渗透率场敏感系数的数值计算

由渗流微分方程定解问题，利用格林互易定理从理论上导出了渗透率场敏感系数 的计算公式，并借助于数值积分和差分方法给出了渗透率场敏感系数的离散形式. 由Peacem an方程建立了井压渗透率场敏感系数与网格压力渗透率场敏感系数的关系. 对理论模型利用 三维不均匀非稳定渗流场的压强数值解计算了井压对渗透率场的敏感系数，并用敏感系数的 直接计算方法进行了验证，结果表明本文的计算方法与直接计算方法相吻合.     

裂缝-孔隙介质储层渗透率表征及其影响因素分析

含裂缝多孔介质渗透率预测是非常规油气资源勘探开发的一个紧迫问题.现有多孔介质岩石物理模型通常利用圆形孔管模拟宏观岩石孔隙空间,难以定量描述软孔隙/裂缝在压力作用下的闭合情况,缺乏裂缝/孔隙间流量交换的连通机制.本文提出含三维裂缝/软孔隙网络多孔介质模型,将储层岩石裂缝/软孔隙表示为椭圆截面微管,建立了周期性压力作用下微...     

A fractal model for the starting pressure gradient for Bingham fluids in porous media embedded with randomly distributed fractal-like tree networks

An analytical expression is derived for the starting pressure gradient for Bingham fluids in porous media embedded with randomly distributed fractal-like tree networks based on fractal theory and technique. The proposed model relates the flow rate and the starting pressure gradient to the structural parameters of porous media and microstructural parameters of fractal-like tree networks, the yield stress and fractal dimensions of porous media and maximum mother diameter of randomly distributed fractal-like tree networks. The results show that the starting pressure gradient decreases with the increase of porosity of matrix material, fractal dimension for mother diameters, diameter ratio and permeability, and the starting pressure gradient increases with the increase of the length ratio and the yield stress. The model predictions from the present model for the starting pressure gradient are in good agreement with the available expression.     

Application of GA in optimization of pore network models generated by multi-cellular growth algorithms

In pore network modeling, the void space of a rock sample is represented at the microscopic scale by a network of pores connected by throats. Construction of a reasonable representation of the geometry and topology of the pore space will lead to a reliable prediction of the properties of porous media. Recently, the theory of multi-cellular growth (or L-systems) has been used as a flexible tool for generation of pore network models which do not require any special information such as 2D SEM or 3D pore space images. In general, the networks generated by this method are irregular pore network models which are inherently closer to the complicated nature of the porous media rather than regular lattice networks. In this approach, the construction process is controlled only by the production rules that govern the development process of the network. In this study, genetic algorithm has been used to obtain the optimum values of the uncertain parameters of these production rules to build an appropriate irregular lattice network capable of the prediction of both static and hydraulic information of the target porous medium.     

Numerical analysis of bacterial transport in saturated porous media

A mathematical model to describe bacterial transport in saturated porous media is presented. Reversible/irreversible attachment and growth/decay terms were incorporated into the transport model. Additionally, the changes of porosity and permeability due to bacterial deposition and/or growth were accounted for in the model. The predictive model was used to fit the column experimental data from the literature, and the fitting result showed a good match with the data. Based on the parameter values determined from the literature experimental data, numerical experiments were performed to examine bacterial sorption and/or growth during bacterial transport through saturated porous media. In addition, sensitivity analysis was performed to investigate the impact of key model parameters for bacterial transport on the permeability and porosity of porous media. The model results show that the permeability and porosity of porous media could be altered due to bacterial deposition and growth on the solid matrix. However, variation of permeability due to bacterial growth was trivial compared with natural permeability variation. Copyright © 2005 John Wiley & Sons, Ltd.     

Variable-density groundwater flow and solute transport in irregular 2D fracture networks

Numerical simulations of variable-density flow and solute transport have been conducted to investigate dense plume migration for various configurations of 2D fracture networks. For orthogonal fractures, simulations demonstrate that dispersive mixing in fractures with small aperture does not stabilize vertical plume migration in fractures with large aperture. Simulations in non-orthogonal 2D fracture networks indicate that convection cells form and that they overlap both the porous matrix and fractures. Thus, transport rates in convection cells depend on matrix and fracture flow properties. A series of simulations in statistically equivalent networks of fractures with irregular orientation show that the migration of a dense plume is highly sensitive to the geometry of the network. If fractures in a random network are connected equidistantly to the solute source, few equidistantly distributed fractures favor density-driven transport. On the other hand, numerous fractures have a stabilizing effect, especially if diffusive transport rates are high. A sensitivity analysis for a network with few equidistantly distributed fractures shows that low fracture aperture, low matrix permeability and high matrix porosity impede density-driven transport because these parameters reduce groundwater flow velocities in both the matrix and the fractures. Enhanced molecular diffusion slows down density-driven transport because it favors solute diffusion from the fractures into the low-permeability porous matrix where groundwater velocities are smaller. For the configurations tested, variable-density flow and solute transport are most sensitive to the permeability and porosity of the matrix, which are properties that can be determined more accurately than the geometry and hydraulic properties of the fracture network, which have a smaller impact on density-driven transport.     

FracKfinder: A MATLAB Toolbox for Computing Three-Dimensional Hydraulic Conductivity Tensors for Fractured Porous Media

Fractures in porous media have been documented extensively. However, they are often omitted from groundwater flow and mass transport models due to a lack of data on fracture hydraulic properties and the computational burden of simulating fractures explicitly in large model domains. We present a MATLAB toolbox, FracKfinder, that automates HydroGeoSphere (HGS), a variably saturated, control volume finite-element model, to simulate an ensemble of discrete fracture network (DFN) flow experiments on a single cubic model mesh containing a stochastically generated fracture network. Because DFN simulations in HGS can simulate flow in both a porous media and a fracture domain, this toolbox computes tensors for both the matrix and fractures of a porous medium. Each model in the ensemble represents a different orientation of the hydraulic gradient, thus minimizing the likelihood that a single hydraulic gradient orientation will dominate the tensor computation. Linear regression on matrices containing the computed three-dimensional hydraulic conductivity (K) values from each rotation of the hydraulic gradient is used to compute the K tensors. This approach shows that the hydraulic behavior of fracture networks can be simulated where fracture hydraulic data are limited. Simulation of a bromide tracer experiment using K tensors computed with FracKfinder in HGS demonstrates good agreement with a previous large-column, laboratory study. The toolbox provides a potential pathway to upscale groundwater flow and mass transport processes in fractured media to larger scales.     

Prediction of diffusion coefficients in porous media using tortuosity factors based on interfacial areas

Determination of aqueous phase diffusion coefficients of solutes through porous media is essential for understanding and modeling contaminant transport. Prediction of diffusion coefficients in both saturated and unsaturated zones requires knowledge of tortuosity and constrictivity factors. No methods are available for the direct measurement of these factors, which are empirical in their definition. In this paper, a new definition for the tortuosity factor is proposed, as the real to ideal interfacial area ratio. We define the tortuosity factor for saturated porous media (tau5) as the ratio S/S(o) (specific surface of real porous medium to that of an idealized capillary bundle). For unsaturated media, tortuosity factor (tau(a)) is defined as a(aw)/a(aw),o (ratio of the specific air-water interfacial area of real and the corresponding idealized porous medium). This tortuosity factor is suitably measured using sorptive tracers (e.g., nitrogen adsorption method) for saturated media and interfacial tracers for unsaturated media. A model based on this new definition of tortuosity factors, termed the interfacial area ratio (IAR) model, is presented for the prediction of diffusion coefficients as a function of the degree of water saturation. Diffusion coefficients and diffusive resistances measured in a number of saturated and unsaturated granular porous media, for solutes in dilute aqueous solutions, agree well with the predictions of the IAR model. A comparison of permeability of saturated sands estimated based on tau(s) and the same based on the Kozeny-Carman equation confirm the usefulness of the tau(s) parameter as a measure of tortuosity.     

Coefficient of permeability determined by measurable parameters

Selection of a coefficient of permeability or the hydraulic conductivity in practical situations has required a fair amount of judgment and the expectation of accuracy can range over an order of magnitude. A determination of the coefficient based on measurable characteristics would greatly reduce the judgment aspect. The coefficient of permeability of a porous media can be derived from the following measurable parameters: the density and viscosity of the permeating fluid, the porosity of the media, the average hydraulic radius of the pores, and the gravitation constant. The hydraulic radius is calculated from the grain size distribution analysis of the porous media assuming spherical particles and a factor to account for the shape of the particles. The shape factor ranges from 1.0 to 1.4 in the extreme but, for common porous media, ranges only from 1.0 to about 1.1. All of the variables, except the shape factor, are measurable by standard test procedures. An important advantage of using this procedure for determining permeability is that the factors can be visualized as part of the physical flow process. The procedure assumes laminar flow and applies equally to liquid or gas fluids.     

Simulation of gaseous diffusion in partially saturated porous media under variable gravity with lattice Boltzmann methods

Liquid distributions in unsaturated porous media under different gravitational accelerations and corresponding macroscopic gaseous diffusion coefficients were investigated to enhance understanding of plant growth conditions in microgravity. We used a single-component, multiphase lattice Boltzmann code to simulate liquid configurations in two-dimensional porous media at varying water contents for different gravity conditions and measured gas diffusion through the media using a multicomponent lattice Boltzmann code. The relative diffusion coefficients (D rel) for simulations with and without gravity as functions of air-filled porosity were in good agreement with measured data and established models. We found significant differences in liquid configuration in porous media, leading to reductions in D rel of up to 25% under zero gravity. The study highlights potential applications of the lattice Boltzmann method for rapid and cost-effective evaluation of alternative plant growth media designs under variable gravity.     

The effect of microporosity on transport properties in porous media

Sizeable amounts of connected microporosity with various origins can have a profound effect on important petrophysical properties of a porous medium such as (absolute/relative) permeability and capillary pressure relationships. We construct pore-throat networks that incorporate both intergranular porosity and microporosity. The latter originates from two separate mechanisms: partial dissolution of grains and pore fillings (e.g. clay). We then use the reconstructed network models to estimate the medium flow properties. In this work, we develop unique network construction algorithms and simulate capillary pressure–saturation and relative permeability–saturation curves for cases with inhomogeneous distributions of pores and micropores. Furthermore, we provide a modeling framework for variable amounts of cement and connectivity of the intergranular porosity and quantifying the conditions under which microporosity dominates transport properties. In the extreme case of a disconnected inter-granular network due to cementation a range of saturations within which neither fluid phase is capable of flowing emerges. To our knowledge, this is the first flexible pore scale model, from first principles, to successfully approach this behavior observed in tight reservoirs.     

A fractal model for streaming potential coefficient in porous media

Streaming potential is the result of coupling between a fluid flow and an electric current in porous rocks. The modified Helmholtz–Smoluchowski equation derived for capillary tubes is mostly used to determine the streaming potential coefficient of porous media. However, to the best of our knowledge, the fractal geometry theory is not yet applied to analyse the streaming potential in porous media. In this article, a fractal model for the streaming potential coefficient in porous media is developed based on the fractal theory of porous media and on the streaming potential in a capillary. The proposed model is expressed in terms of the zeta potential at the solid?liquid interface, the minimum and maximum pore/capillary radii, the fractal dimension, and the porosity of porous media. The model is also examined by using another capillary size distribution available in published articles. The results obtained from the model using two different capillary size distributions are in good agreement with each other. The model predictions are then compared with experimental data in the literature and those based on the modified Helmholtz–Smoluchowski equation. It is shown that the predictions from the proposed fractal model are in good agreement with experimental data. In addition, the proposed model is able to reproduce the same result as the Helmholtz–Smoluchowski equation, particularly for high fluid conductivity or large grain diameters. Other factors influencing the streaming potential coefficient in porous media are also analysed.     

含湿孔隙岩石有效热导率的数值分析

本文采用有限元方法研究含湿孔隙岩石的有效热导率,即随机划分网格并指定材料性质,建立三维含湿孔隙岩石的有限元模型,模型的上下表面施加不同的温度,侧面绝热,计算出总热流,然后结合上下表面的温度梯度计算出岩石的有效热导率.考虑到单个随机模型不一定具有代表性,对给定的孔隙率和饱和度均生成了200种矿物、水、空气随机分布的岩石模型,进行Monte Carlo实验和统计分析,统计分析结果与前人实验结果吻合良好.数值分析结果表明,孔隙岩石的有效热导率与岩石的孔隙率、饱和度、固体矿物组分及孔隙的分布情况有关,数值计算的误差随着网格数目的增加而减小.此有限元方法可以用来估算岩石的有效热导率,在已知组分性质的多矿物岩石物性计算方面有广阔应用前景.     

渗流方程的三维粗化算法

本文提出了地下流体渗流问题的三维解粗化算法,在粗网格内流体压强分布用直接解法求解三维渗流方程,用这些解计算粗网格的等效渗透率,在流体流速大的区域仍采用精细网格的计算方法.用所得等效渗透率计算了粗化网格的渗流场的压强分布,结果表明渗流方程的三维粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了100多倍.     

A transport phase diagram for pore-level correlated porous media

Transport in porous media is often characterized by the advection–dispersion equation, with the dispersion coefficient as the most important parameter that links the hydrodynamics to the transport processes. Morphological properties of any porous medium, such as pore size distribution, network topology, and correlation length control transport. In this study we explore the impact of correlation length on transport regime using pore-network modelling. Earlier direct simulation studies of dispersion in carbonate and sandstone rocks showed larger dispersion compared to granular homogenous sandpacks. However, in these studies, isolation of the impact of correlation length on transport regime was not possible due to the fundamentally different pore morphologies and pore-size distributions. Against this limitation, we simulate advection–dispersion transport for a wide range of Péclet numbers in unstructured irregular networks with “different” correlation lengths but “identical” pore size distributions and pore morphologies. Our simulation results show an increase in the magnitudes of the estimated dispersion coefficients in correlated networks compared to uncorrelated ones in the advection-controlled regime. The range of the Péclet numbers which dictate mixed advection–diffusion regime considerably reduces in the correlated networks. The findings emphasize the critical role of correlation length which is depicted in a conceptual transport phase diagram and the importance of accounting for the micro-scale correlation lengths into predictive stochastic pore-scale modelling.     

Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

三维层状孔隙介质中弹性波的一种积分表达式I：理论

孔隙介质弹性波传播理论在地球物理勘探、地震工程和岩土动力学等领域有着广泛的应用.而孔隙介质中的弹性波受孔隙度、渗透率、流体黏滞系数等参数的影响,因此研究波场的传播特征将有助于分析和提取这些信息.本文在Biot理论的基础上,针对三维层状孔隙介质模型,利用在合成理论地震图的研究中已经被证实具有稳定、高效且适用范围较广的Luco-Apsel-Chen（LAC）广义反透射方法,给出了弹性波场的一种积分形式的半解析解,可通过数值方法高效、准确地计算层状孔隙介质中的理论波场,所以该积分形式的半解析解可为三维层状孔隙介质波场传播特征的理论数值模拟研究提供一种新的途径和手段.     

Bayesian uncertainty quantification for flows in heterogeneous porous media using reversible jump Markov chain Monte Carlo methods

In this paper, we study the uncertainty quantification in inverse problems for flows in heterogeneous porous media. Reversible jump Markov chain Monte Carlo algorithms (MCMC) are used for hierarchical modeling of channelized permeability fields. Within each channel, the permeability is assumed to have a log-normal distribution. Uncertainty quantification in history matching is carried out hierarchically by constructing geologic facies boundaries as well as permeability fields within each facies using dynamic data such as production data. The search with Metropolis–Hastings algorithm results in very low acceptance rate, and consequently, the computations are CPU demanding. To speed-up the computations, we use a two-stage MCMC that utilizes upscaled models to screen the proposals. In our numerical results, we assume that the channels intersect the wells and the intersection locations are known. Our results show that the proposed algorithms are capable of capturing the channel boundaries and describe the permeability variations within the channels using dynamic production history at the wells.     

基于三相孔隙弹性理论的火山岩气层岩石物理响应特征分析

Unlike previous theories with velocity and/or elastic modulus averaging, we use a three-phase porous rock physics model developed by Santos for analyzing the seismic response of two immiscible fluids in saturated porous media. Considering reservoir reference pressure and coupling drag of two fluids in pores, the effects of frequency, porosity, and gas saturation on the phase velocities of the P-and S-waves are discussed in detail under field conditions. The effects of porosity and gas saturation on Vp/Vs are also provided. The data for our numerical experiments are from a sample of deep volcanic rock from Daqing. The numerical results show that the frequency dispersion effect can be ignored for deep volcanic rocks with low porosity and low permeability. It is concluded that for deep volcanic rocks the effect of gas content in pores on Vp/Vs is negligible but the effect of porosity is significant when there is a certain amount of water contained in the pores. The accurate estimate of lithology and porosity in this case is relatively more important.