非饱和多孔介质中混合元法的化学-热-渗流-力学耦合的本构模拟

在文献[1]中建立的多孔介质中化学-热-渗流-力学(CTHM)本构模型基础上,针对文献[2]建立的非饱和多孔介质中热-渗流-力学耦合分析的混合有限元方法,发展了非饱和多孔介质中混合元的化学-热-渗流-力学(CTHM)耦合本构模拟算法。采用非关联流动多重屈服准则模拟非饱和多孔介质的材料非线性行为。推导了u-pw-pa-T形式的包含了耦合率本构方程积分的向后欧拉映射算法和一致性弹塑性切线模量矩阵(单元刚度矩阵)的混合元一致性算法。本文给出了临界状态线(CSL)和状态边界面(SBS)两个屈服准则的一致性算法。数值结果显示了本文所发展的混合元耦合本构模拟算法在模拟由热、化学、力学荷载共同引起的多孔介质中化学-热-渗流-力学(CTHM)耦合行为的能力和有效性。     

Numerical Algorithms for Network Modeling of Yield Stress and other Non-Newtonian Fluids in Porous Media

Many applications involve the flow of non-Newtonian fluids in porous, subsurface media including polymer flooding in enhanced oil recovery, proppant suspension in hydraulic fracturing, and the recovery of heavy oils. Network modeling of these flows has become the popular pore-scale approach for understanding first-principles flow behavior, but strong nonlinearities have prevented larger-scale modeling and more time-dependent simulations. We investigate numerical approaches to solving these nonlinear problems and show that the method of fixed-point iteration may diverge for shear-thinning fluids unless sufficient relaxation is used. It is also found that the optimal relaxation factor is exactly equal to the shear-thinning index for power-law fluids. When the optimal relaxation factor is employed it slightly outperforms Newton??s method for power-law fluids. Newton-Raphson is a more efficient choice (than the commonly used fixed-point iteration) for solving the systems of equations associated with a yield stress. It is shown that iterative improvement of the guess values can improve convergence and speed of the solution. We also develop a new Newton algorithm (Variable Jacobian Method) for yield-stress flow which is orders of magnitude faster than either fixed-point iteration or the traditional Newton??s method. Recent publications have suggested that minimum-path search algorithms for determining the threshold pressure gradient (e.g., invasion percolation with memory) greatly underestimate the true threshold gradient when compared to numerical solution of the flow equations. We compare the two approaches and reach the conclusion that this is incorrect; the threshold gradient obtained numerically is exactly the same as that found through a search of the minimum path of throat mobilization pressure drops. This fact can be proven mathematically; mass conservation is only preserved if the true threshold gradient is equal to that found by search algorithms.     

Fractals — An overview of potential applications to transport in porous media

We present an overview of the potential applicability of fractal concepts to various aspects of transport phenomena in heterogeneous porous media. Three examples of phenomena where a fractal approach should prove illuminating are presented. In the first example we consider pore level heterogeneities as typified by pore surface roughness. We suggest that roughness may be usefully modelled by fractal curves and surfaces and also cite experimental evidence for regarding pores as fractals. In the second example we consider a fractal network approach to modelling large-scale heterogeneities. The presence of features on all length scales in simple fractal models should capture the essential role played by the presence of heterogeneities on many scales in natural reservoirs. Studies of transport phenomena in such models may yield valuable insights into the problems of macroscopic dispersion. The final example concerns dispersion in multiphase flow. Here the fractal character is attributed to the distribution of the fluid phases rather than the porous medium itself. Again studies of transport phenomena in simple fractal models should help to clarify various problems associated with the corresponding phenomena in real reservoirs.     

On the Validity of Darcy's Law for Stable High-Concentration Displacements in Granular Porous Media

Recently, it has been suggested that Darcy's Law might not be applicable for modelling miscible, density-dependent flow in porous media. To investigate this, three sets of careful laboratory column experiments were performed on coarse and medium sands, consisting of upward displacement of water by sodium chloride solutions with concentrations ranging from 5 to 200g/l. Data on salt concentrations and water pressures were collected in horizontal transects along the flow direction. Salt concentration data were also collected in the influent and exit lines. The experimental data were analysed using a simplified approach based on Darcy's Law alone, applied with the assumption of a sharp interface. Darcy's Law was used to estimate porous medium permeability by fitting predictions to experimental data. Consistent estimates of permeability were obtained for each set of experiments. The results indicate that Darcy's Law adequately describes high concentration displacements through saturated coarse- and medium-grained porous media.     

Geometrical interpretation of the multi‐point flux approximation L‐method

In this paper, we first investigate the influence of different Dirichlet boundary discretizations on the convergence rate of the multi‐point flux approximation (MPFA) L‐method by the numerical comparisons between the MPFA O‐ and L‐method, and show how important it is for this new method to handle Dirichlet boundary conditions in a suitable way. A new Dirichlet boundary strategy is proposed, which in some sense can well recover the superconvergence rate of the normal velocity. In the second part of the work, the MPFA L‐method with homogeneous media is studied. A systematic concept and geometrical interpretations of the L‐method are given and illustrated, which yield more insight into the L‐method. Finally, we apply the MPFA L‐method for two‐phase flow in porous media on different quadrilateral grids and compare its numerical results for the pressure and saturation with the results of the two‐point flux approximation method. Copyright © 2008 John Wiley & Sons, Ltd.     

双相介质参数反演的遗传算法

研究遗传算法在双相介质材料参数反演中的作用。将一维双相介质在动载荷下的表面位移响应的计算值与实测值进行拟合，以最小方差作为目标函数，把双相介质参数反演问题归结为非线性多峰函数的最优化问题。全局最优解的求解采用多点并行搜索的遗传算法，克服了传统梯度爬山法难于求得全局最优的困难。算例表明了遗传算法的可行性和稳健性。     

Rheo-PIV of a yield-stress fluid in a capillary with slip at the wall

An analysis of the yielding and flow behavior of a model yield-stress fluid, 0.2 wt% Carbopol gel, in a capillary with slip at the wall has been carried out in the present work. For this, a study of the flow kinematics in a capillary rheometer was performed with a two-dimensional particle image velocimetry (PIV) system. Besides, a stress-controlled rotational rheometer with a vane rotor was used as an independent way to measure the yield stress. The results in this work show that in the limit of resolution of the PIV technique, the flow behavior agrees with the existence of a yield stress, but there is a smooth solid?Cliquid transition in the capillary flow curve, which complicates the determination of the yield stress from rheometrical data. This complication, however, is overcome by using the solely velocity profiles and the measured wall shear stresses, from which the yield-stress value is reliably determined. The main details of the kinematics in the presence of slip were all captured during the experiments, namely, a purely plug flow before yielding, the solid?Cliquid transition, as well as the behavior under flow, respectively. Finally, it was found that the slip velocity increases in a power-law way with the shear stress.     

Modelling fluid flow in fractured-porous rock masses by finite-element techniques

One of the major difficulties of modelling fluid flow processes in hard-rock geologies is the complex nature of the porosity systems. Hydraulic behaviour in these rock masses is characterized by both porous and fractured interflow zones. Traditionally, fractured-porous rocks have been modelled as an equivalent porous medium or as a system of fractures separated by impermeable blocks. A new method is proposed that unifies these two approaches for modelling fluid flow processes in fractured-porous media. The basic idea is to use a combination of isoparametric elements for the porous zones and line elements for the fractures. The coupling between the governing equations for each element type is achieved using the superposition principle. The effectiveness of the new approach is demonstrated by comparing numerical solutions with known solutions for problems of flow and solute transport in fractured-porous media.     

Thermodynamically Consistent Limiting Forced Convection Heat Transfer in a Asymmetrically Heated Porous Channel: An Analytical Study

Fluid transport and the associated heat transfer through porous media is of immense importance because of its numerous practical applications. In view of the widespread applications of porous media flow, the present study attempts to investigate the forced convective heat transfer in the limiting condition for the flow through porous channel. There could be many areas, where heat transfer through porous channel attain some limiting conditions, thus, the analysis of limiting convective heat transfer is far reaching. The primary aim of the present study is focused on the limiting forced convection analysis considering the flow of Newtonian fluid between two asymmetrically heated parallel plates filled with saturated porous media. Utilizing a few assumptions, which are usually employed in the literature, an analytical methodology is executed to obtain the closed-form expression of the temperature profile, and in the following the expression of the limiting Nusselt numbers. The parametric variations of the temperature profile and the Nusselt numbers in different cases have been shown highlighting the influential role of different performance indexing parameters, like Darcy number, porosity of the media, and Brinkman number of forced convective heat transfer in porous channel. In doing so, the underlying physics of the transport characteristics of heat has been delineated in a comprehensive way. Moreover, a discussion has been made regarding an important feature like the onset of point of singularity as appeared on the variation of the Nusselt number from the consideration of energy balance in the flow field, and in view of second law of thermodynamics.     

Network model evaluation of permeability and spatial correlation in a real random sphere packing

In principle, network models can replicate exactly the microstructure of porous media. In practice, however, network models have been constructed using various assumptions concerning pore structure. This paper presents a network model of a real, disordered porous medium that invokes no assumptions regarding pore structure. The calculated permeability of the model agrees well with measured permeabilities, providing a new and more rigorous confirmation of the validity of the network approach. Several assumptions commonly used in constructing network models are found to be invalid for a random packing of equal spheres. In addition, the model permits quantification of the effect of pore-scale correlation (departure from randomness) upon permeability. The effect is comparable to reported discrepancies between measured permeabilities and predictions of other network models. The implications of this finding are twofold. First, a key assumption of several theories of transport in porous media, namely that pore dimensions are randomly distributed upon a network, may be invalid for real porous systems. Second, efforts both to model and to measure pore-scale correlations could yield more accurate predictions of permeability.     

On the modelling conditions of mass transfer in porous media presenting capacitance effects by a dispersion-convection equation for the mobile fluid and a diffusion equation for the stagnant fluid

The modelling of mass transfer in porous media presenting capacitance effects by a dispersion-convection equation for the mobile fluid and a diffusion equation for the stagnant fluid has been shown (Piquemal, 1992) to be erroneous in the general case, because it is assumed that the mean concentration of the flowing fluid equals the point concentration at the boundary of the stagnant fluid. This boundary condition cannot be realized. This paper gives the conditions that allows the use of this kind of model with an acceptable approximation. The problem has been solved in the case of two schematic structures: the first is a cylindrical tube with stagnant pockets in its wall, the second a stratified medium. The characteristic lengths of the mobile and immobile domain must obey a criterion in which the porous medium characteristics and the flow velocity appear.     

Dynamic Network Model of Mobility Control in Emulsion Flow Through Porous Media

Modeling the flow of emulsion in porous media is extremely challenging due to the complex nature of the associated flows and multiscale phenomena. At the pore scale, the dispersed phase size can be of the same order of magnitude of the pore length scale and therefore effective viscosity models do not apply. A physically meaningful macroscopic flow model must incorporate the transport of the dispersed phase through the porous material and the changes on flow resistance due to drop deformation as it flows through pore throats. In this work, we present a dynamic capillary network model that uses experimentally determined pore-level constitutive relationships between flow rate and pressure drop in constricted capillaries to obtain representative transient macroscopic flow behavior emerging from microscopic emulsion flow at the pore level. A parametric analysis is conducted to study the effect of dispersed phase droplet size and capillary number on the flow response to both emulsion and alternating water/emulsion flooding in porous media. The results clearly show that emulsion flooding changes the continuous-phase mobility and consequently flow paths through the porous media, and how the intensity of mobility control can be tuned by the emulsion characteristics.     

An adaptive finite element method for viscoplastic flows in a square pipe with stick–slip at the wall

This paper presents the numerical solution of non-linear yield stress phenomena by using a new mixed anisotropic auto-adaptive finite element method. The Poiseuille flow of a Bingham fluid with slip yield boundary condition at the wall is considered. Despite its practical interest, for instance for pipeline flows of yield-stress fluids such as concrete and cements, this problem has not been addressed yet to our knowledge. The case of a pipe with a square section has been investigated in detail. The computations cover the full range of the two main dimensionless numbers and exhibit complex flow patterns: all the different flow regimes are completely identified.     

Linking Morphology of Porous Media to Their Macroscopic Permeability by Deep Learning

Flow, transport, mechanical, and fracture properties of porous media depend on their morphology and are usually estimated by experimental and/or computational methods. The precision of the computational approaches depends on the accuracy of the model that represents the morphology. If high accuracy is required, the computations and even experiments can be quite time-consuming. At the same time, linking the morphology directly to the permeability, as well as other important flow and transport properties, has been a long-standing problem. In this paper, we develop a new network that utilizes a deep learning (DL) algorithm to link the morphology of porous media to their permeability. The network is neither a purely traditional artificial neural network (ANN), nor is it a purely DL algorithm, but, rather, it is a hybrid of both. The input data include three-dimensional images of sandstones, hundreds of their stochastic realizations generated by a reconstruction method, and synthetic unconsolidated porous media produced by a Boolean method. To develop the network, we first extract important features of the images using a DL algorithm and then feed them to an ANN to estimate the permeabilities. We demonstrate that the network is successfully trained, such that it can develop accurate correlations between the morphology of porous media and their effective permeability. The high accuracy of the network is demonstrated by its predictions for the permeability of a variety of porous media.

     

Pressure-driven and electroosmotic non-Newtonian flows through microporous media via lattice Boltzmann method

The lattice Boltzmann method is developed to simulate the pressure-driven flow and electroosmotic flow of non-Newtonian fluids in porous media based on the representative elementary volume scale. The flow through porous media was simulated by including the porosity into the equilibrium distribution function and adding a non-Newtonian force term to the evolution equation. The non-Newtonian behavior is considered based on the Herschel–Bulkley model. The velocity results for pressure-driven non-Newtonian flow agree well with the analytical solutions. For the electroosmotic flow, the influences of porosity, solid particle diameter, power law exponent, yield stress and electric parameters are investigated. The results demonstrate that the present lattice Boltzmann model is capable of modeling non-Newtonian flow through porous media.     

SPH-based numerical treatment of the interfacial interaction of flow with porous media

In this paper, the macroscopic equations of mass and momentum are developed and discretized based on the smoothed particle hydrodynamics (SPH) formulation for the interaction at an interface of flow with porous media. The theoretical background of flow through porous media is investigated to highlight the key constraints that should be satisfied, particularly at the interface between the porous media flow and the overlying free flow. The study aims to investigate the derivation of the porous flow equations, computation of the porosity, and treatment of the interfacial boundary layer. It addresses weak assumptions that are commonly adopted for interfacial flow simulation in particle-based methods. As support to the theoretical analysis, a two-dimensional weakly compressible SPH model is developed based on the proposed interfacial treatment. The equations in this model are written in terms of the intrinsic averages and in the Lagrangian form. The effect of particle volume change due to the spatial change of porosity is taken into account, and the extra stress terms in the momentum equation are approximated by using Ergun's equation and the subparticle scale model to represent the drag and turbulence effects, respectively. Four benchmark test cases covering a range of flow scenarios are simulated to examine the influence of the porous boundary on the internal, interface, and external flows. The capacity of the modified SPH model to predict velocity distributions and water surface behavior is fully examined with a focus on the flow conditions at the interfacial boundary between the overlying free flow and the underlying porous media.     

The nature of flows through porous media

The present paper points out that the pressure drop of a porous media flow is only due to a small extent to the shear force term usually employed to derive the Kozeny—Darcy law. For a more correct derivation, additional shear terms have to be taken into account since the fluid is also exposed to elongational forces when it passes through the porous media matrix. These are usually not taken into account in the conventional theoretical treatment of flow through porous media as is explained in the literature. This explains why the available theoretical derivations of the Kozeny—Darcy relationship, which are based on one part of the shear-caused pressure drop only, require an adjustment of the constant in the theoretically derived equation to be applicable to experimental results. Details of this derivation are given in this paper and existing derivations are extended to yield better agreement with experiments.To verify experimentally some of the results of the theoretical derivation provided, porous media flows of dilute polymer solutions are studied experimentally. It is shown that the addition of small amounts of high molecular weight polymers to a solvent with Newtonian flow properties causes drastic pressure drop increases if the flow rate exceeds an onset flow rate corresponding to a critical Deborah number of the porous matrix-polymer solution system. This can only be explained if the flow field in the porous medium is exposed to shear and elongational strain. The extent of this interaction is deduced from experimental findings.     

On the existence of diffusive waves in some non-linear unsteady flows of non-Newtonian fluids through porous media

We investigate the unsteady flow of power law fluids through porous media. We determine the pressure and velocity distributions when fluid is injected into a porous medium of infinite extend. We obtain solutions of progressive-wave type by means of a translation. We determine the necessary conditions for the existence of this type of solution regarding the prescribed pressure of injection and the initial pressure and velocity distributions in the porous medium. Similarity solutions are also obtained for the cases of a prescribed time dependent pressure of injection and a prescribed constant flow rate of injection. In the latter case the resulting ordinary differential equation is solved numerically. Point source solutions are also obtained for the case when an amount of fluid is instantaneously injected into the porous media. In all cases the rheological effects are presented and analyzed.     

多孔介质中的双稳热对流

对矩形横截面多孔介质中热对流的复杂分岔行为──二次分岔进行研究．使用Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ约化并充分利用问题本身的对称性，研究了于最低的两个不同临界Ｒａｙｌｅｉｇｈ数处从平凡的静态传热解产生的热对流主分岔解之间的相互作用；揭示了主分岔解的二次分岔并给出了主分岔解及二次分岔解的渐近展开．稳定性分析表明从第二临界Ｒａｙｌｅｉｇｈ数产生的主分岔解经二次分岔后由不稳定变得稳定，从而与由最小临界Ｒａｙｌｅｉｇｈ数产生的主分岔解组成双稳定热对流．文中理论分析可较恰当地解释已有的数值模拟结果．