In this work, we propose an approach for computing the positive solution of a fully fuzzy linear system where the coefficient matrix is a fuzzy n×n matrix. To do this, we use arithmetic operations on fuzzy numbers that introduced by Kaffman in [18] and convert the fully fuzzy linear system into two n × n and 2n × 2n crisp linear systems. If the solutions of these linear systems don't satisfy in positive fuzzy solution condition, we introduce the constrained least squares problem to obtain optimal fuzzy vector solution by applying the ranking function in given fully fuzzy linear system. Using our proposed method, the fully fuzzy linear system of equations always has a solution. Finally, we illustrate the efficiency of proposed method by solving some numerical examples.

We present a comparison of three different hydraulic fracture models as well as an anisotropic diffusivity model with the observed microseismic data from shale gas reservoirs in the Horn River Basin of Canada. We investigated the validity of these models in the prediction of hydraulic fracture geometries using tempo-spatial extension of microseismic data. In the study area, ten horizontal wells were drilled and hydraulically fractured in multiple stages in the Muskwa, Otter Park, and Evie shale gas formations in 2013. The treatments were monitored by downhole microseismic measurements. We integrated microseismic analyses, geomechanical information extracted from well logs, and fracturing treatment parameters performed in the area. We compared fracture geometry predicted by Perkins-Kern-Nordgren (PKN), Khristianovic-Geertsma-de Klerk (KGD), and a Pseudo-3D (P3D) fracturing models as well as an anisotropic diffusivity model with actual fracture geometries derived from microseismic records in more than one hundred fracturing stages. For the study area, we find that there are no barriers to hydraulic fracture vertical growth between the Muskwa, Otter Park and Evie shales. Therefore, the fracture height to length ratio is higher than unity in many stages. Large fracturing heights suggest that the PKN model might be more suitable for fracture modeling than the KGD model. However, our analyses show that the fracture length predicted by the KGD model is closer to, but still far less than the fracture length illustrated by microseismic events. Pseudo 3D model also predicts fracture lengths which are slightly larger than the modeled fracture lengths by the KGD and PKN equations and still significantly smaller than the microseismic fracture lengths. These differences are observed throughout all stages suggesting that these methods are not able to perfectly predict the hydraulic fracturing behavior in the study wellpad. Vertical extension of microseismic data with linear patterns into the Keg River formation below the shale formations suggests the presence of natural fractures in the study area. This study presents a distinctive insight into the complex hydraulic fracture modeling of shales in the Horn River basin and suggests that diffusivity mapping is a simple, but powerful tool for hydraulic fracture modeling in these formations. Observed microseismic fracture lengths are significantly higher than lengths predicted by the geomechanical models and closer to diffusivity models, which suggests the possibility of increasing well-spacing in future development using diffusivity equation for improving treatment design.

In order to attenuate the migration artifacts and increase the special resolution of the subsurface reflectivity, conventional migration may be replaced by the least squares migration (LSM). However, this is a costly procedure. To reduce the cost, the feasibility of using the multigrid methods in solving the linear system of prestack Kirchhoff LSM equation is investigated. This study showed that the conventional method of multigrid is not viable to solve Kirchhoff LSM equation for at least two reasons. The main reason is that the Hessian matrix is not a diagonally dominant matrix. Therefore, the conventional iterative solvers of the multigrid are not effective. The performance of Conjugate Gradient (CG) multigrid is discussed. It is shown that since CG does not have a smoothing property, it should not be considered as an effective multigrid iterative solver. Using the CG as an iterative solver for the multigrid may slightly reduces the number of iterations for the same rate of convergence in the CG itself. However, it does not reduce the total computational cost.

Summary We present an integrated interpretation of microseismic, treatment, and production data from hydraulic-fracturing jobs carried out in two adjacent wellpads in the Horn River Basin, northeast British Columbia, Canada. We conclude that poor correlation coefficients (R2) in crossplots of normalized production rate vs. the product of stimulated reservoir volume (SRV) and porosity and total organic carbon (TOC) (SRV×ϕ×TOC) indicate pressure interference between wells or wellpads. Good correlation coefficients in the same crossplots indicate lack of interference. The SRV×ϕ×TOC product reflects the hydrocarbon pore SRV because there is a relationship between TOC and hydrocarbon saturation in shales (Lopez and Aguilera 2018). Our results suggest that natural-fracture networks have an important effect on well connectivity and on the spatial distribution of microseismic data. Connectivity between wellpads occurs through a network of pre-existing natural fractures, which are approximately perpendicular to the least principal compressive stress in the area. This conclusion is supported by data analysis from Wellpads I and II in the Horn River Basin. Wellpad I includes eight wells that were drilled and fractured in the Muskwa and Otter Park formations (four wells in each formation) in 2010. Wellpad II includes three wells drilled and fractured in 2011 in each of the three shale formations, Muskwa, Otter Park, and Evie. There is a 1-year interval between fracturing on the first and second wellpads. The data analysis includes evaluation of magnitudes, b-values, moment-tensor inversion (MTI), and the spatial and temporal distributions of three-component microseismic events recorded during more than 200 stages of fracturing by multiwell downhole arrays. We analyzed Gutenberg-Richter frequency/magnitude graphs for each fracturing stage, and with proper integration of b-values, fracture-complexity index (FCI), MTI information, and treatment data, we distinguished hydraulic-fracturing-related events and events associated with slip along the surface of natural fractures. The results are compared with 5- and 4-year gas-production data in Wellpads I and II, respectively. Identification of natural fractures and information about interactions between hydraulically fractured wells are both essential for optimal well placement and completion, reservoir characterization, SRV calculation, and reservoir simulation. This study presents a distinctive insight into the integrated interpretation of microseismic events and production data to identify the activation of natural fractures and interference between the hydraulically fractured wells. The methodology developed in this study is thus related to production engineering, but examines it from the point of view of microseismic data.

A hybrid hydraulic fracture (HHF) model composed of (1) complex discrete fracture networks (DFN) and (2) planar fractures is proposed for modeling the stimulated reservoir volume (SRV). Modeling the SRV is complex and requires a synergetic approach between geophysics, petrophysics, and reservoir engineering. The objective of this paper is to characterize and evaluate the SRV considering the initial hydraulic fracturing efficiency, fracture network complexity, mechanics, and microseismicity distribution along 145 stimulated stages in a multilateral horizontal well on the Muskwa, Otter Park and Evie Formations in the Horn River Shale in Canada. Hydraulic fracturing jobs in shale reservoirs are designed with a view to achieve economic production by exploiting fracture network complexity. The task involves significant challenges in modeling and forecasting, which complicates the examination of operations to enhance their performance, including refracturing or infill drilling. In this study, an HHF is run in a numerical simulation model to evaluate the SRV performance in planar and complex fracture networks using microseismicity data collected during 75 stages of hydraulic fracturing in the Horn River shale. Post-fracturing production is appraised with Rate Transient Analysis (RTA) for determining effective permeability under flowing conditions, compare to the numerical simulation and the hydraulic fracturing design. Fracturing stages with larger fracture patch sizes, associated with the microseismic events in a fixed stress drop, correspond to higher stimulated areas, fracture conductivity, and gas production. Several microseismic events are observed in the heel of the laterals that are aligned to the far field NE stresses, indicated a loss of efficiency along the wellbore lateral during hydraulic fracturing. The hydraulic propagation modeling revealed increment of the leak-off coefficient, related to the natural fractures and communication with other stages. The production performance is evaluated in the numerical model, to measure interference between stages. The SRV, modeled with HHF networks, is able to match the post-fracturing production history. Fracture mechanics is important in order to understand the flowing performance of the reservoir. The inclusion of propagating models and RTA allowed to characterize possible fracture geometries in the reservoir and to observe limitations inherent to large dispersion and uncertainty of the microseismicity cloud. Also, to observe areas where the stimulation may have propped natural fractures totally or partially, which will benefit the production of gas. This study presents a better understanding and characterization of the SRV in shale gas reservoirs, especially in those cases where microseismicity dispersion is problematic and where the SRV is not easily delimited.