MIM model into conservative form, which is suitable to deal with sources and bounded domains. Mathematical proofs are illustrated by comparing the discretized fractal p.d.e. with Monte Carlo simulations.

Adjoint state method for fractional diffusion: Parameter identification

Fractional partial differential equations provide models for sub-diffusion, among which the fractal Mobile-Immobile Model (fMIM) is often used to represent solute transport in complex media. The fMIM involves four parameters, among which we have the order of an integro-differential operator that accounts for the possibility for solutes to be sequestered during very long times. To guess fMIM parameters from experiments, an accurate method consists in optimizing an objective function that measures how much model solutions deviate from data. We show that solving an adjoint problem helps accurate computing of the gradient of such an objective function, with respect to the parameters. We illustrate the method by applying it on experimental data issued from tracing tests in porous media.

Identifying Space-Dependent Coefficients and the Order of Fractionality in Fractional Advection–Diffusion Equation

Tracer tests in natural porous media sometimes show abnormalities that suggest considering a fractional variant of the advection–diffusion equation supplemented by a time derivative of non-integer order. We are describing an inverse method for this equation: It finds the order of the fractional derivative and the coefficients that achieve minimum discrepancy between solution and tracer data. Using an adjoint equation divides the computational effort by an amount proportional to the number of freedom degrees, which becomes large when some coefficients depend on space. Method accuracy is checked on synthetical data, and applicability to actual tracer test is demonstrated.

Stability of uniform vertical flow through a close porous filter in the presence of solute immobilization

Abstract.In the present paper we consider slow filtration of a mixture through a close porous filter. The heavy solute penetrates slowly into the porous filter due to the external vertical filtration flow and diffusion. This process is accompanied by the formation of the domain with heavy fluid near the upper boundary of the filter. The developed stratification, at which the heavy fluid is located above the light fluid, is unstable. When the mass of the heavy fluid exceeds the critical value, one can observe the onset of the Rayleigh-Taylor instability. Due to the above peculiarities we can distinguish between two regimes of vertical filtration: 1) homogeneous seepage and 2) convective filtration. When considering the filtration process it is necessary to take into account the diffusion accompanied by the immobilization effect (or sorption) of the solute. The immobilization is described by the linear MIM (mobile/immobile media) model. It has been shown that the immobilization slows down the process of forming the unstable stratification. The purpose of the paper is to find the stability conditions for homogeneous vertical seepage of he solute into the close porous filter. The linear stability problem is solved using the quasi-static approach. The critical times of instability are estimated. The stability maps are plotted in the space of system parameters. The applicability of quasi-static approach is substantiated by direct numerical simulation of the full nonlinear equations.Graphical abstract

Laplace-Transform Based Inversion Method for Fractional Dispersion

Partial differential equations with memory are challenging models for mass transport in porous media where fluid and tracer may be stored by the solid matrix, and then released. Moreover, integral transforms (generalizing time moments) of solutions to such models are linked to the corresponding transport parameters. Inverting that link provides a method to determine model parameters on the basis of solutions. It is checked using numerically generated profiles before passing to experimental data.

The Effect of Solute Immobilization on a Stability of a Diffusion Front in Porous Media Under Gravity Field

In the present paper, we consider the effect of solute immobilization on a stability of planar diffusion front in gravity field. The case when heavy admixture diffuses from upper boundary into a bulk of semi-infinite domain of porous medium is investigated. In this situation, the base state formed by the diffusion corresponds to the layer of heavier liquid located above the lighter one. This unsteady base state is potentially unstable to the Rayleigh–Taylor fingering instability of front. The instability conditions are examined using the quasi-static method within the fMIM diffusion model. It is found that the immobilization results in the increase in the critical time for instability significantly, whereas the effect of immobilization on the critical wave number is weak. The stability maps in the parameter space are obtained.

Two-dimensional thermal convection in porous enclosure subjected to the horizontal seepage and gravity modulation

Coupled effect of horizontal seepage and gravity modulation on the onset and nonlinear regimes of two-dimensional thermal buoyancy convection in horizontal fluid-saturated porous cylinder of rectangular cross section with perfectly conductive boundaries is studied. It is shown that gravity modulation makes destabilizing effect. Null-dimensional dynamical system describing supercritical convective regimes is derived. Conditions for existence of stable periodical regimes are defined. It is found that system demonstrates dynamics on a torus which can be either resonant or non-resonant depending on the parameters. Synchronization domains which correspond to the resonant torus existence in the parameter space are determined by the rotation number technique. It is found that at certain values of the ratio of the cross section height to width, degeneracy takes place. In this case different stable periodical regimes forming one-parametric family coexist. Linear stability and nonlinear dynamics of the system at fi...

The Effect of Sorption on Linear Stability for the Solutal Horton–Rogers–Lapwood Problem

This investigation is devoted to linear stability analysis within the solutal analog of Horton–Rogers–Lapwood (HRL) problem with sorption of solid particles. The solid nanoparticles are treated as solute within the continuous approach. Therefore, the infinite horizontal porous layer saturated with mixture (carrier fluid and solute) is considered. The solute concentration difference between the layer boundaries is assumed as constant. The solute sorption is simulated in accordance with the linear mobile/immobile media model. In the first part, the instability of steady net mass flow through this layer is studied analytically. The critical values of parameters have been found. It is known that for the HRL problem the seepage makes the critical mode oscillatory, but the stability threshold remains unchanged. In contrast, if the sorption is taken into account, the stability threshold varies. In the latter case, the critical value of solutal Rayleigh–Darcy number increases versus that for the standard HRL problem. The second part is devoted to investigation of instability for modulated in time horizontal net mass flow. The ordinary differential equation has been obtained for description of the behavior near the convection instability threshold. This equation is analyzed numerically by the Floquet method; the parametric excitation of convection is observed.

Effect of solute immobilization on the stability problem within the fractional model in the solute analog of the Horton-Rogers-Lapwood problem

Abstract.The paper is devoted to the linear stability analysis within the solute analogue of the Horton-Rogers-Lapwood (HRL) problem. The solid nanoparticles are treated as solute within the continuous approach. Therefore, we consider the infinite horizontal porous layer saturated with a mixture (carrier fluid and solute). Solute transport in porous media is very often complicated by solute immobilization on a solid matrix of porous media. Solute immobilization (solute sorption) is taken into account within the fractal model of the MIM approach. According to this model a solute in porous media immobilizes within random time intervals and the distribution of such random variable does not have a finite mean value, which has a good agreement with some experiments. The solute concentration difference between the layer boundaries is assumed as constant. We consider two cases of horizontal external filtration flux: constant and time-modulated. For the constant flux the system of equations that determines the frequency of neutral oscillations and the critical value of the Rayleigh-Darcy number is derived. Neutral curves of the critical parameters on the governing parameters are plotted. Stability maps are obtained numerically in a wide range of parameters of the system. We have found that taking immobilization into account leads to an increase in the critical value of the Rayleigh-Darcy number with an increase in the intensity of the external filtration flux. The case of weak time-dependent external flux is investigated analytically. We have shown that the modulated external flux leads to an increase in the critical value of the Rayleigh-Darcy number and a decrease in the critical wave number. For moderate time-dependent filtration flux the differential equation with Caputo fractional derivatives has been obtained for the description of the behavior near the convection instability threshold. This equation is analyzed numerically by the Floquet method; the parametric excitation of convection is observed.Graphical abstract