A practical model for fluid flow in discrete-fracture porous media by using the numerical manifold method

In this study, a numerical manifold method (NMM) model is developed to analyze flow in porous media with discrete fractures in a non-conforming mesh. This new model is based on a two-cover-mesh system with a uniform triangular mathematical mesh and boundary/fracture-divided physical covers, where local independent cover functions are defined. The overlapping parts of the physical covers are elements where the global approximation is defined by the weighted average of the physical cover functions. The mesh is generated by a tree-cutting algorithm. A new model that does not introduce additional degrees of freedom (DOF) for fractures was developed for fluid flow in fractures. The fracture surfaces that belong to different physical covers are used to represent fracture flow in the direction of the fractures. In the direction normal to the fractures, the fracture surfaces are regarded as Dirichlet boundaries to exchange fluxes with the rock matrix. Furthermore, fractures that intersect with Dirichlet or Neumann boundaries are considered. Simulation examples are designed to verify the efficiency of the tree-cutting algorithm, the calculation's independency from the mesh orientation, and accuracy when modeling porous media that contain fractures with multiple intersections and different orientations. The simulation results show good agreement with available analytical solutions. Finally, the model is applied to cases that involve nine intersecting fractures and a complex network of 100 fractures, both of which achieve reasonable results. The new model is very practical for modeling flow in fractured porous media, even for a geometrically complex fracture network with large hydraulic conductivity contrasts between fractures and the matrix.     

An equivalent medium model for wave simulation in fractured porous rocks

Seismic wave propagation in reservoir rocks is often strongly affected by fractures and micropores. Elastic properties of fractured reservoirs are studied using a fractured porous rock model, in which fractures are considered to be embedded in a homogeneous porous background. The paper presents an equivalent media model for fractured porous rocks. Fractures are described in a stress‐strain relationship in terms of fracture‐induced anisotropy. The equations of poroelasticity are used to describe the background porous matrix and the contents of the fractures are inserted into a matrix. Based on the fractured equivalent‐medium theory and Biot's equations of poroelasticity, two sets of porosity are considered in a constitutive equation. The porous matrix permeability and fracture permeability are analysed by using the continuum media seepage theory in equations of motion. We then design a fractured porous equivalent medium and derive the modified effective constants for low‐frequency elastic constants due to the presence of fractures. The expressions of elastic constants are concise and are directly related to the properties of the main porous matrix, the inserted fractures and the pore fluid. The phase velocity and attenuation of the fractured porous equivalent media are investigated based on this model. Numerical simulations are performed. We show that the fractures and pores strongly influence wave propagation, induce anisotropy and cause poroelastic behaviour in the wavefields. We observe that the presence of fractures gives rise to changes in phase velocity and attenuation, especially for the slow P‐wave in the direction parallel to the fracture plane.     

A Method to Represent a Well in a Three-Dimensional Discrete Fracture Network Model

In discrete fracture network (DFN) modeling, fractures are randomly generated and placed in the model domain. The rock matrix is considered impermeable. Small fractures and isolated fractures are often ignored to reduce computational expense. As a result, the rock matrix between fractures could be large and intersections may not be found between a well introduced in the model and the hydraulically connected fracture networks (fracture backbones). To overcome this issue, this study developed a method to conceptualize a well in a three-dimensional (3D) DFN using two orthogonal rectangular fractures oriented along the well's axis. Six parameters were introduced to parameterize the well screen and skin zone, and to control the connectivity between the well and the fracture backbones. The two orthogonal fractures were discretized using a high-resolution mesh to improve the quality of flow and transport simulations around and along the well. The method was successfully implemented within dfnWorks 2.0 (Hyman et al. 2015) to incorporate a well in a 3D DFN and to track particles leaving an injection well and migrating to a pumping well. Verification of the method against MODFLOW/MODPATH found a perfect match in simulated hydraulic head and particle tracking. Using three examples, the study showed that the method ensured the connectivity between wells and fracture backbones, and honored the physical processes of flow and transport along and around wells in DFNs. Recommendations are given for estimating the values of the six introduced well parameters in a real-world case study.     

Seismic dispersion and attenuation in layered porous rocks with fractures of varying orientations

Wave‐induced fluid flow plays an important role in affecting the seismic dispersion and attenuation of fractured porous rocks. While numerous theoretical models have been proposed for the seismic dispersion and attenuation in fractured porous rocks, most of them neglect the wave‐induced fluid flow resulting from the background anisotropy (e.g. the interlayer fluid flow between different layers) that can be normal in real reservoirs. Here, according to the theories of poroelasticity, we present an approach to study the frequency‐dependent seismic properties of more realistic and complicated rocks, i.e. horizontally and periodically layered porous rock with horizontal and randomly orienting fractures, respectively, distributed in one of the two periodical layers. The approach accounts for the dual effects of the wave‐induced fluid flow between the fractures and the background pores and between different layers (the interlayer fluid flow). Because C₃₃ (i.e., the modulus of the normally incident P‐wave) is directly related to the P‐wave velocity widely measured in the seismic exploration, and its comprehensive dispersion and attenuation are found to be most significant, we study mainly the effects of fracture properties and the stiffness contrast between the different layers on the seismic dispersion and attenuation of C₃₃. The results show that the increasing stiffness contrast enhances the interlayer fluid flow of the layered porous rocks with both horizontal and randomly orienting fractures and weakens the wave‐induced fluid flow between the fractures and the background pores, especially for the layered porous rock with horizontal fractures. The modelling results also demonstrate that for the considered rock construction, the increasing fracture density reduces the interlayer fluid flow while improves the dispersion and attenuation in the fracture‐relevant frequency band. Increasing fracture aspect ratio is found to reduce the dispersion and attenuation in the fracture‐relevant frequency band only, especially for the layered porous rock with horizontal fractures.     

A new scheme for numerical modelling of flow and transport processes in 3D fractured porous media

The objective of this work is to develop a new numerical approach for the three-dimensional modelling of flow and transient solute transport in fractured porous media which would provide an accurate and efficient treatment of 3D complex geometries and inhomogeneities. For this reason, and in order to eliminate as much as possible the number of degrees of freedom, the fracture network, fractures and their intersections, are solved with a coupled 2D–1D model while the porous matrix is solved independently with a 3D model. The interaction between both models is accounted for by a coupling iterative technique. In this way it is possible to improve efficiency and reduce CPU usage by avoiding 3D mesh refinements of the fractures. The approach is based on the discrete-fracture model in which the exact geometry and location of each fracture in the network must be provided as an input. The formulation is based on a multidimensional coupling of the boundary element method-multidomain (BEM-MD) scheme for the flow and boundary element dual reciprocity method-multidomain (BE-DRM-MD) scheme for the transport. Accurate results and high efficiency have been obtained and are reported in this paper.     

Anisotropic permeability in fractured reservoirs from frequency‐dependent seismic Amplitude Versus Angle and Azimuth data

Attempts have previously been made to predict anisotropic permeability in fractured reservoirs from seismic Amplitude Versus Angle and Azimuth data on the basis of a consistent permeability‐stiffness model and the anisotropic Gassmann relations of Brown and Korringa. However, these attempts were not very successful, mainly because the effective stiffness tensor of a fractured porous medium under saturated (drained) conditions is much less sensitive to the aperture of the fractures than the corresponding permeability tensor. We here show that one can obtain information about the fracture aperture as well as the fracture density and orientation (which determines the effective permeability) from frequency‐dependent seismic Amplitude Versus Angle and Azimuth data. Our workflow is based on a unified stiffness‐permeability model, which takes into account seismic attenuation by wave‐induced fluid flow. Synthetic seismic Amplitude Versus Angle and Azimuth data are generated by using a combination of a dynamic effective medium theory with Rüger's approximations for PP reflection coefficients in Horizontally Transversely Isotropic media. A Monte Carlo method is used to perform a Bayesian inversion of these synthetic seismic Amplitude Versus Angle and Azimuth data with respect to the parameters of the fractures. An effective permeability model is then used to construct the corresponding probability density functions for the different components of the effective permeability constants. The results suggest that an improved characterization of fractured reservoirs can indeed be obtained from frequency‐dependent seismic Amplitude Versus Angle and Azimuth data, provided that a dynamic effective medium model is used in the inversion process and a priori information about the fracture length is available.     

Effect of fracture fill on frequency‐dependent anisotropy of fractured porous rocks

In fractured reservoirs, seismic wave velocity and amplitude depend on frequency and incidence angle. Frequency dependence is believed to be principally caused by the wave‐induced flow of pore fluid at the mesoscopic scale. In recent years, two particular phenomena, i.e., patchy saturation and flow between fractures and pores, have been identified as significant mechanisms of wave‐induced flow. However, these two phenomena are studied separately. Recently, a unified model has been proposed for a porous rock with a set of aligned fractures, with pores and fractures filled with two different fluids. Existing models treat waves propagating perpendicular to the fractures. In this paper, we extend the model to all propagation angles by assuming that the flow direction is perpendicular to the layering plane and is independent of the loading direction. We first consider the limiting cases through poroelastic Backus averaging, and then we obtain the five complex and frequency‐dependent stiffness values of the equivalent transversely isotropic medium as a function of the frequency. The numerical results show that, when the bulk modulus of the fracture‐filling fluid is relatively large, the dispersion and attenuation of P‐waves are mainly caused by fractures, and the values decrease as angles increase, almost vanishing when the incidence angle is 90° (propagation parallel to the fracture plane). While the bulk modulus of fluid in fractures is much smaller than that of matrix pores, the attenuation due to the “partial saturation” mechanism makes the fluid flow from pores into fractures, which is almost independent of the incidence angle.     

Vertically fractured transversely isotropic media: dimensionality and deconstruction

A vertically fractured transversely isotropic (VFTI) elastic medium is one in which any number of sets of vertical aligned fractures (each set has its normal lying in the horizontal x₁, x₂‐plane) pervade the medium and the sets of aligned fractures are the only features of the medium disturbing the axi‐symmetry about the x₃‐axis implying that in the absence of fractures, the background medium is transversely isotropic (TI). Under the assumptions of long wavelength equivalent medium theory, the compliance matrix of a fractured medium is the sum of the background medium's compliance matrix and a fracture compliance matrix. For sets of parallel rotationally symmetric fractures (on average), the fracture compliance matrix is dependent on 3 parameters − its normal and tangential compliance and its strike direction. When one fracture set is present, the medium is orthorhombic and the analysis is straightforward. When two (non‐orthogonal) or more sets are present, the overall medium is in general elastically monoclinic; its compliance tensor components are subject to two equalities yielding an 11 parameter monoclinic medium. Constructing a monoclinic VFTI medium with n embedded vertical fracture sets, requires 5 TI parameters plus 3×n fracture set parameters. A deconstruction of such an 11 parameter monoclinic medium involves using its compliance tensor to find a background transversely isotropic medium and several sets of vertical fractures which, in the long wavelength limit, will behave exactly as the original 11 parameter monoclinic medium. A minimal deconstruction, would be to determine, from the 11 independent components, the transversely isotropic background (5 parameters) and two fracture sets (specified by 2 × 3 = 6 parameters). Two of the background TI medium's compliance matrix components are known immediately by inspection, leaving nine monoclinic components to be used in the minimal deconstruction of the VFTI medium. The use of the properties of a TI medium, which are linear relations on its compliance components, allows the deconstruction to be reduced to solving a pair of non‐linear equations on the orientations of two fracture sets. A single root yielding a physically meaningful minimum deconstruction yields a unique minimal representation of the monoclinic medium as a VFTI medium. When no such root exists, deconstruction requires an additional fracture set and uniqueness is lost. The boundary between those monoclinic media that have a unique minimal representation and those that do not is yet to be determined.     

Frequency-dependent PP and PS reflection coefficients in fractured media

When a porous layer is permeated by mesoscale fractures, wave-induced fluid flow between pores and fractures can cause significant attenuation and dispersion of velocities and anisotropy parameters in the seismic frequency band. This intrinsic dispersion due to fracturing can create frequency-dependent reflection coefficients in the layered medium. In this study, we derive the frequency-dependent PP and PS reflection coefficients versus incidence angle in the fractured medium. We consider a two-layer vertical transverse isotropy model constituted by an elastic shale layer and an anelastic sand layer. Using Chapman's theory, we introduce the intrinsic dispersion due to fracturing in the sand layer. Based on the series coefficients that control the behaviour of velocity and anisotropy parameters in the fractured medium at low frequencies, we extend the conventional amplitude-versus-offset equations into frequency domain and derive frequency-dependent amplitude-versus-offset equations at the elastic–anelastic surface. Increase in fracture length or fracture density can enlarge the frequency dependence of amplitude-versus-offset attributes of PP and PS waves. Also, the frequency dependence of magnitude and phase angle of PP and PS reflection coefficients increases as fracture length or fracture density increases. Amplitude-versus-offset type of PP and PS reflection varies with fracture parameters and frequency. What is more, fracture length shows little impact on the frequency-dependent critical phase angle, while the frequency dependence of the critical phase angle increases with fracture density.     

Bayesian seismic inversion for estimating fluid content and fracture parameters in a gas-saturated fractured porous reservoir

Understanding the effects of in situ fluid content and fracture parameters on seismic characteristics is important for the subsurface exploration and production of fractured porous rocks. The ratio of normal-to-shear fracture compliance is typically utilized as a fluid indicator to evaluate anisotropy and identify fluids filling the fractures, but it represents an underdetermined problem because this fluid indicator varies as a function of both fracture geometry and fluid content. On the bases of anisotropic Gassmann's equation and linear-slip model, we suggest an anisotropic poroelasticity model for fractured porous reservoirs. By combining a perturbed stiffness matrix and asymptotic ray theory, we then construct a direct relationship between the PP-wave reflection coefficients and characteristic parameters of fluids(P-and S-wave moduli) and fractures(fracture quasi-weaknesses), thereby decoupling the effects of fluid and fracture properties on seismic reflection characterization.By incorporating fracture quasi-weakness parameters, we propose a novel parameterization method for elastic impedance variation with offset and azimuth(EIVOA). By incorporating wide-azimuth observable seismic reflection data with regularization constraints, we utilize Bayesian seismic inversion to estimate the fluid content and fracture parameters of fractured porous rocks. Tests on synthetic and real data demonstrate that fluid and fracture properties can be reasonably estimated directly from azimuthal seismic data and the proposed approach provides a reliable method for fluid identification and fracture characterization in a gas-saturated fractured porous reservoir.     

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.     

Diffraction imaging of sub‐vertical fractures and karst with full‐resolution 3D Ground‐Penetrating Radar

Vertical fractures with openings of less than one centimetre and irregular karst cause abundant diffractions in Ground‐Penetrating Radar (GPR) records. GPR data acquired with half‐wavelength trace spacing are uninterpretable as they are dominated by spatially undersampled scattered energy. To evaluate the potential of high‐density 3D GPR diffraction imaging a 200 MHz survey with less than a quarter wavelength grid spacing (0.05 m × 0.1 m) was acquired at a fractured and karstified limestone quarry near the village of Cassis in Southern France. After 3D migration processing, diffraction apices line up in sub‐vertical fracture planes and cluster in locations of karstic dissolution features. The majority of karst is developed at intersections of two or more fractures and is limited in depth by a stratigraphic boundary. Such high‐resolution 3D GPR imaging offers an unprecedented internal view of a complex fractured carbonate reservoir model analogue. As seismic and GPR wave kinematics are similar, improvements in the imaging of steep fractures and irregular voids at the resolution limit can also be expected from high‐density seismic diffraction imaging.     

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.     

Fluid-dependent shear-wave splitting in a poroelastic medium with conjugate fracture sets

The dependence of shear‐wave splitting in fractured reservoirs on the properties of the filling fluid may provide a useful attribute for identifying reservoir fluids. If the direction of wave propagation is not perpendicular or parallel to the plane of fracturing, the wave polarized in the plane perpendicular to the fractures is a quasi‐shear mode, and therefore the shear‐wave splitting will be sensitive to the fluid bulk modulus. The magnitude of this sensitivity depends upon the extent to which fluid pressure can equilibrate between pores and fractures during the period of the deformation. In this paper, we use the anisotropic Gassmann equations and existing formulations for the excess compliance due to fracturing to estimate the splitting of vertically propagating shear waves as a function of the fluid modulus for a porous medium with a single set of dipping fractures and with two conjugate fracture sets, dipping with opposite dips to the vertical. This is achieved using two alternative approaches. In the first approach, it is assumed that the deformation taking place is quasi‐static: that is, the frequency of the elastic disturbance is low enough to allow enough time for fluid to flow between both the fractures and the pore space throughout the medium. In the second approach, we assume that the frequency is low enough to allow fluid flow between a fracture set and the surrounding pore space, but high enough so that there is not enough time during the period of the elastic disturbance for fluid flow between different fracture sets to occur. It is found that the second approach yields a much stronger dependence of shear‐wave splitting on the fluid modulus than the first approach. This is a consequence of the fact that at higher wave frequencies there is not enough time for fluid pressure to equilibrate and therefore the elastic properties of the fluid have a greater effect on the magnitude of the shear‐wave splitting.     

Low-frequency anisotropy in fractured and layered media

Computing effective medium properties is very important when upscaling data measured at small scale. In the presence of stratigraphic layering, seismic velocities and anisotropy parameters are scale and frequency dependent. For a porous layer permeated by aligned fractures, wave-induced fluid flow between pores and fractures can also cause significant dispersion in velocities and anisotropy parameters. In this study, we compare the dispersion of anisotropy parameters due to fracturing and layering at low frequencies. We consider a two-layer model consisting of an elastic shale layer and an anelastic sand layer. Using Chapman's theory, we introduce anisotropy parameters dispersion due to fractures (meso-scale) in the sand layer. This intrinsic dispersion is added to anisotropy parameters dispersion induced by layering (macro-scale) at low frequencies. We derive the series coefficients that control the behaviour of anisotropy parameters at low frequencies. We investigate the influences of fracture length and fracture density on fracturing effect, layering effect and combined effect versus frequency and volume fraction of sand layer. Numerical modelling results indicate that the frequency dependence due to layering is not always the dominant effect of the effective properties of the medium. The intrinsic dispersion is not negligible compared with the layering effect while evaluating the frequency-dependent properties of the layered medium.     

Numerical analysis of seismic wave propagation in fluid-saturated porous multifractured media

Elastic wave propagation and attenuation in porous rock layers with oriented sets of fractures, especially in carbonate reservoirs, are anisotropic owing to fracture sealing, fracture size, fracture density, filling fluid, and fracture strike orientation. To address this problem, we adopt the Chapman effective medium model and carry out numerical experiments to assess the variation in P-wave velocity and attenuation, and the shear-wave splitting anisotropy with the frequency and azimuth of the incident wave. The results suggest that velocity, attenuation, and anisotropy vary as function of azimuth and frequency. The azimuths of the minimum attenuation and maximum P-wave velocity are nearly coincident with the average strike of the two sets of open fractures. P-wave velocity is greater in sealed fractures than open fractures, whereas the attenuation of energy and anisotropy is stronger in open fractures than sealed fractures. For fractures of different sizes, the maximum velocity together with the minimum attenuation correspond to the average orientation of the fracture sets. Small fractures affect the wave propagation less. Azimuth-dependent anisotropy is low and varies more than the other attributes. Fracture density strongly affects the P-wave velocity, attenuation, and shear-wave anisotropy. The attenuation is more sensitive to the variation of fracture size than that of velocity and anisotropy. In the seismic frequency band, the effect of oil and gas saturation on attenuation is very different from that for brine saturation and varies weakly over azimuth. It is demonstrated that for two sets of fractures with the same density, the fast shear-wave polarization angle is almost linearly related with the orientation of one of the fracture sets.     

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.     

Seismic characterization of reservoirs with multiple fracture sets using velocity and attenuation anisotropy data

Knowledge about the spatial distribution of the fracture density and the azimuthal fracture orientation can greatly help in optimizing production from fractured reservoirs. Frequency-dependent seismic velocity and attenuation anisotropy data contain information about the fractures present in the reservoir. In this study, we use the measurements of velocity and attenuation anisotropy data corresponding to different seismic frequencies and azimuths to infer information about the multiple fracture sets present in the reservoir. We consider a reservoir model with two sets of vertical fractures characterized by unknown azimuthal fracture orientations and fracture densities. Frequency-dependent seismic velocity and attenuation anisotropy data is computed using the effective viscoelastic stiffness tensor and solving the Christoffel equation. A Bayesian inversion method is then applied to measurements of velocity and attenuation anisotropy data corresponding to different seismic frequencies and azimuth to estimate the azimuthal fracture orientations and the fracture densities, as well as their uncertainties. Our numerical examples suggest that velocity anisotropy data alone cannot recover the unknown fracture parameters. However, an improved estimation of the unknown fracture parameters can be obtained by joint inversion of velocity and attenuation anisotropy data.     

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.     

Observations of fluid‐dependent shear‐wave splitting in synthetic porous rocks with aligned penny‐shaped fractures‡

P‐ and S‐wave velocity and attenuation coefficients (accurate to ±0.3% and ±0.2 dB/cm, respectively) were measured in synthetic porous rocks with aligned, penny‐shaped fractures using the laboratory ultrasonic pulse‐echo method. Shear‐wave splitting was observed by rotating the S‐wave transducer and noting the maximum and minimum velocities relative to the fracture direction. A block of synthetic porous rock of fracture density 0.0201 ± 0.0068 and fracture size 3.6 ± 0.38 mm (measured from image analysis of X‐ray CT scans) was sub‐sampled into three 20–30 mm long, 50 mm diameter core plugs oriented at 0°, 45° and 90° to the fracture normal (transversely isotropic symmetry axis). Full waveform data were collected over the frequency range 500–1000 kHz for both water and glycerin saturated cores to observe the effect of pore fluid viscosity at 1 cP and 100 cP, respectively. The shear‐wave splitting observed in the 90° core was 2.15 ± 0.02% for water saturated and 2.39 ± 0.02% for glycerin saturated, in agreement with the theory that suggests that the percentage splitting should be 100 times the fracture density and independent of the saturating fluid. In the 45° core, by contrast, splitting was 0.00 ± 0.02% for water saturation and ?0.77 ± 0.02% for glycerin saturation. This dependence on fracture orientation and pore fluid viscosity is consistent with the poro‐visco‐elastic theory for aligned, meso‐scale fractures in porous rocks. The results suggest the possible use of shear‐ or converted‐wave data to discriminate between fluids on the basis of viscosity variations.