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We investigate transient heat and solute transfers in liquid-saturated porous media. The macroscopic equivalent models are obtained by a homogenization process from the pore-scale description. The large value of the Lewis number in liquid mixtures introduces a possible separation of scales between the heat and solute diffusion wavelengths \(L^{\mathrm{heat}}\) and \(L^{\mathrm{solute}}\) at a given excitation time. Thus, two separations of scales could be present between the pore scale of characteristic length l and \(L^{\mathrm{heat}}\) and \(L^{\mathrm{solute}}\). Depending on the ratio of these two separations of scales results in different equivalent macroscopic models or in non-homogenizable situations. Among numerous possibilities, we present two different homogenizable problems and a non-homogenizable situation. As the characteristic excitation time is decreased, the solute pseudo-wavelength and the heat pseudo-wavelength increase; a unique separation of scales between both solute and heat transfers and the pore scale appears, and we recover the classical macroscopic thermodiffusion model.

The flow induced above an impermeable membrane undergoing orthogonal linear stretching beneath a quiescent fluid-saturated porous medium is investigated. The problem is governed by two parameters, a stretching ratio $$\alpha $$ and a dimensionless permeability parameter $$\beta $$ . Numerical solutions of the nonlinearly coupled governing ordinary differential equations are given for selected values of $$\beta $$ over a range of $$\alpha $$ and also for the case of radial stretching $$\alpha = 1$$ over a range of $$\beta $$ . Results are plotted for the wall shear stresses and the velocity induced into the boundary layer from the far field. Asymptotics for large $$\alpha $$ at $$\beta = 0$$ and large $$\beta $$ at $$\alpha = 1$$ are presented. Sample velocity profiles at $$\beta = 0,2$$ are presented as are profiles for a radially stretching disk at $$\alpha = 1$$ .

Knowledge of colloid straining mechanism in porous media is of importance for protecting groundwater from being contaminated by biocolloids (e.g., bacteria and protozoa) and by contaminants whose transport can be facilitated by mobile particles. This study examined effects of flow velocity on colloid straining in porous media under unfavorable chemical conditions. Saturated column experiments were conducted using glass beads as collector and a $$3\,\mu \text{ m}$$ carboxylate-modified polystyrene latex microsphere as model colloid. To unambiguously examine colloid straining mechanisms, attachment was minimized by extensively cleaning the collectors and adopting deionized water as solution. Results show that increasing flow velocity decreases colloid straining under unfavorable chemical conditions, in agreement with to theoretical finding in literature. This study additionally examined effects of nonionic surfactant (Triton X-100) on colloid straining in porous media under unfavorable chemical conditions. Results show that the addition of Triton X-100 decreases colloid straining and the decrease is enhanced by increasing the concentration of Triton X-100.

We study numerically and experimentally the displacement of a liquid by a gas in a two-dimensional model porous medium. In contrast with previous pore network studies on drainage in porous media, the gas pressurization is fully taken into account. The influence of the gas injection rate on the displacement pattern, breakthrough time and the evolution of the pressure in the gas phase due in part to gas compressibility are investigated. A good agreement is found between the simulations and the experiments as regards the invasion patterns. The agreement is also good on the drainage kinetics when the dynamic liquid films are taken into account.

The effect of vertical throughflow on the onset of convection in a rectangular box occupied by a saturated porous medium uniformly heated from below, is studied using linear stability theory. It is found that, for small values of the throughflow, the stabilizing effect of the throughflow and the stabilizing effect of the confining lateral walls of the box are approximately independent of each other.

Bacterial transport is heavily influenced by chemical gradients and interfaces that exist in the subsurface. The main aim of this article is to describe a method of simulating the propagation of a traveling bacterial wave in a contaminated region and the resulting degradation of the contaminant. The presence of the chemotactic term and the relatively small bacterial diffusion means that the wave contains a very sharp wavefront. We, therefore, use an upwind conservative numerical scheme to obtain accurate and numerically stable solutions. The accuracy of the method is verified by comparisons with an exact one-dimensional solution of a simplified problem to give the same wavespeed. The method is then used to simulate the propagation of a realistic chemotactic wave in one dimension. We then use adaptive mesh refinement (AMR) to compute the propagation of chemotactic waves in two dimensions using the simplified model calibrated to give the same wavespeed as the full model.

A large number of papers are adopting the Beavers and Joseph (BJ) condition (Beavers and Joseph, J Fluid Mech 30(1):197–207, 1967) for describing the boundary condition between a saturated porous medium and a free fluid, in place of the adherence condition. The aim of the paper is to bring some insight into the domain of validity of the BJ condition. After a short review of some papers on the subject, we point out that the experimental conditions of BJ do not show a good separation of scales. That makes the BJ condition not transposable to different macroscopic situations. When the separation of scales is good, an intrinsic boundary condition is obtained by using the homogenization technique of multiple scale asymptotic expansions. As in the BJ condition and other theoretical works, e.g., Jager and Mikelic (Transp Porous media, 2009, to appear) we obtain the adherence condition of the free fluid at the first order approximation. However, the corrector to the adherence condition is \({\mathcal{O}(\varepsilon^2)}\) whereas it is \({\mathcal{O}(\varepsilon)}\) in the BJ condition, where e is the separation of scales parameter.

The effect of the connectivity of high-conductivity elements on the value the effective hydraulic conductivity of a saturated porous medium is investigated. The increase in effective conductivity due to the connectivity is estimated on the basis of a simple combinatorial model.

The effect of heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium is discussed. The distinction between moderate and high heterogeneity of permeability is emphasized. Topics discussed include the limitations of the use of the log–normal distribution for permeability, the different effects of vertical and horizontal heterogeneity, and the use of an electrical resistor analogy applied to an illustrative quartered-square example. In a postscript, a new approximate criterion for instability applicable to the case of strong heterogeneity is proposed.

The problem of fully developed forced convection in a parallel-plates channel filled with a saturated porous medium (involving a Brinkman model for the momentum equation), with the effect of viscous dissipation (involving a Brinkman number), is discussed. Some general matters relating to the possibility of fully developed convection are also discussed.

In this note the problem of the onset of bioconvection in a horizontal layer occupied by a saturated porous medium is analyzed. Gyrotactic effects are incorporated in the analysis. The Darcy flow model is employed, and it is assumed that the bioconvection Peclet number is not greater than unity. Critical values of the bioconvection Rayleigh number and the corresponding critical Rayleigh number are obtained for various values of the bioconvection Peclet number, the gyrotaxis number and the cell eccentricity.

Despite a large number of publications on bioconvection in suspensions of motile microorganisms, bioconvection in a fluid saturated porous medium is a relatively new area of research. This paper is motivated by experimental research by Kessler (1986) who established that a porous medium prevents the development of convection instability in algal suspensions. This suggests that there may exist a critical value of the permeability of a porous medium. If the permeability is smaller than critical, the system is stable and bioconvection does not develop. If the permeability is larger than critical, bioconvection may develop. This paper presents a model of bioconvection of gyrotactic motile microorganisms in a fluid saturated porous medium. The focus of this research is the determination of the critical value of permeability of a porous medium by a linear stability analysis. A simple but elegant analytical solution for the critical Darcy number is obtained.

Groundwater contamination by dense nonaqueous phase liquids (DNAPLs) has received increasing attention in the last decade. The fingering process of DNAPL migration in porous me- dia remains an incompletely understood subject. The main reason is that natural porous media are opaque and hence very few visualizations are available. This paper presents the visual results of two-dimensional experiments in a glass tank in which DNAPL penetrated into water-saturated homogeneous porous media. The results provide a clear reference for conceptual models of DNAPL finger development due to immiscible flow instabilities. The fingering process can be divided into two stages, that is, the finger initiation stage and the finger elongation stage. At the finger initiation stage, many DNAPL protuberances appear at the interface between tetrachloroethylene (PCE) and water along the surface of the porous media. During the finger elongation stage, some relatively larger protuberances develop into primary fingers. Secondary fingers may develop on the existing primary fingers. The fingers grew downwardly in a winding manner with the mechanisms of shielding, tip- growing, splitting, and coalescing. The fingers grew linearly with time and had similar growth rate at the finger elongation stage. The average wavelength (space between fingers) of the primary fingers was 0.051 m and the average PCE content in the region with fingers was 2.5% (7.0% saturation).

In this work we develop the volume averaged form of the frequency-dependent governing equations for electrohydrodynamics in a saturated porous medium. The concept of local electrical equilibrium is identified, and when this condition is valid we obtain a one-equation model describing the coupled transport of momentum and electric charge. When local electrical equilibrium is not valid, separate forms of Maxwell's equations must be developed for both the fluid and solid phases.

Investigation suggests that certain assumptions must be implicitly made whenever the wellknown and much admired Richards-Hassler methodology (for investigating flow phenomena in multiphase saturated porous media) is being employed. Whenever the propositions that are being taken for granted are not fully stated, however, they can take on the appearance of being presumptions. The analysis presented in this essay, therefore, is intended to clarify the differences between what historically has been said, and what still needs to be said, about the rationale that underlies the sense of the always intriguing but sometimes misunderstood Richards-Hasslet conceptualizations.

The macroscopic description of the dynamics of two immiscible fluids flowing through a deformable porous medium is obtained from the description of the pore scale using the homogenization theory of periodic structures. The result is a generalization of the description of saturated porous media with a BIOT structure. The homogenization process permits the complete determination of the effective parameters and the clarification of the couplings between the different phases. Two simple examples are presented for the generalized Darcy flows. Special care is taken concerning the significance of the volume averaged stress, as provided by the homogenization process. It is shown that the physical stress is different from the volume averaged stress.

It is shown that, on the Brinkman model, spin-up is confined to boundary layers whose thickness is of order k 1/2, and the spin-up is established in a time of order ϱk/eμ, where k, ϱ, e and μ denote permeability, density, porosity and dynamic viscosity, respectively.