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The paper considers a mathematical model of the joint motion of two immiscible incompressible fluids in a poroelastic medium. This model is a generalization of the classical Musket-Leverett model, in which porosity is considered to be a given function of the spatial coordinate. The model under study is based on the mass conservation equations for liquids and the porous skeleton, Darcy's law for liquids, which takes into account the movement of the porous skeleton, the Laplace formula for capillary pressure, the Maxwell-type rheological equation for porosity, and the "system as a whole" equilibrium condition. In the thin layer approximation, the original problem is reduced to the successive determination of the porosity of the solid skeleton and its velocity. Then an elliptic-parabolic system is derived for the “reduced pressure” and saturation of the wetting phase. Its solution is understood in a generalized sense due to the degeneration on the solution of the equations of the system. The proof of the existence theorem is carried out in four stages: regularization of the problem, proof of the physical maximum principle for saturation, construction of Galerkin approximations, passage to the limit in regularization parameters based on the method of compensated compactness., В работе рассматривается математическая модель совместного движения двух несмешивающихся несжимаемых жидкостей в пороупругой среде. Данная модель является обобщением классической модели Маскета-Леверетта, в которой пористость считается заданной функцией пространственной координаты. В основе изучаемой модели лежат уравнения сохранения массы жидкостей и пористого скелета, закон Дарси для жидкостей, учитывающий движение пористого скелета, формула Лапласа для капиллярного давления, реологическое уравнение для пористости типа Максвелла и условие равновесия «системы в целом». В приближении тонкого слоя исходная задача сводится к последовательному определению пористости твердого скелета и его скорости, а затем выводится эллиптико-параболическая система для «приведенного давления» и насыщенности смачивающей фазы. В связи с вырождением на решении уравнений системы ее решение понимается в обобщенном смысле. Доказательство теоремы существования осуществляется в четыре этапа: регуляризация задачи, доказательство физического принципа максимума для насыщенности, построение галеркинских приближений, предельный переход по параметрам регуляризации на основе метода компенсированной компактности.

An a priori analysis for a generalized local projection stabilized finite element approximation of the Stokes, and the Darcy flow equations are presented in this paper. A first-order conforming P 1 c finite element space is used to approximate both the velocity and pressure. It is shown that the stabilized discrete bilinear form satisfies the inf-sup condition in the generalized local projection norm. Moreover, a priori error estimates are established in a mesh-dependent norm as well as in the L 2 -norm for the velocity and pressure. The optimal and quasi-optimal convergence properties are derived for the Stokes and the Darcy flow problems. Finally, the derived estimates are numerically validated with appropriate examples.

Fracture has significant effect on the production of hydrogen–carbon compound reservoirs. Underground fluid always flows through different spaces which involves different flow patterns. In this study, the embedded discrete fracture method (EDFM) to couple the flow in matrix and fracture is introduced. The Darcy flow is considered in fracture while the Pre-Darcy flow is employed in matrix. The model of two phase flow problem is established. The governing equations are coupled through multiple spatial scales. The structured mesh is applied to discrete the whole matrix domain. The fracture grids are embedded to the matrix meshes. The verification shows good accuracy of the fully implicit method for solving the model. Based on the numerical simulation, the results of 1D, 2D flow problems considering Pre-Darcy flow are presented. The characteristic of pressure and saturation is analyzed. Some numerical experiment examples are conducted to demonstrate the capability of the method including multiple wells system, complex fracture problem, and hydraulic fractured horizontal well system. Furthermore, we extend the model to 3D transport problem. The newly developed numerical simulation method improves the limitations of EDFM in Pre-Darcy flow. The newly developed simulation method can be taken as an effective tool for hydrocarbon production analysis when Pre-Darcy flow happens in fractured porous media.

Darcy's law has been widely used to study the groundwater drainage process within an aquitard. However, non-Darcy flow is frequently encountered in laboratory and in situ investigations. With consideration of a sudden drop in boundary hydraulic heads, aquitard compaction characteristics and their sensitivities to the non-Darcy flow control variables were analyzed. The non-Darcy flow was found to retard groundwater drainage, and the retardation effects were much more significant at early-to-intermediate time points. Under this specific boundary condition, the time–compaction curve in a log–log graph at early time points was found to be close to a straight line, whose slope can be used to indirectly evaluate the extent of the non-Darcy effect. A non-Darcy flow-based type curve method was developed for estimating aquitard hydraulic conductivity (K) and skeletal specific storage ( S s ), and this method was used to interpret the time–compaction data recorded in a laboratory experiment. The tested aquitard was determined to be associated with non-Darcy flow due to the fact that the time–compaction curve deviated from the Darcy's law-based theoretical curve. Darcy's law resulted in an underestimated K. In contrast, the estimated S s was almost unaffected by the flow state, if the observation lasted long enough to reach final steady compaction.

Several recent papers have addressed modelling of the tissue growth by the multi-phase models where the velocity is related to the pressure by one of the physical laws (Stoke's, Brinkman's or Darcy's). While each of these models has been extensively studied, not so much is known about the connection between them. In the recent paper (arXiv:2303.10620), assuming the linear form of the pressure, the Authors connected two multi-phase models by an inviscid limit: the viscoelastic one (of Brinkman's type) and the inviscid one (of Darcy's type). Here, we prove that the same is true for a nonlinear, power-law pressure. The new ingredient is that we use relation between the pressure $p$ and the Brinkman potential $W$ to deduce compactness in space of $p$ from the compactness in space of $W$., Comment: 13 pages + appendix + bibliography; in the new version a missing reference in the Appendix was added

The conventional and a generalized form of Darcy’s law for the absolute permeability of porous media at low flow rates has been derived using the Langevin equation. It can account for the conventional viscous resistance and additionally a term referred to as thermal resistance. The latter is hypothesized to stem from the relationship between the mean and molecular random velocity along the mean flow direction. The key to note when using the Langevin equation as the starting point for the derivation is that the frictional resistance occurring are the same whether the fluid or the object is moving relative to each other. Hence, a spherical particle moving through a stagnant fluid will experience the same friction as a fluid flowing past a stagnant spherical particle. Experimental data for absolute permeability at different temperatures indicate a temperature dependency, which can be accounted for by the thermal resistance term, which so far has been neglected. A procedure for matching and predicting pressure variation and absolute permeability vs. temperature is described. The generalized form of Darcy’s law presented should be of interest in all sciences where fluid transport in porous media occurs at varying temperatures, e.g., in soil science, hydrology, geothermal, chemical and petroleum engineering. It has also been discussed in relation to the Kozeny–Carman equation.

To improve the heat efficiency base fluids (water, engine oil, glycol), the interaction of nanoparticles (nanotubes, droplets, nanowires, metals and non-metals) into such traditional liquids is the most frequent mechanism and attained the researchers attention, especially in current decade. The nanofluid is a suspension of submerged solid particles in base fluids. The nano-materials convinced the applications in the field of nanotechnology, thermal engineering, industrial and bio-engineering. Following to such motivating applications in mind, current research reports the stagnation point flow of radiative micropolar nanofluid over an off centered rotating disk with applications of motile microorganisms. The novel dynamic of thermal radiation and activation energy are also incorporated. The appropriate transformations are utilized to reduce the partial differential equations into dimensionless forms. A numerical shooting scheme is used to obtain the approximate solution with MATLAB software. The effects of prominent parameter on velocity profile, nanofluid temperature, concentration of nanoparticles and microorganism profile are physically incorporated.

A common way to develop and exploit unconventional tight reservoirs is to use horizontal wells with multiple fractures. It has been postulated that in unconventional tight reservoirs as a result of hydraulic fracturing the stimulated region can be created near the hydraulic fracture. Recently, as practical alternatives to computationally intensive numerical models, analytical multi-linear flow models for unconventional tight reservoirs have been developed. In these models, fluid flow is assumed to obey Darcy's law. However, as unconventional tight gas reservoirs are characterized by widespread micro/nanoscale pores, non-Darcy flow typically occurs in the unstimulated region. In many unconventional tight reservoirs, despite their low permeability and the existence of the stimulated region, unstimulated regions can also affect the production significantly and thus it is necessary to consider the non-Darcy flow in unstimulated regions for modeling the reservoir production. In this paper, an analytical multi-linear flow model considering non-Darcy flow effect for unstimulated regions is proposed. Pseudo-time including pressure-dependent apparent permeability is introduced to derive the model. The pseudo-time is evaluated at the average pressure within the region of influence. The applicability of the model has been discussed. The model was validated by comparison with numerical simulation. It was found that the model’s results are in reasonable agreement with the numerical simulation results and the average relative error is less than 15%. The model can be used to simulate the multistage fractured horizontal well production in unconventional tight gas reservoirs where the non-Darcy flow effect is significant for practical purposes.

The aim of this work is the simulation of volumetric strain of tuberous crop during drying. We propose a poroelastic model for predicting the drying kinetics and volume loss of potato cubes during convective drying. The Biot’s theory of poroelasticity was used, which considers the Lamé parameters, Young’s modulus and Poisson’s ratio. Drying kinetics and volumetric strain were modeled and compared versus experimental data. An X-ray microtomograph coupled with image analysis was used to visualize the shape and size of the samples during drying. Drying experiments were conducted at 50, 60 and 70 °C, 20% RH, with an air velocity of 1 and 2 m/s. The drying process was interrupted several times to perform tomographic acquisitions. We found a period of ideal shrinkage, nevertheless, the volumetric strain reveals a kinetic behavior over time. The model computes the volumetric strain, which describes correctly the experimental data obtained by microtomography.

The present investigation centers around a numerical study of the impact of the presence of phototactic microorganisms on the thermo-bioconvection inside a wavy-walled horizontal porous annulus that is saturated with a Newtonian fluid. Consequently, an equation describing the movement of the microorganism cells is added to the mathematical model of natural convection. The thermal equilibrium, Oberbeck–Boussinesq approximations and Darcy law are taken into consideration. The transformed dimensionless partial differential equations with the temperature-concentration-stream function formulation are discretized applying the alternate direction implicit finite difference scheme. Results confirm that in the absence of microorganisms the Nusselt number decreases with the increase of the amplitude of wall waviness. It was also observed that the wall waviness influence the thermo-bioconvective instabilities and the flow regime structure.

We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [26], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. Here, we describe the model together with its numerical discretization, and we prove a convergence result assuming the non-degeneracy of the phase mobilities and that the discrete solutions remain physical in the sense that, roughly speaking, the porosity does not vanish and the fractures remain open. This is, to our knowledge, the first convergence result for this type of models taking into account two-phase flows in fractured porous media and the non-linear poromechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity [36, 37]. Numerical tests employing the Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and $\mathbb P_2$ finite elements for the mechanical deformation are also provided to illustrate the behavior of the solution to the model.

The article presents the development of reference materials of gas permeability of rocks (imitators) with traceability to Get 210–2019: State Primary Standard of Units of Specific Adsorption of Gases, Specific Surface Area of Pores, Specific Volume of Pores, Dimension of Pores, Open Porosity, and Coefficient of Gas Permeability of Solid Substances and Fabricated Materials. At the time of the beginning of this study, in the system of metrological assurance of gas permeability coefficient measurements, there were only reference materials of an approved type with traceability to the measurement results obtained in an interlaboratory experiment using gas permeability analyzers that were calibrated using the same reference materials, and reference materials with traceability to the State Primary Standard were absent. As a result, in the metrological practice of gas permeability measurement, there was no stable basis for comparison. This fact prompted the authors to research work in this area. The article considers the key stages of the process of reference materials development: analysis of similar reference materials of an approved type, selection of the starting material for reference materials, performing experimental research, and determining, on the basis of this research, the metrological characteristics of reference materials. Ceramic cylinders based on aluminum oxide were used as the starting material for reference materials; the method to measure the gas permeability coefficient is based on the method of stationary filtration. Measurements of the gas permeability coefficient were carried out on samples with different gas permeability using nitrogen and helium gases. Based on the results of these measurements, for each sample, the gas permeability coefficients were calculated at the given reverse pore pressure and the absolute gas permeability coefficient. As a result of the study, a set of reference materials of gas permeability of rocks (imitators) GSO 11546–2020 / GSO 11550–2020 was approved, the range of certified values is (0.1 10 –3 – 5) μ m 2 , and the expanded uncertainty of certified values Uo (k = 2, P = 0.95), is 3 %. The authors believe that the GSO 11546–2020 / GSO 11550–2020 dataset will ensure metrological traceability and reliability of gas permeability coefficient measurements. As a result, it will bring practical relevance to a petrophysical research laboratory.

Clarifying the flow laws of shale gas under high temperature and high pressure is the prerequisite to accurately predicting the productivity of deep shale gas wells. In this paper, a self-diffusion flow model of flow field and temperature field coupling (referred to as self-diffusion flow and heat coupling model) was established based on the previously proposed self-diffusion flow model, while considering the influence of the temperature field change. Then, its calculation result was compared with that of the flow model based on Darcy's law and Knudsen diffusion (referred to as modified Darcy flow model). Based on the self-diffusion flow and heat coupling model, the self-diffusion flow behaviors of deep shale gas under the influence of temperature field change were analyzed, and the influence of bottomhole temperature on the degree of reserve recovery of deep shale gas was discussed. Finally, the self-diffusion flow and heat coupling model was applied to simulate the production of one shale-gas horizontal well in the Upper Ordovician Wufeng Formation–Lower Silurian Longmaxi Formation in the Changning Block of the Sichuan Basin. And the following research results were obtained. First, at the same parameters, the shale gas production calculated by the self-diffusion flow and heat coupling model is higher than the result calculated by the modified Darcy flow model. Second, when temperature field change is taken into consideration, the self–dviffusion coefficient profile presents a peak, the gas density profile presents a valley and the data points corresponding to the peak/valley move synchronously to the internal formation, which indicates that the self–diffusion coefficient influences the gas mass transfer rate, and the influence range of near well low temperature on gas self-diffusion increases continuously as the production continues. Third, when the bottomhole temperature is lower than the formation temperature, the self–diffusion coefficient of the gas near the well decreases and the gas is blocked near the well, which reduces the gas well production. Fourth, the production simulation result of the case well shows that the self-diffusion flow and heat coupling model can predict the production of deep shale gas more accurately if temperature field change is taken into consideration. In conclusion, the self-diffusion flow and heat coupling model established in this paper is of higher reliability and accuracy and can be used for productivity simulation and prediction of deep shale gas wells. The conclusion of this paper has certain guiding significance for deep shale gas production and gas well productivity prediction.

We consider the homogenization limit of the compressible barotropic Navier–Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is decreasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy’s law. The range of sizes of particles we consider is exactly the same which leads to Darcy’s law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovskiĭ operator. This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents [Formula: see text].

The contradiction between supply and demand of energy leads to more and more attention on the large-scale energy storage technology; Compressed Air Energy Storage (CAES) technology is a new energy storage technology that is widely concerned in the world. The research of coupled heat transfer and seepage in fractured surrounding rocks is the necessary basis to evaluate the operation safety and effectiveness of CAES. Current studies point to the possibility of cracking in concrete liner seals, but the thermodynamic processes and leakage characteristics of compressed air in the presence of cracking and the heat transfer characteristics of seepage have not been addressed and reported. In order to investigate the leakage, the gas seepage and heat transfer law in fractured rock when the hard rock CAES gas reservoir seal cracks, the COMSOL fracture Darcy module, and the non-Darcy Forchheimer model are used as the constitutive seepage. The global ODE is used to calculate the thermodynamic process of compressed air in gas storage with coupled seepage and heat transfer process. The pressure and temperature of compressed air are obtained as the unsteady boundary of the seepage heat transfer model. A program for calculating the seepage and heat transfer characteristics of fractured surrounding rock in the CAES gas reservoir is established. On this basis, with the proposed Suichang CAES cavern as the background, the seepage and heat transfer characteristics of the fractured surrounding rock of the gas storage are studied. The results showed that when there are fewer cracks in the lining and surrounding rock of the air reservoir, the air pressure decreases due to a small amount of air leakage after 30 operation cycles, and the leakage rate of each cycle is 0.7% of the gas storage capacity, but it still meets the engineering requirements. If the plant is operating under these conditions, the charging rate will need to be increased by 1.2 kg/s per cycle charging stage. In the discharging and power generation phase, the high-pressure air that previously percolated into the rock mass cracks could flow back into the air storage through the lining cracks. Therefore, it is incorrect and unreliable to consider the gas which flows out from the inner surface of the lining as unusable. When the lining crack width is less than 0.3 mm, the seepage flow is Darcy flow and the non-Darcy effect can be ignored; when the lining crack width is greater than 0.5 mm, the non-Darcy effect of seepage cannot be ignored. The gas velocity in the surrounding rock fracture medium is on the order of 0.01 m/s with an influence range of over 100 m, and the gas velocity in the pore medium is on the order of 10-6 m/s with an influence range of 50 m. The findings of this study contribute to a better understanding of the interaction between the thermodynamic properties of compressed air and the seepage heat transfer process in compressed air storage underground reservoirs, as well as the gas leakage process in the event of liner seal cracking.

Existing models are built upon to develop new ones. As a foundational model in porous media flow, the Darcy flow model has been built upon by many researchers. The groundwater flow equation evolved from the Darcy equation. Dupuit–Forchheimer built upon it to develop the simplified forms for both the confined and unconfined aquifers flow, usable for studying groundwater flow into wells. Aside from height, permeability, and availability of water in the aquifers, other factors influence groundwater flow into wells, as enshrined in the flow equation. This paper investigates the roles of storability, hydraulic conductivity, and source/sink strength in both confined and unconfined groundwater flow into wells using the Dupuit–Forchheimer assumption. In this model, the Dupuit–Forchheimer pressure assumption is substituted into the groundwater flow equations and solved using the Bessel form for separation of variable technique, and Mathematica 11.2 computational software to obtain the expressions for the pressure, which are computed and presented quantitatively. The results show that an increase in the hydraulic conductivity and storability have no effect on the flow pressures in the confined and unconfined aquifers but cause fluctuation in the pressure structure in the unconfined aquifer; the source/sink strength factor causes fluctuation in the pressure structures in both confined and unconfined aquifers flow. However, in both confined/unconfined aquifers the pressures increase as the radii of the wells increase. Importantly, the fluctuation in the pressure structures causes a loss of energy for groundwater flow into the wells.

Liquid permeability of sedimentary rocks is relevant in several contexts, but gas permeability is easier to measure, so liquid permeability is typically estimated from gas permeability via empirical or semiempirical correction procedures. A frequently used and trusted procedure is the well-known Klinkenberg correction, which is based on the pressure dependence of gas permeability. However, from gaseous and liquid flow-through experiments on a series of Fontainebleau, Castlegate, Bentheim, and Obernkirchen sandstones, this study indicates that the equivalent liquid permeability derived from gas permeability via the Klinkenberg correction only compares with liquid permeability, when the gaseous flow adheres to Darcy’s law. The lower and upper limits to Darcy flow are defined by the Knudsen and Reynolds numbers, respectively. Both numbers can be estimated from porosity and pore-throat distribution, so from these properties, it is possible to assess the flow and pressure limits for the applicability of the Klinkenberg correction. For the studied sandstones, non-Darcy flow is indicated for the largest pores with diameters above approximately 10 μm, causing an erroneous Klinkenberg correction. Knudsen diffusion takes place in pores smaller than approximately 0.1 μm, but the contribution to the overall gas permeability of these small pores is, however, insignificant in these sandstones. Liquid permeability modeled from contributions from each pore size by using Kozeny’s equation and surface relaxation times from nuclear magnetic resonance data shows that the largest pores have no positive effect on permeability because of the existence of pore throats; instead, they may have a negative effect on permeability because of turbulence.

The article is devoted to the development of the theory of non-equilibrium pressure and temperature fields in two-phase thermodynamic systems simulating natural oil and gas reservoirs. Such systems are classified as highly dissipative, since the flow process is accompanied by the predominant action of internal friction forces. The studies made it possible to significantly clarify the concept of thermodynamic processes and establish new physical regularities governing the formation of temperature effects during fluid filtration in real porous media. It is shown that the maximum values of the temperature effect do not reach the value of the integral Joule–Thomson effect for any values of the medium and liquid parameters, and in real oil and gas reservoirs, temperature variations are several orders of magnitude smaller than those predicted on the basis of the equilibrium theory.

This paper deals with a numerical study of free convection in a horizontal cylindrical annulus filled with a fluidsaturated porous medium in the presence of a transverse magnetic field and the heat generation or absorption effect. It is assumed that the inner and outer walls of the cylindrical annulus are maintained at constant temperatures Ti and To , respectively, as Ti > To . In addition to the heat equation, the model consists of the equation of motion under the Darcy law and Boussinesq approximation. The system of equations is solved numerically by the alternatingdirection implicit finite difference method. This investigation concerns the effects of the magnetic field inclination angle, Hartmann number, and the heat generation or absorption coefficient on heat transfer and the flow pattern. The results demonstrate that the heat transfer rate and flow regime depend mainly on the characteristics mentioned. The obtained data are presented graphically in terms of the streamlines and isotherms.

Orientated-type flow channels of proton exchange membrane fuel cells cause non-Darcy effect occurring in flow regions. Therefore, the species transportation is affected by inertial effect. However, how the inertial force affects convection and diffusion of different species has not been discussed before. Thus, a modified two-dimensional, non-isothermal, two-phase and steady state model considering non-Darcy effect is employed in this study, and reactants and products transportations through diffusion and convection under inertial effects are quantitatively analyzed for the first time. Simulation results reveal that the convective transportation of reactants increases more under the influence of inertial force; water vapor transportation through convection increases the water content in porous regions. At the same time, liquid water expels more rapidly from gas diffusion layers under baffle regions, and enlarging baffle volumes increases the regions where the liquid water is rapidly removed under the inertial effect.

This study numerically presents the transient behaviour of complete evaporation process inside porous channel. Based on the modified enthalpy formulation under non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions, solutions have been obtained utilizing the finite volume method for various relevant parameters. The results have been validated with the experiments and they show good agreement between them. The results demonstrate that LTNE model should be employed while modelling the transient complete evaporation process in porous media. The non-Darcy effects have considerable influences on the locations of the initiation and termination of the phase change processes as compared to Darcy flow model. Therefore, the behaviours of the two-phase and the superheated vapour zones, when the steady-state solutions is reached, are substantially changed for non-Darcy flow as compared to Darcy model due this assumption provides an additional mechanism for heat transfer that represented by the heat exchange between the solid and fluid phases. It has been observed that the two-phase zone is considerably extended in the axial direction and this zone has not been occupied a large size towards transverse direction due to the axial velocity is noticeably higher than the transverse velocity, caused by reducing the diffusive energy through the two-phase region. The results indicate that non-Darcian effects leads to considerably extend the size of superheated region towards the outlet of the channel, which is not the case for Darcy model. The non-Darican effects become more pronounced for high Reynolds number and heat flux. The influences of varying in Reynolds number, heat flux, porosity, and solid thermal conductivity are substantial near the heated surface, whereas Darcy number has a minor impact on the solution of complete evaporation process. It can be emphasised from the present predictions that the non-Darcy flow along with LTNE condition is indispensable model for the complete evaporation process, especially under high mass flowrate as well heat flux.

The goal of this paper is to present a different approach to the homogenization of the Dirichlet boundary value problem in porous medium. Unlike the standard energy method or the method of two-scale convergence, this approach is not based on the weak formulation of the problem but on the very weak formulation. To illustrate the method and its advantages we treat the stationary, incompressible Navier-Stokes system with the non-homogeneous Dirichlet boundary condition in periodic porous medium. The nonzero velocity trace on the boundary of a solid inclusion yields a non-standard addition to the source term in the Darcy law. In addition, the homogenized model is not incompressible.

A computation analysis is performed to study double-diffusive natural convection in a right-angle trapezoidal cavity packed with a porous medium. The horizontal top and bottom boundaries are insulated and impermeable. The vertical left sidewall is kept at a constant heat flux and high concentration, whereas the temperature and concentration of the mixture at the inclined sidewall are held at lower temperatures and lower concentration. The dimensionless nonlinear system is solved by employing finite difference method along Successive Under Relaxation technique. The findings are compared and validated with the existing literature for the Darcy flow driven through a single buoyancy effect (difference in density is only due to temperature variations) in a porous square enclosure. The numerical results are expressed in the form of dimensionless temperature, concentration, streamlines, isotherms and iso-concentrations, local and average Nusselt and Sherwood numbers. It is demonstrated that the Rayleigh number and buoyancy parameter enhance both surface heat and concentration rates.

Introduction. In this article, the problem of temperature distribution in an oil-bearing formation with a hydraulic fracture and a vertical injection well is numerically modeled. Materials and Methods. To describe the process of temperature distribution in the formation under the action of the fluid injected into the formation, the Fourier-Kirchhoff equation of convective heat transfer is used. To solve this equation, the discontinuous Galerkin method on staggered unstructured grids is used. To describe the process of pressure change in the formation under the action of the injection well, an equation is used that is obtained based on the continuity equation and Darcy’s law. To solve it, the discontinuous Galerkin method on an unstructured triangular grid is used. To parallelize the numerical algorithm, the MPI library is used. Results. The article presents a numerical algorithm and the results of modeling the dynamics of the temperature fields in an oil reservoir with a hydraulic fracture and a vertical injection well. Discussion and Conclusion. A numerical algorithm based on the discontinuous Galerkin method for math modeling of the temperature and pressure fields in a oil-bearing formation with a hydraulic fracture and injection well was developed and implemented. The results obtained for the distribution of temperature and pressure in the fracture are adequate and in good agreement with the specified initial-boundary conditions. Further work in this direction involves modeling on tetrahedral unstructured meshes for a more accurate study of the ongoing processes.

A flow erosion model of water inrush was developed that couples the Darcy, Forchheimer, and Navier–Stokes fields under the theory of continuum mechanics. Water flow and fluidized particles were regarded as single-phase mixed fluids based on the fundamentals of flow transition (aquifer laminar flow to turbulent flow) being the main cause of mine water inrush; the effects of rock disintegration and the coupled effects of flow and erosion were incorporated. The water source of the aquifer, water-inrush channel of the fracture network, and flow path of the slope roadway export were organically connected in a unified flow field. The weak integral forms of the Darcy, Forchheimer, and Navier–Stokes equations were constructed according to the principle of virtual displacement. The convective terms were discretized by the finite volume method and the other terms were discretized by the finite element method; the calculation source program was developed based on the finite element language and its compiler (FELAC). The source program system was used to numerically simulate water inrush in a fractured zone in the Jiangjiawan Mine. It reproduced the entire dynamic process of a water inrush and revealed the distribution and variation characteristics of the pressure field, velocity field, porosity, and concentration, as well as the mechanisms underlying the sudden change in water flow.

Gas permeability of moulds and cores is an important factor to consider in the casting process. In foundries, gas permeability is measured by using instruments which give dimensionless numbers. Thi...

Assessment of the long term possibilities and risks related to geological storage requires insight in the deep groundwater flow systems. The objective of this paper is to show the relevance of the deep creep flow of the earth’s viscous upper mantle and crust as a complement to the groundwater flow. The paper presents an approach based on Fourier decomposition of the topography. The creep flow equations are solved analytically, which results in simple indices like penetration depth and relaxation time characterizing the gravity-driven creep flow. Thanks to the very high effective viscosity of the Earth’s subsurface a Darcy-like equation is obtained in which the ‘creep conductivity’ is Fourier mode dependent, which allows for simple comparison with the hydraulic conductivity for groundwater flow. Order of magnitude calculations indicate that for horizontal length scales of 100–1000 km the subsurface creep velocities are 0.3–30 mm/year, respectively, which shows that creep velocities in the deep subsurface are significant with respect to deep groundwater velocities.

A clear understanding of osmosis’s influence on Darcy flow is essential in the context of petroleum industry. Thus, in this study, osmosis’s influence on Darcy’s law is investigated theoretically b...

As an important indicator, permeability can predict the gas drainage yield and prevent the mine gas disasters. We first reviewed our previous inversion method to investigate permeability coefficient of gas in coal particles; however, the relationship between permeability and adsorption pressure had not been summarized and explained theoretically. Here, a permeability evolution model including two crucial parameters of initial permeability and deformation coefficient was developed. The inversion gas permeability coefficients were converted into permeability, and then, the permeability of the same coal sample was fitted according to the evolution model. The results show that (i) the modeled results are matched reasonably well with the inversion permeability data, and thus, this model has been validated; and (ii) as volatile matter content increases, the initial permeability decreases exponentially, but the deformation coefficient basically grows in a linear trend. These two parameters are coupling to lead to a negative exponential decrease in permeability as gas pressure rises.

Background/Objectives: Atmospheric visibility is predominantly affected by aerosols and it is improved by reducing the effects of concentration of smog and haze. The model is developed suitably to study the effects of smog and haze through the concentration of aerosols in the presence and absence of chemical reactions. Methods: The atmospheric fluid in the inclined channel flow is assumed to be incompressible, laminar, and presented through the solution of Navier Stokes equation for velocity and advectiondiffusion equation for the concentration of aerosols. The governing equations of the study are attained as two-dimensional partial differential equations and are solved using the perturbation technique with appropriate boundary conditions. The solution leads to detailed graphs for velocity components and concentration of aerosols. Findings: This study examined the effects of concentration of smog and haze semi analytically in a two-dimensional channel bounded by porous beds together with the effects of the electric and magnetic field. The derived results are depicted graphically for different parameters. The obtained results of concentration clearly show that there is no considerable influence of porous, gravity and magnetic field effect in reducing smog and haze concentration, whereas the increasing electric field reduces the haze concentration. Keywords: Aerosols; Darcy law; haze; perturbation method; porous beds;smog

Hydraulic conductivity plays a vital role in the studies encompassing explorations on flow and porous media. The study investigates the compaction characteristics of a river sand (Beas, Sutlej, and Ghaggar rivers) and fly ash mix in different proportions and evaluates four empirical equations for estimating hydraulic conductivity. Experiments show that an increase in the fly ash content results in a decrease in the maximum dry density (MDD) and an increase in the corresponding optimum moisture content (OMC) of sand–fly ash samples. MDD at optimum fly ash content was achieved at low water content, which resulted in less dry unit weight than that of typical conventional fill. In Beas, Sutlej, and Ghaggar sands the optimum fly ash content up to which the hydraulic conductivity value reduced uniformly was found to be 30, 45, and 40%, respectively. Any further increase in the fly ash content results in a negligible decrease in hydraulic conductivity value. The observed hydraulic conductivity of sand–fly ash mix lies in the range of silts, which emboldens the use of sand–fly ash mix as embankment material. Further, the evaluation of empirical equations considered in the study substantiates the efficacy of the Terzaghi equation in estimating the hydraulic conductivity of river sand-fly ash mix.

Permeability functionalities are noticeable in the wood drying and impregnation processes because it has an effect on the bulk flow of a fluid through the wood structure. The aim of this work was to measure the radial and tangential specific gas permeability of Eucalyptus nitens wood for 10 different provenances. Furthermore, the effect of the radial position (defined as inner-wood, middle-wood, and outer-wood) on the measurements was investigated. The specific gas permeability of E. nitens ranged in order of magnitude from 10−20 to 10−16 m2. The results showed that the radial position had a statistically significant effect on the specific permeability in both flow directions and, in general, the permeability values increased from inner-wood to outer-wood. Moreover, the scanning electron microscopy images showed obstruction of vessels by tyloses which explained the lower permeability values obtained in the inner-wood of E. nitens. Differences in the magnitude of the permeability values could explain the diverse drying and impregnation behavior according to the wood zones, the low permeability values in the internal wood-zone could suggest an opportunity for improving it, through chemical modification or structural changes to introduce new functionalities of this radial wood-zone in Eucalyptus nitens.

In this paper we discussedincompressiblefluid flow problem through free and porous areas by using Darcy's law and continuity equation, by apply the boundary conditions required to specify the solutio

Seepage failure test means the foundation soil instability caused by underground water perturbation, which is one of the most common methods to evaluate a site seepage resistance, but it is time consuming. High water pressure is always applied to shorten the test time. However, significant error is generated as calculating the seepage failure slope (SPS), the gradient, due to the nonlinear characteristics as the pressure increases. The Darcy law and the permeability coefficient are no longer applicable. Here, an analytical method for confined aquifer is proposed to investigate the nonlinear permeability coefficient and SPS. The in situ tests were conducted on the dislocation interfaces at the reservoir banks of the Baihetan hydropower station to verify the proposed method. The test results indicate that the SPS calculated values by nonlinear equation is much smaller than that by linear equation, which means that the conventional linear calculation results will overestimate the resistance of the site seepage failure. Moreover, the results calculated by steady and nonsteady nonlinear equations are very close, while the storativity is between 1 × 10−3 and 1 × 10−5, which is the ordinary storativity of most aquifers in this region. Hence, in general, the steady nonlinear equation can be used to calculate the site SPS to simplify the calculation.

The description of the lower limit of the application of Darcy’s law is given, which is due to the influence of a number of anomalous factors that arise during the filtration flow of liquids through low-permeability finely dispersed media. The influence of such factors as the action of the forces of intermolecular interaction is considered; boundary layers and surface wettability; concentration and electric potential gradients; the presence of impurities in the liquid; gas saturation and vaporization; changes in the structure of the porous medium, separately or in the aggregate, leading to a violation of Darcy’s law.

The accurate characterization of fabrics used in vacuum assisted resin transfer molding (VARTM) is essential in order to model the flow through these porous preforms. A wide range of these fabrics are available for composite manufacturing through VARTM and thus brings about a need to opt a methodology which characterizes the in-plane permeability of these preforms. These permeability values can then be used in simulations that can track the flow front progression and mold filling time. This work identifies the permeability of an E-glass fabric based on Darcy's law. Woven fabric having areal weight of 200 grams per square meter (gsm) is under consideration. The experiments are conducted at constant pressure conditions using 2D Radial flow method. Stereo microscopy of the preform material is done for detailed study of the weaving pattern. It is concluded that plain woven fabric exhibits anisotropic behavior when tested for in-plane permeability. Permeability is found to be higher in a direction which offers more interspacing between adjacent fibers threads causing more resin to flow in this direction.

The paper describes the filtration curve up to the filtration rate at which the nonlinear filtration appears due to the inertial component of the filtration flow structure, that is, the critical value of the filtration rate υcr, which determines the upper limit of the possible application of Darcy’s law.

Investigating the liquid transport kinetics of solid catalysts is of great importance for gaining a better understanding of the manufacturing and performance of such catalysts during reaction upon contact with the liquid. Terahertz pulsed imaging (TPI) coupled with a newly designed flow cell was used to quantify the rate of water ingress into α-alumina pellets with a range of different porosities. A wide range of compaction forces (cold compaction, 7–58 kN) and sintering conditions (no firing and sintering at 1200 and 1300 °C) was investigated to explore the optimal pellet microstructure, i.e. mechanically strong but sufficiently porous for fast liquid transport kinetics. The results confirm that both the microstructure characteristics, particularly porosity, as well as the surface properties, i.e. wettability, influence the liquid transport kinetics. Fitting the TPI penetration rates with a power law shows that the type of observed mass transport characteristics is consistent with Darcy flow. The Lucas–Washburn equation was used to calculate the hydraulic radius based on the transport data. In summary, the results demonstrate that TPI has great potential to study the liquid transport kinetics of porous ceramic catalysts and catalyst supports and that can comfortably quantify transport processes at rates of 250 μm s−1 and beyond in such substrates for better quality control and optimised design and performance of such materials.

Composite industry increasingly uses natural fibres because of their environment-friendly advantages. These natural fibres may swell during the mould filling process when they absorb resin, and this swelling reduces the porosity and permeability of the preform. Hence, computational modelling of the flow in swelling porous media would be useful to model the different mould filling processes with the swelling effect. This paper demonstrates the possibility of using computational fluid dynamics to study the effect of swelling on liquid composite moulding mould filling in isotropic and orthotropic porous media. An empirical relation for local permeability changes is used to model the flow of resin under constant volume flow rate and constant injection pressure conditions. The flow front locations and inlet pressure predicted by the computational fluid dynamics simulations are in good agreement with the experimental data for 1D rectilinear flow case. Further, to capture the flow patterns, two different arrangements employing point injection are considered. It was observed that the volume fraction of resin in swelling porous medium is 6% less than rigid porous medium at any given time. It was also observed that the location of the inlet and outlet has a considerable effect on the flow front advancement.

We consider a micropolar fluid flow in a media perforated by periodically distributed obstacles of size e. A non-homogeneous boundary condition for microrotation is considered: the microrotation is assumed to be proportional to the rotation rate of the velocity on the boundary of the obstacles. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when e tends to zero, an analogue of the classical micropolar Darcy law in the theory of porous media is derived.

Magnetic field plays an important role in numerous fields such as biological, chemical, mechanical and medical research. In clinical and medical research the high field magnets are extremely important to create 3D images of anatomical and diagnostic importance from nuclear magnetic resonance signals. In view of these applications, the purpose of present work is to explore the impact of an external magnetic field on the viscoelastic fluid flow in the existence of electroosmosis, porous medium and slip boundary conditions. The governing equation is modified under the suitable dimensionless quantities. The resulting non-dimensional differential equation is evaluated by analytical as well as numerical (finite difference and cubic B-spline) methods. The convergence analysis is also presented for the numerical methods. The variations of sundry parameters on velocity, volume flow rate and skin friction are presented through graphical representations. The current analysis depicts that, the higher velocities are noticed in viscoelastic fluid as compared with Newtonian fluid. The velocity enhances with rising of slip and Darcy parameters. Volume flow rate rises with the slip and viscoelastic parameters. Skin friction is a decreasing function of zeta potential, Darcy number and Hall current parameter. The limiting solutions can be captured for the Newtonian fluid model by setting the viscoelastic parameter to zero.

Radiative behavior of nanomaterial with different shapes of nanoparticles within a permeable domain is simulated with an innovative approach. To predict the characteristics of Al2O3–H2O, shape factor impact has been considered. Outputs were extracted for various buoyancy, Hartmann and radiation. Outcomes indicate that domination of conduction mode occurs in existence of magnetic force. Augmenting temperature gradient can be seen for greater Rd. CVFEM has been involved to obtain and discuss impacts of all the governing parameters. The graphical outcomes demosntare that Nuave reduces owing to a significant augmentation in Lorentz term. A converse behavior has been observed due to an augment in Ra and Rd. No-slip case has been considered to generate the numerical and graphical results. The present results show many new behaviors that warrant further study especially non-Newtonian fluid models.

Groundwater flow modeling is an important tool for understanding and computing hydrology and water availability of an aquifer zone. However, an accurate representation of boundaries and their initial conditions are vital for simulation of the groundwater flow phenomena. In this study, efforts have been made to develop a GIS based methodology for estimating flux across boundaries of the study area using Darcy flow tool. The spatial maps of topography, bore log, transmissivity, hydraulic conductivity, porosity and groundwater levels for the study area were created in ArcGIS 9.3.1 using krigging method. A buffer zone of 1×1 km2 cell size was created on inner and outer side of the boundaries and Darcy flow model was used to estimate specified flux across boundaries. The groundwater behavior of the study area was simulated with specified flux boundary condition (Neumann boundary condition) and no flow boundary condition to assess importance and estimation accuracy of estimated flux. Darcy model output indicates that flux across the boundaries contributed about 36.20 mm in average annual change in groundwater table depth. With estimated specified flux, simulation accuracy of groundwater flow model (R2) increased to 0.97 from 0.90. The satisfactory level (R2=0.97) of simulation accuracy reveals that developed methodology can be used for estimating flux across boundaries in the absence of physical boundaries.

The solvability of the initial-boundary value problem is proved for the system of equations of one-dimensional unsteady fluid motion in a heat-conducting viscous porous medium

Nanomaterials are quite significant in the physiological and engineering processes. Such materials have motivated the recent scientists in view of their enhanced thermophysical characteristics than conventional liquids. With this in mind, the intention here is to address the peristaltic activity of Sisko nano-liquid subject to Hall and Ohmic heating effects. Fluid-saturated porous space is modeled using the modified Darcy’s law. Thermal radiation is also accounted. In addition, the analysis carried out is subject to thermal jump, velocity slip and zero mass flux condition. Numerical solution to the resulting nonlinear problem through the lubrication approach is developed. Solution analysis is made by pointing out the impacts of sundry variables. The temperature profile increases by increasing the Hartman number due to the Joule heating effect. However, temperature decreases by increasing the Hall parameter.

In the present study, gravity wave interaction with a series of submerged artificial permeable reefs is analysed within the framework of linearised water wave theory. For wave past porous walls of the artificial reefs, a quadratic pressure drop is assumed to account for the wave energy dissipation due to the changes in wave height. The physical problem is handled for a solution using a numerical model based on the iterative multi-domain boundary element method and the developed numerical model is validated with known results in the literature. The iterative model is compared with the numerical model based on linear pressure drop boundary condition (i.e., Darcy law). The study reveals that the wave transmission reduces with the increase in the number of reef units. It is demonstrated that the transmission coefficient can be reduced to less than 0.5 when the number of reef units is greater than or equal to three for a relative height greater than 0.7, reef porosities less than 20% and for 0.40h