Investigation of Self-Similar Interface Evolution in Carbon Dioxide Sequestration in Saline Aquifers

In this work, we investigate numerically the injection of supercritical carbon dioxide into a deep saline reservoir from a single well. We analyze systematically the sharp-interface evolution in different flow regimes. The flow regimes can be parameterized by two dimensionless numbers, the gravity number, \(\Gamma \) and the mobility ratio, \(\lambda \) . Numerical simulations are performed using the volume of fluid method, and the results are compared with the solutions of the self-similarity equation established in previous works, which describes the evolution of the sharp interface. We show that these theoretical solutions are in very good agreement with the results from the numerical simulations presented over the different flow regimes, thereby showing that the theoretical and simulation models predict consistently the spreading and migration of the created \(\hbox {CO}_{2}\) plume under complex flow behavior in porous media. Furthermore, we compare the numerical results with known analytic approximations in order to assess their applicability and accuracy over the investigated parametric space. The present study indicates that the self-similar solutions parameterized by the dimensionless numbers \(\lambda , \Gamma \) are significant for examining effectively injection scenarios, as these numbers control the shape of the interface and migration of the \(\hbox {CO}_{2}\) plume. This finding is essential in assessing the storage capacity of saline aquifers.     

Measurement and Correlation of Pressure Drop Characteristics for Air Flow Through Sintered Metal Porous Media

Sintered metal porous media are currently used to replace conventional orifices as restrictors in air-bearing systems. The flow properties in porous media are generally approximated by Darcy and Forchheimer regimes in different flow regions. In this study, an ISO expanded expression is proven defective when it is used to represent flow properties through porous media under slight pressure drops ( ${<}10$  kPa). A modified Forchheimer equation is therefore developed to correlate the pressure drop with flow rate, including compressibility and inertial effects. Experimental and theoretical investigations on pressure drop characteristics are conducted with a series of metal-sintered porous media. Permeability is first determined in a strict Darcy region with $Re<0.1$ , followed by the inertial coefficient with $Re>0.1$ , rather than determining these two simultaneously. The theoretical mass flow rate in terms of the modified Forchheimer equation provides close approximations to the experimental data.     

Coupled Water and Salt Transport in Porous Materials: Rapid Determination of a Varying Diffusion Coefficient from Experimental Data

A direct method for an accurate and rapid evaluation of a varying salt diffusion coefficient, \(D\) , from experimental data is proposed for a coupled water and salt transport in porous materials. The evaluation uses data on the moisture and salt concentration profiles and is based on a formula obtained from the Boltzmann-Matano method. The coupled transport is described by the diffusion-advection model of Bear and Bachmat. A simple expression for \(D\) in the center of the concentration interval is deduced from the formula to provide a rapid estimate on \(D\) . Possible extensions of this analytical approach are pointed out, suggesting that it can serve as a convenient general tool in engineering calculations. The theoretical results are applied to a laboratory experiment in which a coupled moisture and chloride transport had been investigated in a lime plaster, and the chloride diffusion coefficient had been obtained numerically in dependence on the chloride concentration. The agreement with the numerical results is shown to be rather good, except at low concentrations where our analytical results should be more reliable. It is also shown that the unusually high value of the calculated chloride diffusion coefficient—about three orders of magnitude higher than for free chloride ions in water—cannot be explained by possible inaccuracies in the measurements and/or numerical calculations. The reason is that changes in the measured profiles’ data could cause a change in \(D\) of just the same order of magnitude. This shows that, besides diffusion and advection, additional mechanisms take part in the considered chloride transport.     

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number $10^{-5}\le \hbox {Da}\le 10^{-3}$ , porous layer height ratio $0\le d/L\le 1$ , thermal conductivity ratio $1\le k_{1,3}\le 20$ , and dimensionless time $0\le \tau \le 1000$ on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.     

Determination of Nanoparticle Macrotransport Coefficients from Pore Scale Processes

Nanoparticle transport in porous media is modeled using a hierarchical set of differential equations corresponding to pore scale and macroscale. At the pore scale, movement and interaction of a single particle with a solid matrix is modeled using the advection–dispersion–sorption equation. A single nanoparticle entering the space encounters viscous, diffusion and surface forces. Surface forces (electrostatic and van der Waals forces) between nanoparticles and mineral grains appear as sorption propensity on solid matrix boundary condition. These local events are then transformed into a macroscale continuum by imposing periodic boundary conditions for contiguous unit cells representing porous media and using a scheme of moment analysis. At the macroscale, propagation and retention of particles are characterized by three position-independent coefficients: mean nanoparticle velocity vector \({\bar{\mathbf{U}}}^*\), macroscopic dispersion coefficient \({\bar{\mathbf{D}}}^*\), and mean nanoparticle retention rate constant \({\bar{K}}^*\). The modeling results are validated with a set of nanoparticle transport tests in porous microchips. We also present simulations of realistic porous media, where an actual image of sandstone samples is processed into binary tones. The representative unit cells are constructed from the resulting binary image by searching for areas within the sample with maximum similarities to the whole sample in terms of porosity and specific surface area, which are found to show strong correlations with the resulting \({\bar{\mathbf{U}}}^*\) and \({\bar{K}}^*\), respectively.     

Steady States of Fokker–Planck Equations: III. Degenerate Diffusion

This is the third paper in a series concerning the study of steady states of a Fokker–Planck equation in a general domain in \({\mathbb R}^n\) with \(L^{p}_{loc}\) drift term and \(W^{1,p}_{loc}\) diffusion term for any \(p>n\). In this paper, we give some existence results of stationary measures of the Fokker–Planck equation under Lyapunov conditions which allow the degeneracy of diffusion.     

Micro-structural Impact of Different Strut Shapes and Porosity on Hydraulic Properties of Kelvin-Like Metal Foams

Various ideal periodic isotropic structures of foams (tetrakaidecahedron) with constant ligament cross section are studied. Different strut shapes namely circular, square, diamond, hexagon, star, and their various orientations are modeled using CAD. We performed direct numerical simulations at pore scale, solving Navier–Stokes equation in the fluid space to obtain various flow properties namely permeability and inertia coefficient for all shapes in the porosity range, \(0.60<\varepsilon <0.95\) for wide range of Reynolds numbers, \(10^{-6} . We proposed an analytical model to obtain pressure drop in metallic foams in order to correlate the resulting macroscopic pressure and velocity gradients with the Ergun-like approach. The analytical results are fully compared with the available numerical data, and an excellent agreement is observed.     

Numerical Simulation of Forced Convective Heat Transfer Past a Square Diamond-Shaped Porous Cylinder

Fluid flow and heat transfer around and through a porous cylinder is an important issue in engineering applications. In this paper a numerical study is carried out for simulating the fluid flow and forced convection heat transfer around and through a square diamond-shaped porous cylinder. The flow is two-dimensional, steady, and laminar. Conservation laws of mass, momentum, and heat transport equations are applied in the clear region and Darcy–Brinkman–Forchheimer model for simulating the flow in the porous medium has been used. Equations with the relevant boundary conditions are numerically solved using a finite volume approach. In this study, Reynolds and Darcy numbers are varied within the ranges of $1<Re<45$ and $10^{-6}<Da<10^{- 2}$ , respectively. The porosity $(\varepsilon )$ is 0.5. This paper presents the effect of Reynolds and Darcy numbers on the flow structure and heat transfer characteristics. Finally, these parameters are compared among solid and porous cylinder. It was found that the drag coefficient decreases and flow separation from the cylinder is delayed with increasing Darcy number. Also the size of the thermal plume decreases by decreasing Darcy number.     

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability简

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.     

Advection–Dispersive Mechanisms of Electrolyte Species in Porous Matrix

Advection–dispersive phenomena inside two geometries of porous media, corresponding to a structured and unstructured network of minichannels, are studied from a combination of analytical, numerical, and electrodiffusion techniques. The instantaneous limiting diffusion current, connected to the concentration of electroactive species flowing in the porous matrix, is recorded and measured during flow experimentation. Transport phenomena investigation consists of the step injection of a tracer of higher concentration than the bulk flow in order to characterize the mixing of electrolyte species inside two arrangements of network, so called \(\times \) _network and T_network. The experimental results are supported by 2D-numerical simulations performed in the \(\times \) _network. A pore model is proposed in order to predict the pore velocity, which is used within the resolution of the diffusion–convection balance. The numerical simulations, based on a second-order finite difference scheme, give rise to a good agreement in terms of mixing index and the methodology employed for the numerical injection concentration appears suitable. The numerical experiments are quite-well representative in laminar regime below the critical Reynolds number ( \(Re_{\mathrm{crit}}\approx \,200\) ). Above \(Re_{\mathrm{crit}}\) , the inertial effects are not negligible and the momentum transfer needs to be taken into account. This latter phenomenon is analyzed at the pore-scale in term of local skin friction measured at the channels crossing, then compared and discussed from analytical solutions.     

Bounded Domain Problem for the Modified Buckley–Leverett Equation

The focus of the present study is the modified Buckley–Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley–Leverett (BL) equation by including a balanced diffusive–dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers a non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this paper, we first show that for the MBL equation, the solution of the finite interval \([0,L]\) boundary value problem converges to that of the half line \([0,+\infty )\) boundary value problem exponentially as \(L\rightarrow +\infty \) . This result provides a justification for the use of the finite interval in numerical studies for the half line problem [Y. Wang and C.-Y. Kao, Central schemes for the modified Buckley–Leverett equation, J. Comput. Sci. 4(1–2), 12 – 23, 2013]. Furthermore, we numerically verify that the convergence rate is consistent with the theoretical derivation. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.     

Unsteady MHD double-diffusive convection boundary-layer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink

This paper presents an analytical study of the unsteady MHD free convective heat and mass transfer flow of a viscous, incompressible, gray, absorbing-emitting but non-scattering, optically-thick and electrically conducting fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot vertical plate with constant velocity. We employ a Darcian viscous flow model for the porous medium the Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted in mass diffusion equation. The governing equations are solved in closed form by Laplace-transform technique. A parametric study of all involved parameters is conducted and representative set of numerical results for the velocity, temperature, concentration, shear stress function $\frac{\partial u}{\partial y} \vert_{y=0}$ , temperature gradient $\frac{\partial \theta }{ \partial y}\vert_{y=0}$ , and concentration gradient $\frac{ \partial \phi }{\partial y}\vert_{y=0}$ is illustrated graphically and physical aspects of the problem are discussed.     

Combined Effect of Stress, Pore Pressure and Temperature on Methane Permeability in Anthracite Coal: An Experimental Study

The effects of particle-size distribution on the longitudinal dispersion coefficient ( $D_{\mathrm{L}})$ D L ) in packed beds of spherical particles are studied by simulating a tracer column experiment. The packed-bed models consist of uniform and different-sized spherical particles with a ratio of maximum to minimum particle diameter in the range of 1–4. The modified version of Euclidian Voronoi diagrams is used to discretize the system of particles into cells that each contains one sphere. The local flow distribution is derived with the use of Laurent series. The flow pattern at low particle Reynolds number is then obtained by minimization of dissipation rate of energy for the dual stream function. The value of $D_{\mathrm{L}}$ D L is obtained by comparing the effluent curve from large discrete systems of spherical particles to the solution of the one-dimensional advection–dispersion equation. Main results are that at Peclet numbers above 1, increasing the width of the particle-size distribution increases the values of $D_{\mathrm{L}}$ D L in the packed bed. At Peclet numbers below 1, increasing the width of the particle-size distribution slightly lowers $D_{\mathrm{L}}$ D L .     

The Onset of Prandtl–Darcy–Prats Convection in a Horizontal Porous Layer

We consider the effect of finite Prandtl–Darcy numbers of the onset of convection in a porous layer heated isothermally from below and which is subject to a horizontal pressure gradient. A dispersion relation is found which relates the critical Darcy–Rayleigh number and the induced phase speed of the cells to the wavenumber and the imposed Péclet and Prandtl–Darcy numbers. Exact numerical solutions are given and these are supplemented by asymptotic solutions for both large and small values of the governing nondimensional parameters. The classical value of the critical Darcy–Rayleigh number is $4\pi ^2$ 4 π 2 , and we show that this value increases whenever the Péclet number is nonzero and the Prandtl–Darcy number is finite simultaneously. The corresponding wavenumber is always less than $\pi $ π and the phase speed of the convection cells is always smaller than the background flux velocity.     

Chaotic and Periodic Natural Convection for Moderate and High Prandtl Numbers in a Porous Layer subject to Vibrations

The analysis of natural convection for moderate and high Prandtl numbers in a fluid-saturated porous layer heated from below and subject to vibrations is presented with a twofold objective. First, it aims at investigating the significance of including a time derivative term in Darcy’s equation when wave phenomena are being considered. Second, it is dedicated to reporting results related to the route to chaos for moderate and high Prandtl number convection. The results present conclusive evidence indicating that the time derivative term in Darcy’s equation cannot be neglected when wave phenomena are being considered even when the coefficient to this term is extremely small. The results also show occasional chaotic “bursts” at specific values (or small range of values) of the scaled Rayleigh number, $R$ , exceeding some threshold. This behavior is quite distinct from the case without forced vibrations, when the chaotic solution occupies a wide range of $R$ values, interrupted only by periodic “bursts.” Periodic and chaotic solution alternate as the value of the scaled Rayleigh number varies.     

The Forchheimer equation: A theoretical development

In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by $$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$ The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.     

An asymptotic approach of Brownian deposition of nanofibres in pipe flow

An asymptotic approach is considered for the transport and deposition of nanofibres in pipe flow. Convection and Brownian diffusion are included, and Brownian diffusion is assumed to be the dominant mechanism. The fibre position and orientation are modelled with a probability density function for which the governing equation is a Fokker–Planck equation. The focus is set on dilute fibres concentrations implying that interaction between individual fibres is neglected. At the entrance of the pipe, a fully developed velocity profile is set and it is assumed that the fibres enter the pipe with a completely random orientation and position. A small parameter ${\varepsilon =l/a}$ is introduced, where l is the fibre half-length and a is the pipe radius. The probability density function is expanded for small ${\varepsilon}$ and the solution turns out to be multi-structured with three areas, consisting of one outer solution and two boundary layers. For the deposition of fibres on the wall, it is found that for parabolic flow, and for the lowest order, the deposition can be obtained with a simplified angle averaged convective-diffusion equation. It is suggested that this simplification is valid also for more complex flows like when the inflow boundary condition yields a developing velocity profile and flows within more intricate geometries than here studied. With the model fibre, deposition rates in human respiratory airways are derived. The results obtained compare relatively well with those obtained with a previously published model.     

An Exact Result for the Macroscopic Response of Porous Neo-Hookean Solids

Making use of the particulate microgeometries of Idiart (J. Mech. Phys. Solids 56:2599–2617, 2008), we derive an exact and closed-form result for the macroscopic response of porous Neo-Hookean solids with random microstructures. The stored-energy function is a solution to a Hamilton-Jacobi equation with initial porosity and macroscopic deformation gradient playing the roles of time and space. The main theoretical and practical aspects of the result are discussed.     

Random Attractors for Degenerate Stochastic Partial Differential Equations

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate $p$ -Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate $p$ -Laplace equations we prove that the deterministic, $\infty $ -dimensional attractor collapses to a single random point if enough noise is added.     

On equilibrium and primary variables in transport in porous media

Thermodynamic equilibrium, which involves mechanical, thermal, and chemical equilibria, in a multiphase porous medium, is defined and discussed, both at the microscopic level, and at the macroscopic one. Conditions are given for equilibrium in the presence of forces between the surface of the solid matrix and the fluid phases. The concept ofapproximate thermodynamic equilibrium is introduced and discussed, employing the definition of athermodynamic potential. This discussion serves as the basis for the methodology of determining the number of degrees of freedom in models of phenomena of transport (of mass, energy, and momentum) in porous media. Equilibrium and nonequilibrium cases are considered. The proposed expressions for the number of degrees of freedom in macroscopic transport models, represent the equivalent ofGibbs phase rule in thermodynamics. Based on balance considerations and thermodynamic relationships, it is shown that the number of degrees of freedom, NF, in a problem of transport in a deformable porous medium, involving NP fluid phases and NC components, under nonisothermal conditions, with equilibrium among all phases and components, is $${\text{NF = NC + NP + 4}}{\text{.}}$$ Under nonequilibrium conditions among the phases, the rule takes the form $${\text{NF = NC }} \times {\text{ NP + 2NP + NC + 4}}{\text{.}}$$ In both cases, when fluid phase velocities are determined by Darcy's law, NF is reduced by NP. When the solid matrix is nondeformable, NF is reduced by 3. The number of degrees of freedom is also determined for conditions of approximate chemical and thermal equilibria, and for conditions of equilibrium that prevail only among some of the phases present in the system. Examples of particular cases are presented to illustrate the proposed methodology.