Fully coupled simulation of a hydraulic fracture interacting with natural fractures with a hybrid discrete‐continuum method

A hybrid discrete‐continuum numerical scheme is developed to study the behavior of a hydraulic fracture crossing natural fractures. The fully coupled hybrid scheme utilizes a discrete element model for an inner domain, within which the hydraulic fracture propagates and interacts with natural fractures. The inner domain is embedded in an outer continuum domain that is implemented to extend the length of the hydraulic fracture and to better approximate the boundary effects. The fracture is identified to propagate initially in the viscosity‐dominated regime, and the numerical scheme is calibrated by using the theoretical plane strain hydraulic fracture solution. The simulation results for orthogonal crossing indicate three fundamental crossing scenarios, which occur for various stress ratios and friction coefficients of the natural fracture: (i) no crossing, that is, the hydraulic fracture is arrested by the natural fracture and makes a T‐shape intersection; (ii) offset crossing, that is, the hydraulic fracture crosses the natural fracture with an offset; and (iii) direct crossing, that is, the hydraulic fracture directly crosses the natural fracture without diversion. Each crossing scenario is associated with a distinct net pressure history. Additionally, the effects of strength contrast and stiffness contrast of rock materials and intersection angle between the hydraulic fracture and the natural fracture are also investigated. The simulations also illustrate that the level of fracturing complexity increases as the number and extent of the natural fractures increase. As a result, we can conclude that complex hydraulic fracture propagation patterns occur because of complicated crossing behavior during the stimulation of naturally fractured reservoirs. Copyright © 2017 John Wiley & Sons, Ltd.     

岩体水力学基础(二)─—岩体水力学的基础理论

单裂隙水流立方定律是岩体水力学中最基本的．也是最重要的定理。本文从不变形平直等宽单裂隙水流运动定理开始，讨论了受应力作用岩体单裂隙、一组平行裂隙、一组正交裂隙以及岩体裂隙系统中水流运动规律，分析了单裂隙中水流渗透压力与裂隙变形之关系。运用水力学原理推导了岩溶单管道流定理，为岩体水力学模型研究奠定了基础。     

非达西裂隙流对渗透性基岩中流场及溶质羽的影响

以基岩渗流方向与裂隙轴向呈45度角为例,探讨了当裂隙轴向与基岩水流斜交时,裂隙流的非达西程度对流场及溶质运移的影响.使用Comsol Multiphysics多物理场仿真软件构建了一个在中部包含水平单裂隙的正方形多孔介质模型,裂隙中的非达西流用Izbash方程去刻画,裂隙的上游基岩中存在持续的溶质源.随着裂隙水流的非达西程度逐渐增强,流场及污染物分布表现出如下特征:（1）裂隙中水流流速逐渐增大;流线在裂隙与基岩界面处的折射逐渐偏离折射定律;（2）裂隙水流的流向逐渐偏向基岩水流的渗流方向;（3）溶质羽宽度变宽但对称性逐渐降低;（4）溶质在水平方向上的浓度峰值逐渐降低,右侧浓度逐渐升高;（5）裂隙产生的回弥散对溶质运移作用逐渐增强,使裂隙中更多的溶质运移到了上层基岩中.总体而言,裂隙流的非达西程度对流场及污染物分布有着显著的影响.      

Some possibilities of digital and analogue evaluation of satellite dataAnaloge und digitale methoden zur auswertung von satellitendatenQuelques possibilités d'interprétation digitale et analogue des données fournies par des satellites

The high data rate and information flow of remote sensing systems call for the implementation of automatic analogue and digital interpretation techniques. Based on ERTS-1 data representing typical European scenes digital and analogue image evaluation was performed. A statistical analysis of the data, as first step of the evaluation process, showed high correlation between both infrared channels. Feature identification was also performed by a digital quicklook method which takes into consideration the different reflection properties of surface. The possibility of data reduction for operational applications is discussed. The results obtained digitally are compared to those obtained by the analogue evaluation method of density slicing.     

Flow Deflection in Intersected Tubes

ＩＮＴＲＯＤＵＣＴＩＯＮＴｈｅｔｈｅｏｒｅｔｉｃａｌｃａｌｃｕｌａｔｉｏｎｓｏｆｗａｔｅｒｍｉｇｒａｔｉｏｎ ,ｗｈｅｔｈｅｒｉｎａｎｉｎｄｉｖｉｄｕａｌｆｒａｃｔｕｒｅｏｒｉｎａｆｉｓｓｕｒｅｎｅｔｗｏｒｋ ,ｈａｖｅａｌｌｂｅｅｎｂａｓｅｄｏｎｔｈｅｅｑｕａｔｉｏｎｆｏｒｖｉｓｃｏｕｓｆｌｏｗｂｅｔｗｅｅｎｐａｒａｌｌｅｌｐｌａｔｅｓ.ＷｉｌｓｏｎａｎｄＷｉｔｈｅｒ ｓｐｏｏｎ (1976 )ｆｉｒｓｔａｃｃｏｍｐｌｉｓｈｅｄａｈｙｄｒａｕｌｉｃｓｉｍｕｌａｔｉｏｎｏｆｆｒａｃｔｕｒｅｗａｔｅｒｃｒｏｓｓ…     

The estimation of dispersion behavior in discrete fractured networks of andesite in Lan-Yu Island,Taiwan

This study was based on the discrete fracture model to investigate the influence of fracture parameters on the solute transport in the fractured rocks of andesite in Lan-Yu island, Taiwan. In the simulation cases, the centers of fractures, fracture lengths and apertures were assumed to have Poisson’s distribution, negative exponential distribution and lognormal distribution, respectively. With the above assumptions, constructing the discrete fracture model became practicable. Using the mass-balance equation with specified boundary conditions, the flow field in the rock was solved. Then particles were released under the flow field. Monte Carlo method was used assuming that the amount of particles was proportional to the flow rates to get the particle accumulated percentage breakthrough curve and to estimate the dispersion coefficient. On the basis of the discrete fracture model, it was possible to evaluate the property of dispersion behavior of andesite in Lan-Yu Island with flow and transport mechanism. Properties of the dispersion behavior such as the relation between distance and traveling-time (ln〈r ²〉 and ln 〈t〉), anisotropic behavior, and the overall dispersion coefficient in a fracture network were characterized: the slope value of ln〈r ²〉 and ln〈t〉 was 1.64 an indication of non-Fickian dispersion, the particles dispersion along the flow (D11) was bigger than that perpendicular to the flow (D22), and the dispersion coefficient by this study was 0.91 m comparing the value 1 m from Sauty’s method.     

Hydromechanical properties and alteration of natural fracture surfaces in the Soultz granite (Bas-Rhin, France)

Natural fractures are characterized by rough surfaces and complex fluid flows. A large distribution of apertures (residual voids) within their walls and the presence of contact points (in situ normal loads) produce heterogeneous flows (channeling). The resulting permeabilities, porosities or fluid–rock exchange surfaces cannot be realistically modeled by parallel and smooth plate models. Four natural fractures are sampled at different depths and degrees of alteration in the Soultz sandstone and granite (EPS1 drillhole, Soultz-sous-Forêts, Bas-Rhin, France). The fracture surfaces are measured with mechanical profilometry and maps of asperity heights (XYZ). Resulting local apertures (XYe) are then calculated. A statistical study of the surface profiles (XZ) show that the fractures are more or less rough and tortuous according to the types of alteration. Altered samples are characterized by smoother surfaces of fractures. Such differences imply that (i) the average fracture aperture is not representative for the whole fracture and that (ii) the different local apertures should be integrated in hydraulic and mechanical models. A hydraulic model (finite difference calculations) of fluid flow, taking into account the elastic closure (Hertz contact theory) of fractures with depth, is used. Maps of contact points and relative local loads within the fracture planes are compared to flow maps. They show different channeling of fluid flows. Strongly altered fractures are characterized by homogeneous fluxes despite the presence of numerous contact zones during the closure of fracture. By contrast, fresh fractures develop, increasing fluid flow channels with depth.Fracture closure (increasing normal stress) does not systematically increase the channeling of fluid flow. There is evidence for a general smoothing out of the irregularities of the fracture walls due to precipitation of secondary minerals, indicating that the cubic law can be commonly valid, also at great crustal depth but this validity depends on the degree of fracture alteration. Mineralogical and geochemical observations, thus, should be taken into account to perform more accurate permeability calculations and models of fluid circulation in fracture networks.     

岩体裂隙网络渗流变水温影响分析

考虑岩体裂隙渗流变水温影响,推导单裂隙变水温水流近似解析解和有限元解,在此基础上分别建立裂隙二维网络变水温渗流数值求解方程,分别对应裂隙网络变水温渗流分析的近似解析法和子结构法。分析变温水流运动规律发现：（1）单裂隙内水流水头与水力坡降成非线性关系,当水流由高温区向低温区流动时,水头分布曲线为凸曲线,此时按线性渗流简化水头整体偏小;当由低温区向高温区流动时,水头分布曲线为凹曲线,此时按线性渗流简化水头整体偏大。（2）单裂隙内,高水温处水力坡降小,低水温处水力坡降大;裂隙平均水温越高,流速越快;裂隙网络内存在与裂隙宽度相似的温度偏流效应,即交叉节点水流有偏向水流温度高的裂隙流动的趋势。在温度较高和温度梯度较大的区域,应该考虑水流温度变化对渗流场的影响。     

基于X形真实裂隙通道的煤层瓦斯渗流模拟

使用常规X射线或CT扫描进行图像处理,研究裂隙结构时,主裂隙不易提取,其特征不明显,不能直接反映主裂隙构造与流体速度关系。运用高清相机拍摄井下煤壁X形真实裂隙进行数字化处理,并运用AutoCAD软件提取裂隙特征,将图片矢量化导入Comsol Mutiphysics仿真模拟软件进行计算,模拟得到瓦斯分布压力场和渗流速度场云图。结果表明:含X形裂隙煤样中,瓦斯自入口开始,渗流压力从左至右递减,裂隙通道内瓦斯压力均匀分布,是一段压力缓冲区;非裂隙区渗流速度场分布不均匀,X形裂隙支流处较汇流处瓦斯渗流更为活跃,瓦斯的流向和裂隙走向的夹角对渗流速度有明显的影响,瓦斯自下边界流入时的最大渗流速度是自左边界流入的29.5倍;裂隙通道内的渗流速度与裂隙的尺度成单调递减对数函数关系,裂缝尺度越大,达西渗流速度越低,当裂隙尺度为0.68~1.23 mm时对渗流速度影响最明显。研究成果可直观地了解煤裂隙内瓦斯渗流特征。      

A Connectivity Index for Discrete Fracture Networks

Connectivity is an important measure for assessing flow transport in rock, especially through fractures. In this paper, rock fracture systems are modelled by a discrete fracture model simulated by a marked point process. A connectivity index is then introduced to quantify the connectivity between any two points in space. Monte Carlo simulation is used to evaluate the connectivity index for stationary cases and relationships between the connectivity index and the parameters of the discrete fracture model are analysed. The average number of intersections per fracture, X_f, and the fracture intensity, P12 (P32), are calculated and the relationships between these parameters and the connectivity index are investigated, concluding that X_f is the more suitable parameter for the classification of rock mass flow properties. The relationships between the connectivity index and the percolation state of the fractured medium are also discussed. An edge correction is briefly discussed and a practical example is used to demonstrate the method of computing the connectivity index.     

Flow behaviour and physical chemistry of bouncing putties and related polymers in view of tectonic laboratory applications

The initiation of analogue studies of rock flow is stimulated by improving our knowledge of suitable model materials. Bouncing Putties and “Plasticines” are the most frequently used model materials in analogue studies of flow instabilities in deforming rocks. Polydimethyl-siloxane (PDMS) and polyborondimethylsiloxane (PBDMS), both substrates of Bouncing Putty, are introduced as convenient geological model materials. The chemistry of PDMS, PBDMS, Bouncing Putties and “Plasticines” is reviewed. A comprehensive set of instructions and graphs is provided for the manipulation of these model materials.In particular, a high viscosity PDMS produced as an intermediate compound under the code name SGM36 by Dow Corning (Great Britain) opens exciting possibilities for analogue studies of rock flow, because it is perfectly transparent. This allows continuous observation of three-dimensional strain markers during an experiment.The polymeric flow mechanisms are compared with the flow behaviour and crystal plasticity theory of rocks. The flow of natural rocks is taken to be of Reiner-Rivlin type with powers n varying between 1 and 10.Flow curves have been constructed for Bouncing Putties, Plastilinas (cf. Plasticines) and SGM36 (cf. PDMS). These original curves are supplemented with extensive data on similar materials compiled from the literature. The combined data reveal a consistent flow curve pattern for each group of model materials considered.Strain-rate softening of commercially available Bouncing Putties and “Plasticines” at low strain rates can be attributed to the solid filler concentration. The power n, which describes the departure from Newtonian flow, appears to be dependent on the angular filler volume concentration c and is governed by the preliminary equation n = 1−11c + 48c². This finding provides a technique for manipulating liquid polymers to simulate natural rock flow with various powers of n.The (T, P) dependence of the viscosity and thermal properties of PDMS are outlined to stimulate modelling which includes natural (T, P) gradients.     

煤层气储层裂隙阵列侧向测井响应数值模拟与分析

裂隙是煤层气储层重要的流体渗流通道,是煤层气有效储集体形成的重要条件。研究裂隙的测井响应特征是确定煤层裂隙参数的关键。利用三维有限元方法,确定不同裂隙条件的阵列侧向测井响应。研究表明:阵列侧向测井修正的视电导率与裂隙孔隙度、裂隙流体电导率基本为线性关系;随裂隙倾角增大,阵列侧向视电阻率升高;随基块电阻率增大,阵列侧向视电阻率增加;高角度裂隙(组)的阵列侧向测井深浅视电阻率呈现正差异,而基块电阻率较大的低角度裂隙(组)的响应可呈现负差异特征;相同裂隙孔隙度下,裂隙密度增加可能使裂隙组的阵列侧向视电阻率增加,当裂隙密度足够大时,测井响应基本不随裂隙密度发生变化;交叉裂隙的测井响应与平行裂隙组差异明显,且不同倾角组合的交叉裂隙的测井响应之间差异明显。      

Influence of three-dimensional roughness of rock fracture on seepage characteristics based on the digital image technology

Digital image processing technology can objectively reflect the surface roughness of coal and rock mass fractures and turn physical properties such as fluctuation height to physical data of the rock fracture surface to be used for numerical analyses; thus, it can effectively be applied to the seepage flow analysis of rock fractures. First, clear digital images of the fracture surface height are obtained under the same camera conditions, and then characteristic function values specific to various structure are normalized. Subsequently, the fluctuation height distribution of the rock surface is restored according to the measured fluctuation heights of local points. Finally, according to the surface fluctuation of fractures, invasive restructuring is carried out for three-dimensional fractures in the same coordinate system. Obtained physical parameters on the three-dimensional fractures of the rock are inputted into the COMSOL Multiphysics software to obtain the characteristic three-dimensional model of the rock fracture surface. The numerical analysis results are compared with experimental data on fracture seepage obtained from a seepage coupling true triaxial test system, and the relevance between the simulated result and the physical experiment is higher than 90%, which confirms that the integration of the digital image technology and the numerical analysis method can effectively simulate seepage in rough fractures. The hydraulic gradient of rough fractures and seepage velocity are also consistent with Forchheimer flow characteristics.     

Self‐similar solution of a plane‐strain fracture driven by a power‐law fluid

This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable, linear elastic medium. The fracture is driven by injection of an incompressible, viscous fluid with power‐law rheology and behaviour index n?0. The opening of the fracture and the internal fluid pressure are related through the elastic singular integral equation, and the flow of fluid inside the crack is modelled using the lubrication theory. Under the additional assumptions of negligible toughness and no lag between the fluid front and the crack tip, the problem is reduced to self‐similar form. A solution that describes the crack length evolution, the fracture opening, the net fluid pressure and the fluid flow rate inside the crack is presented. This self‐similar solution is obtained by expanding the fracture opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a numerical minimization procedure. The influence of the fluid index n in the crack propagation is also analysed. Copyright © 2002 John Wiley & Sons, Ltd.     

Occurrence of springs in massifs of crystalline rocks,northern Portugal

An inventory of artesian springs emerging from fractures (fracture springs) was conducted in the Pinh?o River Basin and Morais Massif, northern Portugal, comprising an area of approximately 650 km². Over 1,500 springs were identified and associated with geological domains and fracture sets. Using cross-tabulation analysis, spring distributions by fracture sets were compared among geological environments, and the deviations related to differences in rock structure and, presumably, to differences in deformational histories. The relation between spring frequencies and rock structures was further investigated by spectral determination, the model introduced in this study. Input data are the spring frequencies and fracture lengths in each geological domain, in addition to the angles between fracture strikes and present-day stress-field orientation (θ). The model's output includes the so-called intrinsic densities, a parameter indexing spring occurrence to factors such as fracture type and associated deformational regime and age. The highest densities (12.2 springs/km of lineament) were associated with young shear fractures produced by brittle deformation, and the lowest (0.1) with old tensional and ductile fractures. Spectral determination also relates each orientation class to a dominant structural parameter: where spring occurrence is controlled by θ, the class is parallel to the present-day stress-field orientation; where the control is attributed to the length of fractures, the spring occurrence follows the strike of large-scale normal faults crossing the region. Electronic Publication     

水力裂缝穿越非连续面扩展时的断裂过程研究

非连续面发育是非常规油气储层的显著地质特征之一,水力裂缝能否穿越非连续面扩展会关系到压裂的改造效果。为研究水力裂缝穿越非连续面扩展时断裂过程区（fracture process zone,简称FPZ）发育特征,采用自主设计的可视化压裂试验装置对含预制摩擦界面的砂岩平板试件开展水力压裂试验。基于数字图像相关法实时监测了水力裂缝正交穿越界面扩展过程中的位移及应变场特征。试验结果表明,水力裂缝穿越界面扩展之前,断裂过程区已经开始跨越界面发育;裂缝能否穿越界面扩展在FPZ的初始发育阶段已经注定,不受FPZ内应力软化过程影响。基于Renshaw-Pollard准则建立了考虑FPZ边界范围的裂缝穿越非连续面扩展准则,并通过前人及文中试验数据进行了可靠性验证。相比而言,改进准则更准确地考虑了裂缝前端线弹性断裂力学的适用范围。研究发现FPZ长宽比对裂缝穿越界面扩展准则有显著影响,相同条件下,FPZ长宽比越大,裂缝正交穿越界面扩展所需要的摩擦系数下限值越小。     

Finite element simulation of fluid flow in fractured rock media

Fluid dynamics models are used by the petroleum industry to model single- and/or multi-phase flow within fractured rock formations, in order to facilitate extraction of fluids such as oil and natural gas, and in other areas of engineering to study groundwater flow, as well as to estimate contaminant seepage and transport. In this paper, the numerical modelling software Comsol is used to simulate air and water flow through a specimen of granite with a single vertical fracture subjected to triaxial loading conditions. The intent of the model is to simulate triaxial test findings on a rock specimen with a natural fracture. Fluid flow is simulated at various confining and inlet pressures using the cubic law. Model results were in good agreement with laboratory findings. Pressure distribution along the fracture and across the specimen are as expected with a near linear pressure distribution along the length of the fracture. A drawdown effect on pressure distribution across the specimen in the vicinity of the fracture is also observed. Pressure gradient was largely uniform; however, some localised zones of high gradient along the fracture are observed.     

Modeling spatial fracture intensity as a control on flow in fractured rock

Spatial fracture intensity (P ₃₂, fracture area by volume) is an important characteristic of a jointed rock mass. Although it can hardly ever be measured, P ₃₂ can be modeled based on available geological information such as spatial data of the fracture network. Flow in a mass composed of low-permeability hard rock is controlled by joints and fractures. In this article, models were developed from a geological data set of fractured andesite in LanYu Island (Taiwan) where a site is investigated for possible disposal of low-level and intermediate-level radionuclide waste. Three different types of conceptual models of spatial fracture intensity distribution were generated, an Enhanced Baecher’s model (EBM), a Levy–Lee Fractal model (LLFM) and a Nearest Neighborhood model (NNM). Modeling was conducted on a 10 × 10 × 10 m synthetic fractured block. Simulated flow was forced by a 1% hydraulic gradient between two vertical x–z faces of the cube (from North to South) with other boundaries set to no-flow conditions. Resulting flow vectors are very sensitive to spatial fracture intensity (P ₃₂). Flow velocity increases with higher fracture intensity (P ₃₂). R-squared values of regression analysis for the variables velocity (V/V _max) and fracture intensity (P ₃₂) are 0.293, 0.353, and 0.408 in linear fit and 0.028, 0.08, and 0.084 in power fit. Higher R ² values are positively linked with structural features but the relation between velocity and fracture intensity is non-linear. Possible flow channels are identified by stream-traces in the Levy–LeeFractal model.     

Algorithmic Simulation of Flow in Rock Fractures

The two sides of a rock fracture are geometrically often very different. The difference or heterogeneity can be captured using aperture. This is also practically very significant because locally low or zero aperture in a rock fracture impedes flow. The aperture measurements of each of five paired rock blocks are discretely digitized (0–9). Each resulting weakly continuous two-dimensional matrix is then traversed by the nflow (new-flow) flow algorithm, once from each entry at one end to the other end. The paths are either incomplete and end surrounded by zeros or are completed at the other (exit) end. The numerical digit distribution is held constant from block to block. Fracture through-flow is defined as the average percentage of complete paths from one end to the other for that block. Fracture roughness is negatively correlated with fracture through-flow (r ~ −0.893). A rough fracture will only allow little flow compared with a smooth fracture with a smoother aperture distribution. Channeling of complete paths toward exit points is discussed for all blocks. Channeling is not related to roughness. A long complete sample path in a very rough fracture is shown. The distribution of numbers of complete paths is also investigated.     

Heterogeneous seepage at the Nopal I natural analogue site, Chihuahua, Mexico

A study of seepage occurring in an adit at the Nopal I uranium mine in Chihuahua, Mexico, was conducted as part of an integrated natural analogue study to evaluate the effects of infiltration and seepage on the mobilization and transport of radionuclides. An instrumented seepage collection system and local automated weather station permit direct correlation between local precipitation events and seepage. Field observations recorded between April 2005 and December 2006 indicate that seepage is highly heterogeneous with respect to time, location, and quantity. Seepage, precipitation, and fracture data were used to test two hypotheses: (1) that fast flow seepage is triggered by large precipitation events, and (2) that an increased abundance of fractures and/or fracture intersections leads to higher seepage volumes. A few zones in the back adit recorded elevated seepage volumes immediately following large (>20 mm/day) precipitation events, with transit times of less than 4 h through the 8-m thick rock mass. In most locations, there is a 1?C6 month time lag between the onset of the rainy season and seepage, with longer times observed for the front adit. There is a less clear-cut relation between fracture abundance and seepage volume; processes such as evaporation and surface flow along the ceiling may also influence seepage.