Thawing in saturated porous media

A mathematical model for thawing in a saturated porous medium is considered. The well-posedness of the corresponding mathematical problem is proved and similarity solutions are found.

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Sommario Si considera un modello matematico per to scongelamento in un mezzo poroso saturo. Viene dimostrata la buona posizione del corrispondente problema matematico e si trovano soluzioni di similarità.

     

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Nonlinear flow in fractured porous media

Conventional models of filtration in fractured porous bodies involve certain unwarrantable assumptions related to the definition of basic equations and the underestimation of a connection between the effective properties of a body and both the stress system and the pressure of a flowing fluid. A new theory is developed with the help of reconsidering those underlying assumptions and of a conception of the body being subject to elastic deformations. The theory is illustrated by means of studying a particular problem of stationary filtration.     

Approximate analytical solutions for solute transport in two-layer porous media

Mathematical models for transport in layered media are important for investigating how restricting layers affect rates of solute migration in soil profiles; they may also improve the analysis of solute displacement experiments. This study reports an (approximate) analytical solution for solute transport during steady-state flow in a two-layer medium requiring continuity of solute fluxes and resident concentrations at the interface. The solutions were derived with Laplace transformations making use of the binomial theorem. Results based on this solution were found to be in relatively good agreement with those obtained using numerical inversion of the Laplace transform. An expression for the flux-averaged concentration in the second layer was also obtained. Zero- and first-order approximations for the solute distribution in the second layer were derived for a thin first layer representing a water film or crust on top of the medium. These thin-layer approximations did not perform as well as the binomial solution, except for the first-order approximation when the Peclet number,P, of the first layer, was low (P<5). Results of this study indicate that the ordering of two layers will affect the predicted breakthrough curves at the outlet of the medium. The two-layer solution was used to illustrate the effects of dispersion in the inlet or outlet reservoirs using previously published data on apparatus-induced dispersion.The U.S. Government right to retain a non-exclusive, royalty free licence in and to any copyright is acknowledged.     

Nonlinear fluid flow laws for orthotropic porous media

Nonlinear fluid flow laws for orthotropic porous media are written in invariant tensor form. As usual in the theory of fluid flow through porous media [1, 2], the equations contain the flow velocity up to the second power. Expressions that determine the nonlinear resistances to fluid flow are presented and it is shown that, on going over from linear to nonlinear flow laws, the asymmetry effect may manifest itself, that is, the fluid flow characteristics may differ along the same straight line in the positive and negative directions. It is shown that, as compared with the linear fluid flow law for orthotropic media when for three symmetry groups a single flow law is sufficient, in nonlinear laws the anisotropy manifestations are much more variable and each symmetry group must be described by specific equations. A system of laboratory measurements for finding the nonlinear flow characteristics for orthotropic porous media is considered.     

Advective fluxes in multiphase porous media under nonisothermal conditions

We consider the case in which more than one fluid phase occupies the void space of a porous medium. The advective flux law is formulated for a fluid phase, under nonisothermal conditions and with the presence of solutes in the fluid phases. The derivation of the flux laws is based on an approximated version of the averaged balance equation for linear momentum. Taking into account momentum transfer through the interface between the fluid phases, leads to coupling between the flow in adjacent phases. Fluxes are also shown to depend on the surface tension at the interface between the adjacent fluid phases. Since the latter depends on temperature and solute concentration in the two phases, the advective flux is shown to depend on both temperature and solute concentration gradients in the two phases. A preliminary order of magnitude analysis gives conditions under which the coupling phenomena are not negligible. The approach is applied to the unsaturated zone, as a typical example of a multiphase porous medium.The main conclusion is that the well known Darcy law for single phase flow, may have to be modified for a multi fluid phase system, especially when temperature and solute concentration are not uniform.     

双相介质参数反演的混沌搜索方法

混沌运动能在一定的范围内按自身的规律不重复地遍历所有状态。利用这个特点 ,本文将混沌运动引入到双相介质参数反问题的研究中。首先利用边界元方法实现了由介质参数到地表位移的非线性映射 ,然后通过建立合成位移与实测位移的相关函数将参数识别问题归结为优化问题 ,最后利用混沌运动指导优化搜索求得介质参数。算例结果表明了混沌搜索方法用于双相介质参数反演问题的可行性和有效性     

Deletion of nondominant effects in modeling transport in porous media

A methodology for eliminating nondominant effects in models that describe transport phenomena in porous media is presented. The methodology is based on the introduction of dimensionless numbers and on a proper evaluation of the order of magnitude of terms. These dimensionless numbers are redefined as characteristics of transport and transformation phenomena in porous media. It is shown that different time scales and different length scales may have to be employed for different variables. A method for evaluating the order of magnitude of the error of prediction when terms are deleted, is presented.     

Fluid flow through heterogeneous porous media in a rotating square channel

An analytical three-dimensional solution to the fluid flow problem through heterogeneous porous media in a rotating square channel is presented. The permeability of the fluid saturated porous domain varies in the vertical direction, thus affecting the imposed main flow in the channel. As a result of Coriolis acceleration, secondary circulation in a plane perpendicular to the main flow direction is created. A particular example of a monotonic distribution of the permeability function is analyzed leading to a single vortex secondary circulation. Nevertheless, multiple vortex secondary flow solutions are possible depending on the particular variation of the permeability in the vertical direction. No secondary motion is expected for isothermal flows in homogeneous porous media.     

Fluid flow in fractured and fracturing porous media: A unified view

Fluid flow in fractures that pre-exist or propagate in a porous medium can have a major influence on the deformation and flow characteristics. With the aim of carrying out large-scale calculations at reasonable computing costs, a sub-grid scale model has been developed. While this model was originally embedded in extended finite element methods, thereby exploiting some special properties of the enrichment functions, we will herein show that, using proper micro–macro relations, in particular for the mass balance, sub-grid scale models can be coupled to a range of discretisation methods at the macroscopic scale, from standard interface elements to isogeometric finite element analysis.     

An overview on nonlinear porous flow in low permeability porous media

This paper gives an overview on nonlinear porous flow in low permeability porous media, reveals the microscopic mechanisms of flows, and clarifies properties of porous flow fluids. It shows that, deviating from Darcy's linear law, the porous flow characteristics obey a nonlinear law in a low-permeability porous medium, and the viscosity of the porous flow fluid and the permeability values of water and oil are not constants. Based on these characters, a new porous flow model, which can better describe low permeability reservoir, is established. This model can describe various patterns of porous flow, as Darcy's linear law does. All the parameters involved in the model, having definite physical meanings, can be obtained directly from the experiments.     

Parameters inversion of fluid-saturated porous media based on neural networks

The multi-layers feedforward neural network is used for inversion of material constants of fluid-saturated porous media. The direct analysis of fluid-saturated porous media is carried out with the boundary element method. The dynamic displacement responses obtained from direct analysis for prescribed material parameters constitute the sample sets training neural network. By virtue of the effective L-M training algorithm and the Tikhonov regularization method as well as the GCV method for an appropriate selection of regularization parameter, the inverse mapping from dynamic displacement responses to material constants is performed. Numerical examples demonstrate the validity of the neural network method. Project supported by the National Natural Science Foundation of China (Nos. 19872002 and 10272003) and Climbing Foundation of Northern Jiaotong University.     

Effects of an impermeable wall in dissipative dynamics of saturated porous media

A phase transition model for porous media in consolidation is studied. The model is able to describe the phenomenon of fluid-segregation during the consolidation process, i.e., the coexistence of two phases differing on fluid content inside the porous medium under static load. Considering pure Darcy dissipation, the dynamics is described by a Cahn–Hilliard-like system of partial differential equations (PDE). The goal is to study the dynamics of the formation of stationary fluid-rich bubbles. The evolution of the strain and fluid density profiles of the porous medium is analysed in two physical situations: fluid free to flow through the boundaries of the medium and fluid flow prevented at one of the two boundaries. Moreover, an analytic result on the position of the interface between the two phases is provided.     

Variable density flow in porous media. A study by means of pore level numerical simulations

This paper is devoted to the modelling of isothermal low Reynolds and Mach numbers transient compressible flow through porous media. Traditionally, this type of flow at the macroscopic level is described by the classical Darcy's law combined with a mass balance that includes the transient term. This model is called the ‘classic model’. The aim of this paper is to explore the validity of this classic model. To this end, the flow of an ideal gas is considered within two‐dimensional model porous media. The flow is due to the imposed pressure variations at the outlet of the fluid domain. At the microscopic level, the flow is computed by solving the full compressible Navier–Stokes equations in two dimensions. Special attention is given to the outlet boundary conditions. The analysis is based on the comparison between the macroscopic data, obtained on the one hand by spatially averaging the microscopic results, and on the other hand by solving the problem directly at the macroscopic level. Situations for which a good agreement is found between the two series of data and situations for which discrepancies are observed are exhibited. These various behaviours are discussed in terms of the various time scales controlling the flow and are explained by analysing the flow structure at pore level. Copyright © 1999 John Wiley & Sons, Ltd.     

弹塑性多孔介质流固耦合新理论:混合耦合理论

在全球气候变化和双碳政策的大背景下,多孔介质中固体的变形和流体的输运问题变得尤为重要。然而,在多孔介质中建立流固耦合模型仍面临的挑战之一是需要考虑跨越宏观尺度到纳米尺度的耦合作用。本文利用基于非平衡热力学的混合耦合理论,提出了一个弹塑性多孔介质流固耦合新模型,在同一个理论框架内研究了弹性变形、塑性变形和液体渗流之间跨尺度的耦合,考虑了耗散过程中的熵产,并利用Helmholtz自由能来连接宏观尺度上的力学变形和纳米尺度上的液体输运之间的相互作用。在应力-应变关系中采用了弹塑性刚度系数以反映塑性的影响。同时,经典的达西定律扩展为考虑固体的塑性变形。通过与文献中模型的比较,验证了该模型的有效性。最后,数值分析表明在多孔介质的流固耦合中塑性变形具有比较显著的影响。     

Propagation of nitromethane detonations in porous media

The characteristics of the propagation of a detonation in chemically sensitized nitromethane in a dense porous medium are investigated. By introducing liquid NM+15% (by weight) DETA into densely packed beds of solid spherical glass beads 66μm to 2.4 mm in diameter, a highly heterogeneous explosive mixture is obtained. The critical (i.e., failure) charge diameter of this mixture is systematically measured in unconfined charges over a wide range of bead sizes. Velocity measurements are also made for the various charges. It is found that there exists a critical bead size above which the critical diameter decreases with increasing bead size and below which it decreases with decreasing bead size. This result indicates an abrupt change in the mechanism of propagation at the critical bead size. Velocity measurements further support this by emphasizing the different behavior above and below the critical point.     

An extended theory to predict the onset of viscous instabilities for miscible displacements in porous media

Many enhanced oil recovery schemes involve the displacement of oil by a miscible fluid. Whether a displacement is stable or unstable has a profound effect on how efficiently a solvent displaces oil within a reservoir. That is, if viscous fingers are present, the displacement efficiency and, hence, the economic return of the recovery scheme is seriously impaired bacause of macroscopic bypassing of the oil. As a consequence, it is of interest to be able to predict the boundary which separates stable displacements from those which are unstable.This paper presents a dimensionless scaling group for predicting the onset of hydrodynamic instability of a miscible displacement in porous media. An existing linear perturbation analysis was extended in order to obtain the scaling group. The new scaling group differs from those obtained in previous studies because it takes into account a variable unperturbed concentration profile, both transverse dimensions of the porous medium, and both the longitudinal and the transverse dispersion coefficient.It has been shown that stability criteria derived in the literature are special cases of the general condition given here. Therefore, the stability criterion obtained in this study should be used for a displacement conducted under arbitrary conditions. The stability criterion is verified by comparing it with miscible displacement experiments carried out in a Hele-Shaw cell. Moreover, a comparison of the theory with some porous medium experiments from the literature also supports the validity of the theory.Nomenclature c solvent concentration - C _g fractional glycerine volume - D molecular diffusion coefficient, cm²/s - D _L longitudinal dispersion coefficient, cm²/s - D _T transverse dispersion coefficient, cm²/s - g gravitational acceleration, cm/s² - h distance between the plates, cm - I _sr dimensionless scaling group - k permeability, cm² - L _x width of the porous medium, cm - L _y height of the porous medium, cm - t time, s - u velocity in thex direction, cm/s - v velocity in they direction, cm/s - V displacement velocity, cm/s - w velocity in thez direction, cm/s - z length of the graded viscosity bank, cm - eigenvalue in thex direction - eigenvalue in they direction - wave number - viscosity, poise - density, g/cc - time constant, s^-1 - porosity