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1.
The purpose of this study is to compare the reliability of various methods of estimating normal rock fracture compliance from elastic wave measurements. We compare ultrasonic through‐transmission laboratory measurements for a smooth fracture in a Westerly granite specimen with numerical simulations and analytical solutions. The focus is on deriving compliance from time delays. The influence of specimen and source transducer width was constrained using numerical wave simulations. We find that measured ultrasonic phase delays are better suited to estimate the fracture compliance than group delays. Using the frequency domain instead of the time domain increases the accuracy of the fracture compliance estimates. We further show that for cases where precise phase delay measurements are unavailable, employing first break times in conjunction with numerical simulations can be considered as an alternative. 相似文献
2.
Elastic properties of fluid saturated porous media with aligned fractures can be studied using the model of fractures as linear-slip interfaces in an isotropic porous background. Such a medium represents a particular case of a transversely isotropic (TI) porous medium, and as such can be analyzed with equations of anisotropic poroelasticity. This analysis allows the derivation of explicit analytical expressions for the low-frequency elastic constants and anisotropy parameters of the fractured porous medium saturated with a given fluid. The five elastic constants of the resultant TI medium are derived as a function of the properties of the dry (isotropic) background porous matrix, fracture properties (normal and shear excess compliances), and fluid bulk modulus. For the particular case of penny-shaped cracks, the expression for anisotropy parameter ε has the form similar to that of Thomsen [Geophys. Prospect. 43 (1995) 805]. However, contrary to the existing view, the compliance matrix of a fluid-saturated porous-fractured medium is not equivalent to the compliance matrix of any equivalent solid medium with a single set of parallel fractures. This unexpected result is caused by the wave-induced flow of fluids between pores and fractures. 相似文献
3.
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. 相似文献
4.
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. 相似文献
5.
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. 相似文献
6.
Joint PP‐ and PSV‐wave amplitudes versus offset and azimuth inversion for fracture compliances in horizontal transversely isotropic media 
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下载免费PDF全文 We propose a robust approach for the joint inversion of PP‐ and PSV‐wave angle gathers along different azimuths for the elastic properties of the homogeneous isotropic host rock and excess compliances due to the presence of fractures. Motivated by the expression of fluid content indicator in fractured reservoirs and the sensitivity of Lamé impedances to fluid type, we derive PP‐ and PSV‐wave reflection coefficients in terms of Lamé impedances, density, and fracture compliances for an interface separating two horizontal transversely isotropic media. Following a Bayesian framework, we construct an objective function that includes initial models. We employ the iteratively reweighted least‐squares algorithm to solve the inversion problem to estimate unknown parameters (i.e., Lamé impedances, density, and fracture compliances) from PP‐ and PSV‐wave angle gathers along different azimuths. Synthetic tests reveal that the unknown parameters estimated using the joint inversion approach match true values better than those estimated using a PP‐wave amplitude inversion only. A real data test indicates that reasonable results for subsurface fracture detection are obtained from the joint inversion approach. 相似文献
7.
We use linear slip theory to evaluate seismic reflections at non‐welded interfaces, such as faults or fractures, sandwiched between general anisotropic media and show that at low frequencies the real parts of the reflection coefficients can be approximated by the responses of equivalent welded interfaces, whereas the imaginary parts can be related directly to the interface compliances. The imaginary parts of low frequency seismic reflection coefficients at fault zones can be used to estimate the interface compliances, which can be related to fault properties upon using a fault model. At normal incidence the expressions uncouple and the complex‐valued P‐wave reflection coefficient can be related linearly to the normal compliance. As the normal compliance is highly sensitive to the infill of the interface, it can be used for gas/fluid identification in the fault plane. Alternatively, the tangential compliance of a fault can be estimated from the complex‐valued S‐wave reflection coefficient. The tangential compliance can provide information on the crack density in a fault zone. Coupling compliances can be identified and quantified by the observation of PS conversion at normal incidence, with a comparable linear relationship. 相似文献
8.
Fractures are pervasive features within the Earth’s crust and have a significant influence on the multi-physical response of the subsurface. The presence of coherent fracture sets often leads to observable seismic scattering enabling seismic techniques to remotely locate and characterise fracture systems. In this study, we confirm the general scale-dependence of seismic scattering and provide new results specific to shear-wave propagation. We do this by generating full waveform synthetics using finite-difference wave simulation within an isotropic background model containing explicit fractures. By considering a suite of fracture models having variable fracture density and fracture size, we examine the widening effect of wavelets due to scattering within a fractured medium by using several different approaches, such as root-mean-square envelope analysis, shear-wave polarisation distortion, differential attenuation analysis and peak frequency shifting. The analysis allows us to assess the scattering behavior of parametrised models in which the propagation direction is either normal or parallel to the fracture surfaces. The quantitative measures show strong observable deviations for fractures size on the order of or greater than the dominant seismic wavelength within the Mie and geometric scattering regime for both propagation normal and parallel to fracture strike. The results suggest that strong scattering is symptomatic of fractures having size on the same order of the probing seismic wave. 相似文献
9.
Although it is believed that natural fracture sets predominantly have near‐vertical orientation, oblique stresses and some other mechanisms may tilt fractures away from the vertical. Here, we examine an effective medium produced by a single system of obliquely dipping rotationally invariant fractures embedded in a transversely isotropic with a vertical symmetry axis (VTI) background rock. This model is monoclinic with a vertical symmetry plane that coincides with the dip plane of the fractures. Multicomponent seismic data acquired over such a medium possess several distinct features that make it possible to estimate the fracture orientation. For example, the vertically propagating fast shear wave (and the fast converted PS‐wave) is typically polarized in the direction of the fracture strike. The normal‐moveout (NMO) ellipses of horizontal reflection events are co‐orientated with the dip and strike directions of the fractures, which provides an independent estimate of the fracture azimuth. However, the polarization vector of the slow shear wave at vertical incidence does not lie in the horizontal plane – an unusual phenomenon that can be used to evaluate fracture dip. Also, for oblique fractures the shear‐wave splitting coefficient at vertical incidence becomes dependent on fracture infill (saturation). A complete medium‐characterization procedure includes estimating the fracture compliances and orientation (dip and azimuth), as well as the Thomsen parameters of the VTI background. We demonstrate that both the fracture and background parameters can be obtained from multicomponent wide‐azimuth data using the vertical velocities and NMO ellipses of PP‐waves and two split SS‐waves (or the traveltimes of PS‐waves) reflected from horizontal interfaces. Numerical tests corroborate the accuracy and stability of the inversion algorithm based on the exact expressions for the vertical and NMO velocities. 相似文献
10.
Fluid flow in many hydrocarbon reservoirs is controlled by aligned fractures which make the medium anisotropic on the scale of seismic wavelength. Applying the linear‐slip theory, we investigate seismic signatures of the effective medium produced by a single set of ‘general’ vertical fractures embedded in a purely isotropic host rock. The generality of our fracture model means the allowance for coupling between the normal (to the fracture plane) stress and the tangential jump in displacement (and vice versa). Despite its low (triclinic) symmetry, the medium is described by just nine independent effective parameters and possesses several distinct features which help to identify the physical model and estimate the fracture compliances and background velocities. For example, the polarization vector of the vertically propagating fast shear wave S1 and the semi‐major axis of the S1‐wave normal‐moveout (NMO) ellipse from a horizontal reflector always point in the direction of the fracture strike. Moreover, for the S1‐wave both the vertical velocity and the NMO velocity along the fractures are equal to the shear‐wave velocity in the host rock. Analysis of seismic signatures in the limit of small fracture weaknesses allows us to select the input data needed for unambiguous fracture characterization. The fracture and background parameters can be estimated using the NMO ellipses from horizontal reflectors and vertical velocities of P‐waves and two split S‐waves, combined with a portion of the P‐wave slowness surface reconstructed from multi‐azimuth walkaway vertical seismic profiling (VSP) data. The stability of the parameter‐estimation procedure is verified by performing non‐linear inversion based on the exact equations. 相似文献
11.
Michael A. Schoenberg 《Geophysical Prospecting》2009,57(2):169-185
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. 相似文献
12.
Angular and Frequency-Dependent Wave Velocity and Attenuation in Fractured Porous Media 总被引：1，自引：0，他引：1
José M. Carcione Boris Gurevich Juan E. Santos Stefano Picotti 《Pure and Applied Geophysics》2013,170(11):1673-1683
Wave-induced fluid flow generates a dominant attenuation mechanism in porous media. It consists of energy loss due to P-wave conversion to Biot (diffusive) modes at mesoscopic-scale inhomogeneities. Fractured poroelastic media show significant attenuation and velocity dispersion due to this mechanism. The theory has first been developed for the symmetry axis of the equivalent transversely isotropic (TI) medium corresponding to a poroelastic medium containing planar fractures. In this work, we consider the theory for all propagation angles by obtaining the five complex and frequency-dependent stiffnesses of the equivalent TI medium as a function of frequency. We assume that the flow direction is perpendicular to the layering plane and is independent of the loading direction. As a consequence, the behaviour of the medium can be described by a single relaxation function. We first consider the limiting case of an open (highly permeable) fracture of negligible thickness. We then compute the associated wave velocities and quality factors as a function of the propagation direction (phase and ray angles) and frequency. The location of the relaxation peak depends on the distance between fractures (the mesoscopic distance), viscosity, permeability and fractures compliances. The flow induced by wave propagation affects the quasi-shear (qS) wave with levels of attenuation similar to those of the quasi-compressional (qP) wave. On the other hand, a general fracture can be modeled as a sequence of poroelastic layers, where one of the layers is very thin. Modeling fractures of different thickness filled with CO2 embedded in a background medium saturated with a stiffer fluid also shows considerable attenuation and velocity dispersion. If the fracture and background frames are the same, the equivalent medium is isotropic, but strong wave anisotropy occurs in the case of a frameless and highly permeable fracture material, for instance a suspension of solid particles in the fluid. 相似文献
13.
Seismic amplitude variation with offset and azimuth (AVOaz) inversion is well known as a popular and pragmatic tool utilized to estimate fracture parameters. A single set of vertical fractures aligned along a preferred horizontal direction embedded in a horizontally layered medium can be considered as an effective long-wavelength orthorhombic medium. Estimation of Thomsen’s weak-anisotropy (WA) parameters and fracture weaknesses plays an important role in characterizing the orthorhombic anisotropy in a weakly anisotropic medium. Our goal is to demonstrate an orthorhombic anisotropic AVOaz inversion approach to describe the orthorhombic anisotropy utilizing the observable wide-azimuth seismic reflection data in a fractured reservoir with the assumption of orthorhombic symmetry. Combining Thomsen’s WA theory and linear-slip model, we first derive a perturbation in stiffness matrix of a weakly anisotropic medium with orthorhombic symmetry under the assumption of small WA parameters and fracture weaknesses. Using the perturbation matrix and scattering function, we then derive an expression for linearized PP-wave reflection coefficient in terms of P- and S-wave moduli, density, Thomsen’s WA parameters, and fracture weaknesses in such an orthorhombic medium, which avoids the complicated nonlinear relationship between the orthorhombic anisotropy and azimuthal seismic reflection data. Incorporating azimuthal seismic data and Bayesian inversion theory, the maximum a posteriori solutions of Thomsen’s WA parameters and fracture weaknesses in a weakly anisotropic medium with orthorhombic symmetry are reasonably estimated with the constraints of Cauchy a priori probability distribution and smooth initial models of model parameters to enhance the inversion resolution and the nonlinear iteratively reweighted least squares strategy. The synthetic examples containing a moderate noise demonstrate the feasibility of the derived orthorhombic anisotropic AVOaz inversion method, and the real data illustrate the inversion stabilities of orthorhombic anisotropy in a fractured reservoir. 相似文献
14.
当介质中存在按照一定方向排列的裂隙时，会在该区域形成接触程度不好的界面.当有地震波传播时，界面两侧的质点不再满足位移连续的条件，位移差与该点的应力有关.从而引起质点间的滑动，形成粘滞性界面.界面的这种特性可以用切向柔量和法向柔量来表示，柔量值越大，界面的接触程度和质点位移的连续性都越差.文中详细推导了平面波在粘滞性界面上的反射和透射系数公式，通过数值算例证明了粘滞性界面上的反射系数与入射波频率有关的性质.并说明了界面的粘滞性会在一定程度上影响到反射波的AVO特性.且当平面P波入射时，法向柔量对反射P波和转换波的影响均大于切向柔量的影响. 相似文献
15.
The hydrocarbon industry is moving increasingly towards tight sandstone and shale gas resources – reservoirs that require fractures to be produced economically. Therefore, techniques that can identify sets of aligned fractures are becoming more important. Fracture identification is also important in the areas of coal bed methane production, carbon capture and storage (CCS), geothermal energy, nuclear waste storage and mining. In all these settings, stress and pore pressure changes induced by engineering activity can generate or reactivate faults and fractures. P‐ and S‐waves are emitted by such microseismic events, which can be recorded on downhole geophones. The presence of aligned fracture sets generates seismic anisotropy, which can be identified by measuring the splitting of the S‐waves emitted by microseismic events. The raypaths of the S‐waves will have an arbitrary orientation, controlled by the event and geophone locations, meaning that the anisotropy system may only be partly illuminated by the available arrivals. Therefore to reliably interpret such splitting measurements it is necessary to construct models that compare splitting observations with modelled values, allowing the best fitting rock physics parameters to be determined. Commonly, splitting measurements are inverted for one fracture set and rock fabrics with a vertical axis of symmetry. In this paper we address the challenge of identifying multiple aligned fracture sets using splitting measured on microseismic events. We analyse data from the Weyburn CCS‐EOR reservoir, which is known to have multiple fracture sets, and from a hydraulic fracture stimulation, where it is believed that only one set is present. We make splitting measurements on microseismic data recorded on downhole geophone arrays. Our inversion technique successfully discriminates between the single and multiple fracture cases and in all cases accurately identifies the strikes of fracture sets previously imaged using independent methods (borehole image logs, core samples, microseismic event locations). We also generate a synthetic example to highlight the pitfalls that can be encountered if it is assumed that only one fracture set is present when splitting data are interpreted, when in fact more than one fracture set is contributing to the anisotropy. 相似文献
16.
Colin M. Sayers 《Geophysical Prospecting》2009,57(2):187-192
Natural fractures in reservoirs play an important role in determining fluid flow during production and knowledge of the orientation and density of fractures is required to optimize production. Variations in reflection amplitude versus offset (AVO) are sensitive to the presence of fractures but current models used to invert the seismic response often make simplified assumptions that prevent fractured reservoirs from being characterized correctly. For example, many models assume a single set of perfectly aligned fractures, whereas most reservoirs contain several fracture sets with variable orientation within a given fracture set. In addition, many authors only consider the azimuthal variation in the small offset amplitude versus offset and azimuth response (the variation in AVO gradient with azimuth), while the effect of fractures on amplitude versus offset and azimuth increases with increasing offset. In this paper, the variation in the reflection coefficient of seismic P -waves as a function of azimuth and offset due to the presence of multiple sets of fractures with variable orientation within any fracture set is used to determine the components of a second-rank fracture compliance tensor α ij . The variation in the trace of this tensor as a function of position in the reservoir can be used to estimate the variation in fracture density with position in the reservoir and may be used to choose the location of infill wells in the field. The principal axes of α ij reveal the most compliant direction within the reservoir and may be used to optimize the trajectory of deviated wells. The determination of the principal axes of α ij requires wide azimuth acquisition and the use of the small-offset amplitude versus offset and azimuth (the azimuthal variation of the AVO gradient) may give misleading results. 相似文献
17.
Variable-density groundwater flow and solute transport in porous media containing nonuniform discrete fractures 总被引：1，自引：0，他引：1
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. 相似文献
18.
We present a method for inversion of fracture compliance matrix components from wide‐azimuth noisy synthetic PS reflection data and quantitatively show that reflection amplitude variations with offset and azimuth for converted PS‐waves are more informative than P‐waves for fracture characterization. We consider monoclinic symmetry for fractured reservoir (parameters chosen from Woodford Shale), which can be formed by two or more sets of vertical fractures embedded in a vertically transverse isotropic background. Components of effective fracture compliance matrices for a medium with monoclinic symmetry are related to the characteristics of the fractured medium. Monte Carlo simulation results show that inversion of PS reflection data is more robust than that of PP reflection data to uncertainties in our a priori knowledge (vertically transverse isotropic parameters of unfractured rock) than PP reflection data. We also show that, while inversion of PP reflections is sensitive to contrasts in elastic properties of upper and lower media, inversion of PS reflections is robust with respect to such contrasts. 相似文献
19.
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. 相似文献
20.
Hydraulic fractures generated by fluid injection in rock formations are often mapped by seismic monitoring. In many cases, the microseismicity is asymmetric relative to the injection well, which has been interpreted by stress gradient along the direction of the hydraulic fracture. We present a mathematical model of asymmetric hydrofracture growth based on relations between the solid‐phase stress and the fracture hydraulics. For single fracture and single injection point, the model has three parameters, hydraulic conductivities of the fracture wings, and normalised stress gradient and predicts the positions of the fracture tips as functions of time. The model is applied to a set of microseismic event locations that occurred during and after an injection process. Two different methods are suggested that make it possible to delineate the fracture tips from the set of microseismic events. This makes it possible to determine the model parameters and to check the agreement between the model prediction and the measured data. The comparison of the measured and modelled growth of fracture wings supports both the assumption of the non‐zero stress gradient and the existence of the post‐injection unilateral growth. 相似文献