Fracture Compliance Research Articles

Elastic least-squares migration for quantitative reflection imaging of fracture compliances

To quantitatively image fractures with high resolution, we have developed an elastic least-squares migration (LSM) algorithm coupled with linear-slip theory, which accurately addresses seismic wave interaction with thin structures. We derive a linearized waveform inversion using the Born approximation to the boundary integral equation for scattered waves, including linear-slip interfaces for P-SV and SH wavefields. Numerical modeling tests assuming a laboratory-scale fracture where a 20 cm long fracture is illuminated by waves with a 50 kHz center frequency show that our LSM successfully estimates fracture compliances. Furthermore, due to the presence of coupling compliances at the fracture, the results using our LSM show better images than those using the conventional LSM estimating the Lamé constants. We also numerically illustrate that our LSM can be successfully applied to dipole acoustic borehole logging data with 3 kHz center frequency for single-well reflection imaging of a 10 m long, dipping fracture embedded in a random background. Finally, we apply LSM to laboratory experimental data, measuring PP reflections from a fluid-filled fracture. We confirm that the estimated fracture compliances correspond well to those estimated by earlier amplitude variation with offset inversion. Furthermore, the LSM resolves the spatially varying fracture compliances due to local filling of water in the fracture. Because the linear-slip theory can be applied to thin structures in a wide range of scales, high-resolution imaging results and estimated fracture compliance distributions will be crucial to further address small-scale properties at fractures, joints, and geologic faults.

Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium

In shale gas development, fracture density is an important lithologic parameter to properly characterize reservoir reconstruction, establish a fracturing scheme, and calculate porosity and permeability. The traditional methods usually assume that the fracture reservoir is one set of aligned vertical fractures, embedded in an isotropic background, and estimate some alternative parameters associated with fracture density. Thus, the low accuracy caused by this simplified model, and the intrinsic errors caused by the indirect substitution, affect the estimation of fracture density. In this paper, the fractured rock of monoclinic symmetry assumes two non-orthogonal vertical fracture sets, embedded in a transversely isotropic background. Firstly, assuming that the fracture radius, width, and orientation are known, a new form of P-wave reflection coefficient, in terms of weak anisotropy (WA) parameters and fracture density, was obtained by substituting the stiffness coefficients of vertical transverse isotropic (VTI) background, normal, and tangential fracture compliances. Then, a linear amplitude versus offset and azimuth (AVOA) inversion method, of WA parameters and fracture density, was constructed by using Bayesian theory. Tests on synthetic data showed that WA parameters, and fracture density, are stably estimated in the case of seismic data containing a moderate noise, which can provide a reliable tool in fracture prediction.

Modeling the seismic response of individual hydraulic fracturing stages observed in a time-lapse distributed acoustic sensing vertical seismic profiling survey

In 2017, distributed acoustic sensing (DAS) technology was deployed in a horizontal well to conduct a vertical seismic profiling survey before and after each of 78 hydraulic fracturing stages. From two vibroseis source locations at the surface, time shifts of P- and S-waves were observed but decayed over days. Some stages also showed waves scattered off the stimulated rock volume. We have used 2D finite difference elastic wavefield modeling to understand these observations and connect them to underlying properties of the stimulated rock. We have developed an effective medium model of vertical fractures that close exponentially with time as fluid leaks off into the formation can match the distribution of P- and S-wave time shifts along the well. This has enabled estimates of the height, normal and tangential fracture compliance values, and decay time of the stimulated rock volume. Additionally, the kinematics of scattered waves observed in the data have been found to be consistent with PS conversion across the stimulated rock volume from an individual stage. With higher quality DAS data, stage-by-stage inversion for height, fracture compliance, and decay time attributes may be possible for characterizing variations in the effectiveness of hydraulic fracturing.

The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium

To enable a mathematical description, geologic fractures are considered as infinitely thin planes embedded in a homogeneous medium. These fracture structures satisfy linear slip boundary conditions, namely, a discontinuous displacement and continuous stress. The general finite-difference (FD) method described by the elastic wave equations has challenges when attempting to simulate the propagation of waves at the fracture interface. The FD method expressed by velocity-stress variables with the explicit application of boundary conditions at the fracture interface facilitates the simulation of wave propagation in fractured discontinuous media that are described by elastic wave equations and linear slip interface conditions. We have developed a new FD scheme for horizontal and vertical fracture media. In this scheme, a fictitious grid is introduced to describe the discontinuous velocity at the fracture interface and a rotated staggered grid is used to accurately indicate the location of the fracture. The new FD scheme satisfies nonwelded contact boundary conditions, unlike traditional approaches. Numerical simulations in different fracture media indicate that our scheme is accurate. The results demonstrate that the reflection coefficient of the fractured interface varies with the incident angle, wavelet frequency, and normal and tangential fracture compliances. Our scheme and conclusions from this study will be useful in assessing the properties of fractures, enabling the proper delineation of fractured reservoirs.

Matrix-fluid-fracture decoupled-based elastic impedance variation with angle and azimuth inversion for fluid modulus and fracture weaknesses

The ratio of normal-to-shear fracture compliances, approximate to the product of two ratios of shear-to-compressional (S-to-P) wave velocity and normal-to-tangential fracture weakness, has been treated as an indicator of fluid content for an oil-bearing reservoir with aligned fractures. However, this indicator depends on both fluid saturation and fracturing strength of rocks, while the effective fluid bulk modulus as a sensitive indicator decouples the double effects of rock porosity and pore fluid. Moreover, an incorrect value of S- and P-wave velocity ratio may have a significant effect on the estimation of normal-to-shear fracture compliances. Based on the anisotropic poroelasticity amplitude variation with offset and azimuth (AVOAz) theory, we use the Born formalism and a first-order perturbation assumption to derive a matrix-fluid-fracture decoupled-based linearized PP-wave reflection coefficient in an oil-bearing fractured porous medium. Our derived formula reveals that the combined effects of background matrix, pore fluids and fracture strength on seismic amplitudes are separated, more applicable to implement the fluid identification and fracture detection in a liquid-filled porous reservoir with aligned fractures. Following a Bayesian framework, we develop a Bayesian inversion method to reasonably obtain the fluid modulus and saturated fracture weaknesses. The implementation methodology includes two steps: a model-based least-squares inversion for azimuthal elastic impedances and an iterative Bayesian seismic inversion for model parameters. Tests on synthetic data corrupted with moderate random noise indicate that these model parameters can be estimated reasonably and accurately when the oil-saturated rock is permeated with a single fracture system. Test on a real data set validates that the reliable estimates of fluid and fracture parameters can be used to realize the fluid identification and fracture detection in an oil-bearing fractured porous reservoir in a more direct manner than previous approach.

Risk factors related to contralateral hip fracture following hip fracture in elderly patients: a meta analysis and review

Objective To evaluate the factors related to contralateral hip fracture in the elderly patients with hip fracture by meta analysis. Methods Pubmed, Cochrane, CBMdisc, CNKI Chinese Journal Full Text Database and Wan Fang Database were searched for publications at home and abroad from January 2005 to April 2018 on factors related to contralateral hip fracture after hip fractures in the elderly. The publication quality was strictly evaluated before the data were extracted concerning gender and age(>65 years) of the patients, concomitant osteoporosis (Singh sign ≥4), primary fracture type, concomitant Parkinson disease, concomitant stroke, concomitant senile dementia, concomitant cataract, concomitant rheumatoid arthritis, concomitant diabetes, type of internal fixation for primary fracture and therapeutic compliance. Revman5.0 was used to perform the statistical analysis and the OR value and 95% CI were calculated fore each index. Results A total of 17 studies were included involving 1,504 patients with contralateral hip fracture among 13,717 elderly patients with hip fracture. The factors related to the refracture of the contralateral hip were the age of the patients (OR=-3.55, 95% CI:-5.60~-1.50, P<0.001), osteoporosis (OR=2.38, 95% CI: 1.36~4.17, P=0.002), Parkinson disease (OR=4.54, 95% CI: 2.74~7.53, P<0.001), stroke (OR=0.33, 95% CI: 0.18~0.59, P<0.001), senile dementia (OR=0.43, 95% CI: 0.29~0.62, P<0.001), cataract (OR=0.37, 95% CI: 0.22~0.63, P< 0.001), rheumatoid arthritis (OR=0.32, 95% CI: 0.21~0.50, P<0.001), diabetes (OR=0.65, 95% CI: 0.47~0.91, P=0.01), type of internal fixation for primary fracture (OR=0.51, 95% CI: 0.30~0.85, P=0.01), and therapeutic compliance (OR=0.36, 95% CI: 0.21~0.64, P<0.001). However, the refracture of the contralateral hip was not related to gender (OR=1.07, 95% CI: 0.45~2.56, P=0.88), smoking (OR=0.86, 95% CI: 0.40~1.86, P=0.70), fracture type (OR=0.97, 95% CI: 0.60~1.57, P=0.90), or hypertension (OR=0.70, 95% CI: 0.41~1.21, P=0.20). Conclusions In elderly patients with hip fracture, the risks for contralateral hip fracture may be advanced age, concomitant osteoporosis, Parkinson disease, stroke, senile dementia, cataract, rheumatoid arthritis and diabetes, type of internal fixation for primary fracture, and poor therapeutic compliance. However, no sufficient evidence has suggested that gender, smoking, type of hip fracture or concomitant hypertension might be associated with the contralateral hip fracture. Key words: Elderly; Hip fracture; Meta-analysis; Contralateral hip fracture; Related factors

Interaction of a normally-incident plane wave with a nonlinear poroelastic fracture.

While it has been recognized that a large amplitude incident wave upon a dry fracture can exhibit nonlinear seismic wave scattering due to its stress-dependent mechanical compliance, the impact of pore fluid in the fracture and a fluid-filled poroelastic background medium-features common for fractures in the Earth-are not well understood. As a first step toward an understanding of the nonlinear poroelastic response of elastic waves in fractured media, analytical approximate formulas are used for the amplitude and phase of a normally incident plane wave using a perturbation method, assuming a fluid-filled, highly compliant nonlinear interface embedded in a linear poroelastic solid. The stress-closure behavior of the fracture is modeled by nonlinear, poroelastic displacement-discontinuity boundary conditions (a linear-slip interface). The theory predicts that the static ("Direct current," or DC) and higher-order-harmonic waves produced by the nonlinear scattering can be greatly reduced by the presence of fluid in the fracture. This, however, depends upon a number of parameters, including fracture compliance, fluid properties (compressibility and viscosity), and the permeability of the background medium, as well as environmental parameters such as the initial fluid pressure and stress acting on the fracture. The static effect produces low-frequency fluid pressure pulses when a finite-duration wave is incident upon the fracture-behavior unique to fluid-filled fractures within a poroelastic medium.

Shock-induced Stoneley waves in carbonate rock samples

To assess the productivity of oil- and gas-bearing carbonate reservoirs, acoustic waves are used to extract information on the reservoir properties. We performed shock-induced broadband linear acoustic wave experiments on fractured carbonate samples with different fracture apertures. Samples were prepared from a full-diameter carbonate rock core in which the aperture of a single fracture could be varied by means of spacers. Microseismograms were constructed from multiple wave experiments. We measured the wave speeds of different wave modes in the annulus along the cylindrical core sample and the transmission coefficient of the Stoneley wave over the fracture. We found that the transmission coefficient of the annulus Stoneley wave can directly be related to the fracture aperture and the fracture length. Comparison with a straightforward quasi-1D theory yielded best-fit values for fracture aperture and length that were in agreement with the actual values, for the fracture aperture determined by the size of the spacers. The values of the fracture length were not accurately predicted probably due to the underestimation of the fracture compliance in the experiment. The inversion method can be optimized for carbonate reservoir conditions to extract fracture data from Stoneley wave transmission and reflection coefficients.

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

ABSTRACTFractures 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.

Seismic AVAZ inversion for orthorhombic shale reservoirs in the Longmaxi area, Sichuan

Seismic AVAZ inversion method based on an orthorhombic model can be used to invert anisotropy parameters of the Longmaxi shale gas reservoir in the Sichuan Basin.. As traditional seismic inversion workflow does not sufficiently consider the influence of fracture orientation, we predict fracture orientation using the method based on the Fourier series to correct pre-stacked azimuth gathers to guarantee the accuracy of input data, and then conduct seismic AVAZ inversion based on the VTI constraints and Bayesian framework to predict anisotropy parameters of the shale gas reservoir in the study area.We further analyze the rock physical relation between anisotropy parameters and fracture compliance and mineral content for quantitative interpretation of seismic inversion results. Research results reveal that the inverted anisotropy parameters are related to P- and S-wave respectively, and thus can be used to distinguish the effect of fracture and fluids by the joint interpretation. Meanwhile high values of anisotropy parameters correspond to high values of fracture compliance, so the anisotropy parameters can reflect the development of fractures in reservoir. There is two sets of data from different sources, including the content of brittle mineral quartz obtained from well data and the anisotropy parameters inverted from seismic data, also show the positive correlation. This further indicates high content of brittle mineral makes fractures developing in shale reservoir and enhances seismic anisotropy of the shale reservoir. The inversion results demonstrate the characterization of fractures and brittleness for the Longmaxi shale gas reservoir in the Sichuan Basin.

A Novel Approach for Estimating Formation Permeability and Revisiting After-Closure Analysis of Diagnostic Fracture-Injection Tests

Summary Estimating reservoir-flow capacity is crucial for production estimation, hydraulic-fracturing design, and field development. Laboratory experiments can be used to measure the permeability of rock samples, but the results might not be representative at a field scale because of reservoir heterogeneity and pre-existing natural-fracture systems. Diagnostic fracture-injection tests (DFITs) have now become standard practice to estimate formation pore pressure and formation permeability. However, in low-permeability reservoirs, after-closure radial flow is often absent and this can result in significant uncertainties in interpreting DFIT data. In addition, the established methods for analyzing DFIT data make two oversimplified assumptions: Carter leakoff and constant fracture compliance (or stiffness) during fracture closure. However, both assumptions are violated during fracture closure; therefore, G-function-based models and subsequent related work can lead to an incorrect interpretation and are not capable of consistently fitting both before- and after-closure data coherently. Moreover, current after-closure analysis relies on classic well-test solutions with a constant injection rate. In reality, a “constant injection rate” does not equal “constant leakoff rate into the formation,” because more than 90% of the injected fluid stays inside the fracture at the end of pumping instead of leaking into the formation. The variable leakoff rate clearly violates the constant-rate boundary condition used in existing well-test solutions. In this study, we extend our previous work and derive time-convolution solutions to pressure-transient behavior of a closing fracture with infinite and finite fracture conductivity. We show that the G-function and the square-root-of-time models are only special cases of our general solutions. In addition, we found that after-closure linear-flow and bilinear-flow analysis can be used to infer pore pressure reliably but fail to estimate other parameters correctly. Most importantly, we present a new approach to history match the entire duration of DFIT data to estimate formation-flow capacity, even without knowing closure stress and the roughness properties of the fracture surface. Our approach adds significant value to DFIT interpretation and uncertainty analysis, especially in unconventional reservoirs where the absence of after-closure radial flow is the norm. Two representative field cases are also presented and discussed.

Amplitude variation with offset and azimuth inversion for fluid indicator and fracture weaknesses in an oil-bearing fractured reservoir

Fluid identification and fracture discrimination play an important role in the exploration and development of an oil-bearing fractured reservoir. The most common fluid indicator in fractured reservoirs, the normal-to-shear fracture compliance ratio, is influenced by the fluid content and the fracture intensity. To reduce the ambiguities in the discrimination of the fluid and fracture parameters, we have aimed to extend the scattering theory to implement the fluid identification and fracture detection by incorporating the azimuthal data in an oil-bearing fractured reservoir via the proposed Bayesian amplitude variation with offset and azimuth (AVOAz) inversion approach. The background medium is, as far as the scattering theory is concerned, an isotropic medium without fractures, and the fractured medium is corresponding to a perturbed medium. The elastic parameters of a saturated anisotropic medium can be parameterized as a perturbation over a homogeneous isotropic background medium. We used the scattering theory to derive a generalized AVOAz approximation that provided the iterative estimates of hydrocarbon fluid indicator, shear modulus, density, and fracture weaknesses in a Bayesian scheme. The inversion algorithm is based on a convolutional model and a weak-contrast and small-weakness PP-wave reflection coefficient. The approach is applied to an oil-bearing field data set from a fractured marlstone reservoir. We observe that reasonable estimates of fluid indicator and fracture weaknesses are inverted, which can be used to perform the discrimination of fluid and fracture parameters. We conclude that the proposed approach provides us a potentially powerful tool to estimate the reservoir fluid and fracture properties in a more straightforward and efficient manner than those previous methods.

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.

Fracture detection and fluid identification based on anisotropic Gassmann equation and linear-slip model

Detection of fracture and fluid properties from subsurface azimuthal seismic data improves our abilities to characterize the saturated porous reservoirs with aligned fractures. Motivated by the fracture detection and fluid identification in a fractured porous medium, we have developed a feasible approach to perform a rock physics model-based amplitude variation with offset and azimuth (AVOAz) inversion for the fracture and fluid parameters in a horizontal transversely isotropic (HTI) medium using the PP-wave angle gathers along different azimuths. Based on the linear-slip model, we first use anisotropic Gassmann’s equation to derive the expressions of saturated stiffness components and their perturbations of first-order approximation in terms of elastic properties of an isotropic porous background and fracture compliances induced by a single set of rotationally invariant fractures. We then derive a linearized PP-wave reflection coefficient in terms of fluid modulus, dry-rock matrix term, porosity, density, and fracture compliances or quasi-compliances for an interface separating two weakly HTI media based on the Born scattering theory. Finally, we solve the AVOAz inverse problems iteratively constrained by the Cauchy-sparse regularization and the low-frequency regularization in a Bayesian framework. The results demonstrate that the fluid modulus and fracture quasi-compliances are reasonably estimated in the case of synthetic and real seismic data containing moderate noise in a gas-filled fractured porous reservoir.

Compliance estimation and multiscale seismic simulation of hydraulic fractures

Hydraulic fracturing is a common stimulation technique in unconventional reservoirs to create fractures systems and allow hydrocarbon production. Proppant (granular material) is normally injected during hydraulic fracturing to keep open the fracture network and enhance hydrocarbon production performance. Proppant has a strong influence on fracture compliance and therefore will affect the characteristics of the generated seismic wavefield. To account for the effect of proppant in fracture compliance, we have developed new analytical formulations to obtain normal and tangential compliance for the case of dry and fluid-saturated fractures. We derive these expressions based on Hertz-Mindlin contact theory. Results from the compliance sensitivity analyses provide insights into the effects of proppant distribution and mechanical properties on fracture compliance. We also applied the innovative generalized multiscale finite-element method (GMsFEM) to simulate wave propagation through discrete hydraulic fractures filled with proppant. The GMsFEM approach represents individual fractures on a finely discretized mesh; this fine mesh is used to capture fracture properties by generating quantities (basis functions) that are used for modeling wave propagation on a much coarser grid. This methodology reduces the size of the computational problem, allowing faster results. Simulation results indicate the changes of the scattered wavefield as the proppant placement varies in different parts of the fractures and as the number of fracture stages increases.

Comprehensive before-closure model and analysis for fracture calibration injection falloff test

The fracture calibration injection falloff test, also known as mini-frac and diagnostic fracture injection test (DFIT), is widely used to quantify essential input parameters for the following main fracture treatment design. Over time, data presentations used to facilitate the analysis have revealed trends in the before-closure falloff transient data that have been labeled as abnormal closure. This paper provides new models for all known before-closure leakoff behaviors and a comprehensive model that enables a match with the entire before-closure transient response.The before-closure leakoff behaviors in the comprehensive model include wellbore storage (WBS), friction dissipation, tip extension, pressure dependent leakoff area (PDLA), multiple apparent closures, transferred storage (also known as transverse storage), height recession and variable fracture compliance. Besides the parameters such as closure pressure, leakoff coefficient, and the hydraulic fracture geometry that can be quantified by commonly known analysis techniques, the new model enables quantification of several new parameters.Two diagnostic plots are used to facilitate identification and quantification of the standard and newly modeled parameters: the Bourdet log-log plot of pressure change and derivative and the composite G-function plot. Tip extension is more easily identified on the G-function plot, and the model match with this behavior quantifies the tip extension distance after shut-in and the fracture minimum propagation pressure. So-called pressure dependent leakoff and transferred storage may be related to opening of secondary fractures, and may be seen as multiple closure events easily identified on either diagnostic plot. Such effects enable quantification of closure stress and surface areas for the secondary fracture sets. Alternatively, when the behavior that looks like transferred storage is explained as fracture height recession, the closure stress of overlying or underlying layers can be estimated along with the recessed fracture area. Variable fracture compliance has similar appearance with transferred storage as well in the diagnostic plots. When it occurs, the way to determine fracture closure stress and other parameters are provided.Matches with field data reflect highly enhanced understanding of the nature of hydraulic fracturing and natural fracture flow mechanisms that have not been possible with other models and that greatly refine well design strategies.

Estimating Unpropped-Fracture Conductivity and Fracture Compliance From Diagnostic Fracture-Injection Tests

Summary A new method is proposed to estimate the compliance and conductivity of induced unpropped fractures as a function of the effective stress acting on the fracture from diagnostic-fracture-injection-test (DFIT) data. A hydraulic-fracture resistance to displacement and closure is described by its compliance (or stiffness). Fracture compliance is closely related to the elastic, failure, and hydraulic properties of the rock. Quantifying fracture compliance and fracture conductivity under in-situ conditions is crucial in many Earth-science and engineering applications but is very difficult to achieve. Even though laboratory experiments are used often to measure fracture compliance and conductivity, the measurement results are influenced strongly by how the fracture is created, the specific rock sample obtained, and the degree to which it is preserved. As such, the results may not be representative of field-scale fractures. During the past 2 decades, the DFIT has evolved into a commonly used and reliable technique to obtain in-situ stresses, fluid-leakoff parameters, and formation permeability. The pressure-decline response across the entire duration of a DFIT reflects the process of fracture closure and reservoir-flow capacity. As such, it is possible to use these data to quantify changes in fracture conductivity as a function of stress. In this paper, we present a single, coherent mathematical framework to accomplish this. We show how each factor affects the pressure-decline response, and the effects of previously overlooked coupled mechanisms are examined and discussed. Synthetic and field-case studies are presented to illustrate the method. Most importantly, a new specialized plot (normalized system-stiffness plot) is proposed, which not only provides clear evidence of the existence of a residual fracture width as a fracture is closing during a DFIT, but also allows us to estimate fracture-compliance (or stiffness) evolution, and infer unpropped fracture conductivity using only DFIT pressure and time data alone. It is recommended that the normalized system-stiffness plot (NS plot) be used as a standard practice to complement the G-function or square-root-of-time plot and log-log plot because it provides very valuable information on fracture-closure behavior and the properties of fracture-surface roughness at a field-scale, information that cannot be obtained by any other means.

3D seismic characterization of fractures with random spacing using the double-beam method

Obtaining information on the spatial distribution of subsurface natural and induced fractures is critical in the production of geothermal or hydrocarbon fluids. Traditional seismic characterization methods for subsurface fractures are based on the assumption of effective anisotropy medium theory, which may not be true in reality when the fracture distribution is random. We have tested the recently proposed double-beam method to characterize nonuniformly distributed fractures. We built a 3D layered reservoir model; the reservoir layer was geometrically irregular, and it contained a set of randomly spaced fractures with spatially varying fracture compliances. We used an elastic full-wave finite-difference method to model the wavefield, where we treat the fractures as linear-slip boundaries and the data include all elastic multiple scattering. Taking the surface seismic data as input, the double-beam method forms a focusing source beam and a focusing receiver beam toward the fracture target. The fracture information is derived from the interference pattern of these two beams, which includes fracture orientation, fracture spacing, and fracture compliance as a function of spatial location. The fracture orientation parameter is the most readily determined parameter even for multiple nonorthogonal coexistent fracture sets. The beam-interference amplitude depends on the fracture spacing and compliance in a local average sense for random fractures. The beam-interference amplitude is large when there are many fractures or the compliance value is large, which is important in the interpretation of the fluid-transport properties of a reservoir.

Estimation of fluid indicator and dry fracture compliances using azimuthal seismic reflection data in a gas-saturated fractured reservoir

Amplitude variation with offset and azimuth (AVOA) inversion is a powerful tool used to predict the fluid and fracture information by utilizing the observable seismic reflection amplitudes. Understanding the effect of in situ fluid and fracture characteristics on seismic response is significant in a fractured reservoir. Our goal is to demonstrate an AVOA inversion approach to utilize the observable azimuthal seismic reflection data to predict the fluid and fracture properties in a gas-saturated fractured reservoir. Starting from Gassmann’s anisotropic poroelasticity theory and the linear-slip model, we first derive a linearized PP-wave reflection coefficient in terms of fluid, rigidity, density, and dry fracture compliance parameters based on the scattering function and perturbations in dry stiffness matrix of a horizontal transversely isotropic (HTI) rock containing a system of aligned vertical fractures. The derived equation builds a linearized relationship between the characteristics of fluid and fracture and the seismic response of azimuthal seismic reflection amplitudes. Incorporating the convolution model and Bayesian inversion, we propose a feasible Bayesian AVOA inversion method to estimate the fluid and fracture parameters directly. Cauchy and Gaussian probability distribution are used for the a priori information of model parameters and the likelihood function, respectively, and the nonlinear iteratively reweighted least squares (IRLS) strategy is finally utilized to solve the maximum a posteriori solutions of model parameters. The synthetic examples containing a moderate noise demonstrate the feasibility of inversion method, and the real data illustrates the inversion stabilities of fluid and fracture parameters in a gas-saturated fractured reservoir.

Detection of Natural Fractures from Observed Surface Seismic Data Based on a Linear-Slip Model

Natural fractures play an important role in migration of hydrocarbon fluids. Based on a rock physics effective model, the linear-slip model, which defines fracture parameters (fracture compliances) for quantitatively characterizing the effects of fractures on rock total compliance, we propose a method to detect natural fractures from observed seismic data via inversion for the fracture compliances. We first derive an approximate PP-wave reflection coefficient in terms of fracture compliances. Using the approximate reflection coefficient, we derive azimuthal elastic impedance as a function of fracture compliances. An inversion method to estimate fracture compliances from seismic data is presented based on a Bayesian framework and azimuthal elastic impedance, which is implemented in a two-step procedure: a least-squares inversion for azimuthal elastic impedance and an iterative inversion for fracture compliances. We apply the inversion method to synthetic and real data to verify its stability and reasonability. Synthetic tests confirm that the method can make a stable estimation of fracture compliances in the case of seismic data containing a moderate signal-to-noise ratio for Gaussian noise, and the test on real data reveals that reasonable fracture compliances are obtained using the proposed method.