Optimal Hydraulic Fracture Research Articles

Design and field evaluation of hydraulic fracturing boreholes for terminal mining faces

The terminal mining is an essential part of fully mechanized mining, and the reasonable length of the flap roof during the terminal mining is an important basis for the smooth operation of the terminal mining. With the research background of the 213 working face of the Longde Coal Mine, the contribution of this study is the influence law of hydraulic fracturing on the terminal mining mechanical behavior of the working face. Four different numerical calculation models of terminal mining hydraulic fracturing are established, and the optimal hydraulic fracturing site plan is determined according to the length of the terminal mining flap top and the working resistance of hydraulic support. The effect of hydraulic fracturing construction and the law of mechanical behavior in the terminal mining stage in the field test are analyzed. The results show that through hydraulic fracturing technology, the control of the length of the flap roof of the working face during the terminal mining period has been realized.

Efficient optimization of fracturing parameters with consideration of fracture propagation and heterogeneity in tight gas reservoirs

The parameters of fractures, such as fracture geometries, placement, and conductivity, are critical for the production of horizontal wells in tight gas reservoirs. Current studies on fracture parameter optimization, especially those using the optimization algorithms, are often based on the assumptions that the fractures are uniformly placed with equal length and conductivity. However, non-uniform designs of fracture spacing have been proved to be effective in reducing fracture interference as well as increasing the well production, especially with strong rock heterogeneity. In addition, most of the current literature focuses on identifying optimal hydraulic fracture geometries without considering the fracture propagation process, which may oversimplify the evaluation of completion cost and result in fracture geometries that cannot be realized in practice.In this paper, an automatic optimization process, which could efficiently couple the fracture propagation, is proposed to optimize the fracture parameters. The genetic algorithm (GA) with penalty function method is used to optimize the constraint and non-linear problem. A modified PKN fracture propagation model is established to quickly predict the hydraulic fracture geometries. Fracture locations and fluid injection volume are sequentially optimized with consideration of heterogeneous rock properties. Model parameters are taken from typical tight gas horizontal wells in Jinqiu gas field in Sichuan Province, China. The results show the importance of permeability on both fracture length and production in tight gas well. The lower the permeability, the more significant the production increase after fracture location optimization. More fractures tend to be placed in area with small permeability and fewer fractures are required in highly permeable regions. With an increase in rock heterogeneity, the distribution of fractures becomes more uneven. Conversely, increasing the number of fractures can lead to a more even distribution of fractures. More fluid should be injected in low-permeability regions according to the optimization results. The optimization method is applied to a practical tight gas well in Sichuan Basin and the cumulative gas production can be increased by 6.5% to 8% when compared with the initial uniform designs.

A New Fracturing Method to Improve Stimulation Effect of Marl Tight Oil Reservoir in Sichuan Basin

China’s argillaceous limestone reservoir has a lot of oil and gas resources, and hydraulic fracturing of the argillaceous limestone reservoir faces many difficulties. The first problem is that the heterogeneity of the argillaceous limestone reservoir is strong, and it is difficult to optimize fracturing parameters. The second problem is that there are a lot of natural fractures in the argillaceous limestone reservoir, which leads to a lot of fracturing fluid loss. The third problem is that the closure pressure of the argillaceous limestone reservoir is high, and the conductivity of fractures decreases rapidly under high closure pressure. The last problem is that the fracture shape of the argillaceous limestone reservoir is complex, and the law of proppant migration is unclear. The main research methods in this paper include reservoir numerical simulation, fluid-loss-reducer performance evaluation, flow conductivity tests and proppant migration visualization. To solve the above problems, this paper establishes the fracturing productivity prediction model of complex lithology reservoirs and defines the optimal hydraulic fracturing parameters of the argillous limestone reservoir in the Sichuan Basin. The 70/140 mesh ceramide was selected as the fluid loss additive after an evaluation of the sealing properties of different mesh ceramides. At the same time, the hydraulic fracture conductivity test is carried out in this paper, and it is confirmed that the fracture conductivity of 70/140 mesh and 40/70 mesh composite particle-size ceramics mixed according to the mass ratio of 5:5 is the highest. When the closure pressure is 40 MPa, the conductivity of a mixture of 70/140 mesh ceramic and 40/70 mesh ceramic is 35.6% higher than that of a mixture of 70/140 mesh ceramic and 30/50 mesh ceramic. The proppant migration visualization device is used to evaluate the morphology of the sand dike formed by the ceramsite, and it is clear that the shape of the sand dike is the best when the mass ratio of 70/140 mesh ceramsite and 40/70 mesh ceramsite is 6:4. The research results achieved a good stimulation effect in the SC1 well. The daily oil production of the SC1 well is 20 t, and the monitoring results of the wide-area electromagnetic method show that the fracturing fracture length of the SC1 well is up to 129 m.

A Hybrid Data-Driven/Physics-Based Modeling Approach for Hydraulic Fracture Initiation and Early-Phase Propagation in Shale

Optimal hydraulic fracture (HF) initiation and early-phase propagation results in minimal near-wellbore tortuosity, decreasing the likelihood of screenouts and maximizing the resultant well productivity. While most predictive models for the HF geometry produced in a treatment consider the far-field region, the near-well vicinity should be an integral part of a properly-engineered reservoir stimulation strategy. In this work, a novel hybrid data-driven/physics-based approach is elaborated for modeling HF initiation and early-phase propagation from perforated horizontal wells.A treatment-optimization scheme via oriented perforating is presented using the developed hybrid model, considering the orientation of the induced HF initiation (longitudinal or transverse with respect to a well drilled along the minimum horizontal in-situ principal stress) and the resultant formation breakdown pressure (FBP). Transverse HF initiation and early-phase propagation is ideal for wells drilled in low-permeability “tight” formations, while FBP minimization suppresses the overall on-site hydraulic horsepower (HHP) requirements. The demonstrated optimization scheme is applied separately to the in-situ stress states of seven prolific shale plays from the U.S. and Argentina, suggesting oriented-perforating strategies that target the promotion of transverse HF initiation in two of these (Barnett and Marcellus), while suggesting oriented-perforating strategies that target FBP minimization in the remaining five (Bakken, Fayetteville, Haynesville, Niobrara, and Vaca Muerta). The effectiveness of oriented-perforating can potentially be compromised by fracturing fluid leakages around the casing’s circumference, which is shown to hinder transverse HF initiation.The hybrid-modeling approach is also used to estimate fracture initiation pressure (FIP) values for the seven shale plays studied, indicating significant discrepancies with analytical expressions traditionally used to approximate these FIPs in modern HF computational simulations. This sets the basis for expanding this hybrid-modeling approach over a range of in-situ stress states, incorporating aggregate data-driven (numerically-derived) correction factors in order to compensate for inaccuracies in the analytical approximations, which comprise the physics-based core of the proposed hybrid model.

Modeling the mechanical properties of the geological environment to create am optimal hydraulic fracture

Modeling the mechanical properties of the geological environment to create am optimal hydraulic fracture

An iteratively coupled model for flow, deformation, and fracture propagation in fractured unconventional reservoirs

The accurate and efficient modeling of hydraulic fracture propagation is required to design optimal hydraulic fracture jobs in fractured tight rocks. To this end, we propose and demonstrate the first fixed-stress coupling of pEDFM with XFEM to model hydraulic fracture propagation in naturally fractured reservoirs. This addresses the limitation of EDFM to low-conductivity fractures and is much faster than DFM and fully coupled schemes, which have mostly been applied to the modeling of fracture propagation in fractured reservoirs. The validation studies presented indicate the accuracy of our model at reproducing the analytical solutions to coupled geomechanics and fracture propagation problems. We show that the iterative coupling of pEDFM with XFEM accounts for the interaction between the propagating hydraulic fracture and low-conductivity natural fractures in its vicinity, whereas EDFM does not. This is important when modeling hydraulic fracturing and the subsequent production from multiply fractured hydraulic wells. The iterative coupling approach used in this work provides the flexibility and simplicity needed to model complex fluid and rock behaviors in unconventional oil and gas reservoirs.

Triaxial testing on water permeability evolution of fractured shale.

A sound understanding of the water permeability evolution in fractured shale is essential to the optimal hydraulic fracturing (reservoir stimulation) strategies. We have measured the water permeability of six fractured shale samples from Qiongzhusi Formation in southwest China at various pressure and stress conditions. Results showed that the average uniaxial compressive strength (UCS) and average tensile strength of the Qiongzhusi shale samples were 106.3 and 10.131 MPa, respectively. The nanometre-sized (tiny) pore structure is the dominant characteristic of the Qiongzhusi shale. Following this, we proposed a pre-stressing strategy for creating fractures in shale for permeability measurement and its validity was evaluated by CT scanning. Shale water permeability increased with pressure differential. While shale water permeability declined with increasing effective stress, such effect dropped significantly as the effective stress continues to increase. Interestingly, shale permeability increased with pressure when the pressure is relatively low (less than 4 MPa), which is inconsistent with the classic Darcy's theory. This is caused by the Bingham flow that often occurs in tiny pores. Most importantly, the proposed permeability model would fully capture the experimental data with reasonable accuracy in a wide range of stresses.

Determination of the effect of resin-coating on ceramic proppant settlement for optimal hydraulic fracturing applications

Resin-coated ceramic particles are a new type of proppants being tested in hydraulic fracturing operations, and this study investigates the drag coefficients of resin-coated ceramic particles in water. Eight resin-coated ceramic particles with diameters between 450 and 924 μm are selected in the tests. Using the high-resolution images obtained from a computer tomography (CT) scan, we measure the bulk volume, mean diameter, and volumetric fractions of three constituents making up each particle (resin coating, ceramic body, and air pockets). Compared to other methods, CT scan method leads to the minimum discrepancy between the estimated densities and the ideal particle densities. We measure the settling velocity of resin-coated ceramic particles in water in order to determine the drag coefficients, finding that the Roos and Willmarth's correlation (1971) is more appropriate for calculating the drag coefficients of resin-coated ceramic particles than the other four correlations.

Experimental study of the fracturing-wetting effect of VES fracturing fluid for the coal seam water injection

To improve the efficiency of coal seam water injection, a viscoelastic surfactant (VES) fracturing fluid that is widely used for the coal-bed methane mining was studied for the coal mine water injection. The composition and gel breaking strategy of the VES fracturing fluid with cationic viscoelastic surfactant as the major component were optimized and evaluated upon viscosity, shear stability, cleanness of gel breaking and wettability to achieve the non-backflow fracturing-strong wetting effect for coal seam water injection. The optimal hydraulic fracturing fluid exhibits high viscosities up to 300 mPa·s and good shear stability for at least 2 h, which meets the industry standards for practical applications. Lubricant, a mixture of hydrocarbons including alkanes, cycloalkanes, aromatics, etc., was found to be an excellent gel breaker, which left <50 mg/l gel residue after the gel breaking, and thus endowed the hydraulic fracturing fluid the cleanness properties for coal seam water injection. The high wettability of the gel breaking solution on coal seam was demonstrated with its contact angle of 20° smaller than that of water on average. Our study has provided a scientific foundation for the application of VES fracturing fluids to coal seam water injection.

Effect of injection rate on hydraulic fracturing in naturally fractured shale formations: a numerical study

This paper studies the effect of fluid injection rate on hydraulic fracturing in pre-existing discrete fracture network (DFN) formations. A flow–stress–damage coupling approach has been used in an initial attempt toward how reservoir responses to injection rate under different DFN connected configuration states. The simulation results show that injection rate has an significant influence on the hydraulic fractures (HF) and DFN interaction and hydraulic fracturing effectiveness, which can be characterized by the total interaction area, stimulated DFN length, stimulated HF length and leak-off ratio. For the sparse DFN model, stimulated HF length increases with increasing injection rate and the stimulated DFN length decreases with the increasing injection rate. For the medium DFN model, stimulated HF and DFN length both increase with increasing of injection rate. For the dense DFN model, length of stimulated HF deceases with increasing injection rate; however, the stimulated DFN length increases with the increasing injection rate. The effect of injection rate on hydraulic fracturing is closely related to formation characteristics, which are strongly affected by the DFN connected configuration. For the studied fracture network, the sparse DFN model gets the optimal hydraulic fracturing effectiveness with lower injection rate; however, the dense DFN model has the best hydraulic fracturing effectiveness with higher injection rate. This work strongly links the production technology and hydraulic fracturing effectiveness evaluation and aids in the understanding and optimization of hydraulic fracturing simulations in naturally fractured reservoirs.

Mixed integer simulation optimization for optimal hydraulic fracturing and production of shale gas fields

Optimal development of shale gas fields involves designing a most productive fracturing network for hydraulic stimulation processes and operating wells appropriately throughout the production time. A hydraulic fracturing network design—determining well placement, number of fracturing stages, and fracture lengths—is defined by specifying a set of integer ordered blocks to drill wells and create fractures in a discrete shale gas reservoir model. The well control variables such as bottom hole pressures or production rates for well operations are real valued. Shale gas development problems, therefore, can be mathematically formulated with mixed-integer optimization models. A shale gas reservoir simulator is used to evaluate the production performance for a hydraulic fracturing and well control plan. To find the optimal fracturing design and well operation is challenging because the problem is a mixed integer optimization problem and entails computationally expensive reservoir simulation. A dynamic simplex interpolation-based alternate subspace (DSIAS) search method is applied for mixed integer optimization problems associated with shale gas development projects. The optimization performance is demonstrated with the example case of the development of the Barnett Shale field. The optimization results of DSIAS are compared with those of a pattern search algorithm.

Finding Optimal Hydraulic-Fracture Angles in Productivity-Maximized Shale Well Design

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 170965, “Optimal Hydraulic-Fracture Angle in Productivity-Maximized Shale Well Design,” by Nadav Sorek, SPE, Jose A. Moreno, Ryan Rice, Guofan Luo, and Christine Ehlig-Economides, SPE, Texas A&amp;M University, prepared for the 2014 SPE Annual Technical Conference and Exhibition, Amsterdam, 27–29 October. The paper has not been peer reviewed. In general, hydraulic fractures propagate perpendicular to the horizontal-well axis whenever drilling is parallel to the minimum principal-stress plane. However, operators frequently drill horizontal wells parallel to lease boundaries, resulting in slanted hydraulic-fracture planes at angles less than 90° from the well axis. This study provides a model for the inclined fracture case. It applies and further extends the unified-fracture-design approach for rectangular drainage areas, relating the dimensionless proppant number to the maximum productivity index in pseudosteady-state conditions. Introduction Industry experience suggests that horizontal shale-gas development is enhanced by drilling in the direction parallel to the local minimum principal horizontal stress. Because US mineral leases frequently are rectangular areas with north/south and east/west boundaries, operators often drill parallel to the lease boundaries, prioritizing well saturation over optimum fracture-length propagation. This practice leads to the creation of hydraulic-fracture planes that are slanted at an angle less than 90° with respect to the well axis. Extensive research has investigated the effects of angled fractures on well productivity. Theoretical and empirical data have been used to address the effect of wellbore azimuth on well performance in the Marcellus shale. Results showed that, for each degree a well was suboptimal to minimum horizontal stress, the estimated ultimate recovery decreased by 7.25 Mscf per foot of effective lateral length. Unified-fracture-design methodology indicates the hydraulic- fracturing treatment design that maximizes well productivity for any set of reservoir and proppant properties and a given injected-proppant mass. This methodology introduces the concept of proppant number, which describes the weighted ratio of propped-fracture volume to a square reservoir volume. This study applies and further extends the unified-fracture-design approach to show that, for any given set of reservoir and proppant properties along with a given proppant mass, as long as the created fractures define the same stimulated rock volume, there exists a well direction resulting in maximized well productivity that is not parallel to the minimum-stress direction. First, a correlation is established between the proppant number and the optimum drainage-area aspect ratio. Then, this correlation is used to express what the optimum fracture angle is for a specific proppant number.

Borehole Imaging Technology Visualizes Photorealistically in Oil-Based Muds

Technology Update To estimate reserves, optimize well recovery, and place future development wells more accurately, operators require increasingly detailed understanding of complex oil and gas reservoirs. High-resolution geological data from the borehole— core samples and wireline microresistivity images—are essential to fully characterize reservoir architecture, guide and constrain reservoir models, and make timely decisions with precision and confidence. In unconventional resource plays, for example, geologists need to observe complex natural fracture systems, measure fracture density and direction, determine in-situ stresses, and calculate pressures required to initiate and propagate optimal hydraulic fracturing. In high-cost deepwater environments, exploration teams need to interpret a wide variety of sedimentary facies. Reservoirs often consist of thinly laminated sands or channels, which demand high-definition methods for visualization and interpretation. Typically, deepwater formations exhibit very low resistivities— from 1 ohm-m in shales to as low as 0.2 ohm-m in water-bearing sands. Operators often use whole cores to identify subtle sedimentary facies, but cores can be expensive and time-consuming to acquire. Thus, core samples are usually obtained only in select wells and in the most critical intervals. The resulting lack of geological detail can adversely affect reservoir analysis, interpretation, and modeling. On the other hand, microelectrical borehole images can be acquired continuously over the openhole interval at any depth or in any formation, with relative ease. Until recently, high-definition borehole imaging has been possible only in electrically conductive water-based muds (WBM). However, most deepwater wells and many unconventional shale wells today are drilled with high-performance, nonconductive oil-based muds (OBM). Legacy OBM borehole imaging tools, which were introduced in the last decade, are a significant technical advancement over the preceding dipmeter tools but still exhibit certain limitations. For example, spatial image resolution in OBM is nowhere near the quality obtained from WBM imagers. The lack of circumferential coverage of the borehole leaves large gaps that must be filled by inference.

Unified Fracture Design for very low permeability reservoirs

Unified Fracture Design (UFD) is a methodology for the design of optimal hydraulic fracturing treatments to maximize well performance. Many successful applications of UFD have been demonstrated for reservoirs with low to high permeability. However, UFD has not been systematically applied to reservoirs with very low (nanodarcy) permeability, such as shales. For such reservoirs, massive hydraulic fracturing combined with horizontal drilling has enabled the production of significant amounts of hydrocarbons in recent years. Yet a method for the design of optimal hydraulic fracture treatments for these reservoirs is currently lacking, with present practice relying mainly on rules of thumb and trial and error. This paper fills this gap by developing a design methodology through extension of UFD that addresses two important elements: (a) nanodarcy permeability reservoirs, and (b) fractures with highly elongated drainage areas, that are necessary because of the extremely small reservoir permeability. Simple explicit functions are developed that can be used for efficient computation of optimal fracture dimensions and the productivity index for large values of Proppant Number and highly elongated drainage area around each fracture. A case study is presented to demonstrate how the developed UFD extension can be used in practice and to illustrate the qualitative differences between standard UFD and the developed extension.

Analytical Failure Prediction of Inclined Boreholes

AbstractPredicting when, where, and how pressure-induced borehole failures occur is of fundamental importance in borehole stability analysis and hydraulic fracture design. Numerical and statistical methods are not suited for system analysis, while existing analytical solutions depend on failure location, neglect shear stresses, or only consider vertical boreholes. This paper derives explicit analytical relations between far-field stresses and when, where, and how borehole failures occur in arbitrarily oriented boreholes. These relations reveal that when, where, and how failures occur varies characteristically with the borehole orientation. Further, the optimal drilling- and optimal hydraulic fracturing orientations are determined, together with two simple methods for determining the stress regime based on where or how failures occur. It was also found that failure prediction is more sensitive to noise when the formation is more anisotropic and/or closer to failure.

Applying Science and Best Practices To Increase Production and Optimize Economics in a West Texas Gas Field

This article, written by Assistant Technology Editor Karen Bybee, contains highlights of paper SPE 117538, "Applying Science and Best Practices To increase Production and Optimize Economics in a West Texas Gas Field - A Canyon-Sand Case Study," by Kelly Blackwood, SPE, (now with Encana Oil and Gas), and Kaylene Williamson, SPE, Highmount Energy; Terry Palisch, SPE, and Mark Chapman, SPE, Carbo Ceramics; and Mike Vincent, SPE, Insight Consulting, originally prepared for the 2008 SPE Eastern Regional/AAPG Eastern Section Joint Meeting, Pittsburgh, Pennsylvania, 11-15 Octo ber. The paper has not been peer reviewed. Hydraulic fracturing is the most popular and successful stimulation treatment in the petroleum industry. Fracturing technology has opened numerous unconventional hydrocarbon frontiers that, were it not for this process, would be uneconomic to develop. However, designing the optimal hydraulic fracture is anything but an exact science. The industry relies on many simplifying assumptions that assist in making expedient decisions but can prevent reaching a truly optimized treatment. The full-length paper outlines the approach taken by one Canyon sand operator to apply sound science and best practices to meet their financial. Introduction A field trial comparing the performance of 20/40 economy-lightweight-ceramic (ELWC) proppant to 20/40 Brady sand was completed in 2006 and 2007 in the Canyon sand, Sutton County, Texas. The trial was initiated after modeling suggested that the current hydraulic fractures were conductivity limited. Twenty-one wells in close proximity to each other were included in the study. Production analyses were performed after all wells had at least 1 year of post-fracture production. Both raw production and production normalized to reservoir quality were analyzed, and both indicate that the wells stimulated with ELWC proppant significantly outperformed the offset wells that used Brady sand. Several economic yardsticks then were evaluated in the full-length paper. Canyon Sand The Canyon sandstone is productive in several fields, including the Sonora, Sawyers, Shurley, and Aldwell fields, with first production occurring in 1952. The Canyon-sandstone trend occurs basinward of the western margin of the Eastern shelf in west Texas. The sandstones were deposited from the Eastern shelf into the Val Verde basin. The reservoir is oil prone adjacent to the Midland basin and gas prone adjacent to the Val Verde basin. The vast majority of the sandstone members are fine- to medium-grained quartz lithic arenites.

Efficient global optimization for hydraulic fracturing treatment design

This paper presents a methodology for the optimal hydraulic fracture treatment design. The methodology includes the construction of a “fast surrogate” of an objective function whose evaluation involves the execution of a time-consuming computational model, based on neural networks, DACE modeling, and adaptive sampling. Using adaptive sampling, promising areas are searched considering the information provided by the surrogate model and the expected value of the errors. The proposed methodology provides a global optimization method, hence avoiding the potential problem of convergence to a local minimum in the objective function exhibited by the commonly Gauss–Newton methods. Furthermore, it exhibits an affordable computational cost, is amenable to parallel processing, and is expected to outperform other general purpose global optimization methods such as simulated annealing and genetic algorithms. The methodology is evaluated using two case studies corresponding to formations differing in rock and fluid properties, and geometry parameters. From the results, it is concluded that the methodology can be used effectively and efficiently for the optimal design of hydraulic fracture treatments.

MULTIVARIATE OPTIMISATION OF HYDRAULIC FRACTURE DESIGN

This paper presents a detailed description of a true multivariate optimisation routine for the design of hydraulic fractures. Using the Sequential Unconstrained Minimisation Technique, the optimisation routine intelligently searches for the optimum combination of operator-controllable hydraulic fracture treatment parameters, as opposed to existing simple iterative schemes. Net Present Value (NPV) was used as a measure of the economic impact of hydraulic fracture design on incremental reservoir production revenue. The effects of individual treatment parameters on incremental NPV were investigated for a hypothetical well. The results showed that there was an optimum value for each of the treatment parameters: injection period, injection rate and proppant concentration. For the same hypothetical well, contours of incremental NPV vs two fracture treatment parameters (injection period/injection rate and injection period/proppant concentration) were also constructed. These contours were smooth and continuous and did not show multiple extreme value points. These observations imply that there should be no obvious barriers to the estimation of the optimal hydraulic fracture design using the optimisation routine. Indeed, through visual inspection of the contours in the case of 2 treatment parameters, the optimisation routine consistently arrived at the optimum solution for a given set of inputs. The optimisation routine was extended to 3 treatment parameters (injection period, rate, and proppant concentration) and again consistently arrived at the same optimum solution with different starting positions. The optimisation routine can be extended to any number of operator controllable treatment parameters.

Finite Element Model Simulations Associated With Hydraulic Fracturing

Abstract Recently emphasis has been placed on the development and testing of innovative well stimulation techniques for the recovery of unconventional gas resources. The design of optimal hydraulic fracturing treatments for specified reservoir conditions requires sophisticated models for predicting the induced fracture geometry and interpreting governing mechanisms. This paper presents methodology and results pertinent to hydraulic fracture modeling for the U.S. DOE's Eastern Gas Shales Program (EGSP). The presented finite-element model simulations extend available modeling efforts and provide a unified framework for evaluation of fracture dimensions and associated responses. Examples illustrating the role of multilayering, in-situ stress, joint interaction, and branched cracks are given. Selected comparisons and applications also are discussed.

Productivity of Wells in Vertically Fractured, Damaged Formations

Abstract One primary purpose of hydraulic fracturing as a well stimulation technique is to overcome formation damage. The literature provides ways of designing fracture treatments and evaluating their results but not of incorporating formation damage in vertically fractured wells. Results of an investigation of this problem are presented in this paper. Prediction of stimulation ratios in vertically fractured, damaged wells is accomplished with a mathematical model relating stimulation ratio to relative conductivity of fractures whose lengths are not more than about half the drainage radius of the well. Comparison of results from the new model to results in published predictions verify its utility; these results also show that the range of stimulation ratios that can he predicted for undamaged wells is extended to include relative conductivities of less than 300. This extension is important when using fracturing to stimulate wells with high production rates and high native formation permeabilities. For example, large increases in daily oil production rate can be obtained with stimulation ratio increases as low as 1.25. The importance of complete fracture fill-up (uniform proppant packing) is shown through use of the mathematical model. If, at the mouth of a fracture, only a small fraction (1/2 percent) of the total fracture length is not packed with proppant, nearly all the potential stimulation increase is lost. Proppant crushing, compaction and embedment have been shown in laboratory studies to be responsible for low fracture conductivities in wells producing from highly stressed formations. Equipment and methods for testing the effect of stress (overburden) on conductivity of fractures in consolidated and unconsolidated sands are discussed in this paper. Laboratory tests with simulated fractures in cores from both types of formations showed that crushing, compaction and embedment seriously affect conductivity. Results indicate that similar laboratory tests should be made when accurate knowledge of fracture conductivity is needed to assure good stimulation results for important wells. The chief factor in stimulation ratio reduction was found to be overburden pressure, but the size and type of proppant and the hardness of the formation have significant effects. Fracture conductivity reductions of up to 50 percent were observed with sand propping fractures in consolidated cores; a reduction of 83 percent was measured for an unconsolidated core. The range of effective overburden pressures for which conductivities were measured was from 100 to 5,000 psi. An example fracture design and evaluation problem indicates the usefulness of considering formation damage in planning well stimulation jobs. When damage exists, but its extent and the degree of permeability reduction are not estimated from diagnostic tests, the formation permeability used in planning the job may lead to underdesigning. As the example shows, too low a target stimulation ratio can lead to much lower production rates (by half) than could be attained otherwise. Solutions of equations representing several special cases of the mathematical model are presented in graphical form and details of the derivations of the equations are given in the Appendix. Introduction Since its inception in 1947, hydraulic fracturing has proven to be an effective and widely accepted stimulation technique. In the past 18 years the ability to execute a successful hydraulic fracturing treatment has been substantially increased. The development of design and evaluation procedures has been one of the major contributions to this increased skill. However, as the art of hydraulic fracturing has moved closer to a science, new problems concerning the design and evaluation of the optimal hydraulic fracturing treatment have arisen. Three questions are pertinent to these problems. How is a fracturing job evaluated in a damaged well? What is the effect on the stimulation ratio of not filling the fracture in the vicinity of the wellbore. What is the effect of overburden pressure on fracture conductivity and, consequently, the stimulation ratio? A primary objective of fracturing a well is to stimulate production by overcoming wellbore damage. Presently, however, there is no rational basis for designing or evaluating the optimal fracturing treatment in a damaged well. All present fracture design and evaluation techniques assume that proppants can be uniformly placed in fractures. This assumption may be in serious error, particularly for the portion of a fracture directly adjacent to the wellbore. In this area, turbulence of the injected fluid can cause the proppant to be swept farther into the fracture. JPT P. 120ˆ