Two-dimensional Porous Media Research Articles

Well-posedness of the Muskat problem in subcritical Lp-Sobolev spaces

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

Experimental study of the characteristics of gas-injection barrier in two-dimensional porous media

The inflow of groundwater into the oil shale in-situ pyrolysis zone reduces the heating efficiency and has negative impacts on soil ecology and drinking water safety. Thus, we propose a sealing system for oil shale in-situ pyrolysis exploitation with a gas-injection barrier. In this study, the characteristics of gas-injection barrier were investigated comprehensively in the two-dimensional porous media. First, the evolution and distribution of hydraulic conductivity were studied. Results showed that gas injection promoted water flow in the central subregion when the gas flow rate was less than 50 mL/min. With a further increase in the gas flow rate, a significant inhibition effect on the water flow within all subregions was achieved. Then, the gas-injection barrier mechanism was quantitatively analyzed, showing that a pressure-balanced state in the extended front of the upstream gas flow served as one mechanism when a larger gas flow rate was reached, which was verified by experiments for the first time. Finally, the distribution of the gas flow fitted by a Gaussian curve was explored. In addition, the sensitivity of hydraulic conductivity was discussed; with increasing water flow rate and the coarsening of glass beads, the relative hydraulic conductivity improved comprehensively. All results obtained are conducive to a comprehensive understanding and application of the gas-injection barrier.

Uniqueness of solution of a heterogeneous evolution dam problem associated with a compressible fluid flow through a rectangular porous medium

This paper is concerned with an uniqueness of solution of the weak formulation of an evolution dam problem related to a compressible fluid flow through a two-dimensional, rectangular and heterogeneous porous medium. Note that our problem associated with the equation a(x_1)(u_{x_2}+\chi)_{x_2}-(u+\chi)_t=0. Our technique is based on the idea that we transform the weak form of this equation into a similar situation to the proof of the uniqueness in the incompressible case (see [12]). It is also difficult to adapt the proof obtained in [12] by using some properties of the solutions as in [12, Sect. 2].

Mathematical modelling of the agglomeration in a reactive porous medium with variable permeability

When processing low-grade fuels, such as waste and biomass, the problems associated with bed agglomeration often arise. In this work, one of the variants of the agglomeration model is proposed, in which local permeability decreases as a result of the physicochemical process caused by heating, and the agglomeration centers (particles of the melting material) are randomly distributed in a two-dimensional porous medium. A relatively simple model allows the study of the development of thermohydrodynamic inhomogeneities in the fixed granular bed and evaluation of its hydraulic resistance at different fractions of melting material.

Analysis of thermal responses in a two-dimensional porous medium caused by pulse heat flux

In this article, the generalized model for thermoelastic waves with two relaxation times is utilized to compute the increment of temperature, the displacement components, the stress components, and the changes in the volume fraction field in a two-dimensional porous medium. By using the Fourier-Laplace transform and the eigenvalue method, the considered variables are obtained analytically. The derived approach is estimated with numerical outcomes which are applied to the porous media with a geometrical simplification. The numerical results for the considered variables are performed and presented graphically. Finally, the outcomes are represented graphically to display the difference among the classical dynamical (CD) coupled, the Lord-Shulman (LS), and the Green-Lindsay (GL) models.

Rayleigh–Taylor instabilities in miscible fluids with initially piecewise linear density profiles

The stability of some simple density profiles in a vertically orientated two-dimensional porous medium is considered. The quasi-steady-state approximation is made so that the stability of the system can be approximated. As the profiles diffuse in time, the instantaneous growth rates evolve in time. For an initial step function density profile, the instantaneous growth rate was numerically found to decay like $$T^{-1/2}$$ for large times T, and the corresponding eigenfunctions scale with $$\mathrm{e}^{\omega \sqrt{T}}$$ where $$\omega $$ is a constant. For density profiles initially corresponding to a finite layer, the instantaneous growth rate eventually decayed like $$T^{-1}$$. This corresponds to an instability with algebraic growth, and the eigenfunctions scale with $$T^p$$ (where p is a constant) for large time. For a species initially linearly distributed in a finite layer, when the concentration has an increasing gradient in the downwards direction, the stability of the system was similar to that found for a uniformly distributed finite layer. However, when the concentration had a decreasing gradient in the downwards direction, the growth rates remained constant for a long period time, but eventually decayed in the same way as found in a uniformly distributed finite layer, for very large times. Numerical simulations were performed to validate the predictions made by the linear stability analysis.

Generalized thermoelastic interaction in a two-dimensional porous medium under dual phase lag model

Purpose The purpose of this study is to use the generalized model for thermoelastic wave under the dual phase lag (DPL) model to compute the increment of temperature, the components of displacement, the changes in volume fraction field and the stress components in a two-dimensional (2D) porous medium. Design/methodology/approach Using Fourier and Laplace transformations with the eigenvalue technique, the exact solutions of all physical quantities are obtained. Findings The derived method is evaluated with numerical results, which are applied to the porous medium in a simplified geometry. Originality/value Finally, the outcomes are graphically represented to show the difference among the models of classical dynamical coupled, the Lord and Shulman and DPL.

Onset of Darcy–Bénard convection under throughflow of a shear-thinning fluid

We present an investigation on the onset of Darcy–Benard instability in a two-dimensional porous medium saturated with a non-Newtonian fluid and heated from below in the presence of a uniform horizontal pressure gradient. The fluid is taken to be of power-law nature with constant rheological index and temperature-dependent consistency index . A two-dimensional linear stability analysis in the vertical plane yields the critical wavenumber and the generalised critical Rayleigh number as functions of dimensionless problem parameters, with a non-monotonic dependence from and with maxima/minima at given values of , a parameter representing the effects of consistency index variations due to temperature. A series of experiments are conducted in a Hele-Shaw cell of aspect ratio to provide a verification of the theory. Xanthan Gum mixtures (nominal concentration from 0.10 % to 0.20 %) are employed as working fluids with a parameter range and . The experimental critical wavenumber corresponding to incipient instability of the convective cells is derived via image analysis for different values of the imposed horizontal velocity. Theoretical results for critical wavenumber favourably compare with experiments, systematically underestimating their experimental counterparts by 10 % at most. The discrepancy between experiments and theory is more relevant for the critical Rayleigh number, with theory overestimating the experiments by a maximum factor less than two. Discrepancies are attributable to a combination of factors: nonlinear phenomena, possible subcritical bifurcations, and unaccounted-for disturbing effects such as approximations in the rheological model, wall slip, ageing and degradation of the fluid properties.

Thermomechanical response in a two-dimension porous medium subjected to thermal loading

PurposeThe purposes of this study, a mathematical model of generalized thermoelastic theory subjected to thermal loading is presented to study the wave propagation in a two-dimensional porous medium.Design/methodology/approachBy using Fourier–Laplace transforms with the eigenvalue approach, the physical quantities are analytically obtained.FindingsThe derived method is evaluated with numerical results, which are applied to the porous medium in simplified geometry.Originality/valueNumerical outcomes for all the physical quantities considered are implemented and represented graphically. The variations of temperature, the changes in volume fraction field, the displacement components and the stress components have been depicted graphically.

Fractional order thermoelastic wave assessment in a two-dimension medium with voids

In this article, the generalized thermoelastic theory with fractional derivative is presented to estimate the variation of temperature, the components of stress, the components of displacement and the changes in volume fraction field in two-dimensional porous media. Easily, the exact solutions in the Laplace domain are obtained. By using Laplace and Fourier transformations with the eigenvalues method, the physical quantities are obtained analytically. The numerical results for all the physical quantities considered are implemented and presented graphically. The results display that the present model with the fractional derivative is reduced to the Lord and Shulman (LS) and the classical dynamical coupled (CT) theories when the fractional parameter is equivalent to one and the delay time is equal to zero and respectively.

Optimal control over inserted point source intensity for humidification of a two-dimensional porous medium

Optimal control over inserted point source intensity for humidification of a two-dimensional porous medium

Experimental study of the dryout of a 2D heat-generating model porous medium

Dryout experiments are performed in a quasi two-dimensional heat-generating model porous medium, with bottom flooding. The test section is composed of an array of heating cylinders placed between two ceramic plates. It is held vertically and a liquid (HFE-7000) is injected from bottom at a controlled flow rate, pressure and temperature. Dryout incipient power is investigated by applying an increasing thermal power to a bundle of heating cylinders until a dry zone is detected. The liquid inlet mass fluxes range from 1.0kgm-2s-1 to 10.5kgm-2s-1 and the heat fluxes are between 195kWm-2 and 1086kWm-2. Two kinds of dryout phenomenology are observed. First, at low liquid injection rate (below 4.2kgm-2s-1 inlet mass flux), reaching the dryout power results into a liquid front receding from the top of the test section to the upper limit of the heated zone, while downstream the heated zone, the porous medium is vapour-saturated. Second, at higher flow rate (over 5.2kgm-2s-1 inlet mass flux), the boiling crisis happens at the surface of a single heating element, resulting in a local film boiling, whereas a two-phase flow still go through the whole test section. In both cases, high-speed visualizations allow characterizing 8 boiling/flow regimes, depending on the inlet mass flux, the power released by the heated zone, and the location inside the porous medium. In particular, when reaching the dryout incipient power, the existence of a pulsed flow regime is highlighted. Such a characterization of boiling/flow regimes might be helpful in improving dryout models.

Stable and unstable miscible displacements in layered porous media

The effect of permeability heterogeneities and viscosity variations on miscible displacement processes in porous media is examined using high-resolution numerical simulations and reduced theoretical modelling. The planar injection of one fluid into a fluid-saturated, two-dimensional porous medium with a permeability that varies perpendicular to the flow direction is studied. Three cases are considered, in which the injected fluid is equally viscous, more viscous or less viscous than the ambient fluid. In general it is found that the flow in each case evolves through three regimes. At early times, the flow exhibits the concentration evolves diffusively, independent of both the permeability structure and the viscosity ratio. At intermediate times, the flow exhibits different dynamics including channelling and fingering, depending on whether the injected fluid is more or less viscous than the ambient fluid, and depending on the relative magnitude of the viscosity and permeability variations. Finally, at late times, the flow becomes independent of the viscosity ratio and dominated by shear-enhanced (Taylor) dispersion. For each of the regimes identified above, we develop reduced-order models for the evolution of the transversely averaged concentration and compare them to the full numerical simulations.

A comprehensive model for the prediction of fluid compositional gradient in two-dimensional porous media

Compositional gradient can be described as changes in the composition of components both vertically and horizontally in a hydrocarbon reservoir. In the present work, two-dimensional compositional gradient in multi-component gas and oil mixtures is modeled. A thermodynamic model is developed based on molecular diffusion coefficients in mass diffusive flux. A combination of Sigmund and Bird correlations is considered to estimate molecular diffusion coefficients for gas mixtures. Implementing this comprehensive developed model on a gas condensate sample shows interesting differences not only in trends of component compositions but also in gradient magnitude. A real set of data from a supergiant gas condensate field is used to validate the model. It is perceived that the developed model reduces relative absolute errors to about 50%. In the next step, a comprehensive study was conducted to understand the cross effects of molecular diffusion and natural convection in gas condensate and volatile oil samples. Gas and oil samples are selected to investigate if natural convection has the same effects in different samples. It is observed that increase in natural convection causes to reduce horizontal and vertical composition gradients. This effect is more significant in gas reservoir, as methane composition varies by more than 5.2 mol% diagonally in gas condensate sample, whereas this value is about 0.45 mol% in volatile oil sample. Lower density and higher bulk velocity in gas sample cause more disturbances in flow streams of gas mixture. Evaluation of the developed model shows that the model is reliable for reservoir studies and management programs.

Exact steady-state solution to air pumping from underground partly covered by an impermeable tarp

A pipe is located in a semi-infinite two-dimensional porous medium partially covered by a tarp. A constant pressure gradient may be imposed at infinity on the medium while air is pumped into the pipe. The governing Laplace equation is solved by extending the famous Keldysh–Sedov formulae to doubly connected domains by reduction of the mixed problem to a Riemann–Hilbert problem with discontinuous coefficients; the solution is obtained by combining factorization and functional equations. Then, the influence of the various geometrical and flow parameters is presented and discussed.

Preliminary tests on a microfluidic device to study pore clogging during biocementation

Soil improvement using ureolytic bacteria or other biological agents is a promising technique currently under investigation. It is based on the precipitation of calcium carbonate (biocement) due to the enzymatic hydrolysis of urea. The biocement produced clogs the soil pores, consequently bonding the soil grains and increasing overall strength and stiffness while reducing permeability. This study focused mainly on pore clogging effects. The effect of the enzyme and feeding solution concentrations was studied in small test tubes to find the maximum amount of precipitate found when changing the concentrations of both. Based on it, selected concentrations of enzyme and feeding solution were tested in a microfluidic device conceived to mimic a two-dimensional uniform porous size medium. Qualitatively, the amount of precipitate was proportional to that of the concentrations used. The location of the precipitate was clearly related with the direction of fluid flow during inoculation. These preliminary results highlight the fact that the use of alternative testing devices such as the one developed is a potential tool for the study of clogging phenomena occurring during this treatment.

Non-invasive image processing method to map the spatiotemporal evolution of solute concentration in two-dimensional porous media

A color visualization-based image processing method is developed in this paper to quantify the concentration evolution of the Brilliant Blue FCF transport through a two-dimensional homogeneous porous medium. A series of images are recorded at known time intervals, then the spatial distribution is estimated using a calibration curve, linking the gray pixel value to the solute concentration. Using a multi-dimensional concentration distribution map extraction technique the longitudinal and transverse concentration distributions could be observed with the physical model. The image-processed concentrations are then compared directly with the measured concentrations sampled at the outlet end. The tracer breakthrough curves sampled at multiple points along the central line of the medium are also compared with the solutions from the standard advection–dispersion equation model. It is shown that the non-invasive image processing method may be used to map the spatiotemporal evolution of a solute’s concentration without disturbing the flow or the transport dynamics, although the measured solute breakthrough curves feature some non-Fickian dynamics that cannot be efficiently captured by the standard transport model.

Viscous displacement in porous media: the Muskat problem in 2D

We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the mathematical model as an evolution problem for the sharp interface between the fluids, we show that the Muskat problem with surface tension is a quasilinear parabolic problem, whereas, in the absence of surface tension effects, the Rayleigh–Taylor condition identifies a domain of parabolicity for the fully nonlinear problem. Based upon these aspects, we then establish the local well-posedness for arbitrary large initial data in H s H^s , s > 2 s>2 , if surface tension is taken into account, respectively for arbitrary large initial data in H 2 H^2 that additionally satisfy the Rayleigh–Taylor condition if surface tension effects are neglected. We also show that the problem exhibits the parabolic smoothing effect and we provide criteria for the global existence of solutions.

Effect of hydrate on permeability in porous media: Pore-scale micro-simulation

The permeability variation caused by hydrate dissociation will affect the further exploitation of hydrate resources during the development of natural gas hydrate. To study the effect of hydrate on seepage characteristics in hydrate-bearing sediments, the micro-flow simulation was carried out by the lattice Boltzmann method. The effect of factors related to the pore-scale distribution of mineral particle and hydrate on permeability variation was analyzed in the two-dimensional porous media. Results show that the mineral particles arrangement has no effect on permeability variation for the two-dimensional homogeneous porous media, the permeability variation is sensitive to the hydrate-formation habit and hydrate-distribution morphology in the pore center. The concept of “control seepage channel” is proposed through comparative analysis of velocity field distribution. It controls the change characteristics of permeability. Based on the control seepage channel size and the control seepage channel tortuosity, a new model is presented to measure the permeability in porous media with gas hydrate. Considering two hydrate-formation habits: grain-coating and pore-filling, the relationships between relative permeability and hydrate saturation are proposed to describe permeability variation. The relationship of pore-filling case agrees well with the experimental results. It shows that hydrate grows mainly in the pore center under the experimental conditions.

A lattice Boltzmann investigation of steady-state fluid distribution, capillary pressure and relative permeability of a porous medium: Effects of fluid and geometrical properties

Abstract Simulations of simultaneous steady-state two-phase flow in the capillary force-dominated regime were conducted using the state-of-the-art Shan–Chen multi-component lattice Boltzmann model (SCMC-LBM) based on two-dimensional porous media. We focused on analyzing the fluid distribution (i.e., WP fluid-solid, NP fluid-solid and fluid-fluid interfacial areas) as well as the capillary pressure versus saturation curve which was affected by fluid and geometrical properties (i.e., wettability, adhesive strength, pore size distribution and specific surface area). How these properties influenced the relative permeability versus saturation relation through apparent effective permeability and threshold pressure gradient was also explored. The SCMC-LBM simulations showed that, a thin WP fluid film formed around the solid surface due to the adhesive fluid-solid interaction, resulting in discrete WP fluid distributions and reduction of the WP fluid mobility. Also, the adhesive interaction provided another source of capillary pressure in addition to capillary force, which, however, did not affect the mobility of the NP fluid. The film fluid effect could be enhanced by large adhesive strength and fine pores in heterogeneous porous media. In the steady-state infiltration, not only the NP fluid but also the WP fluid were subjected to the capillary resistance. The capillary pressure effect could be alleviated by decreased wettability, large average pore radius and improved fluid connectivity in heterogeneous porous media. The present work based on the SCMC-LBM investigations elucidated the role of film fluid as well as capillary pressure in the two-phase flow system. The findings have implications for ways to improve the macroscopic flow equation based on balance of force for the steady-state infiltration.