scholarly journals The Effect of Hydraulic Fracture Geometry on Well Productivity in Shale Oil Plays with High Pore Pressure

Energies ◽  
2021 ◽  
Vol 14 (22) ◽  
pp. 7727
Author(s):  
Daniela A. Arias Ortiz ◽  
Lukasz Klimkowski ◽  
Thomas Finkbeiner ◽  
Tadeusz W. Patzek
Keyword(s):  
Pore Pressure ◽  
Hydraulic Fracture ◽  
Hydraulic Fractures ◽  
Shale Oil ◽  
Production Forecast ◽  
Fracture Geometry ◽  
Hydrocarbon Production ◽  
Reservoir Simulations ◽  
Bedding Planes

We propose three idealized hydraulic fracture geometries (“fracture scenarios”) likely to occur in shale oil reservoirs characterized by high pore pressure and low differential in situ stresses. We integrate these geometries into a commercial reservoir simulator (CMG-IMEX) and examine their effect on reservoir fluids production. Our first, reference fracture scenario includes only vertical, planar hydraulic fractures. The second scenario has stimulated vertical natural fractures oriented perpendicularly to the vertical hydraulic fractures. The third fracture scenario has stimulated horizontal bedding planes intersecting the vertical hydraulic fractures. This last scenario may occur in mudrock plays characterized by high pore pressure and transitional strike-slip to reverse faulting stress regimes. We demonstrate that the vertical and planar fractures are an oversimplification of the hydraulic fracture geometry in anisotropic shale plays. They fail to represent the stimulated volume geometric complexity in the reservoir simulations and may confuse hydrocarbon production forecast. We also show that stimulating mechanically weak bedding planes harms hydrocarbon production, while stimulated natural fractures may enhance initial production. Our findings reveal that stimulated horizontal bedding planes might decrease the cumulative hydrocarbon production by as much as 20%, and the initial hydrocarbon production by about 50% compared with the reference scenario. We present unique reservoir simulations that enable practical assessment of the impact of varied hydraulic fracture configurations on hydrocarbon production and highlight the importance of constraining present-day in situ stress state and pore pressure conditions to obtain a realistic hydrocarbon production forecast.

High Accuracy Estimation of Hydraulic Fracture Geometry Using Crosswell Electromagnetics

2021 ◽  
Author(s):  
Shubham Mishra ◽  
Vinil Reddy
Keyword(s):  
Hydraulic Fracture ◽  
Horizontal Wells ◽  
Hydraulic Fractures ◽  
Calibration Data ◽  
Formation Water ◽  
Fracturing Fluid ◽  
Production Forecast ◽  
Fracture Geometry ◽  
Field Development ◽  
Well Completion

Abstract Unconventional resources, which are typically characterized by poor porosity and permeability are being economically developed only after the introduction of hydraulic fracturing (HF) technology, which is required to stimulate the hydrocarbon flow from these impermeable/tight reservoir rocks. Since 1960, HF has been extensively used in the industry. HF is the process of (1) injecting viscous gel fluids through the wellbore into the subterranean hydrocarbon formation, at high pressures sufficient enough to exceed tensile strength of the rock and hydraulically induce cracks/fractures (2) followed by injecting proppant-laden fluid into the open fractures and packing up the fracture with proppant pack, after the injected fluid leaks off into formation. The resultant proppant pack keeps the induced fracture propped open and thus creates a highly conductive flow path for the hydrocarbon to flow from the far-field subterranean formation into the wellbore. Most the modern wells in unconventional reservoirs are horizontal/near-horizontal wells that are completed with large multiple HF treatments across the entire length of the horizontal wellbore (lateral), to increase the reservoir contact per well. Productivity of these wells is dictated by the stimulated reservoir volume (SRV), which is dependent on the number of fractures and conductive hydraulic fracture surface area of each fracture that is propped open. Therefore, estimation of the hydraulic fracture geometry (HFG) dimensions has become very critical for any unconventional field development. Key dimensions are hydraulic fracture length, height, and orientation, which are required to assess the optimum configuration of fracturing, well completion, and reservoir management strategy to achieve maximum production. Designs can be assessed based on HFG observations, and infill well trajectories, spacing, etc. can be planned for further field development. This workflow proposes a method to estimate and model all or at least two parameters of HFG in predominantly horizontal or nearly horizontal wells by use of interwell electromagnetic recordings. The foundation of this workflow is the difference in salinity, or more precisely resistivity, of the fracturing fluid and the resident fluid (hydrocarbon or formation water). The fracturing fluid is usually significantly less resistive than the hydrocarbon that is the dominant resident fluid where fracturing is usually conducted, or less resistive than the formation water in case the HF occurs in high water saturation regions. Therefore, the resistivity contrast between the two fluids will demarcate the boundary of hydraulic fractures and thus help in precisely modeling some or all parameters of HFG. The interwell recordings can be interpreted along a 2D plane between the two wells, one of them bearing the transmitter and the other with the receiver. The interpretations along a 2D plane can be used to calibrate a 3D unstructured HF model, thereby introducing a reliable calibration input that did not exist before. There can be multiple such 2D planes as more than one well can have a receiver, and, in that case, the 3D HF model has more calibration data and is even more precise. The reason this workflow significantly improves precision in HFG estimation and modeling is that it provides the ability to demarcate only the open portion of the HF and not the entire volume where pumping fluid entered, which would include parts that closed too quickly to contribute to the production from the well. Today, the industry, by its best methods, can only see the entire rock volume that broke due to fracturing, although significant parts of that broken volume might not be contributing to the production and thus are irrelevant in the 3D models upon which important decisions such as production forecast and project economics are based.

Predicting Hydraulic Fracturing in a Carbonate Gas Reservoir in Abu Dhabi Using 1D Mechanical Earth Model: Uncertainty and Constraints

2015 ◽  
Author(s):  
Manhal Sirat ◽  
Mujahed Ahmed ◽  
Xing Zhang
Keyword(s):  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Natural Fracture ◽  
Abu Dhabi ◽  
Hydraulic Fractures ◽  
Gas Reservoir ◽  
In Situ Stresses ◽  
Tight Carbonate ◽  
Different Depths

Abstract In-situ stress state plays an important role in controlling fracture growth and containment in hydraulic fracturing managements. It is evident that the mechanical properties, existing stress regime and the natural fracture network of its reservoir rocks and the surrounding formations mainly control the geometry, size and containments of produced hydraulic fractures. Furthermore, the three principal in situ stresses' axes swap directions and magnitudes at different depths giving rise to identifying different mechanical bedrocks with corresponding stress regimes at different depths. Hence predicting the hydro-fractures can be theoretically achieved once all the above data are available. This is particularly difficult in unconventional and tight carbonate reservoirs, where heterogeneity and highly stress variation, in terms of magnitude and orientation, are expected. To optimize the field development plan (FDP) of a tight carbonate gas reservoir in Abu Dhabi, 1D Mechanical Earth Models (MEMs), involving generating the three principal in-situ stresses' profiles and mechanical property characterization with depth, have been constructed for four vertical wells. The results reveal the swap of stress magnitudes at different mechanical layers, which controls the dimension and orientation of the produced hydro-fractures. Predicted containment of the Hydro-fractures within the specific zones is likely with inevitable high uncertainty when the stress contrast between Sv, SHmax with Shmin respectively as well as Young's modulus and Poisson's Ratio variations cannot be estimated accurately. The uncertainty associated with this analysis is mainly related to the lacking of the calibration of the stress profiles of the 1D MEMs with minifrac and/or XLOT data, and both mechanical and elastic properties with rock mechanic testing results. This study investigates the uncertainty in predicting hydraulic fracture containment due to lacking such calibration, which highlights that a complete suite of data, including calibration of 1D MEMs, is crucial in hydraulic fracture treatment.

A Multi-Disciplinary Approach for Well Spacing and Treatment Design Optimization in the Midland Basin Using Lateral Pore Pressure Estimation and Depletion Modeling

2021 ◽  
Author(s):  
Taylor Levon ◽  
Kit Clemons ◽  
Ben Zapp ◽  
Tim Foltz
Keyword(s):  
Pore Pressure ◽  
Hydraulic Fracture ◽  
Initial Study ◽  
Midland Basin ◽  
Well Spacing ◽  
Fracture Modeling ◽  
Original Target ◽  
Treatment Responses ◽  
Treatment Designs

Abstract With a recent trend in increased infill well development in the Midland basin and other unconventional plays, it has been shown that depletion has a significant impact on hydraulic fracture propagation. This is largely because production drawdown causes in-situ stress changes, resulting in asymmetric fracture growth toward the depleted regions. In turn, this can have a negative impact on production capacity. For the initial part of this study, an infill child well was drilled and completed adjacent to a parent well that had been producing for two years. Due to drilling difficulties, the child well was steered to a new target zone located 125 feet above the original target. However, relative to the original target, treatment data from the new zone indicated abnormal treatment responses leading to a study to evaluate the source of these variations and subsequent mitigation. The initial study was conducted using a pore pressure estimation derived from drill bit geomechanics data to investigate depletion effects on the infill child well. The pore pressure results were compared to the child well treatment responses and bottom hole pressure measurements in the parent well. Following the initial study, additional hydraulic fracture modeling studies were conducted on a separate pad to investigate depletion around the infill wells, determine optimal well spacing for future wells given the level of depletion, and optimize treatment designs for future wells in similar depletion scenarios. A depletion model workflow was implemented based on integrating hydraulic fracture modeling and reservoir analytics for future infill pad development. The geomechanical properties were calibrated by DFIT results and pressure matching of the parent well treatments for the in-situ virgin conditions. Parent well fracture geometries were used in an RTA for an analytical approach of estimating drainage area of the parent wells. These were then applied to a depletion profile in the hydraulic fracture model for well spacing analysis and treatment design sensitivities. Results of the initial study indicated that stages in the new, higher interval had higher breakdown pressures than the lower interval. Additionally, the child well drilled in the lower interval had normal breakdown pressures in line with the parent well treatments. This suggests that treatment differences in the wells were ultimately due to depletion of the offset parent well. Based on the modeling efforts, optimal infill well spacing was determined based on the on-production time of the parent wells. The optimal treatment designs were also determined under the same conditions to minimize offset frac hits and unnecessary completion costs. This case study presents the use of a multi-disciplinary approach for well spacing and treatment optimization. The integration of a novel method of estimating pore pressure and depletion modeling workflows were used in an inventive way to understand depletion effects on future development.

scholarly journals Apparent Toughness Anisotropy Induced by “Roughness” of In-Situ Stress: A Mechanism That Hinders Vertical Growth of Hydraulic Fractures and Its Simplified Modeling

SPE Journal ◽  
2019 ◽  
Vol 24 (05) ◽  
pp. 2148-2162 ◽  
Author(s):  
Pengcheng Fu ◽  
Jixiang Huang ◽  
Randolph R. Settgast ◽  
Joseph P. Morris ◽  
Frederick J. Ryerson
Keyword(s):  
Hydraulic Fracture ◽  
High Stress ◽  
In Situ Stress ◽  
Height Growth ◽  
Hydraulic Fractures ◽  
Simple Relationship ◽  
Vertical Growth ◽  
Stress Profiles ◽  
Fracture Height

Summary The height growth of a hydraulic fracture is known to be affected by many factors that are related to the layered structure of sedimentary rocks. Although these factors are often used to qualitatively explain why hydraulic fractures usually have well–bounded height growth, most of them cannot be directly and quantitatively characterized for a given reservoir to enable a priori prediction of fracture–height growth. In this work, we study the role of the “roughness” of in–situ–stress profiles, in particular alternating low and high stress among rock layers, in determining the tendency of a hydraulic fracture to propagate horizontally vs. vertically. We found that a hydraulic fracture propagates horizontally in low–stress layers ahead of neighboring high–stress layers. Under such a configuration, a fracture–mechanics principle dictates that the net pressure required for horizontal growth of high–stress layers within the current fracture height is significantly lower than that required for additional vertical growth across rock layers. Without explicit consideration of the stress–roughness profile, the system behaves as if the rock is tougher against vertical propagation than it is against horizontal fracture propagation. We developed a simple relationship between the apparent differential rock toughness and characteristics of the stress roughness that induce equivalent overall fracture shapes. This relationship enables existing hydraulic–fracture models to represent the effects of rough in–situ stress on fracture growth without directly representing the fine–resolution rough–stress profiles.

Investigation of Rupture and Slip Mechanisms of Hydraulic Fractures in Multiple-Layered Formations

SPE Journal ◽  
2019 ◽  
Vol 24 (05) ◽  
pp. 2292-2307 ◽  
Author(s):  
Jizhou Tang ◽  
Kan Wu ◽  
Lihua Zuo ◽  
Lizhi Xiao ◽  
Sijie Sun ◽  
...  
Keyword(s):  
Fluid Flow ◽  
Hydraulic Fracture ◽  
Fracture Propagation ◽  
Propagation Model ◽  
Rock Deformation ◽  
Hydraulic Fractures ◽  
Fracture Geometry ◽  
Fracture Width ◽  
Hydraulic Fracture Propagation ◽  
Fracture Height

Summary Weak bedding planes (BPs) that exist in many tight oil formations and shale–gas formations might strongly affect fracture–height growth during hydraulic–fracturing treatment. Few of the hydraulic–fracture–propagation models developed for unconventional reservoirs are capable of quantitatively estimating the fracture–height containment or predicting the fracture geometry under the influence of multiple BPs. In this paper, we introduce a coupled 3D hydraulic–fracture–propagation model considering the effects of BPs. In this model, a fully 3D displacement–discontinuity method (3D DDM) is used to model the rock deformation. The advantage of this approach is that it addresses both the mechanical interaction between hydraulic fractures and weak BPs in 3D space and the physical mechanism of slippage along weak BPs. Fluid flow governed by a finite–difference methodology considers the flow in both vertical fractures and opening BPs. An iterative algorithm is used to couple fluid flow and rock deformation. Comparison between the developed model and the Perkins–Kern–Nordgren (PKN) model showed good agreement. I–shaped fracture geometry and crossing–shaped fracture geometry were analyzed in this paper. From numerical investigations, we found that BPs cannot be opened if the difference between overburden stress and minimum horizontal stress is large and only shear displacements exist along the BPs, which damage the planes and thus greatly amplify their hydraulic conductivity. Moreover, sensitivity studies investigate the impact on fracture propagation of parameters such as pumping rate (PR), fluid viscosity, and Young's modulus (YM). We investigated the fracture width near the junction between a vertical fracture and the BPs, the latter including the tensile opening of BPs and shear–displacement discontinuities (SDDs) along them. SDDs along BPs increase at the beginning and then decrease at a distance from the junction. The width near the junctions, the opening of BPs, and SDDs along the planes are directly proportional to PR. Because viscosity increases, the width at a junction increases as do the SDDs. YM greatly influences the opening of BPs at a junction and the SDDs along the BPs. This model estimates the fracture–width distribution and the SDDs along the BPs near junctions between the fracture tip and BPs and enables the assessment of the PR required to ensure that the fracture width at junctions and along intersected BPs is sufficient for proppant transport.

scholarly journals Laboratory Investigation on the -Effect of In-Situ Stresses on Hydraulic Fracture Containment

1982 ◽  
Vol 22 (03) ◽  
pp. 333-340 ◽  
Author(s):  
Norman R. Warpinski ◽  
James A. Clark ◽  
Richard A. Schmidt ◽  
Clarence W. Huddle
Keyword(s):  
Pore Pressure ◽  
Hydraulic Fracture ◽  
Laboratory Experiments ◽  
In Situ Stress ◽  
Enhanced Production ◽  
In Situ Stresses ◽  
Extensive Growth ◽  
Stress Variations ◽  
Fracture Growth

Abstract Laboratory experiments have been conducted to determine the effect of in-situ stress variations on hydraulic fracture containment. Fractures were initiated in layered rock samples with prescribed stress variations, and fracture growth characteristics were determined as a function of stress levels. Stress contrasts of 300 to 400 psi (2 to 3 MPa) were found sufficient to restrict fracture growth in laboratory samples of Nevada tuff and Tennessee and Nugget sandstones. The required stress level was found not to depend on mechanical rock properties. However, permeability and the resultant pore pressure effects were important. Tests conducted at biomaterial interfaces between Nugget and Tennessee sandstones show that the resultant stresses set up near the interface because of the applied overburden stress affect the fracture behavior in the same way as the applied confining stresses. These results provide a guideline for determining the in-situ stress contrast necessary to contain a fracture in a field treatment. Introduction An under-standing of the factors that influence and control hydraulic fracture containment is important for the successful use of hydraulic fracturing technology in the enhanced production of natural gas from tight reservoirs. Optimally, this understanding would provide improved fracture design criteria to maximize fracture surface area in contact with the reservoir with respect to volume injected and other treatment parameters. In formations with a positive containment condition (i.e., where fracturing out of zone is not anticipated), long penetrating fractures could be used effectively to develop the resource. For the opposite case, the options would beto use a small treatment so that large volumes are not wasted in out-of-zone fracturing and to accept a lower productivity improvement, orto reject the zone as uneconomical. These decisions cannot be made satisfactorily unless criteria for vertical fracture propagation are developed and techniques for readily measuring the important parameters are available. Currently, both theoretical and experimental efforts are being pursued to determine the important parameters and their relative effects on fracture growth. Two modes of fracture containment are possible. One is the situation where fracture growth is terminated at a discrete interface. Examples of this include laboratory experiments showing fracture termination at weak or unbonded interfaces and theoretical models that predict that fracture growth will terminate at a material property interface. The other mode may occur when the fracture propagates into the bounding layer, but extensive growth does not take place and the fracture thus is restricted. An example is the propagation of the fracture into a region with an adverse stress gradient so that continued propagation results in higher stresses on the fracture, which limits growth, as suggested by Simonson et al. and as seen in mineback experiments. Another example is the possible restriction caused by propagation into a higher modulus region where the decreased width results in increased pressure drop in the fracture, which might inhibit extensive growth into that region relative to the lower modulus region. Other parameters, such as natural fractures, treatment parameters, pore pressure, etc., may affect either of these modes. Laboratory and mineback experiments have shown that weak interfaces and in-situ stress differences are the most likely factors to contain the fracture, and weak interfaces are probably effective only at shallow depths. Thus, our experiments are being performed to determine the effect of in-situ stresses on fracture containment, both in a uniform rock sample and at material properly interfaces. SPEJ P. 333^

Some Effects of Stress, Friction, and Fluid Flow on Hydraulic Fracturing

1982 ◽  
Vol 22 (03) ◽  
pp. 321-332 ◽  
Author(s):  
M.E. Hanson ◽  
G.D. Anderson ◽  
R.J. Shaffer ◽  
L.D. Thorson
Keyword(s):  
Fluid Flow ◽  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Research Program ◽  
In Situ Stress ◽  
Material Interfaces ◽  
Fracture Geometry ◽  
Funded Research ◽  
Fracturing Process

Abstract We are conducting a U.S. DOE-funded research program aimed at understanding the hydraulic fracturing process, especially those phenomena and parameters that strongly affect or control fracture geometry. Our theoretical and experimental studies consistently confirm the well-known fact that in-situ stress has a primary effect on fracture geometry, and that fractures propagate perpendicular to the least principal stress. In addition, we find that frictional interfaces in reservoirs can affect fracturing. We also have quantified some effects on fracture geometry caused by frictional slippage along interfaces. We found that variation of friction along an interface can result in abrupt steps in the fracture path. These effects have been seen in the mineback of emplaced fractures and are demonstrated both theoretically and in the laboratory. Further experiments and calculations indicate possible control of fracture height by vertical change in horizontal stresses. Preliminary results from an analysis of fluid flow in small apertures are discussed also. Introduction Hydraulic fracturing and massive hydraulic fracturing (MHF) are the primary candidates for stimulating production from tight gas reservoirs. MHF can provide large drainage surfaces to produce gas from the low- permeability formation if the fracture surfaces remain in the productive parts of the reservoir. To determine whether it is possibleto contain these fractures in the productive formations andto design the treatment to accomplish this requires a much broader knowledge of the hydraulic fracturing process. Identification of the parameters controlling fracture geometry and the application of this information in designing and performing the hydraulic stimulation treatment is a principal technical problem. Additionally, current measurement technology may not be adequate to provide the required data. and new techniques may have to be devised. Lawrence Livermore Natl. Laboratory has been conducting a DOE-funded research program whose ultimate goal is to develop models that predict created hydraulic fracture geometry within the reservoir. Our approach has been to analyze the phenomenology of the fracturing process to son out and identify those parameters influencing hydraulic fracture geometry. Subsequent model development will incorporate this information. Current theoretical and stimulation design models are based primarily on conservation of mass and provide little insight into the fracturing process. Fracture geometry is implied in the application of these models. Additionally, pressure and flow initiation in the fractures and their interjection with the fracturing process is not predicted adequately with these models. We have reported previously on some rock-mechanics aspects of the fracturing process. For example, we have studied, theoretically and experimentally, pressurized fracture propagation in the neighborhood of material interfaces. Results of interface studies showed that natural fractures in the interfacial region negate any barrier effect when the fracture is propagating from a lower modulus material toward a higher modulus material. On the other hand, some fracture containment could occur when the fracture is propagating from a higher modulus into a lower modulus material. Effect of moduli changes on the in-situ stress field have to be taken into consideration to evaluate fracture containment by material interfaces. Some preliminary analyses have been performed to evaluate how stress changes when material properties change, but we have not evaluated this problem fully. SPEJ P. 321^

A Study of Inclined Hydraulic Fractures

1973 ◽  
Vol 13 (02) ◽  
pp. 61-68 ◽  
Author(s):  
Abbas Ali Daneshy
Keyword(s):  
Tensile Stress ◽  
Coordinate System ◽  
Hydraulic Fracture ◽  
Isotropic Medium ◽  
Maximum Tensile Stress ◽  
Borehole Wall ◽  
Hydraulic Fractures ◽  
Principal Stresses ◽  
The Subject

Abstract The results of a theoretical and experimental investigation of inclined hydraulic fractures, reported in this paper, indicate that such fractures do not generally initiate perpendicular to the maximum tensile stress induced on the borehole wall. Unlike axial or normal hydraulic fractures, a degree of shear failure seems to be associated with the initiation and extension of almost all inclined hydraulic fractures. These fractures often intersect the borehole along two diametrically opposite axial lines, thus giving it the appearance of an axial fracture. Inclined hydraulic fractures generally change their orientation as they extend away from the wellbore until they become perpendicular to the least compressive in-situ principal stress. Therefore, the borehole trace of such fractures cannot be used for their positive identification. Introduction The process of hydraulic fracturing of a formation essentially consists of injecting a fluid inside the borehole and pressurizing it until the induced stresses exceed the strength of the formation and cause failure. Failure is generally indicated by a sudden major drop in the variations of the borehole fluid pressure with time. In general in an isotropic medium, the over-all plane of a hydraulic fracture is either parallel, plane of a hydraulic fracture is either parallel, inclined, or perpendicular to the axis of the borehole from which it is extending. Accordingly, these fractures will be called axial, inclined or normal, respectively. This classification of hydraulic fractures refers them to the borehole where they are observed rather than the ground surface or the bedding planes. In case of vertical boreholes, axial and normal fractures become identical with vertical and horizontal fractures (which are the terms often used in petroleum industry). In the first comprehensive analysis of the mechanics of hydraulic fracturing, Hubbert and Willis proposed that axial or normal hydraulic fractures initiate when the maximum tensile stress induced around the borehole exceeds the tensile strength of the formation, and that such fractures extend in a plane perpendicular to the least compressive in - situ principal stress. The correctness of this proposal has since been verified by Haimson and Fairhurst, who conducted an extensive series of laboratory experiments on the subject. In their theoretical and experimental work, Haimson and Fairhurst assumed that one of the in-situ principal stresses is parallel to the borehole axis. Under such a condition, one can only create an axial or a normal hydraulic fracture in an isotropic medium. For the case when none of the in-situ principal stresses are parallel to the borehole, Fairhurst derived mathematical expressions for the stress components on the borehole wall, in isotropic and transversely isotropic media. Experimentally, von Schonfeldt and Daneshy independently observed that under such a condition the fracture orientation is influenced by the borehole in its vicinity. The trace of inclined hydraulic fractures at the wellbore was found to be misleading if used for the purpose of determining the over-all fracture orientation. The research reported here is an extension of a previous work on the subject of inclined hydraulic previous work on the subject of inclined hydraulic fractures. It includes the computation of the magnitude and the orientation of the maximum tensile stress induced at the borehole wall, for each experiment, and the resulting fracture shape. Such investigations can, in the course of time, provide means of determining the over-all fracture provide means of determining the over-all fracture type at great depth, which has significant importance in many fields, such as geophysics, petroleum, geological and civil engineering. STRESS DISTRIBUTION AT THE WALL OF THE BOREHOLE Let sigma1, sigma2 and sigma3 be the three in-situ total principal stresses whose values and directions are assumed to remain constant throughout the isotropic porous elastic formation under consideration. No porous elastic formation under consideration. No restriction is imposed upon the direction of any of the in-situ principal stresses, except the mathematical requirement that they should be mutually perpendicular. Consider a coordinate system ox1 x2 x3 perpendicular. Consider a coordinate system ox1 x2 x3 chosen such that ox3 is the borehole axis, and ox1, lies in the plane osigma1 sigma2, Fig. 1. SPEJ P. 61

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