Understanding of hydraulic properties from configurations of stochastically distributed fracture networks

A systematic investigation of the effect of configurations of stochastically distributed fracture networks on hydraulic behaviour for fractured rock masses could provide either quantitative or qualitative correlation between the structural configuration of the fracture network and its corresponding hydraulic behaviour, and enhance our understanding of appropriate application of groundwater flow and contaminant transport modelling in fractured rock masses. In this study, the effect of block sizes, intersection angles of fracture sets, standard deviations of fracture orientation, and fracture densities on directional block hydraulic conductivity and representative elementary volume is systematically investigated in two dimensions by implementing a numerical discrete fracture fluid flow model and incorporating stochastically distributed fracture configurations. It is shown from this investigation that the configuration of a stochastically distributed fracture network has a significant quantitative or qualitative effect on the hydraulic behaviour of fractured rock masses. Compared with the deterministic fracture configurations that have been extensively dealt with in a previous study, this investigation is expected to be more practical and adequate, since fracture geometry parameters are inherently stochastically distributed in the field. Moreover, the methodology and approach presented in this study may be generally applied to any fracture system in investigating the hydraulic behaviours from configurations of the fracture system while establishing a ‘bridge’ from the discrete fracture network flow modelling to equivalent continuum modelling in fractured rock masses. Copyright © 2007 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.     

Constructing a discrete fracture network constrained by seismic inversion data

Rock fractures are of great practical importance to petroleum reservoir engineering because they provide pathways for fluid flow, especially in reservoirs with low matrix permeability, where they constitute the primary flow conduits. Understanding the spatial distribution of natural fracture networks is thus key to optimising production. The impact of fracture systems on fluid flow patterns can be predicted using discrete fracture network models, which allow not only the 6 independent components of the second‐rank permeability tensor to be estimated, but also the 21 independent components of the fully anisotropic fourth‐rank elastic stiffness tensor, from which the elastic and seismic properties of the fractured rock medium can be predicted. As they are stochastically generated, discrete fracture network realisations are inherently non‐unique. It is thus important to constrain their construction, so as to reduce their range of variability and, hence, the uncertainty of fractured rock properties derived from them. This paper presents the underlying theory and implementation of a method for constructing a geologically realistic discrete fracture network, constrained by seismic amplitude variation with offset and azimuth data. Several different formulations are described, depending on the type of seismic data and prior geologic information available, and the relative strengths and weaknesses of each approach are compared. Potential applications of the method are numerous, including the prediction of fluid flow, elastic and seismic properties of fractured reservoirs, model‐based inversion of seismic amplitude variation with offset and azimuth data, and the optimal placement and orientation of infill wells to maximise production.     

Review of Modeling Approaches to Groundwater Flow in Deformed Carbonate Aquifers

We discuss techniques to represent groundwater flow in carbonate aquifers using the three existing modeling approaches: equivalent porous medium, conduit network, and discrete fracture network. Fractures in faulted stratigraphic successions are characterized by dominant sets of sub-vertical joints. Grid rotation is recommended using the equivalent porous medium to match higher hydraulic conductivity with the dominant orientation of the joints. Modeling carbonate faults with throws greater than approximately 100 m is more challenging. Such faults are characterized by combined conduit-barrier behavior. The barrier behavior can be modeled using the Horizontal Flow Barrier Package with a low-permeability vertical barrier inserted to represent the impediment of horizontal flow in faults characterized by sharp drops of the piezometric surface. Cavities can occur parallel to the strike of normal faults generating channels for the groundwater. In this case, flow models need to account for turbulence using a conduit network approach. Channels need to be embedded in an equivalent porous medium due to cavities a few centimeters large, which are present in carbonate aquifers even in areas characterized by low hydraulic gradients. Discrete fracture network modeling enables representation of individual rock discontinuities in three dimensions. This approach is used in non-heavily karstified aquifers at industrial sites and was recently combined with the equivalent porous medium to simulate diffusivity in the matrix. Following this review, we recommend that the future research combines three practiced modeling approaches: equivalent porous medium, discrete fracture network, and conduit network, in order to capture structural and flow aspects in the modeling of groundwater in carbonate rocks.     

A new upscaling method for fractured porous media

We present a method to determine equivalent permeability of fractured porous media. Inspired by the previous flow-based upscaling methods, we use a multi-boundary integration approach to compute flow rates within fractures. We apply a recently developed multi-point flux approximation Finite Volume method for discrete fracture model simulation. The method is verified by upscaling an arbitrarily oriented fracture which is crossing a Cartesian grid. We demonstrate the method by applying it to a long fracture, a fracture network and the fracture network with different matrix permeabilities. The equivalent permeability tensors of a long fracture crossing Cartesian grids are symmetric, and have identical values. The application to the fracture network case with increasing matrix permeabilities shows that the matrix permeability influences more the diagonal terms of the equivalent permeability tensor than the off-diagonal terms, but the off-diagonal terms remain important to correctly assess the flow field.     

Improved Resolution of Ambient Flow through Fractured Rock with Temperature Logs

In contaminant hydrogeology, investigations at fractured rock sites are typically undertaken to improve understanding of the fracture networks and associated groundwater flow that govern past and/or future contaminant transport. Conventional hydrogeologic, geophysical, and hydrophysical techniques used to develop a conceptual model are often implemented in open boreholes under conditions of cross-connected flow. A new approach using high-resolution temperature (±0.001°C) profiles measured within static water columns of boreholes sealed using continuous, water-inflated, flexible liners (FLUTe™) identifies hydraulically active fractures under ambient (natural) groundwater flow conditions. The value of this approach is assessed by comparisons of temperature profiles from holes (100 to 200 m deep) with and without liners at four contaminated sites with distinctly different hydrogeologic conditions. The results from the lined holes consistently show many more hydraulically active fractures than the open-hole profiles, in which the influence of vertical flow through the borehole between a few fractures masks important intermediary flow zones. Temperature measurements in temporarily sealed boreholes not only improve the sensitivity and accuracy of identifying hydraulically active fractures under ambient conditions but also offer new insights regarding previously unresolvable flow distributions in fractured rock systems, while leaving the borehole available for other forms of testing and monitoring device installation.     

Unifying NAPL Drawdown and Transmissivity Testing in Unconfined,Confined, Perched,and Fractured Settings using the Z-Factor and MH Principles

Light nonaqueous phase liquid (LNAPL) flow in in fractured rock is governed by the same physics as porous media, but LNAPL discharge to a well from fractured rock is subject to the unique geometry of the fractures within the rock and the degree of interconnectivity between the factures. Previous conceptualization and definition of drawdown of nonaqueous phase liquids (NAPL) has employed a single drawdown value to represent the entire vertical interval of mobile NAPL. Application of the single drawdown model may result in erroneous calculation of NAPL transmissivity in fractured rock settings. This work illustrates how drawdown in multiphase systems can be variable over the vertical interval of mobile NAPL. In settings with discrete fracture networks, it is clear that consistently applying a single drawdown value will not accurately represent the pressure gradients. This work presents the multiphase head (MH) model, which is proposed as a comprehensive methodology for evaluating NAPL drawdown in fractured rock, and unconsolidated porous media. The MH model utilizes fluid statics and physical principles to accurately represent pressure differences in the formation and convert those into NAPL drawdown for discrete elevations. This first principles approach to describing how drawdown varies with NAPL-production zone elevations and fluid levels, resulting in a more accurate representation of discharge vs. fluid elevation behavior. Application of the MH model to various scenarios has identified that dissimilar scenarios can represent similar behavior during recovery from a NAPL removal event or baildown test. The resulting understanding improves the selection of representative portions of baildown test data to use in NAPL transmissivity analysis. Proper conceptualization of drawdown in bedrock identifies an alternate analysis method, the Z-factor, to estimate NAPL transmissivity. The resulting drawdown calculations and transmissivity analysis method result in a comprehensive approach to calculating NAPL transmissivity in both bedrock and unconsolidated porous media.     

A fracture network model and open fracture analysis of a tight sandstone gas reservoir in Dongpu Depression, Bohaiwan Basin, eastern China

Fracture systems can significantly influence productivity in tight sandstone reservoirs. Understanding the in situ fracture network is of importance in exploration and development of such reservoirs. This paper illustrates that stress-strain analysis, based on geological processes, is a credible prediction method for fracture networks and enables an estimation of which fractures are likely to contribute most to flow based on fracture orientation. This study shows the workflow for fracture modelling and its application in a sandstone reservoir using structural validation tools and interactive fracture simulations constrained with lithology, well log and core data. The fracture network model can help us understand not only what a fractured reservoir looks like but also how the existing fracture network behaves. An important consideration is that we need to understand and constrain the geological history to assess the fracture network properly.     

Grid convergence of variable-density flow simulations in discretely-fractured porous media

Grid convergence in space and time of variable-density flow in fractured-porous rock is systematically assessed. Convergence of the flow simulation is attained using both uniform and adaptive time-stepping. This contrasts to variable-density flow in unfractured porous media where grid convergence variable-density flow problems is almost never achieved. At high discretization levels, the number of fingers in fractured-porous rock is no longer influenced by spatial-temporal grid discretization, which is not the case in unfractured porous media. However, similar to unfractured porous media, the number of fingers in fractured-porous media varies at low discretization levels. Simulated convective pattern and penetration depth of the dense plume in fractured rock depend more on spatial discretization than on temporal discretization. The appropriate spatial-temporal grid is then used to examine some aspects of mixed convection in fractured-porous rock, characterized by the mixed convection number M. The critical mixed convection number M_c = 46 represents the transition between forced and free convection in fractured porous media, which is much higher than M_c = 1 in unfractured porous media. Thus, for mixed convective flow problems, the value of M_c is not a sufficient indicator to predict the convective mode (free convection-forced convection), and the presence of vertical fractures must be included in the prediction of convective flow modes.     

Variable-density groundwater flow and solute transport in porous media containing nonuniform discrete fractures

Variations in fluid density can greatly affect fluid flow and solute transport in the subsurface. Heterogeneities such as fractures play a major role for the migration of variable-density fluids. Earlier modeling studies of density effects in fractured media were restricted to orthogonal fracture networks, consisting of only vertical and horizontal fractures. The present study addresses the phenomenon of 3D variable-density flow and transport in fractured porous media, where fractures of an arbitrary incline can occur. A general formulation of the body force vector is derived, which accounts for variable-density flow and transport in fractures of any orientation. Simulation results are presented that show the verification of the new model formulation, for the porous matrix and for inclined fractures. Simulations of variable-density flow and solute transport are then conducted for a single fracture, embedded in a porous matrix. The simulations show that density-driven flow in the fracture causes convective flow within the porous matrix and that the high-permeability fracture acts as a barrier for convection. Other simulations were run to investigate the influence of fracture incline on plume migration. Finally, tabular data of the tracer breakthrough curve in the inclined fracture is given to facilitate the verification of other codes.     

Predicting Pesticide Attenuation in a Fractured Aquifer Using Lumped‐Parameter Models

Neighboring springs draining fractured‐rock aquifers can display large differences in water quality and flow regime, depending on local variations of the connectivity and the aperture size distribution of the fracture network. Consequently, because homogeneous equivalent parameters cannot be assumed a priori for the entire regional aquifer, the vulnerability to pollution of such springs has to be studied on a case by case basis. In this paper, a simple lumped‐parameter model usually applied to estimate the mean transit time of water (or tracer) is presented. The original exponential piston‐flow model was modified to take land‐use distribution into account and applied to predict the evolution of atrazine concentration in a series of springs draining a fractured sandstone aquifer in Luxembourg, where despite a nationwide ban in 2005, atrazine concentrations still had not begun to decrease in 2009. This persistence could be explained by exponentially distributed residence times in the aquifer, demonstrating that in some real world cases, models based on the groundwater residence time distribution can be a powerful tool for trend reversal assessments as recommended for instance by current European Union guidelines.     

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.     

Studying the Flow Dynamics of a Karst Aquifer System with an Equivalent Porous Medium Model

The modeling of groundwater flow in karst aquifers is a challenge due to the extreme heterogeneity of its hydraulic parameters and the duality in their discharge behavior, that is, rapid response of highly conductive karst conduits and delayed drainage of the low‐permeability fractured matrix after recharge events. There are a number of different modeling approaches for the simulation of the karst groundwater dynamics, applicable to different aquifer as well as modeling problem types, ranging from continuum models to double continuum models to discrete and hybrid models. This study presents the application of an equivalent porous model approach (EPM, single continuum model) to construct a steady‐state numerical flow model for an important karst aquifer, that is, the Western Mountain Aquifer Basin (WMAB), shared by Israel and the West‐Bank, using MODFLOW2000. The WMAB was used as a catchment since it is a well‐constrained catchment with well‐defined recharge and discharge components and therefore allows a control on the modeling approach, a very rare opportunity for karst aquifer modeling. The model demonstrates the applicability of equivalent porous medium models for the simulation of karst systems, despite their large contrast in hydraulic conductivities. As long as the simulated saturated volume is large enough to average out the local influence of karst conduits and as long as transport velocities are not an issue, EPM models excellently simulate the observed head distribution. The model serves as a starting basis that will be used as a reference for developing a long‐term dynamic model for the WMAB, starting from the pre‐development period (i.e., 1940s) up to date.     

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.     

基于方位各向异性弹性阻抗的裂缝岩石物理参数反演方法研究

裂缝储层岩石物理参数的准确获得对地下裂缝预测具有重要意义,而叠前地震反演是获得裂缝岩石物理参数的有效手段.本文从裂缝岩石物理等效模型的构建出发,从测井数据上估测了裂缝岩石物理参数,通过推导含裂缝岩石物理参数的方位各向异性弹性阻抗公式,探讨了基于方位各向异性弹性阻抗的裂缝岩石物理参数地震反演方法.实际工区地震数据应用表明,基于方位各向异性弹性阻抗的裂缝岩石物理参数反演方法合理、可靠,可以降低裂缝岩石物理参数估测的不确定性,为地下裂缝预测提供有力的依据.     

Anisotropic azimuthal surface-wave inversion for a vertically fractured and transverse isotropy medium: Theory and examples from a controlled field experiment

Fractures in elastic media add compliance to a rock in the direction normal to the fracture strike. Therefore, elastic wave velocities in a fractured rock will vary as a function of the energy propagation direction relative to the orientation of the aligned fracture set. Anisotropic Thomson–Haskell matrix Rayleigh-wave equations for a vertically transverse isotropic media can be used to model surface-wave dispersion along the principal axes of a vertically fractured and transversely isotropic medium. Furthermore, a workflow combining first-break analysis and azimuthal anisotropic Rayleigh-wave inversion can be used to estimate P-wave and S-wave velocities, Thomsen's ε, and Thomsen's δ along the principal axes of the orthorhombic symmetry. In this work, linear slip theory is used to map our inversion results to the equivalent vertically fractured and transversely isotropic medium coefficients. We carried out this inversion on a synthetic example and a field example. The synthetic data example results show that joint estimation of S-wave velocities with Thomsen's parameters ε and δ along normal and parallel to the vertical fracture set is reliable and, when mapped to the corresponding vertically fractured and transversely isotropic medium, provides insight into the fracture compliances. When the inversion was carried out on the field data, results indicated that the fractured rock is more compliant in the azimuth normal to the visible fracture set orientation and that the in situ normal fracture compliance to tangential fracture compliance ratio is less than half, which implies some cementation may have occurred along the fractures. Such an observation has significant implications when modelling the transport properties of the rock and its strength. Both synthetic and field examples show the potential of azimuthal anisotropic Rayleigh-wave inversion as the method can be further expanded to a more general case where the vertical fracture set orientation is not known a priori.     

Influence of small-scale heterogeneities on contaminant transport in fractured crystalline rock

We present a sequence of purely advective transport models that demonstrate the influence of small-scale geometric inhomogeneities on contaminant transport in fractured crystalline rock. Special weight is placed on the role of statistically generated variable fracture apertures. The fracture network geometry and the aperture distribution are based on information from an in situ radionuclide retardation experiment performed at Grimsel test site (Swiss Alps). The obtained breakthrough curves are fitted with the advection dispersion equation and continuous-time random walks (CTRW). CTRW is found to provide superior fits to the late-arrival tailing and is also found to show a good correlation with the velocity distributions obtained from the hydraulic models. The impact of small-scale heterogeneities, both in fracture geometry and aperture, on transport is shown to be considerable.     

An efficient numerical model for incompressible two-phase flow in fractured media

Various numerical methods have been used in the literature to simulate single and multiphase flow in fractured media. A promising approach is the use of the discrete-fracture model where the fracture entities in the permeable media are described explicitly in the computational grid. In this work, we present a critical review of the main conventional methods for multiphase flow in fractured media including the finite difference (FD), finite volume (FV), and finite element (FE) methods, that are coupled with the discrete-fracture model. All the conventional methods have inherent limitations in accuracy and applications. The FD method, for example, is restricted to horizontal and vertical fractures. The accuracy of the vertex-centered FV method depends on the size of the matrix gridcells next to the fractures; for an acceptable accuracy the matrix gridcells next to the fractures should be small. The FE method cannot describe properly the saturation discontinuity at the matrix–fracture interface. In this work, we introduce a new approach that is free from the limitations of the conventional methods. Our proposed approach is applicable in 2D and 3D unstructured griddings with low mesh orientation effect; it captures the saturation discontinuity from the contrast in capillary pressure between the rock matrix and fractures. The matrix–fracture and fracture–fracture fluxes are calculated based on powerful features of the mixed finite element (MFE) method which provides, in addition to the gridcell pressures, the pressures at the gridcell interfaces and can readily model the pressure discontinuities at impermeable faults in a simple way. To reduce the numerical dispersion, we use the discontinuous Galerkin (DG) method to approximate the saturation equation. We take advantage of a hybrid time scheme to alleviate the restrictions on the size of the time step in the fracture network. Several numerical examples in 2D and 3D demonstrate the robustness of the proposed model. Results show the significance of capillary pressure and orders of magnitude increase in computational speed compared to previous works.