The Role of Probabilistic Approaches to Transport Theory in Heterogeneous Media

A physical picture of contaminant transport in highly heterogeneous porous media is presented. In any specific formation the associated governing transport equation is valid at any time and space scale. Furthermore, the advective and dispersive contributions are inextricably combined. The ensemble average of the basic transport equation is equivalent to a continuous time random walk (CTRW). The connection between the CTRW transport equation, in a limiting case and the familiar advection–dispersion equation (ADE) is derived. The CTRW theory is applied to the results of laboratory experiments, field observations, and simulations of random fracture networks. All of these results manifest dominant non-Gaussian features in the transport, over different scales, which are accounted for quantitatively by the theory. The key parameter controlling the entire shape of the contaminant plume evolution and breakthrough curves is advanced as a more useful characterization of the transport than the dispersion tensor, which is based on moments of the plume. The role of probabilistic approaches, such as CTRW, is appraised in the context of the interplay of spatial scales and levels of uncertainty. We then discuss a hybrid approach, which uses knowledge of non-stationary aspects of a field site on a larger spatial scale (trends) with a probabilistic treatment of unresolved structure on a smaller scale (residues).     

页岩气和致密砂岩气藏微裂缝气体传输特性

页岩气和致密砂岩气藏发育微裂缝,其开度多在纳米级和微米级尺度且变化大,因此微裂缝气体传输机理异常复杂.本文基于滑脱流动和努森扩散模型,分别以分子之间碰撞频率和分子与壁面碰撞频率占总碰撞频率的比值作为滑脱流动和努森扩散的权重系数,耦合这两种传输机理,建立了微裂缝气体传输模型. 该模型考虑微裂缝形状和尺度对气体传输的影响. 模型可靠性用分子模拟数据验证.结果表明:(1)模型能够合理描述微裂缝中所有气体传输机理,包括连续流动,滑脱流动和过渡流动;(2)模型能够描述不同开发阶段,微裂缝中各气体传输机理对传输贡献的逐渐变化过程;(3)微裂缝形状和尺度影响气体传输,相同开度且宽度越大的微裂缝,气体传输能力越强,且在高压和微裂缝大开度的情况下表现更明显.      

Behavior of Nonaqueous Phase Liquids in Fractured Porous Media under Two‐Phase Flow Conditions

After dense nonaqueous phase liquids (DNAPLs) travel downward through the subsurface, they typically come to rest on fractured bedrock or tight clay layers, which become additional pathways for DNAPL migration. DNAPLs trapped in fractures are continuous sources of groundwater contamination. To decide whether they can be left in place to dissolve or volatilize, or must be removed with active treatment, the movement of DNAPLs in fractured media must be understood at a fundamental level. This work presents numerical simulations of the movements of DNAPLs in naturally fractured media under twophase flow conditions. The flow is modeled using a multiphase network flow model, used to develop predictive capabilities for DNAPL flow in fractures. Capillary pressure–saturation–relative permeability curves are developed for twophase flow in fractures. Comparisons are made between the behavior in crystalline, almost impermeable rocks (e.g. granite) and more permeable rocks like sandstone, to understand the effects of the rock matrix on the displacement of the DNAPLs in the fracture. For capillarydominated flow, displacements occur as a sequence of jumps, as the invading phase overcomes the capillary pressure at downgradient apertures. Preferential channels for the displacement of nonaqueous phase are formed due to high fracture aperture in some regions.     

Transport of Neutrally Buoyant and Dense Variably Sized Colloids in a Two-Dimensional Fracture with Anisotropic Aperture

The transport of monodisperse as well as polydisperse colloid suspensions in a two-dimensional, water saturated fracture with spatially variable and anisotropic aperture is investigated with a particle tracking model. Both neutrally buoyant and dense colloid suspensions are considered. Although flow and transport in fractured subsurface formations have been studied extensively by numerous investigators, the transport of dense, polydisperse colloid suspensions in a fracture with spatially variable and anisotropic aperture has not been previously explored. Simulated snapshots and breakthrough curves of ensemble averages of several realizations of a log-normally distributed aperture field show that polydisperse colloids exhibit greater spreading than monodisperse colloids, and dense colloids show greater retardation than neutrally buoyant colloids. Moreover, it is demonstrated that aperture anisotropy oriented along the flow direction substantially increases colloid spreading; whereas, aperture anisotropy oriented transverse to the flow direction retards colloid movement.     

Effects of the Correlation Length on the Hydraulic Parameters of a Fracture Network

We consider the influences of correlation length and aperture variability on the REV, the equivalent permeability of a fracture network, and the uncertainty in the equivalent permeability using a two-dimensional orthogonal bond percolation model. The percolation threshold, correlation length, effective conductivity, and coefficient of variation of the effective conductivity are investigated over statistically representative multiple realizations with Monte Carlo simulations in 2D fracture networks that have log-normally distributed individual fracture permeabilities. We show that although the aperture variability is large, the REV and the correlation length are similar near the percolation threshold. In contrast, when the fracture density is much larger than the percolation threshold they diverge as the aperture variability increases. We characterize the effects of correlation length and aperture variability on effective conductivity with a simple function. From the coefficient of variation analysis, the correlation length can be a criterion for evaluating which conceptual model is appropriate for describing the flow system for a given fracture network when aperture variability is sufficiently small. However, discrete fracture network models are recommended for flow simulation models because of the difficulty of REV estimation and the uncertainty in equivalent hydraulic parameters when aperture variability is large.     

考虑多重运移机制耦合页岩气藏压裂水平井数值模拟

页岩作为典型的微纳尺度多孔介质,游离气与吸附气共存,传统的达西定律已无法准确描述气体在页岩微纳尺度的运移规律.基于双重介质模型和离散裂缝模型构建页岩气藏分段压裂水平井模型,其中基岩中考虑气体的黏性流、Knudsen 扩散以及气体在基岩孔隙表面的吸附解吸,吸附采用Langmuir等温吸附方程;裂缝中考虑黏性流和Knudsen扩散,在此基础上建立基岩-裂缝双重介质压裂水平井数学模型并采用有限元方法对模型进行求解.结果表明,基岩固有渗透率越小,表面扩散和Knudsen扩散的影响越大,反之则越小;人工裂缝的性质包括条数、开度、半长以及间距,主要影响压裂水平井生产早期,随着人工裂缝参数值的增加,压裂水平井产能增加,累产气量也越大.其次,页岩气藏压裂诱导缝和天然裂缝的发育程度对页岩气藏的产能有很大的影响,水平井周围只有人工裂缝,周围天然裂缝不开启或不发育时,页岩气藏的水平井的产能较低.      

A two scale model of porous rocks with Drucker-Prager matrix: Application to a sandstone

The present paper is devoted to a micromechanical model of porous rocks and its application to a sandstone. This original model takes advantage of a recent homogenization-based macroscopic yield function which couples Drucker-Prager type plasticity of the solid matrix and evolving porosity. Its formulation and implementation are described. Application to a Vosges sandstone shows that, except for very low confining pressures for which the mechanical behavior is quasi-brittle, the model predicts well the ductile behavior at moderate or high confining pressures (for which the pore collapse mechanism is expected to play a dominant role).     

变开度岩体裂隙多相渗流实验与有效渗透率模型

岩体裂隙的有效渗透率是描述岩体非饱和或多相渗流的关键参数,而裂隙开度是影响有效渗透率的重要因素.通过自主研发的粗糙裂隙多相渗流可视化实验平台,针对天然岩体裂隙复制而成的裂隙模型开展变开度条件下的多相渗流可视化实验,研究开度变化对多相渗流流动结构以及有效渗透率的影响.研究表明:非湿润相流体运动通道,在低流量比条件下呈现出气泡流流动结构,而在高流量比条件下呈现较为稳定的通道流流动结构.随着开度的增加,非湿润相流动通道的分支变少、等效宽度增加,两相流体的有效渗透率均增大,流动结构趋于稳定.可视化结果还阐明了柱塞流流动结构下,两相流体交替占据裂隙空间的竞争机制:当非湿润相流体通道由连续转变为不连续时,裂隙进出口压差显著增加;反之,当该通道由不连续转变为连续时,压差显著减小.最后,基于分形理论以及渗透率统计建模方法,建立了考虑开度效应的岩体裂隙多相渗流有效渗透率理论模型,并通过实验测定的有效渗透率数据验证了该模型的正确性与有效性.     

Numerical study on the shear-flow behavior and transport process in rough rock fractures

This paper presents a systematic research for understanding mechanical shearing effects on the fluid flow and the solute transport behavior of rough fractures through a numerical simulation approach. The aperture fields were modeled based on a real rock fracture geometry and the normal displacement obtained from the shear-flow test. The fluid flow through the rough fracture under shear was simulated using a finite element code that solves the Reynolds equation, and the transport behavior through the rough fracture under shear was simulated calculating the advection–dispersion equation. The results show that the fracture apertures increase as the shear displacement increases, with a few major flow channels detected through the fracture. The shear-induced flow channels increase both flow connectivity and transport connectivity, which accelerate the movement of solutes in a particular direction and lead to early breakthrough of the contaminants. Adsorption, acting as a retardation term, has a decisive influence on the transport process. These results can give a basic knowledge of the hydromechanical and solute transport progress through fracture, and will be helpful to safety assessment for high-level radioactive waste disposal facilities.     

水力压裂解析模型裂缝扩展参数敏感性分析

水力压裂形成复杂裂缝网络是致密储层油气开采的重要技术,掌握水压裂缝扩展机理是控制压裂行为和优化压裂效果的关键.水压裂缝动态扩展行为涉及储层岩体、注入压裂液、压裂实施工艺等方面,其中水力压裂扩展时间、压裂液流体动力粘度系数、压裂液流体注入流速、储层岩石剪切模量成为决定裂缝扩展长度和裂缝开度的重要因素.本研究采用KGD、PKN两类等高解析模型对主控因素的参数敏感性进行分析,直观、快速、可靠地获得水压裂缝扩展长度、张开度动态演化行为的量化数值.研究发现,压裂持续开展过程中水压裂缝扩展长度呈线性增长、开度逐渐趋于稳定,高流体动力粘度导致裂缝难扩展、形成较大裂缝开度,通过增加压裂液流体注入流速可同时增加裂缝扩展长度和开度,较高的岩石剪切模量将降低水压裂缝的开度.通过对比两类解析模型在不同参数下的水压裂缝扩展结果,分析压裂参数与裂缝扩展的相关性和敏感系数,讨论水力压裂解析模型的裂缝扩展参数敏感性.     

Experimental Study and Fractal Analysis of Heterogeneity in Naturally Fractured Rocks

Capillary pressure curves of six low porosity and low permeability core samples from The Geysers geothermal field were measured using the mercury-intrusion approach to characterize the heterogeneity of rock. One high permeability Berea sandstone core sample was analyzed similarly, for comparison. The maximum pressure of mercury intruded into the rock was about 200 MPa to reach the extremely small pores. Experimental data showed that the capillary pressure curves of The Geysers rock are very different from that of the Berea sandstone. It was found that the frequently used capillary pressure models could not be used to represent the data from The Geysers rock samples. This might be because of the fractures in the rock. To this end, a fractal technique was proposed to model the features of the capillary pressure curves and to characterize the difference in heterogeneity between The Geysers rock and Berea sandstone. The results demonstrated that the rock from The Geysers geothermal field was fractal over a scaling range of about five orders of magnitude. The values of the fractal dimension of all the core samples (six from The Geysers and one Berea sandstone) calculated using the proposed approach were in the range from 2 to 3. The results showed that The Geysers rock with a high density of fractures had a greater fractal dimension than Berea sandstone which is almost without fractures. This shows that The Geysers rock has greater heterogeneity, as expected.     

Effects of Heterogeneous Fracture Aperture on Multiphase Production from Shale Reservoirs

Production simulation from fractured shale reservoirs is often performed by simplifying the hydraulic fractures as rectangular planes with homogeneous aperture. This study investigates the effects of heterogeneous fracture aperture and proppant distribution in realistic, non-rectangular fractures on the multi-phase production from shales. The heterogeneous hydraulic fractures are generated with the GEOS multiphysics simulator under realistic 3D stress field. These fractures are embedded into the TOUGH+ multi-phase flow simulator for production simulation. The results emphasize the importance of flow barriers within the hydraulic fractures, due both to low-aperture regions caused by the stress-shadow effect and the settling of proppant. The production rate is particularly sensitive to aperture heterogeneity if the flow barriers are close to the wellbore such that a great portion of fracture volume is isolated from the well. A stage-to-stage comparison reveals that production from different stages could vary significantly because the local stress field leads to different fracture area and aperture. The use of proppant prevents fracture closure, but if the propped regions are far from the well, they do not enhance production because flow barriers between these regions and the well act as bottlenecks. The present study highlights the importance of incorporating aperture heterogeneity into production simulation, provides insights on the relationship between flow barriers, proppant concentration, and well production, and proposes a practical method to mitigate numerical difficulties when modeling heterogeneous fractures.

     

A two scale model of porous rocks with Drucker–Prager matrix: Application to a sandstone

The present paper is devoted to a micromechanical model of porous rocks and its application to a sandstone. This original model takes advantage of a recent homogenization-based macroscopic yield function which couples Drucker–Prager type plasticity of the solid matrix and evolving porosity. Its formulation and implementation are described. Application to a Vosges sandstone shows that, except for very low confining pressures for which the mechanical behavior is quasi-brittle, the model predicts well the ductile behavior at moderate or high confining pressures (for which the pore collapse mechanism is expected to play a dominant role).     

Pore-Level Observation of Free Gravity Drainage of Oil in Fractured Porous Media

This work presents results from two sets of experiments conducted to study, in pore level, the role of fracture aperture and tilt angle on the stability of liquid bridges and the shape of a front during free gravity drainage process. Glass micromodels of two different aperture sizes were used to monitor the mechanism of gravity drainage of air?Ccrude oil system, rotating around a bottom corner to create different tilting angles. Oil content within the matrix blocks was determined as a function of time using a series of images obtained during the experiments, from which net drainage rate from the upper and lower matrix blocks is calculated. Liquid bridges are more frequent but less stable at early time of drainage. The liquid bridges, which have widths as thin as 50 ??m, can resist instability to maintain continuity. Liquid bridges formed in stacks with higher tilt angles are more stable, enhancing oil drainage from the upper matrix block and causing higher recoveries. Quantitative analysis of the results shows that a wider fracture aperture increases the oil production rate, but reduces the ultimate recovery. Furthermore, stacks with higher tilt angles present larger ultimate recoveries and smaller production rates. The front geometry in the lower block deviates from linearity due to formation of liquid bridges in the middle fracture. The results of this work can be helpful to better understand the interaction between fractures and matrix blocks.     

Up-Scaling of Double Porosity Fractured Media Using Continuous-Time Random Walks Methods

We present a new application of continuous time random walks (CTRW) methods to model fluid flows in fractured rocks. The proposed method allows large scale equivalent permeability tensors and matrix/fractures exchange function to be computed from high resolution maps of fractured porous media. Knowing these parameters allows us to carry out large scale simulations of flows governed by the dual porosity equations of Warren and Root. A direct connection between the exchange function and the time correlation function of the presence in the fractures of a particle undergoing a suitable Brownian motion over the whole medium is derived. This connection allows us to develop an efficient numerical method to compute the transient exchange term within the complete range of time scales of interest. It also gives an alternative probabilistic interpretation of the Warren and Root model. For the sake of simplicity, in the present paper, the method will only be developed to Cartesian structured grids, although it can be adapted for unstructured grids highly suited to describing complex fracture networks.     

Numerical investigation of dual-porosity model with transient transfer function based on discrete-fracture model

Based on the characteristics of fractures in naturally fractured reservoir and a discrete-fracture model, a fracture network numerical well test model is developed.Bottom hole pressure response curves and the pressure field are obtained by solving the model equations with the finite-element method. By analyzing bottom hole pressure curves and the fluid flow in the pressure field, seven flow stages can be recognized on the curves. An upscaling method is developed to compare with the dual-porosity model(DPM). The comparisons results show that the DPM overestimates the inter-porosity coefficient λ and the storage factor ω. The analysis results show that fracture conductivity plays a leading role in the fluid flow. Matrix permeability influences the beginning time of flow from the matrix to fractures. Fractures density is another important parameter controlling the flow. The fracture linear flow is hidden under the large fracture density.The pressure propagation is slower in the direction of larger fracture density.     

A method to discretize non‐planar fractures for 3D subsurface flow and transport simulations

A method is presented to discretize inclined non‐planar 2D fractures within a 3D finite element grid for subsurface flow and transport simulations. Each 2D fracture is represented as a triangulated surface. Each triangle is then discretized by 2D fracture elements that can be horizontal, vertical or inclined and that can be triangular or rectangular. The 3D grid representing a porous rock formation consists of hexahedra and can be irregular to allow grid refinement. An inclined fracture was discretized by (a) inclined triangles and (b) orthogonal rectangles and flow/transport simulations were run to compare the results. The comparison showed that (i) inclined fracture elements must be used to simulate 2D transient flow, (ii) results of 2D/3D steady‐state and 3D transient flow simulations are identical for both discretization methods, (iii) inclined fracture elements must be used to simulate 2D/3D transport because orthogonal fracture elements significantly underestimate concentrations, and (iv) orthogonal elements can be used to simulate 2D/3D transport if fracture permeability is corrected and multiplied by the ratio of fracture surface areas (orthogonal to inclined). Groundwater flow at a potential site for long‐term disposal of spent nuclear fuel was simulated where a complex 3D fracture network was discretized with this technique. The large‐scale simulation demonstrates that the proposed discretization procedure offers new possibilities to simulate flow and transport in complex 3D fracture networks. The new procedure has the further advantage that the same grid can be used for different realizations of a fracture network model with no need to regenerate the grid. Copyright © 2007 John Wiley & Sons, Ltd.     

Fractal characteristics of rocks fractured under tension

Fractal geometry can be useful for explaining the fracture behavior and rock properties. The fractal properties of rock fracture surface developed under tension were examined. Seven different rock samples were selected for the tests. An automated surface scanning device was used to map the fractured surfaces. Variogram analysis (VA) (for 1D self-affine sets) and power spectral density (PSD) measurement (for 2D self-affine sets) were applied to calculate the fractal dimension. On a comparative basis, there exists a trend between the fractal dimension and loading rate. The profiles in the loading direction yield higher fractal dimensions indicating the anisotropic feature of fractal. The fractal dimensions obtained by PSD and VA display a relationship for grain size and porosity. Higher porosity samples give different fractal dimensions for upper and lower fractures surfaces.