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Abstract: We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is initially at a uniform temperature, and such flows are then induced by suddenly applying either a new constant temperature or a nonzero heat flux to the exterior surface. As time progresses, heat conducts inwards and this may or may not overcome the yield threshold for flow. For the constant temperature case, flow begins immediately should the parameter, Rb, which is a nondimensional yield parameter, be sufficiently large. The ultimate fate, though, is full immobility as the cylinder eventually tends towards a new constant temperature. For the constant heat flux case, the fluid remains immobile but will begin to flow eventually should Rb be sufficiently large. The two cases have different critical values for Rb. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1222-z

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Abstract: Stress dependency of permeability of porous rocks is described by means of a theoretical elastic cylindrical pore-shell model. This model is developed based on a bundle of elastic capillary tubes representation of the preferential flow paths formed in heterogeneous porous rocks. The radial displacement caused in tubes by the pore fluid pressure applied over the surface of the elastic cylindrical flow tubes is expressed by a Lamé-type equation. The radial displacement is incorporated into the Kozeny–Carman relationship to determine the variation of the permeability of porous rocks by variation of the pore fluid pressure. The solution of this equation yields a semi-analytical equation which provides accurate correlations of the stress dependency of the permeability data of porous rocks. The errors associated with the previous formulation of this problem by Zhu et al. (Transp Porous Med 122:235–252, 2018) are explained in view of the present formulation. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1213-0

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Abstract: Wavelet transforms (WTs) constitute a class of powerful tools for spectral and local analysis of data, such as time series, geophysical data, and well logs, as well as modeling of various problems in oil reservoirs, and in particular upscaling their geological models. The application to upscaling has, however, been limited to the reservoir models that are represented by regular computational grids with blocks or cells with regular shapes, such as squares and cubes. The most natural structure of the computational grid is, however, one with irregular and unequal cells, distributed spatially with stochastic orientations. Such a grid is also the ideal computational model for, for example, fractured reservoirs as it allows inclusion of fractures with spatially distributed orientations. In this paper, we propose a generalization of the WT approach to upscaling by developing a new model of a reservoir based on irregular graphs that make it possible to use the WTs for upscaling the geological model highly efficiently. To do so, we first define a computational grid representing a reservoir as a graph and its adjacency matrix and, then, introduce graph WTs using the concept of lifting, utilized in classical signal processing and its extension to graphs. The application of the lifting-based graph WT to upscaling is then developed. The result is an algorithm that may be applied to upscaling of any unstructured geological model represented by a computational grid in which the multiresolution graph WT is applied directly to the spatial distribution of the permeabilities (or other suitable properties). Examples in which the geological model is represented by the Voronoi tessellations are described, and simulation of waterflooding with such graph networks is carried out in order to demonstrate the accuracy and efficiency of the new method. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1219-7

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Abstract: Experimental evidence shows that injecting low-salinity water during the oil recovery process can lead to an increase in the amount of oil recovered. Numerous mechanisms have been proposed to explain this effect, and, in recent years, two which have gained notable support are multicomponent ionic exchange (MIE) and pH increase. Both mechanisms involve ion exchange reactions within the thin film of water separating the oil in a reservoir from the clay minerals on the surface of the reservoir rock. Since the reactions occur on the molecular scale, an upscaled model is required in order to accurately determine the dominant mechanism using centimetre-scale experiments. In this paper, we develop the first stages of this upscaling process by modelling the pore-scale motion of an oil slug through a clay pore throat. We use a law-of-mass-action approach to model the exchange reactions occurring on the oil–water and clay–water interfaces in order to derive expressions for the surface charges as functions of the salinity. By balancing the disjoining pressure in the water film with the capillary pressure across the oil–water interface, we derive an expression for the salinity-dependent film thickness. We compare the two mechanisms by modifying an existing model for the velocity of an oil slug through a pore throat. Numerical results show that the velocity increases as the salinity decreases. The percentage increase is larger for the MIE mechanism, suggesting that MIE may be the dominant causal mechanism; however, this will vary depending on the particular clay and oil being studied. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1220-1

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Abstract: We present a general novel technique to monitor saturation changes on small rock samples of only 15 mm in diameter and 20 mm in length for the purpose of assessing the kinetics of spontaneous imbibition processes. With a fully 3D imbibition configuration involving countercurrent flows through all faces of the sample, the method is based on an NMR technique in which the sole oil phase present within the sample is monitored. The experimental method is fast for two reasons that are (1) the possibility to perform accurate measurements on tiny samples and (2) the adoption of a 3D flow geometry. The kinetics of oil desaturation during spontaneous imbibition is analyzed with the help of an analytical 3D diffusion model, according to which the kinetics is proportional to the value of a “capillary” diffusion coefficient. For the purpose of demonstrating our methodology, we used this technique to compare the spontaneous imbibition of restored sandstone miniplugs from a sandstone reservoir, with and without alkali in the imbibing brine. The imbibition kinetics was quantified as capillary diffusion coefficient values. The studied case results revealed mixed impacts of alkali on the spontaneous imbibition kinetics, involving both a brine–oil interfacial tension change and a wettability alteration of the rock, the latter requiring further investigation beyond the scope of this article. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-01229-z

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Abstract: We present the results of the numerical investigation of the Soret-induced convection in a ternary liquid mixture consisting of dodecane, isobutylbenzene and tetralin, taken in equal portions. The mixture is placed into a square porous cavity with rigid impermeable boundaries heated from below. The lateral boundaries are adiabatic. The problem under consideration is a model of natural hydrocarbon reservoir with porous medium, and the components of mixture are representatives of the main groups of chemical compounds comprising oil. Due to the thermodiffusion effect, dodecane and isobutylbenzene as the lighter components of this mixture with positive separation ratios are accumulated in the warmer domain of the cavity, and the heavy component, tetralin, is accumulated in the colder domain, which may lead to the development of convection. The calculations are performed for the parameters of porous medium close to the real parameters of oil fields and temperature gradient that correspond to geothermal gradient. They provide data on the temporal evolution of the characteristics of the flow and component separation. We also analyze the onset and development of single-vortex and two-vortex instability modes with the growth of the Rayleigh number \(Ra_{\mathrm{{por}}}\) . It is found that at a certain value of the supercriticality, the stationary flow regime is replaced by the oscillatory regime. At even higher values of the Rayleigh number, the chaotic oscillations take place. The transitions between single-vortex and two-vortex flows are also observed. For porosity equal to 0.1, the two oscillatory regimes at different oscillation amplitudes are excited. With the porosity growth, the region of the existence of the oscillatory regimes becomes narrower. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1211-2

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Abstract: The powerful method of eigenfunction superposition is applied to the starting flow in a sector duct filled with a porous medium. Using analytic eigenfunctions and eigenvalues of the Helmholtz equation, the solution can be expressed in a simple series. The properties of the velocity and the transient flow rate are found to depend on the sector geometry and a porous medium factor. The starting solution is then used to construct the solution to arbitrary unsteady flows. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1217-9

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Abstract: Numerical experiment involving both moisture and solute transport predictions is performed to estimate the hydrochemical characteristics of unsaturated porous soil. The moisture and solute transport in the soil are described by the flow and advection–dispersion transport equations. These transport equations are solved by the spectral element method, which is based on Legendre–Gauss–Lobatto quadrature rule and the fully implicit time scheme using the modified Picard iterative procedure constructed with the standard chord slope approximation. The estimation of hydraulic and solute transport parameters has been conducted using the Levenberg–Marquardt method. The goals of the inverse problem were to develop soil hydrochemical characteristics estimation strategies based on combined two of the following functional cost measurements: moisture content, pressure head, hydraulic conductivity, cumulative outflow, and solute concentration. The performance of the inverse algorithm was evaluated using the coefficient of determination, the root-mean-square error, and the relative error analysis which provide an optimal scheme for parameters estimation. The spectral element method was shown to provide good results with negligible error when compared to analytical values. The obtained results indicate excellent agreement of the method for estimating hydraulic and transport parameters with negligible relative error when compared estimated parameters and true values. The choice and the order of combination of objective functions affect crucially the inverse solution especially in case of large hydrochemical parameters estimates. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1216-x

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Abstract: Evaluating the anisotropy of transport parameters in rocks is important for various applications, such as reservoir engineering and rock mechanics. Owing to their anisotropic pore structures, the tortuosity, constrictivity, and pore size distribution of rocks are often anisotropic in nature, which in turn affect the permeability and diffusivity. However, it has still not been determined whether the permeability and diffusivity are anisotropic in the same manner. This study used experiments and numerical modeling to examine the effect of the pore structure on the permeability and diffusivity anisotropies of rocks. The experimental results showed a clear difference in the anisotropy ratios of the permeability (k⊥/k‖) and diffusivity (D e ⊥ /D e ‖ ) for Berea sandstone, which is the de facto standard porous sandstone. The analysis results from micro-focus X-ray computed tomography and simulation with the lattice Boltzmann method supported the experimental difference in anisotropy ratios. In the analysis and simulation, the relation between the minimum cross-sectional porosity area and characteristic pressure gradient was estimated. The analysis results suggest that the minimum cross-sectional porosity areas that influence the permeability anisotropy are too large to physically induce anisotropic NaCl diffusion, and thus, the diffusivity of Berea sandstone is nearly isotropic. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1214-z

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Abstract: In this paper, we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided into two phases: water and biofilm. The biofilm is assumed to consist of four components: water, extracellular polymeric substance (EPS), active bacteria, and dead bacteria. The flow of water is modeled by the Stokes equation, whereas a diffusion–convection equation is involved for the transport of nutrients. At the biofilm–water interface, nutrient transport and shear forces due to the water flux are considered. In the biofilm, the Brinkman equation for the water flow, transport of nutrients due to diffusion and convection, displacement of the biofilm components due to reproduction/death of bacteria, and production of EPS are considered. A segregated finite element algorithm is used to solve the mathematical equations. Numerical simulations are performed based on experimentally determined parameters. The stress coefficient is fitted to the experimental data. To identify the critical model parameters, a sensitivity analysis is performed. The Sobol sensitivity indices of the input parameters are computed based on uniform perturbation by ± 10% of the nominal parameter values. The sensitivity analysis confirms that the variability or uncertainty in none of the parameters should be neglected. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1218-8

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Abstract: Flooding of coastal areas with seawater often leads to density stratification. The stability of the density-depth profile in a porous medium initially saturated with a fluid of density \(\rho _\mathrm{f}\) after flooding with a salt solution of higher density \(\rho _\mathrm{s}\) is analyzed. The standard convection/diffusion equation subject to the so-called Boussinesq approximation is used. The depth of the porous medium is assumed to be infinite in the analytical approaches and finite in the numerical simulations. Two cases are distinguished: the laterally unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and the laterally bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) . The ratio of the diffusivity and the density difference \((\rho _\mathrm{s} - \rho _\mathrm{f})\) induced gravitational shear flow is an intrinsic length scale of the problem. In the unbounded \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) , this geometric length scale is the only length scale and using it to write the problem in dimensionless form results in a formulation with Rayleigh number \(R = 1\) . In the bounded \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) , the lateral geometry provides another length scale. Using this geometrical length scale to write the problem in dimensionless form results in a formulation with a Rayleigh number R given by the ratio of the geometric and intrinsic length scales. For both \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) , the well-known Boltzmann similarity solution provides the ground state. Three analytical approaches are used to study the stability of this ground state, the first two based on the linearized perturbation equation for the concentration and the third based on the full nonlinear equation. For the first linear approach, the surface spatial density gradient is used as an approximation of the entire background density profile. This results in a crude estimate of the \(L^2\) -norm of the concentration showing that the perturbation at first grows, but eventually decays in time. For the other two approaches, the full ground-state solution is used, although for the second linear approach subject to the restriction that the ground state slowly evolves in time (the so-called frozen profile approximation). Just like the ground state, the resulting eigenvalue problems can be written in terms of the Boltzmann variable. The linearized stability approach holds only for infinitesimal small perturbations, whereas the nonlinear, variational energy approach is not subject to such a restriction. The results for all three approaches can be expressed in terms of Boltzmann \(\sqrt{t}\) transformed relationships between the system Rayleigh number and perturbation wave number. The results of the linear and nonlinear approaches are surprisingly close to each other. Based on the system Rayleigh number, this allows delineation of systems that are unconditionally stable, marginally stable, or transiently unstable. These analytical predictions are confirmed by direct two-dimensional numerical simulations, which also show the details of the transient instabilities as function of the wave number for \({{{\mathbf {{\small {\uppercase {case~A}}}}}}}\) and the wave number and Rayleigh number for \({{{\mathbf {{\small {\uppercase {case~B}}}}}}}\) . A numerical example of the effect of a layer with low permeability is also shown. Using typical values of the physical parameters, the analytical and numerical results are interpreted in terms of dimensional length and time scales. In particular, an explicit stability criterion is given for vertical column experiments. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1209-9

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Abstract: The Horton–Rogers–Lapwood problem with strong heterogeneity and anisotropy is examined for a simple case, namely where the heterogeneity is provided by two layers, each of which is homogeneous and isotropic in a horizontal plane. We derived a new hydrodynamic boundary condition at the interface between two different porous media and then formulated and numerically solved the eigenvalue problem to determine the critical values of the wavenumber and Rayleigh number. We found that our approach works and gives the correct result for the homogeneous situation independent of the position of the interface. We also showed that for weak heterogeneity, by using modified anisotropy parameters weighted with the two layer depths, results obtained in Kvernvold and Tyvand (J Fluid Mech 90:609–624, 1979) for a single-layer problem can be used to approximate the critical Rayleigh number for the double-layer problem. We found that the agreement with the two-layer solution is best if a harmonic mean is used to define the mean permeability ratio and an arithmetic mean is utilized to define the mean conductivity ratio. PubDate: 2019-04-01 DOI: 10.1007/s11242-018-1210-3

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Abstract: The classic models describing the hygric mass transfers inside porous materials seem unsuitable in the case of bio-based materials. They are based on the assumption of instantaneous local equilibrium between relative humidity and water content (Künzel in Simultaneous heat and moisture transport in building components—one- and two-dimensional calculation using simple parameters, Fraunhofer IRB Verlag, Stuttgart, 1995). These two parameters evolve according to the diffusive fluxes following the sorption isotherms. This study shows that it leads to predicting much shorter times of stabilization than those experimentally obtained. A new approach is presented here; it is free from the local instantaneous equilibrium introducing a local kinetics to describe the transformation of water from vapor state to liquid state and vice versa. The local kinetics of sorption is coupled with the well-known hysteresis phenomenon. It is adjusted from bibliographic data (Collet et al. in Energy Build 62:294–303, 2013) giving mass evolution of three hemp concretes under adsorption/desorption conditions. 1D cylindrical simulations allow an excellent fitting on the experiments. Finally, a semiempirical model is proposed, allowing to determine the kinetics parameters more easily. PubDate: 2019-03-20 DOI: 10.1007/s11242-019-01272-4

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Abstract: Reactive transport in fractured media is conceptualized as a multi-scale problem that couples a pore-scale component, which comprises Navier–Stokes flow, multi-component transport and aqueous equilibrium in the fracture, and a Darcy-scale component, which comprises multi-component diffusive transport, aqueous equilibrium and mineral reactions in the porous matrix. The model that implements this multi-scale approach builds on an existing pore-scale model and is able to capture complex fracture geometries with the embedded-boundary method. The embedded boundary acts as the interface between pore- and Darcy-scale domains. Adaptive mesh refinement is used to match resolutions at the interface while using coarser resolution away from the interface when not needed in the Darcy-scale domain. The new model is validated and then compared to results from a pore-scale model. Multi-scale model results are shown to be equivalent to pore-scale results under diffusion-controlled reactions in the pore scale and very fast dissolution in the Darcy scale. The multi-scale model provides a more accurate solution for a given resolution as it effectively sets the equilibrium concentrations as boundary conditions. The multi-scale model is capable to capture flow channelization observed in an experimental fractured core and, at the same time, limitations in the dissolution of calcite by diffusive transport through an altered porous layer. Discrepancies in effluent calcium concentrations between the multi-scale results and results from a reduced-dimension Darcy-scale model for this fractured core experiment are attributed to the solution of the flow field and the gradients that develop inside the fracture. Discrepancies in effluent magnesium concentrations exemplify the limitations of the approach because the multi-scale model requires calibration of reactive surface areas as Darcy-scale continuum models. PubDate: 2019-03-20 DOI: 10.1007/s11242-019-01266-2

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Abstract: We describe a simple but effective stochastic method to model the void structure of a single fracture in a form of voxel representation. A fracture void is delineated by two bounding wall surfaces that are separated by some distance (i.e. the local aperture) at each location on the medial surface that lies within the fracture void and serves as a model reference frame. The three surface height fields are generated based on four parameters (mean, standard deviation and two spatial correlation lengths) for each field and two parameters (coefficient and synergistic length) for the spatial correlation between the fracture walls. Testing of generated models demonstrates that not only are the model fracture apertures spatially correlated and characterized as a Gaussian field, but also the two fracture walls are closely correlated, with a similar shape and/or height. With respect to fracture apertures, three quantities, i.e. the mean aperture, roughness and anisotropy, can be derived from the fracture models to describe fracture morphology. The effect of model fractures on fluid flow is investigated in order to establish the relationship between fracture permeability and the three morphological quantities, revealing a way to avoid the significant estimation error associated with the use of the cubic law. PubDate: 2019-03-20 DOI: 10.1007/s11242-019-01271-5

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Abstract: Two distinct but interconnected approaches can be used to model diffusion in fluids; the first focuses on dynamics of an individual particle, while the second deals with collective (effective) motion of (infinitely many) particles. We review both modeling strategies, starting with Langevin’s approach to a mechanistic description of the Brownian motion in free fluid of a point-size inert particle and establishing its relation to Fick’s diffusion equation. Next, we discuss its generalizations which account for a finite number of finite-size particles, particle’s electric charge, and chemical interactions between diffusing particles. That is followed by introduction of models of molecular diffusion in the presence of geometric constraints (e.g., the Knudsen and Fick–Jacobs diffusion); when these constraints are imposed by the solid matrix of a porous medium, the resulting equations provide a pore-scale representation of diffusion. Next, we discuss phenomenological Darcy-scale descriptors of pore-scale diffusion and provide a few examples of other processes whose Darcy-scale models take the form of linear or nonlinear diffusion equations. Our review is concluded with a discussion of field-scale models of non-Fickian diffusion. PubDate: 2019-03-18 DOI: 10.1007/s11242-019-01262-6

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Abstract: The objective of this work is to study the applicability of various machine learning algorithms for the prediction of some rock properties which geoscientists usually define due to special laboratory analysis. We demonstrate that these special properties can be predicted only basing on routine core analysis (RCA) data. To validate the approach, core samples from the reservoir with soluble rock matrix components (salts) were tested within 100 + laboratory experiments. The challenge of the experiments was to characterize the rate of salts in cores and alteration of porosity and permeability after reservoir desalination due to drilling mud or water injection. For these three measured characteristics, we developed the relevant predictive models, which were based on the results of RCA and data on coring depth and top and bottom depths of productive horizons. To select the most accurate machine learning algorithm, a comparative analysis has been performed. It was shown that different algorithms work better in different models. However, two-hidden-layer neural network has demonstrated the best predictive ability and generalizability for all three rock characteristics jointly. The other algorithms, such as support vector machine and linear regression, also worked well on the dataset, but in particular cases. Overall, the applied approach allows predicting the alteration of porosity and permeability during desalination in porous rocks and also evaluating salt concentration without direct measurements in a laboratory. This work also shows that developed approaches could be applied for the prediction of other rock properties (residual brine and oil saturations, relative permeability, capillary pressure, and others), of which laboratory measurements are time-consuming and expensive. PubDate: 2019-03-18 DOI: 10.1007/s11242-019-01265-3

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Abstract: The injection of seawater-like brines alters stiffness, strength and time-dependent deformation rates for water-saturated chalks. This study deals with the mechanical effects and oil production upon brine injection through wettability-altered samples. The results from two test programs are presented: (a) ‘Wettability determination program’ and (b) ‘triaxial test program’. Kansas chalk samples were saturated by a mixture of oil and water and aged over time at 90 °C. The wettability index of the altered samples was estimated using chromatographic separation tests by co-injecting sulphate ions that adsorb on the water-wet mineral surfaces and non-affine tracer. A good repeatability was observed. In the triaxial test program, unaged water-wet and aged mixed-wet samples were hydrostatically loaded to 1.5 times yield stress so stiffness and strength could be determined. The samples were kept at the same stress level over time to monitor the volumetric creep. After a stagnant flow period of 15 days, MgCl2 brine and seawater were flushed through the samples so the oil production and ion concentration of the effluent water could be obtained. The combined observations of the bulk volume, oil volume and estimated solid volume (from effluent analyses) enabled us to calculate pore volume and thereby oil saturation with time. The mixed-wet samples were found to be stiffer and stronger than the water-wet samples, and when the stress was kept at 1.5 times yield the creep curves overlapped. During the flow-through period, the changes in ion composition are insensitive to the presence of oil, and ongoing water weakening for mixed-wet samples is the same as in the water-wet samples. Further, we found that oil was only produced during the first 2–3 pore volumes (PVs) injected. Afterwards, no oil was produced even though the chemical reactions took place and pore volume reduced. PubDate: 2019-03-16 DOI: 10.1007/s11242-019-01269-z

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Abstract: The fourth-order Darcy–Bénard eigenvalue problem for onset of thermal convection in a 2D rectangular porous box is investigated. The conventional type of solution has normal-mode dependency in at least one of the two spatial directions. The present eigenfunctions are of non-normal-mode type in both the horizontal and the vertical direction. A numerical solution is found by the finite element method, since no analytical method is known for this non-degenerate fourth-order eigenvalue problem. All four boundaries of the rectangle are impermeable. The thermal conditions are handpicked to be incompatible with normal modes: The lower boundary and the right-hand wall are heat conductors. The upper boundary has given heat flux. The left-hand wall is thermally insulating. The computed eigenfunctions have novel types of complicated cell structures, with intricate internal cell walls. PubDate: 2019-03-14 DOI: 10.1007/s11242-019-01263-5

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Abstract: In some gas–solid reactions, a new solid substance is produced. The product acts as a shield and prevents the collision between gas and solid reactants which further causes an incomplete reaction. If the molar volume of the new product differs from the solid reactant, the inner structure of porous media is changed as well. In this paper, we discuss such gas–solid reactions in porous media using the two-dimensional lattice gas cellular automata FHP-III model. We simulate the fluid flow and chemical reaction in different porous media. We also show the effects of porosity and morphology of the solid, and reaction probability on the reaction process. Results obtained from the simulations agree closely with the theory of gas–solid reactions and diffusion theories. Hence, the proposed model is a good choice to simulate gas–solid chemical reactions in porous media at the mesoscopic level. PubDate: 2019-03-14 DOI: 10.1007/s11242-019-01259-1