Dependency of flow and transport properties on aperture distributions and compression states

Fluid conductivity and elastic properties in fractures depend on the aperture geometry – in particular, the roughness of fracture surfaces. In this study, we have characterized the surface roughness with a log-normal distribution and investigated the transport and flow behaviour of the fractures with varying roughness characteristics. Numerical flow and transport simulations have been performed on a single two-dimensional fracture surface, whose aperture geometry changes with different variances and correlation lengths in each realization. We have found that conventional measurement of hydraulic conductivity alone is insufficient to determine these two parameters. Transient transport measurements, such as the particle breakthrough time, provide additional constraints to the aperture distribution. Nonetheless, a unique solution to the fracture aperture distribution is still under-determined with both hydraulic conductivity and transport measurements. From numerical simulations at different compression states, we have found that the flow and transport measurements exhibit different rates of changes with respect to changes in compression. Therefore, the fracture aperture distribution could be further constrained by considering the flow and transport properties under various compression states.     

Scale Effects Related to Flow in Rough Fractures

— A numerical fracture flow simulation based on the lubrication approximation is used to investigate the influence of roughness on the flow inside a rough fracture, at low Reynolds number. Facing surfaces are described as self-affine topographies with identical roughness magnitude. Resolution of the Reynolds equation is achieved using two distinct numerical schemes, with consistency. Fracture closure is studied assuming perfect plastic contact between facing surfaces. Long-range correlations are shown to exist in the local aperture field due to the fracture geometry and subsequently in the local fluxes inside the fracture. Flow channeling is the result of these correlations in terms of spatial distribution of the flow, and is responsible for either flow-enhancing or flow-inhibiting behavior of the fracture. Matching between the two surfaces at scales larger than a mismatch scale is studied. The mismatch scale is the upper limit scale for the local apertures scale invariance. It appears to control flow channeling and the related dispersion of the possible behaviors over a large statistics of fractures with identical statistical features. Hydraulic anisotropy of a given fracture is investigated: the dependence of the fracture transmittivity on the pressure drop orientation is proved to be sinusoidal, with an amplitude that is controlled by the mismatch scale.     

A constitutive model for water flow in unsaturated fractured rocks

A conceptual model for describing effective saturation in fractured hard rock is presented. The fracture network and the rock matrix are considered as an equivalent continuum medium where each fracture is conceptualized as a porous medium of granular structure and the rock matrix is assumed to be impermeable. The proposed model is based on the representation of a rough‐walled fracture by an equivalent porous medium, which is described using classical constitutive models. A simple closed‐form equation for the effective saturation is obtained when the van Genuchten model is used to describe saturation inside fractures and fractal laws are assumed for both aperture and number of fractures. The relative hydraulic conductivity for the fractured rock is predicted from a simple relation derived by Liu and Bodvarsson. The proposed constitutive model contains three independent parameters, which may be obtained by fitting the proposed effective saturation curve to experimental data. Two of the model parameters have physical meaning and can be identified with the reciprocal of the air entry pressure values in the fractures of minimum and maximum apertures. Effective saturation and relative hydraulic conductivity curves match fairly well the simulated constitutive relations obtained by Liu and Bodvarsson. Copyright © 2008 John Wiley & Sons, Ltd.     

甘肃北山花岗岩裂隙几何学特征研究及岩石质量初探

文中以甘肃北山花岗岩中发育的构造裂隙(主要指节理)为研究对象,通过野外裂隙调查,应用传统的概率统计方法与分形几何学理论,利用Mapinfo,ArcGIS平台进行裂隙几何学参数(方位、长度、密度等)的统计、计算和裂隙网络的空间结构分析,研究花岗岩岩体中裂隙的几何学特征。并以此为基础,对甘肃北山花岗岩岩体质量优劣进行初步评价。结果表明:在10～200cm范围内,裂隙网络是分形的;5个测点裂隙网络的分维值分别是1.636,1.548,1.596,1.724,1.604。分维数D不仅能刻画岩体中结构面发育的数量,而且能反映结构面在岩体中分布的均匀程度和交切方式。因此,可以表征岩体的质量优劣,对岩体质量进行分级。按照分维所划分的岩体质量分级,北山花岗岩属于裂隙较发育、岩体质量等级一般的岩体     

不同尺度裂缝对弹性波速度和各向异性影响的实验研究

裂缝广泛分布于地球介质中并且具有多尺度的特点,裂缝尺度对于油气勘探和开发有着重要的意义.本文制作了一组含不同长度裂缝的人工岩样,其中三块含裂缝岩样中的裂缝直径分别为2 mm、3 mm和4 mm,裂缝的厚度都约为0.06 mm,裂缝密度大致相同(分别为4.8%、4.86%和4.86%).在岩样含水的条件下测试不同方向上的纵横波速度,实验结果表明,虽然三块裂缝岩样中的裂缝密度大致相同,但是含不同直径裂缝岩样的纵横波速度存在差异.在各个方向上,含数量众多的小尺度裂缝的岩样中纵横波速度都明显低于含少量的大尺度裂缝的岩样中纵横波速度.尤其是对纵波速度和SV波速度,在不同尺度裂缝岩样中的差异更明显.在含数量多的小尺度裂缝的岩样中纵波各向异性和横波各向异性最高,而含少量的大尺度的裂缝的岩样中的纵波各向异性和横波各向异性较低.实验测量结果与Hudson理论模型预测结果进行了对比分析,结果发现Hudson理论考虑到了裂缝尺度对纵波速度和纵波各向异性的影响,但是忽略了其对横波速度和横波各向异性的影响.     

Two-phase flow properties of a horizontal fracture: The effect of aperture distribution

A systematic numerical method has been presented to investigate the constitutive relationships between two-phase flow properties of horizontal fractures and aperture distributions. Based on fractal geometry, single rough-walled fractures are generated numerically by modified successive random addition (SRA) method and then aperture distributions with truncated Gaussian distribution are formed by shear displacement between lower and upper surfaces. (The truncated Gaussian distribution is used to describe aperture evolution under different normal stresses.) According to the assumption of two-dimensional porous media and local parallel plate model, invasion percolation approach is employed to model the two-phase flow displacement (imbibition) in generated horizontal fractures, in which capillary forces are dominant over viscous and gravity forces. For truncated Gaussian distributions, constitutive relationships from numerical simulation are compared to closed-form relationships and a good agreement is obtained. The simulation results indicate strong phase interference with the sum of two phase relative permeability values being less than one in the intermediate saturations. It is found that fracture properties related to residual saturations depend on spatial correlation of aperture distributions. Based on the simulation results, we proposed an empirical relationship between the fracture residual-saturation-rated parameters and the corresponding aperture distributions.     

Comparison of Measured and Modelled Hydraulic Conductivities of Fractured Sandstone Cores

— A new method for characterising the detailed fracture geometry in sandstone cores is presented. This method is based on the impregnation of samples with coloured resin, without significant disturbance of the fractures. The fractures are made clearly visible by the resin, thus allowing the fracture geometry to be examined digitally. In order to model the bulk hydraulic conductivity, the samples are sectioned serially perpendicular to the flow direction. The hydraulic conductivity of individual sections is estimated by summing the contribution of the matrix and each fracture from the digital data. Finally, the hydraulic conductivity of the bulk sample is estimated by a harmonic average in series along the flow path. Results of this geometrical method are compared with actual physical conductivity values measured from fluid experiments carried out prior to sectioning. The predicted conductivity from the fracture geometry parameters (e.g., fracture aperture, fracture width, fracture length and fracture relative roughness all measured using an optical method) is in good agreement with the independent physical measurements, thereby validating the approach.     

Numerical investigation of alternative fracture stiffness measures and their respective scaling behaviours

We study the mechanical deformation of fractures under normal stress, via tangent and specific fracture stiffnesses, for different length scales using numerical simulations and analytical insights. First, we revisit an equivalent elastic layer model that leads to two expressions: the tangent stiffness is the sum of an “intrinsic” stiffness and the normal stress, and the specific stiffness is the tangent stiffness divided by the fracture aperture at current stress. Second, we simulate the deformation of rough fractures using a boundary element method where fracture surfaces represented by elastic asperities on an elastic half‐space follow a self‐affine distribution. A large number of statistically identical “parent” fractures are generated, from which sub‐fractures of smaller dimensions are extracted. The self‐affine distribution implies that the stress‐free fracture aperture increases with fracture length with a power law in agreement with the chosen Hurst exponent. All simulated fractures exhibit an increase in the specific stiffness with stress and an average decrease with increase in length consistent with field observations. The simulated specific and tangent stiffnesses are well described by the equivalent layer model provided the “intrinsic” stiffness slightly decreases with fracture length following a power law. By combining numerical simulations and the analytical model, the effect of scale and stress on fracture stiffness measures can be easily separated using the concept of “intrinsic” stiffness. We learn that the primary reason for the variability in specific stiffness with length comes from the fact that the typical aperture of the self‐affine fractures itself scales with the length of the fractures.     

具有离散裂缝空间分布的二维固体中地震波传播的有限差分模拟

结合有限差分方法和等效介质理论,模拟了离散分布裂缝介质中地震波的传播. 基于等效介质理论,利用二维有限差分实现封闭裂缝的离散分布;裂缝可以处理成固体岩石中的高度柔性界面,并可以用线性滑动或者位移间断模型进行裂缝的物理描述. 对于含有多组裂隙的破裂固体,其有效柔度可以认为是固体骨架背景柔度和裂缝附加柔度之和. 在一阶近似条件下,固体骨架和裂缝参数可以通过有效各向异性系数联系起来,有效各向异性系数决定了各向异性（裂缝效应）对于地震波传播的影响. 通过与射线理论方法的对比检验,说明本文提出的模拟方法的有效性,并通过几个数值算例说明本方法可有效模拟不同的裂缝分布效应. 结果表明,即使在裂缝密度很小的情况下,具有相同裂缝密度的不同的空间分布可以产生不同的波场特征. 同时,也验证了不同裂缝尺度对波长的不同影响,以及裂缝尺度具有幂率分布（分形）时,尺度对波场的影响. 最后得出结论：在运用建立在等效介质理论基础上的地震各向异性概念来描述裂缝固体的特征时,要倍加小心,等效介质理论中尚未合理处理的裂缝尺度和空间分布对波的传播特征具有重要的影响.     

地震非均匀度物理意义的实验研究

资料表明某些强地震前中小地震的地震非均匀度（GL值）参数在强地震孕育进入中期或短期阶段出现明显异常变化, 显示出区域中小地震活动状态发生变化. 本文通过对含有障碍体的平直断层、 挤压型雁列式断层及Ⅲ型剪切裂纹等3种类型的岩石样本变形破坏过程声发射事件时间序列的分析, 讨论了GL值变化的物理意义. 计算结果表明, 这3种标本变形破坏过程中, 在应力应变处于非弹性阶段前夕或在岩石整体破裂失稳前破裂成核期间, GL值出现持续大于１的异常变化. 表明GL值能较好地刻画受压岩石破坏前的应变的非弹性变化, 具有一定的标本破裂指示意义.      

Ground water/surface water interaction in a fractured rock aquifer

In a recent field study of ground water/surface water interaction between a bedrock stream and an underlying fractured rock aquifer, it was determined that the majority of ground water discharge occurred through sparsely located vertical fractures. In this paper, the dominant mechanisms governing ground water/surface water exchange in such an environment are investigated using a numerical model. The study was conducted using several conceptual models based on the field study results. Although the field results provided the motivation for the modeling study, it was not intended to match modeling and field results directly. In addition, the extent of capture zones for discharging or recharging fractures was explored. The results of this study are intended to provide a better understanding of contaminant migration in the vicinity of bedrock streams. Based on the numerical results, the rate of ground water discharge (or recharge) was found to depend on the aperture size of the discharging feature, and on the distribution of hydraulic head with depth within the fracture network. It was determined that the extent of both the capture zone and reverse capture zone for an individual fracture can be extremely large, and will be determined by the height of the stream stage, the fracture apertures of the network, and the hydraulic-head distribution within the network. Because both the stream stage and the hydraulic-head distribution are transient, the size of the capture zone and/or the reverse capture zone for an individual fracture may change significantly over time. As a result, the migration path for contaminants within the fracture network and between the surface and subsurface will also vary significantly with time.     

用正演数值计算方法开展双侧向测井对裂缝的响应研究

采用有限元分析软件ANSYS对裂缝的双侧向测井视电阻率与裂缝孔隙度、泥浆电阻率、裂缝倾角和基岩电阻率的关系进行了计算. 在大量正演数据的基础上得出双侧向测井响应反演公式和裂缝孔隙度计算公式,提出更为精细的裂缝倾角的弹性划分模型,用双侧向视电阻率值近似估算裂缝倾角的方法,提高了利用双侧向测井求裂缝产状与裂缝孔隙度（裂缝宽度）的精度.     

裂缝性储层的电各向异性响应特征研究

以水平裂缝分布的孔隙介质模型为基础,建立了水平裂缝面的平行裂缝分布和带粗糙表面的点接触裂缝模型,导出了两类裂缝模型的水平和垂直电阻率响应关系,分析了两类裂缝模型在没有围压和存在围压条件下的水平电阻率、垂直电阻率变化规律和电各向异性系数的变化特征.讨论了裂缝开度、裂缝密度和裂缝粗糙度等裂缝特征参数对裂缝性储层电各向异性的影响.为简化讨论,所有电性响应特征的分析都忽略了裂缝和孔隙表面的导电性和极化的影响.利用平行分布水平裂缝面模型和带粗糙面的裂缝模型,考察了围压条件下的裂缝性岩石的电各向异性响应特征,得到了对实际应用有意义的结果.     

Capillary Crack Imbibition: A Theoretical and Experimental Study Using a Hele-Shaw Cell

—?We study the filling of horizontal cracks with constant aperture driven by capillary forces. The physical model of the crack consists of a narrow gap between two flat glass plates (Hele-Shaw cell). The liquid enters the gap through a hole in the bottom plate. The flow is driven purely by the force acting on the contact lines between solid, liquid, and gas. We developed a theoretical model for this type of flow on the basis of Darcy's law; it allows for the consideration of different surface conditions.¶We run the experiment for two surface conditions: Surfaces boiled in hydrogen peroxide to remove initial contamination, and surfaces contaminated with 2-propanol after boiling in hydrogen peroxide. The flow rate depends on the gap aperture and on the interaction of the liquid with the air and the solid surfaces: The smaller the aperture, the lower the flow rate due to viscous resistance of the liquid. The flow rate is also reduced when the glass surfaces are contaminated with 2-propanol. The contact line force per unit length is approximately 60% higher on clean glass surfaces than it is on glass surfaces with the 2-propanol contamination. These experimental results are in agreement with our theoretical model and are confirmed by independent measurements of the liquid-solid interaction in capillary rise experiments under static conditions with the same Hele-Shaw cell.¶Another aspect of this study is the distribution of the liquid for the different surface conditions. The overall shape is a circular disk, as assumed in the theoretical model. However, a pronounced contact line roughness develops in case of the surfaces contaminated with 2-propanol, and air bubbles are trapped behind the contact line. A further analysis of the flow regime using the capillary number and the ratio of the viscosities of the involved fluids (water and air) reveals that the experiments take place in the transition zone between stable displacement and capillary fingering, i.e., neither viscous nor capillary fingers develop under the conditions of the experiment. The contact line roughness and the trapped air bubbles in the contaminated cell reflect local inhomogeneities of the surface wettability.     

The fractal geometry of flow paths in natural fractures in rock and the approach to percolation

The distributions of contact areas in single, natural fractures in quartz monzonite (Stripa granite) are found to have fractal dimensions which decrease fromD=2.00 to values nearD=1.96 as stress normal to the fractures is increased from 3 MPa up to 85 MPa. The effect of stress on fluid flow is studied in the same samples. Fluid transport through a fracture depends on two properties of the fracture void space geometry. the void aperture; and the tortuosity of the flow paths, determined through the distribution of contact area. Each of these quantities change under stress and contribute to changes observed in the flow rate. A general flow law is presented which separates these different effects. The effects of tortuosity on flow are largely governed by the proximity of the flow path distribution to a percolation threshold. A fractal model of correlated continuum percolation is presented which quantitatively reproduces the flow path geometries. The fractal dimension in this model is fit to the measured fractal dimensions of the flow systems to determine how far the flow systems are above the percolation threshold.     

基于数字岩心的含裂隙储层砂岩介电性质研究

了解储层岩石的介电特性在石油工业的各个方面都有重要的应用.小尺度裂隙是影响岩石介电性质的地质因素之一,获得裂隙对含裂隙岩石介电性质影响的定量关系具有重要的理论和实践意义.以含裂隙人造砂岩的三维微观数字结构为基础,通过基于三维有限差分算法计算的岩石介电性质与实验数据的对比验证数值计算方法的有效性.在此基础上,通过理论模型获得不同孔隙度基质的介电性质,并在不含裂隙人造砂岩的三维微观数字结构中人为添加以裂隙密度和纵横比为定量表征参数的裂隙,应用验证后的数值算法模拟随频率变化的含裂隙砂岩的介电性质,分析和研究不同孔隙度基质中定向排列裂隙对砂岩介电性质的影响.结果表明,当裂隙孔隙度随裂隙纵横比或裂隙密度发生改变时,含裂隙砂岩的介电性质与裂隙密度以及裂隙纵横比呈正相关关系,而当裂隙孔隙度保持不变时,含裂隙砂岩的介电性质随裂隙纵横比的减小而增大;裂隙参数的改变对不同基质孔隙度的含裂隙砂岩的介电性质的影响趋势较为一致,但随着基质孔隙度的减小,裂隙对砂岩介电性质的影响逐渐增大.裂隙参数和基质孔隙度对含裂隙砂岩介电性质影响的研究结果为基于介电特性的裂缝性油气储层的定量表征提供了依据,在油气勘探开发中具有重要的应用前景.     

Effects of the geometry of two‐dimensional fractures on their hydraulic aperture and on the validity of the local cubic law

Flow through rough fractures is investigated numerically in order to assess the validity of the local cubic law for different fracture geometries. Two‐dimensional channels with sinusoidal walls having different geometrical properties defined by the aperture, the amplitude, and the wavelength of the walls' corrugations, the corrugations asymmetry, and the phase shift between the two walls are considered to represent different fracture geometries. First, it is analytically shown that the hydraulic aperture clearly deviates from the mean aperture when the walls' roughness, the phase shift, and/or the asymmetry between the fracture walls are relatively high. The continuity and the Navier–Stokes equations are then solved by means of the finite element method and the numerical solutions compared to the theoretical predictions of the local cubic law. Reynolds numbers ranging from 0.066 to 66.66 are investigated so as to focus more particularly on the effect of flow inertial effects on the validity of the local cubic law. For low Reynolds number, typically less than 15, the local cubic law properly describes the fracture flow, especially when the fracture walls have small corrugation amplitudes. For Reynolds numbers higher than 15, the local cubic law is valid under the conditions that the fracture presents a low aspect ratio, small corrugation amplitudes, and a moderate phase lag between its walls.     

On the Importance of Gravity in DNAPL Invasion of Saturated Horizontal Fractures

Invasion percolation (IP) models of dense non‐aqueous phase liquid (DNAPL) invasion into saturated horizontal fractures typically neglect viscous and gravity forces, as it is assumed that capillarity dominates in many situations. An IP model simulating DNAPL invasion into saturated horizontal fractures was modified to include gravity as a local effect. The model was optimized using a genetic algorithm, and demonstrated that the inclusion of gravity is important for replicating the architecture of the DNAPL invasion pattern. The optimized gravity‐included simulation showed the DNAPL invasion pattern to be significantly more representative of the experimentally observed pattern (80% accuracy) than did the optimized gravity‐neglected simulation (70% accuracy). Additional simulations of DNAPL invasion in 360 randomly generated fractures were compared with and without gravity forces. These simulations showed that with increasing fracture roughness, the minimum difference between simulations with and without gravity increases to 35% for a standard deviation of the mid‐aperture elevation field (SD_z) of 10 mm. Even for low roughness (SD_z = 0.1 mm), the difference was as high as 30%. Furthermore, a scaled Bond Number is defined which includes data regarding DNAPL type, media type and statistical characteristics of the fracture. The value of this scaled Bond Number can be used to determine the conditions under which gravity should be considered when simulating DNAPL invasion in a macroscopically horizontal fracture. Finally, a set of equations defining the minimum and maximum absolute percentage difference between gravity‐included and gravity‐neglected simulations is presented based on the fracture and DNAPL characteristics.     

Characterization of fractured reservoirs using a consistent stiffness‐permeability model: focus on the effects of fracture aperture

In this paper we propose a method for the characterization of naturally fractured reservoirs by quantitative integration of seismic and production data. The method is based on a consistent theoretical frame work to model both effective hydraulic and elastic properties of fractured porous media and a (non‐linear) Bayesian method of inversion that provides information about uncertainties as well as mean (or maximum likelihood) values. We model a fractured reservoir as a porous medium containing a single set of vertical fractures characterized by an unknown fracture density, azimuthal orientation and aperture. We then look at the problem of fracture parameter estimation as a non‐linear inverse problem and try to estimate the unknown fracture parameters by joint inversion of seismic amplitude versus angle and azimuth data and dynamic production data. Once the fracture parameters have been estimated the corresponding effective stiffness and permeability tensors can be estimated using consistent models. A synthetic example is provided to clearly explain and test the workflow. It shows that seismic and production data complement each other, in the sense that the seismic data resolve a non‐uniqueness in the fracture orientation and the production data help to recover the true fracture aperture and permeability, because production data are more sensitive to the fracture aperture than the seismic data.