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Book Performance Comparison of Transverse and Longitudinal Fractured Horizontal Wells Over Varied Reservoir Permeability

Download or read book Performance Comparison of Transverse and Longitudinal Fractured Horizontal Wells Over Varied Reservoir Permeability written by Fen Yang and published by . This book was released on 2014 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Since the first application in the mid-1980's, multiple fractured horizontal wells have proven to be an effective means of extracting hydrocarbons. These wells require careful consideration of wellbore orientation relative to the horizontal principle stress. Wellbore orientation can lead to transverse fractures which are perpendicular to the wellbore, or longitudinal fractures parallel the wellbore. Questions arise regarding whether one fracture orientation is consistently preferred over the other, or if certain conditions affect the choice. Historical work has examined the impact of horizontal wellbore azimuth in the Barnett and Marcellus Shale where public data was reviewed and statistical well analysis was conducted respectively. Comparison between transverse and longitudinal fracturing in moderate gas reservoirs has been performed with experimental study. This work includes both simulations and actual field cases studies. It compares transverse multiple fractured horizontal wells with longitudinal ones in terms of both well performance and economics. The study covers both gas and oil reservoirs and extends prior work to unconventional resources by extending the reservoir permeability to 0.00005 md. A range of reservoir permeability is identified for the preferable fracture configuration through simulations. Field production history of the Bakken, Barnett, Eagle Ford and Delaware formations are investigated and compared to the simulation results. In addition, this work analyzes the impact of fracture conductivity, lateral length, fracture half-length, completion method and hydrocarbon prices. The conclusions can be used as a reference in decision making on horizontal drilling and hydraulic fracturing for both unconventional and conventional resources"--Abstract, page iii.

Book Computational Fluid Dynamic Analysis in High Permeability Hydraulic Fractured Horizontal Gas Wells

Download or read book Computational Fluid Dynamic Analysis in High Permeability Hydraulic Fractured Horizontal Gas Wells written by Hrithu Vasudevan and published by . This book was released on 2016 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Hydraulic fracturing of horizontal wells in unconventional reservoirs has become the dominant type of well completion performed in the United States. In very low permeability reservoirs (~.00001-.0001 mD), the wellbore is aligned with the minimum horizontal stress, and the completion includes multiple transverse fractures. These fractures may be placed with either open hole sleeve type completion systems (OHMS), or cased hole plug and perf systems (P-n-P). In slightly higher permeability reservoirs (1 to 10 mD) multiple longitudinal fractures have been found to be preferred to completions with transverse fractures. This study presents an evaluation of gas well productivity for both transverse and longitudinal fractured horizontal wells using CFD simulations. The first part of the work includes an evaluation of one and two transverse fractures, over reservoir permeability of 1, 10 and 100 mD. Results, given as fold of increase, are compared to the single transverse fracture model of Augustine (2011). The work includes a parametric study of fracture width, penetration ratio and vertical to horizontal permeability ratio on production rates. The second part of the study includes CFD simulations for a single longitudinal fracture, and compares productivity results of this fracture orientation to transverse fractures in the 1, 10 and 100 mD cases. Results of this study suggest OHMs completions outperform P-n-P completions. The results of the work also corroborate the findings of Yang (2015) and Kassim et al (2016) suggesting that longitudinal fractured wells perform better in the slightly higher permeability reservoirs (1-10 mD)"--Abstract, page iii.

Book Modeling Performance of Horizontal Wells with Multiple Fractures in Tight Gas Reservoirs

Download or read book Modeling Performance of Horizontal Wells with Multiple Fractures in Tight Gas Reservoirs written by Guangwei Dong and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiple transverse fracturing along a horizontal well is a relatively new technology that is designed to increase well productivity by increasing the contact between the reservoir and the wellbore. For multiple transverse fractures, the performance of the well system is determined by three aspects: the inflow from the reservoir to the fracture, the flow from the fracture to the wellbore, and the inflow from the reservoir to the horizontal wellbore. These three aspects influence each other and combined, influence the wellbore outflow. In this study, we develop a model to effectively formulate the inter-relationships of a multi-fracture system. This model includes a reservoir model and a wellbore model. The reservoir model is established to calculate both independent and inter-fracture productivity index to quantify the contribution from all fractures on pressure drop of each fracture, by using the source functions to solve the single-phase gas reservoir flow model. The wellbore model is used to calculate the pressure distribution along the wellbore and the relationship of pressure between neighboring fractures, based on the basic pressure drop model derived from the mechanical energy balance. A set of equations with exactly the same number of fractures will be formed to model the system by integrating the two models. Because the equations are nonlinear, iteration method is used to solve them. With our integrated reservoir and wellbore model, we conduct a field study to find the best strategy to develop the field by hydraulic fracturing. The influence of reservoir size, horizontal and vertical permeability, well placement, and fracture orientation, type (longitudinal and transverse), number and distribution are completely examined in this study. For any specific field, a rigorous step-by-step procedure is proposed to optimize the field.

Book Unconventional Shale Gas Development

Download or read book Unconventional Shale Gas Development written by Rouzbeh G. Moghanloo and published by Gulf Professional Publishing. This book was released on 2022-02-23 with total page 498 pages. Available in PDF, EPUB and Kindle. Book excerpt: Unconventional Shale Gas Development: Lessons Learned gives engineers the latest research developments and practical applications in today’s operations. Comprised of both academic and corporate contributors, a balanced critical review on technologies utilized are covered. Environmental topics are presented, including produced water management and sustainable operations in gas systems. Machine learning applications, well integrity and economic challenges are also covered to get the engineer up-to-speed. With its critical elements, case studies, history plot visuals and flow charts, the book delivers a critical reference to get today’s petroleum engineers updated on the latest research and applications surrounding shale gas systems. Bridges the gap between the latest research developments and practical applications through case studies and workflow charts Helps readers understand the latest developments from the balanced viewpoint of academic and corporate contributors Considers environmental and sustainable operations in shale gas systems, including produced water management

Book Optimization of Fractured Well Performance of Horizontal Gas Wells

Download or read book Optimization of Fractured Well Performance of Horizontal Gas Wells written by Fellipe Vieira Magalhães and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In low-permeability gas reservoirs, horizontal wells have been used to increase the reservoir contact area, and hydraulic fracturing has been further extending the contact between wellbores and reservoirs. This thesis presents an approach to evaluate horizontal well performance for fractured or unfractured gas wells and a sensitivity study of gas well performance in a low permeability formation. A newly developed Distributed Volumetric Sources (DVS) method was used to calculate dimensionless productivity index for a defined source in a box-shaped domain. The unique features of the DVS method are that it can be applied to transient flow and pseudo-steady state flow with a smooth transition between the boundary conditions. In this study, I conducted well performance studies by applying the DVS method to typical tight sandstone gas wells in the US basins. The objective is to determine the best practice to produce horizontal gas wells. For fractured wells, well performance of a single fracture and multiple fractures are compared, and the effect of the number of fractures on productivity of the well is presented based on the well productivity. The results from this study show that every basin has a unique ideal set of fracture number and fracture length. Permeability plays an important role on dictating the location and the dimension of the fractures. This study indicated that in order to achieve optimum production, the lower the permeability of the formation, the higher the number of fractures.

Book Numerical Simulation of Pressure Transient Analysis in Tight Formation and Field Data Categorization and Typical Well Production Data Analysis

Download or read book Numerical Simulation of Pressure Transient Analysis in Tight Formation and Field Data Categorization and Typical Well Production Data Analysis written by Yue Zhu and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Fracturing Horizontal Wells

Download or read book Fracturing Horizontal Wells written by Mohamed Y. Soliman and published by McGraw Hill Professional. This book was released on 2016-04-21 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: Effectively Apply Modern Fracturing Methods in Horizontal Wells Improve productivity and maximize natural gas extraction using the practical information contained in this comprehensive guide. Written by world-renowned experts, Fracturing Horizontal Wells features complete details on the latest fracking tools and technologies. Illustrations, tables, and real-world examples are found throughout. Discover how to handle site selection and testing, build accurate simulations, and efficiently extract energy from horizontal sources, including shale formations. Environmental standards, regulatory compliance, and safety protocols are also included. Fracturing Horizontal Wells covers: • Fracture Stimulation of Horizontal Wells • Transitioning from Vertical to Horizontal Wellbores • Reservoir Engineering Aspects of Horizontal Wells • Reservoir Engineering Aspects of Fractured Horizontal Wells • Fracturing Horizontal Wells: Rock Mechanics Overview • Drilling of Horizontal Wells • Proppant and Proppant Transport • Fracture Diagnostic Testing • Interval Isolation • Horizontal Completion Fracturing Methods and Techniques • Use of Well Logging Measurements and Analysis for Fracturing Design • Fracture Treatment Diagnostics • Environmental Stewardship

Book Proceedings of the Fifth International Technical Symposium on Deepwater Oil and Gas Engineering

Download or read book Proceedings of the Fifth International Technical Symposium on Deepwater Oil and Gas Engineering written by Baojiang Sun and published by Springer Nature. This book was released on with total page 665 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Analytical and Numerical Solutions for the Case of a Horizontal Well with a Radial Power Law Permeability Distribution

Download or read book Analytical and Numerical Solutions for the Case of a Horizontal Well with a Radial Power Law Permeability Distribution written by Ryan Sawyer Broussard and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, I present the development of analytical solutions in the Laplace domain for a fully-penetrating, horizontal well producing at a constant flow rate or constant wellbore pressure in the center of a composite, cylindrical reservoir system with an impermeable outer boundary. The composite reservoir consists of two regions. The cylindrical region closest to the wellbore is stimulated, and the permeability within this region follows a power-law function of the radial distance from the wellbore. The unstimulated outer region has homogeneous reservoir properties. The current norm for successful stimulation of low permeability reservoir rocks is multi-stage hydraulic fracturing. The process of hydraulic fracturing creates thin, high permeability fractures that propagate deep into the reservoir, increasing the area of the rock matrix that is exposed to this low-resistance flow pathway. The large surface area of the high conductivity fracture is what makes hydraulic fracturing so successful. Unfortunately, hydraulic fracturing is often encumbered by problems such as high capital costs and a need for large volumes of water. Therefore, I investigate a new stimulation concept based upon the alteration of the permeability of a large volume around the producing well assembly from its original regime to that following a power-law function. I evaluate the effectiveness of the new concept by comparing it to conventional multi-stage hydraulic fracturing. The results of this investigation show that the power-law permeability reservoir (PPR) has a performance advantage over the multi-fractured horizontal treatment (MFH) only when the fracture conductivity and fracture half-length are small. Most importantly, the results demonstrate that the PPR can provide respectable flow rates and recovery factors, thus making it a viable stimulation concept for ultra-low permeability reservoirs, especially under conditions that may not be conducive to a conventional MHF treatment. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/149312

Book Analytical Modeling of Multi Fractured Horizontal Wells in Heterogeneous Unconventional Reservoirs

Download or read book Analytical Modeling of Multi Fractured Horizontal Wells in Heterogeneous Unconventional Reservoirs written by Jie Zeng and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Current analytical models for multi-fractured horizontal wells (MFHW) generally neglect reservoir heterogeneity, typical seepage characters of unconventional reservoir, partially penetrating fracture and various fracture damage mechanisms. In this thesis, three linear flow models have been developed to facilitate pressure and rate behavior analysis of shale, tight sand and unconventional reservoir with damaged fractures. These models are validated by comparing with KAPPA Ecrin and are more accurate than previous linear flow models in modeling partially penetrating cases. Field data are analyzed and results prove the reliability of these models. The first model is for heterogeneous shale reservoir with multiple gas transport mechanisms. It subdivides the reservoir into seven parts, namely, two upper/lower regions, two outer regions, two inner regions, and hydraulic fracture region. Fracture interference is simulated by locating a no-flow boundary between two adjacent fractures. The locations of these boundaries are determined based on the boundary's pressure to satisfy the no-flow assumption. Adsorption/desorption, gas slippage and diffusion effects are included for rigorous modeling of flow in shale. Sensitivity analysis results suggest that larger desorption coefficient causes smaller pressure and its derivative as a larger proportion of gas is desorbed in formation and contributes to productivity. The influences of other parameters, such as matrix II permeability, matrix block size, secondary fracture permeability, hydraulic fracture conductivity, and fracture pattern are also discussed. The second model is for heterogeneous tight sand reservoir with threshold pressure gradient (TPG). The linear flow sub-regions are the same as those of the first model. TPG and pressure drop within the horizontal wellbore are included. Simulation results suggest that TPG affects middle-late time behaviors. Greater TPG results in higher pressure drop and accelerates production decline. But this influence is marginal when TPG is small. Effects of other parameters, such as formation permeability, fracture length, conductivity, and wellbore storage are also investigated. The third model is for heterogeneous reservoir with various fracture damage. In this model, the following possible fracture damage situations are discussed: (1) choked fracture damage (2) partially propped fracture, (3) fracturing fluid leak-off damage, (4) dual or multiple damage effects. Simulation results indicate that choked frature damage influences early-mid time performance. Partially propped section within fracture dominates formation linear flow regime. Only severe matrix impairment near fracture face can have noticeable effects on pressure and rate response. A new parameter, skin factor ratio, is applied to describe the relative magnitude of multiple damage mechanisms. Reservoir heterogeneity and fracture damage make the pressure and rate behaviors deviate significantly from undamaged one but one can distinguish major damage mechanisms even in heterogeneous reservoir.

Book The Performance of Fractured Horizontal Well in Tight Gas Reservoir

Download or read book The Performance of Fractured Horizontal Well in Tight Gas Reservoir written by Jiajing Lin and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Horizontal wells have been used to increase reservoir recovery, especially in unconventional reservoirs, and hydraulic fracturing has been applied to further extend the contact with the reservoir to increase the efficiency of development. In the past, many models, analytical or numerical, were developed to describe the flow behavior in horizontal wells with fractures. Source solution is one of the analytical/semi-analytical approaches. To solve fractured well problems, source methods were advanced from point sources to volumetric source, and pressure change inside fractures was considered in the volumetric source method. This study aims at developing a method that can predict horizontal well performance and the model can also be applied to horizontal wells with multiple fractures in complex natural fracture networks. The method solves the problem by superposing a series of slab sources under transient or pseudosteady-state flow conditions. The principle of the method comprises the calculation of semi-analytical response of a rectilinear reservoir with closed outer boundaries. A statistically assigned fracture network is used in the study to represent natural fractures based on the spacing between fractures and fracture geometry. The multiple dominating hydraulic fractures are then added to the natural fracture system to build the physical model of the problem. Each of the hydraulic fractures is connected to the horizontal wellbore, and the natural fractures are connected to the hydraulic fractures through the network description. Each fracture, natural or hydraulically induced, is treated as a series of slab sources. The analytical solution of superposed slab sources provides the base of the approach, and the overall flow from each fracture and the effect between the fractures are modeled by applying superposition principle to all of the fractures. It is assumed that hydraulic fractures are the main fractures that connect with the wellbore and the natural fractures are branching fractures which only connect with the main fractures. The fluid inside of the branch fractures flows into the main fractures, and the fluid of the main fracture from both the reservoir and the branch fractures flows to the wellbore. Predicting well performance in a complex fracture network system is extremely challenged. The statistical nature of natural fracture networks changes the flow characteristic from that of a single linear fracture. Simply using the single fracture model for individual fracture, and then adding the flow from each fracture for the network could introduce significant error. This study provides a semi-analytical approach to estimate well performance in a complex fracture network system.

Book Performance Evaluation of Horizontal Wells With Multistage Fractures in Tight Formations by Considering Non Darcy Effect and Fracture Dynamics

Download or read book Performance Evaluation of Horizontal Wells With Multistage Fractures in Tight Formations by Considering Non Darcy Effect and Fracture Dynamics written by Liwu Jiang and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Performance evaluation of fractured wells has attracted increasing attentions since horizontal drilling and hydraulic fracturing technologies have been successfully and efficiently applied in tight formations; however, such an evaluation process is physically complex and mathematically challenging due to the composite effects of non-Darcy flow within fractures and fracture dynamics including complex fracture geometry, fracture network, and stress-sensitive permeability. It is of fundamental and practical importance to evaluate the production performance of a fractured well by taking the aforementioned composite effects into account in a consistent and accurate manner. A new slab source function was developed in the Laplace domain for improving the prediction accuracy on the early time transient pressure response which can take the fracture width into account. The semi-analytical method was used to solve the fluid flow problem through such hydraulic fractures each of which is discretized into small segments, while continuity equations at the interface of such discretized fracture segments were applied to generate the coupled matrix-fracture flow model. The effect of complex fracture geometry and shape-dependent conductivity on the transient pressure response are then examined and discussed. The fracture width is found to affect the early time transient pressure response, while the shape-dependent conductivity imposes a key impact on the transient pressure behaviour for the early and intermediate periods. A coupled matrix-fracture flow model is then proposed and solved in the Laplace domain with consideration of the aforementioned composite effects which may escalate the nonlinearity of the governing equations. An iterative method was developed to deal with the nonlinearity caused by non-Darcy flow in fractures, while the Pedrosa's transform formulation and perturbation technique is used to weaken and linearize the nonlinear effect caused by stress-sensitive permeability. It is found that the non-Darcy effect dominates the early time flow regime, while stress-sensitive effects in fractures and matrix mainly affect the transient pressure response for the intermediate- and late-stage flow regimes. The Laplace-transform difference method was further adopted to extend the current simplified fracture models more closely to reality by taking complex fracture networks into account. The fracture network is first discretized into small segments, and the finite difference method in the Laplace domain can be used to solve the fluid flow problem through such complex fracture systems. Furthermore, the effect of fractal-like discrete fracture network on the transient pressure behaviour can be examined, while the combined effects of complex fracture networks and stress-sensitive permeability can then be analyzed. Transition flow regime is identified for the scenarios with discrete natural fracture networks which can be affected by the stress-sensitive permeability in natural fractures. All the theoretical models have been validated and then extended to field cases. Type curves are generated and beneficial to examine the effect of each factor on the transient behaviour for horizontal wells with multistage fractures in tight formations under various conditions. The discrepancy between the real field pressure data and the traditional type curves without non-Darcy and stress-sensitive effects can be used to evaluate the degree of these two effects. Two new flow regimes including the fluid feed and transition flow regimes occurred in real field pressure tests can be employed to quantify the complexity of the hydraulic and natural fracture networks, respectively.

Book Proceedings

Download or read book Proceedings written by and published by . This book was released on 1997 with total page 690 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Temperature Prediction Model for Horizontal Well with Multiple Fractures in Shale Reservoir

Download or read book Temperature Prediction Model for Horizontal Well with Multiple Fractures in Shale Reservoir written by Nozomu Yoshida and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Fracture diagnostics is a key technology for well performance prediction of a horizontal well in a shale reservoir. The combination of multiple fracture diagnostic techniques gives reliable results, and temperature data has potential to provide more reliability on the results. In this work, we show an application of a temperature prediction model for a horizontal well with multiple hydraulic fractures in order to investigate the possibility of evaluating reservoir and hydraulic fracture parameters using temperature data. The model consists of wellbore model and reservoir model. The wellbore model was formulated based on mass, momentum and energy balance. The reservoir flow model was solved by a numerical reservoir simulation, and the reservoir thermal model was formulated by transient energy balance equation considering viscous dissipation heating and temperature variation caused by fluid expansion besides heat conduction and convection. The reservoir flow and reservoir thermal model were coupled with the wellbore model to predict temperature distribution in a horizontal well considering boundary conditions at the contact of reservoir and wellbore. In the reservoir system, primary hydraulic fractures which are transverse to the horizontal well were modeled with thin grid cells explicitly, and the hydraulically-induced fracture network around the horizontal well was modeled as higher permeable zone to unstimulated matrix zone. The reservoir grids between two primary fractures were logarithmically spaced in order to capture transient flow behavior. We applied the model to synthetic examples: horizontal well with identical five fractures and with different five fractures. The results show two fundamental mechanisms: heat conduction between formation and wellbore fluid at non-perforated zone, and wellbore fluid mixing effect at each fracture. The synthetic example with identical fractures shows that fracture locations affect wellbore temperature distribution because of fluid mixing effect between reservoir inflow and wellbore fluid. And also, the synthetic example with different fractures shows that the fracture heterogeneity causes different magnitude of temperature change due to inflow variation per fracture. In addition, the model was applied to synthetic examples without network fracture region in order to find the effects by the network. It reveals that under constant rate condition, network fracture masks large temperature change due to small pressure change at the contact between fracture and formation, and that under constant BHP condition, network fracture augments temperature change with the increase of flow rate in wellbore and inflow rate from reservoir. Sensitivity studies were performed on temperature distribution to identify influential parameters out of the reservoir and hydraulic fracture parameters including reservoir porosity, reservoir permeability, fracture half-length, fracture height, fracture permeability, fracture porosity, fracture network parameters, and fracture interference between multiple clusters. In this work, in order to find contributions by a target fracture, temperature change sensitivity is evaluated. Single fracture case reveals that fracture permeability, network fracture parameters and fracture geometries are primary influential parameters on temperature change at the fracture location. And also, multiple fractures case shows that temperature change is augmented with the increase of fracture geometry and is decreased with the increase of fracture permeability. These results show the possibility of using temperature to determine these sensitive parameters, and also the quantified parameter sensitivities provide better understandings of the temperature behavior of horizontal well with multiple fractures. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/149366

Book Performance of the Horizontal Wells in a Naturally Fractured Carbonate Reservoir

Download or read book Performance of the Horizontal Wells in a Naturally Fractured Carbonate Reservoir written by Fuad Al-Bayati and published by . This book was released on 2017 with total page 80 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Stress Reorientation in Low Permeability Reservoirs

Download or read book Stress Reorientation in Low Permeability Reservoirs written by Nicolas Patrick Roussel and published by . This book was released on 2011 with total page 792 pages. Available in PDF, EPUB and Kindle. Book excerpt: The acknowledgement of the existence of stress changes in the reservoir due to production from a propped-open fracture has resulted in the development of a new concept: oriented or altered-stress refracturing. By initiating a secondary fracture perpendicular to the initial fracture, refracturing makes it possible to access higher pressurized regions of the reservoir, thus improving the productivity of the well. The redistribution of stresses around a fractured vertical well has two sources: (a) opening of propped fracture (mechanical effects) and (b) production or injection of fluids in the reservoir (poroelastic effects). The coupling of both phenomena is numerically modeled to quantify the extent and timing of stress reorientation around fractured production wells. Guidelines and type-curves are established that allow an operator to choose the timing of the refracture operation in the life of the well, and evaluate the potential increase in well production after refracturing. The selection of candidate wells for refracturing is often very difficult based on the information available at the surface. We propose a systematic methodology, based on dimensionless groups, that allows a field engineer to evaluate a well's potential for refracturing from an analysis of field production data and other reservoir data commonly available. This analysis confirms the crucial role played by stress reorientation in the success of refracturing operations. Another topic of interest is the multi-stage fracturing of horizontal wells. The opening of a propped transverse fracture causes a reorientation of stresses in its neighborhood, which in turn affects the direction of propagation of subsequent fractures. This phenomenon, often referred to as stress shadowing, can negatively impact the efficiency of each fracturing stage. By calculating the trajectory of multiple transverse fractures, we offer some insight on the completion designs that will (a) minimize fracture spacing without compromising the efficiency of each fracturing stage and (b) effectively stimulate natural fractures in the vicinity of the created fracture. In addition, a novel detection method of mechanical interference between multiple transverse fractures is established, based on net fracturing pressure data measured in the field, to calculate the optimum fracture spacing for a specific well.