METHOD FOR PERFORMING WELLBORE FRACTURE OPERATIONS USING FLUID TEMPERATURE PREDICTIONS

[0001] This application is a continuation in part of U.S. patent application Ser. No. 14/126209 filed Jun. 30, 2012, which claims priority to U.S. Provisional Application No. 61/574,130 filed on Jul. 28, 2011 and which is a continuation in part of U.S. patent application Ser. No. 12/479,335, filed on Jun. 5, 2009 and which claims priority to PCT Application No. PCT/US2012/048877 filed Jul. 30, 2012, the entire contents of all four applications are hereby incorporated by reference herein.

[0002] The present disclosure relates generally to methods and systems for performing wellsite operations. More particularly, this disclosure is directed to methods and systems for performing fracture and production operations, such as investigating subterranean formations and characterizing hydraulic fracture networks in a subterranean formation.

[0003] In order to facilitate the recovery of hydrocarbons from oil and gas wells, the subterranean formations surrounding such wells can be hydraulically fractured. Hydraulic fracturing may be used to create cracks in subsurface formations to allow oil or gas to move toward the well. A formation is fractured by introducing a specially engineered fluid (referred to as “fracturing fluid” or “fracturing slurry” herein) at high pressure and high flow rates into the formation through one or more wellbore. Hydraulic fractures may extend away from the wellbore hundreds of feet in two opposing directions according to the natural stresses within the formation. Under certain circumstances, they may form a complex fracture network.

[0004] The fracturing fluids may be loaded with proppants, which are sized particles that may be mixed with the fracturing fluid to help provide an efficient conduit for production of hydrocarbons to flow from the formation/reservoir to the wellbore. Proppant may comprise naturally occurring sand grains or gravel, man-made or specially engineered proppants, e.g. fibers, resin-coated sand, or high-strength ceramic materials, e.g. sintered bauxite. The proppant collects heterogeneously or homogenously inside the fracture to “prop” open the new cracks or pores in the formation. The proppant creates a plane of permeable conduits through which production fluids can flow to the wellbore. The fracturing fluids are preferably of high viscosity, and therefore capable of carrying effective volumes of proppant material. Fluid viscosity may vary with fluid temperature.

[0005] The fracturing fluid may be realized by a viscous fluid, sometimes referred to as “pad” that is injected into the treatment well at a rate and pressure sufficient to initiate and propagate a fracture in hydrocarbon formation. Injection of the “pad” is continued until a fracture of sufficient geometry is obtained to permit placement of the proppant particles. After the “pad,” the fracturing fluid may consist of a fracturing fluid and proppant material. The fracturing fluid may be gel, oil based, water based, brine, acid, emulsion, foam, or any other similar fluid. The fracturing fluid can contain several additives, viscosity builders, drag reducers, fluid-loss additives, corrosion inhibitors and the like. In order to keep the proppant suspended in the fracturing fluid until such time as all intervals of the formation have been fractured as desired, the proppant may have a density close to the density of the fracturing fluid utilized. Sometimes certain type of fibers may be used together with the proppant for various purposes, such as enhanced proppant-carrying, proppant segmenting, selective fracture growth, leakoff prevention, etc.

[0006] Proppants may be comprised of any of the various commercially available fused materials, such as silica or oxides. These fused materials can comprise any of the various commercially available glasses or high-strength ceramic products. Following the placement of the proppant, the well may be shut-in for a time sufficient to permit the pressure to bleed off into the formation or to permit the degradation of fibers, cross-linked gel or filter cake, depending on fluid temperature. The shut-in process causes the fracture to close and exert a closure stress on the propping agent particles. The shut-in period may vary from a few minutes to several days.

[0007] Current hydraulic fracture monitoring methods and systems may map where the fractures occur and the extent of the fractures. Some methods and systems of microseismic monitoring may process seismic event locations by mapping seismic arrival times and polarization information into three-dimensional space through the use of modeled travel times and/or ray paths. These methods and systems can be used to infer hydraulic fracture propagation over time.

[0008] Conventional hydraulic fracture models may also assume a bi-wing type induced fracture. These bi-wing fractures may be short in representing the complex nature of induced fractures in some unconventional reservoirs with preexisting natural fractures. Published models may map the complex geometry of discrete hydraulic fractures based on monitoring microseismic event distribution.

[0009] In some cases, models may be constrained by accounting for either the amount of pumped fluid or mechanical interactions both between fractures and injected fluid and among the fractures. Some of the constrained models may provide a fundamental understanding of involved mechanisms, but may be complex in mathematical description and/or require computer processing resources and time in order to provide accurate simulations of hydraulic fracture propagation.

[0010] Unconventional formations, such as shales, are being developed as reservoirs of hydrocarbon production. Once considered as source rocks and seals, shale formations are now considered as tight-porosity and low-permeability unconventional reservoirs. Hydraulic fracturing of shale formations may be used to stimulate and produce from the reservoir. The effectiveness and efficiency of a fracturing job may ultimately be judged by production from the stimulated reservoir.

[0011] Patterns of hydraulic fractures created by the fracturing stimulation may be complex and form a fracture network as indicated by the distribution of associated microseismic events. Models of complex hydraulic fracture networks (HFNs) have been developed to represent the created hydraulic fractures. Examples of fracture models are provided in U.S. Pat. Nos. 6,101,447, 7,363,162, 7,788,074, 8,498,852, 20080133186, 20100138196, and 20100250215.

[0012] Due to the complexity of HFNs, production from a stimulated shale reservoir may be numerically simulated. Numerical simulation for stimulation job design and post-job analysis may be time-consuming, and it may be inconvenient to construct a numerical model and carry out runs for each of the numerous designs of a stimulation job. Analytical solutions to HFN models and associated calculations for predicting fluid temperature or proppant transport are constantly sought to enhance stimulation job design and post-job analysis.

[0013] The present application discloses methods and systems for characterizing hydraulic fracturing of a subterranean formation based upon inputs from sensors measuring field data in conjunction with a hydraulic fracture network model. The fracture model constrains geometric properties of the hydraulic fractures of the subterranean formation using the field data in a manner that significantly reduces the complexity of the fracture model and thus reduces the processing resources and time required to provide accurate characterization of the hydraulic fractures of the subterranean formation. Such characterization can be generated in real-time to manually or automatically manipulate surface and/or down-hole physical components supplying fracturing fluids to the subterranean formation to adjust the hydraulic fracturing process as desired, such as by optimizing the fracturing plan for the site (or for other similar fracturing sites).

[0014] In some embodiments, the methods and systems of the present disclosure are used to design wellbore placement and hydraulic fracturing stages during the planning phase in order to optimize hydrocarbon production. In some embodiments, the methods and systems of the present disclosure are used to adjust the hydraulic fracturing process in real-time by controlling the flow rates, temperature, compositions, and/or properties of the fracturing fluid supplied to the subterranean formation. In some embodiments, the methods and systems of the present disclosure are used to adjust the hydraulic fracturing process by modifying the fracture dimensions in the subterranean formation in real time.

[0015] The method and systems of the present disclosure may also be used to facilitate hydrocarbon production from a well and from subterranean fracturing (whereby the resulting fracture dimensions, directional positioning, orientation, and geometry, and the placement of a proppant within the fracture more closely resemble the desired results).

[0016] In another aspect, the disclosure relates to a method of performing an oilfield operation about a wellbore penetrating a subterranean formation. The method involves performing a fracture operation. The fracture operation involves generating a plurality of fractures about the wellbore and generating a fracture network about the wellbore. The fracture network includes the fractures and a plurality of matrix blocks positioned thereabout. The fractures are intersecting, partially or fully propped, and hydraulically connected. The matrix blocks are positioned about the fractures. The method also involves generating rate of hydrocarbon flow through the fracture network, generating a hydrocarbon fluid distribution based on the flow rate, and performing a production operation, the production operation comprising generating a production rate from the hydrocarbon fluid distribution.

[0017] In another aspect, the disclosure relates to a method of performing an oilfield operation about a wellbore penetrating a subterranean formation. The method involves performing a fracture operation. The fracture operation involves stimulating the wellbore and generating a fracture network about the wellbore. The stimulating involves injecting fluid into the subterranean formation such that fractures are generated about the wellbore. The fracture network includes the fractures and a plurality of matrix blocks positioned thereabout. The fractures are intersecting and hydraulically connected. The plurality of matrix blocks is positioned about the fractures. The method also involves placing proppants in the fracture network, generating rate of hydrocarbon flow through the fracture network, generating a hydrocarbon fluid distribution based on the flow rate, and performing a production operation. The production operation involves generating a production rate from the hydrocarbon fluid distribution.

[0018] Finally, in another aspect, the disclosure relates to a method of performing an oilfield operation about a wellbore penetrating a subterranean formation. The method involves designing a fracture operation based on job parameters and performing the fracture operation. The fracture operation involves generating a fracture network about the wellbore. The fracture network includes a plurality of fractures and a plurality of matrix blocks. The fractures are intersecting and hydraulically connected. The matrix blocks are positioned about the fractures. The method also involves optimizing the fracture operation by adjusting the fracture operation based on a comparison of a simulated production rate with actual data, generating a rate of hydrocarbon flow through the fracture network, generating a hydrocarbon fluid distribution based on the flow rate, and performing a production operation. The simulated production rate is based on the fracture network. The production operation involves generating a production rate from the hydrocarbon fluid distribution.

[0019] In yet another aspect, the disclosure relates to a method of performing an oilfield operation about a wellbore penetrating a subterranean formation. The method involves performing a fracture operation comprising injecting fluid into the formation and generating fractures about the wellbore. The fractures form a fracture network about the wellbore. The method further involves collecting during the performing data comprising injection temperature and pressure, generating a fluid and proppant distribution through the fracture network by performing real time simulations of the fracture network based on the collected data (the fluid distribution comprising temperature distribution), and performing a production operation comprising generating production from the reservoir embedded with the generated fractures. The method may involve optimizing the fracturing operation during its design stage based on comparison of predicted production corresponding to various fracturing designs with different job parameters. The method may also involve optimizing the fracture operation by adjusting the generating based on a comparison of the predicted production with actual production.

[0020] This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.

[0038] The description that follows includes exemplary systems, apparatuses, methods, and instruction sequences that embody techniques of the subject matter herein. However, it is understood that the described embodiments may be practiced without these specific details.

[0039] The present disclosure relates to techniques for performing fracture operations to predict temperature of fracturing fluid. The fracture operations involve fracture modeling that utilize elliptical wiremesh modeling and proppant transport modeling to estimate production. The techniques may involve viscosity and/or temperature estimations.

[0040] FIGS. 1.1-1.4 depict various oilfield operations that may be performed at a wellsite, and FIGS. 2.1-2.4 depict various information that may be collected at the wellsite. FIGS. 1.1-1.4 depict simplified, schematic views of a representative oilfield or wellsite 100 having subsurface formation 102 containing, for example, reservoir 104 therein and depicting various oilfield operations being performed on the wellsite 100. FIG. 1.1 depicts a survey operation being performed by a survey tool, such as seismic truck 106.1, to measure properties of the subsurface formation. The survey operation may be a seismic survey operation for producing sound vibrations. In FIG. 1.1, one such sound vibration 112 generated by a source 110 reflects off a plurality of horizons 114 in an earth formation 116. The sound vibration(s) 112 may be received in by sensors, such as geophone-receivers 118, situated on the earth's surface, and the geophones 118 produce electrical output signals, referred to as data received 120 in FIG. 1.1.

[0041] In response to the received sound vibration(s) 112 representative of different parameters (such as amplitude and/or frequency) of the sound vibration(s) 112, the geophones 118 may produce electrical output signals containing data concerning the subsurface formation. The data received 120 may be provided as input data to a computer 122.1 of the seismic truck 106.1, and responsive to the input data, the computer 122.1 may generate a seismic and microseismic data output 124. The seismic data output may be stored, transmitted or further processed as desired, for example by data reduction.

[0042] FIG. 1.2 depicts a drilling operation being performed by a drilling tool 106.2 suspended by a rig 128 and advanced into the subsurface formations 102 to form a wellbore 136 or other channel. A mud pit 130 may be used to draw drilling mud into the drilling tools via flow line 132 for circulating drilling mud through the drilling tools, up the wellbore 136 and back to the surface. The drilling mud may be filtered and returned to the mud pit. A circulating system may be used for storing, controlling, or filtering the flowing drilling muds. In this illustration, the drilling tools are advanced into the subsurface formations to reach reservoir 104. Each well may target one or more reservoirs. The drilling tools may be adapted for measuring downhole properties using logging while drilling tools. The logging while drilling tool may also be adapted for taking a core sample 133 as shown, or removed so that a core sample may be taken using another tool.

[0043] A surface unit 134 may be used to communicate with the drilling tools and/or offsite operations. The surface unit may communicate with the drilling tools to send commands to the drilling tools, and to receive data therefrom. The surface unit may be provided with computer facilities for receiving, storing, processing, and/or analyzing data from the operation. The surface unit may collect data generated during the drilling operation and produce data output 135 which may be stored or transmitted. Computer facilities, such as those of the surface unit, may be positioned at various locations about the wellsite and/or at remote locations.

[0044] Sensors (S), such as gauges, may be positioned about the oilfield to collect data relating to various operations as described previously. As shown, the sensor (S) may be positioned in one or more locations in the drilling tools and/or at the rig to measure drilling parameters, such as weight on bit, torque on bit, pressures, temperatures, flow rates, compositions, rotary speed and/or other parameters of the operation. Sensors (S) may also be positioned in one or more locations in the circulating system.

[0045] The data gathered by the sensors may be collected by the surface unit and/or other data collection sources for analysis or other processing. The data collected by the sensors may be used alone or in combination with other data. The data may be collected in one or more databases and/or transmitted on or offsite. All or select portions of the data may be selectively used for analyzing and/or predicting operations of the current and/or other wellbores. The data may be historical data, real time data or combinations thereof. The real time data may be used in real time, or stored for later use. The data may also be combined with historical data or other inputs for further analysis. The data may be stored in separate databases, or combined into a single database.

[0046] The collected data may be used to perform analysis, such as modeling operations. For example, the seismic data output may be used to perform geological, geophysical, and/or reservoir engineering analysis. The reservoir, wellbore, surface, and/or processed data may be used to perform reservoir, wellbore, geological, and geophysical or other simulations. The data outputs from the operation may be generated directly from the sensors, or after some preprocessing or modeling. These data outputs may act as inputs for further analysis.

[0047] The data may be collected and stored at the surface unit 134. One or more surface units may be located at the wellsite, or connected remotely thereto. The surface unit may be a single unit, or a complex network of units used to perform the necessary data management functions throughout the oilfield. The surface unit may be a manual or automatic system. The surface unit 134 may be operated and/or adjusted by a user.

[0048] The surface unit may be provided with a transceiver 137 to allow communications between the surface unit and various portions of the current oilfield or other locations. The surface unit 134 may also be provided with, or functionally connected to, one or more controllers for actuating mechanisms at the wellsite 100. The surface unit 134 may then send command signals to the oilfield in response to data received. The surface unit 134 may receive commands via the transceiver or may itself execute commands to the controller. A processor may be provided to analyze the data (locally or remotely), make the decisions, and/or actuate the controller. In this manner, operations may be selectively adjusted based on the data collected. Portions of the operation, such as controlling drilling, weight on bit, pump rates or other parameters, may be optimized based on the information. These adjustments may be made automatically based on computer protocol, and/or manually by an operator. In some cases, well plans may be adjusted to select optimum operating conditions, or to avoid problems.

[0049] FIG. 1.3 depicts a wireline operation being performed by a wireline tool 106.3 suspended by the rig 128 and into the wellbore 136 of FIG. 1.2. The wireline tool 106.3 may be adapted for deployment into a wellbore 136 for generating well logs, performing downhole tests and/or collecting samples. The wireline tool 106.3 may be used to provide another method and apparatus for performing a seismic survey operation. The wireline tool 106.3 of FIG. 1.3 may, for example, have an explosive, radioactive, electrical, or acoustic energy source 144 that sends and/or receives electrical signals to the surrounding subsurface formations 102 and fluids therein.

[0050] The wireline tool 106.3 may be operatively connected to, for example, the geophones 118 and the computer 122.1 of the seismic truck 106.1 of FIG. 1.1. The wireline tool 106.3 may also provide data to the surface unit 134. The surface unit 134 may collect data generated during the wireline operation and produce data output 135 which may be stored or transmitted. The wireline tool 106.3 may be positioned at various depths in the wellbore to provide a survey or other information relating to the subsurface formation.

[0051] Sensors (S), such as gauges, may be positioned about the wellsite 100 to collect data relating to various operations as described previously. As shown, the sensor (S) is positioned in the wireline tool 106.3 to measure downhole parameters which relate to, for example, porosity, permeability, fluid composition, and/or other parameters of the operation.

[0052] FIG. 1.4 depicts a production operation being performed by a production tool 106.4 deployed from a production unit or Christmas tree 129 and into the completed wellbore 136 of FIG. 1.3 for drawing fluid from the downhole reservoirs into surface facilities 142. Fluid flows from reservoir 104 through perforations in the casing (not shown) and into the production tool 106.4 in the wellbore 136 and to the surface facilities 142 via a gathering network 146.

[0053] Sensors (S), such as gauges, may be positioned about the oilfield to collect data relating to various operations as described previously. As shown, the sensor (S) may be positioned in the production tool 106.4 or associated equipment, such as the Christmas tree 129, gathering network, surface facilities, and/or the production facility, to measure fluid parameters, such as fluid composition, flow rates, pressures, temperatures, and/or other parameters of the production operation.

[0054] While simplified wellsite configurations are shown, it will be appreciated that the oilfield or wellsite 100 may cover a portion of land, sea and/or water locations that hosts one or more wellsites. Production may also include injection wells (not shown) for added recovery or for storage of hydrocarbons, carbon dioxide, or water, for example. One or more gathering facilities may be operatively connected to one or more of the wellsites for selectively collecting downhole fluids from the wellsite(s).

[0055] It should be appreciated that FIGS. 1.2-1.4 depict tools that can be used to measure not just properties of an oilfield, but also properties of non-oilfield operations, such as mines, aquifers, storage, and other subsurface facilities. Also, while certain data acquisition tools are depicted, it will be appreciated that various measurement tools (e.g., wireline, measurement while drilling (MWD), logging while drilling (LWD), core sample, etc.) capable of sensing parameters, such as seismic two-way travel time, density, resistivity, production rate, etc., of the subsurface formation and/or its geological formations may be used. Various sensors (S) may be located at various positions along the wellbore and/or the monitoring tools to collect and/or monitor the desired data. Other sources of data may also be provided from offsite locations.

[0056] The oilfield configuration of FIGS. 1.1-1.4 depict examples of a wellsite 100 and various operations usable with the techniques provided herein. Part, or all, of the oilfield may be on land, water and/or sea. Also, while a single oilfield measured at a single location is depicted, reservoir engineering may be utilized with any combination of one or more oilfields, one or more processing facilities, and one or more wellsites.

[0057] FIGS. 2.1-2.4 are graphical depictions of examples of data collected by the tools of FIGS. 1.1-1.4, respectively. FIG. 2.1 depicts a seismic trace 202 of the subsurface formation of FIG. 1.1 taken by seismic truck 106.1. The seismic trace may be used to provide data, such as a two-way response over a period of time. FIG. 2.2 depicts a core sample 133 taken by the drilling tools 106.2. The core sample may be used to provide data, such as a graph of the density, porosity, permeability or other physical property of the core sample over the length of the core. Tests for density and viscosity may be performed on the fluids in the core at varying pressures and temperatures. FIG. 2.3 depicts a well log 204 of the subsurface formation of FIG. 1.3 taken by the wireline tool 106.3. The wireline log may provide a resistivity or other measurement of the formation at various depts. FIG. 2.4 depicts a production decline curve or graph 206 of fluid flowing through the subsurface formation of FIG. 1.4 measured at the surface facilities 142. The production decline curve may provide the production rate Q as a function of time t.

[0058] The respective graphs of FIGS. 2.1, 2.3, and 2.4 depict examples of static measurements that may describe or provide information about the physical characteristics of the formation and reservoirs contained therein. These measurements may be analyzed to define properties of the formation(s), to determine the accuracy of the measurements and/or to check for errors. The plots of each of the respective measurements may be aligned and scaled for comparison and verification of the properties.

[0059] FIG. 2.4 depicts an example of a dynamic measurement of the fluid properties through the wellbore. As the fluid flows through the wellbore, measurements are taken of fluid properties, such as flow rates, pressures, composition, etc. As described below, the static and dynamic measurements may be analyzed and used to generate models of the subsurface formation to determine characteristics thereof. Similar measurements may also be used to measure changes in formation aspects over time.

[0060] In one aspect, these techniques employ a model for characterizing a hydraulic fracture network as described below. Such a model includes a set of equations that quantify the complex physical process of fracture propagation in a formation driven by fluid injected through a wellbore. In one embodiment, these equations are posed in terms of 12 model parameters: wellbore radius x_w and wellbore net pressure pw−σc, fluid injection rate q and duration t_p, matrix plane strain modulus E, fluid viscosity μ (or other fluid flow parameter(s) for non-Newtonian fluids), confining stress contrast Δσ, fracture network sizes h, a, e, and fracture spacing dx and dy.

[0061] Various fracture networks as used herein may have natural and/or man-made fractures. To facilitate production from a wellbore, the wellbore may be stimulated by performing fracture operations. For example, a hydraulic fracture network can be produced by pumping fluid into a formation. A hydraulic fracture network can be represented by two perpendicular sets of parallel planar fractures. The fractures parallel to the x-axis may be equally separated by distance dy and those parallel to the y-axis are separated by distance dx as illustrated in FIG. 3. Consequently, the numbers of fractures, per unit length, parallel to the x-axis and the y-axis, respectively, are

$\begin{array}{cc}{n}_x=\frac{1}{d_y}\mathrm{and}n_y=\frac{1}{d_x}.& \left(1\right)\end{array}$

[0062] The pumping of fracturing fluid over time produces a propagating fracture network that can be represented by an expanding volume in the form of an ellipse (FIG. 3) subject to stress σ_min with height h, major axis a, minor axis b or aspect ratio e:

$\begin{array}{cc}e=\frac{b}{a}.& \left(2\right)\end{array}$

[0063] The governing equation for mass conservation of the injected fluid in the fractured subterranean formation is given by:

$\begin{array}{cc}2\pi \mathrm{ex}\frac{\partial \left(\mathrm{\phi \rho }\right)}{\partial t}+4\frac{\partial \left(\mathrm{Bx}\rho {\stackrel{\_}{v}}_e\right)}{\partial x}=0,\mathrm{or}& \left(3a\right)\\ \frac{2\pi y}{e}\frac{\partial \left(\phi p\right)}{\partial t}+4\frac{\partial }{\partial y}\left(\frac{\mathrm{By}\rho {\stackrel{\_}{v}}_e}{e}\right)=0,& \left(3b\right)\end{array}$

which for an incompressible fluid becomes respectively

$\begin{array}{cc}2\pi \mathrm{ex}\frac{\partial \phi }{\partial t}+4\frac{\partial \left(\mathrm{Bx}{\stackrel{\_}{v}}_e\right)}{\partial x}=0,\mathrm{or}& \left(3c\right)\\ \frac{2\pi y}{e}\frac{\partial \phi }{\partial t}+\frac{\partial }{\partial y}\left(\frac{\mathrm{By}{\stackrel{\_}{v}}_e}{e}\right)=0,& \left(3d\right)\end{array}$

[0064] where φ is the porosity of the formation,

• • [0065] ρ is the density of injected fluid
  • [0066] v_e is an average fluid velocity perpendicular to the elliptic boundary, and B is the elliptical integral given by

$\begin{array}{cc}B=\frac{\pi }{2}[1-{\left(\frac{1}{2}\right)}^2\left(1-e^2\right)-{\left(\frac{1·3}{2·4}\right)}^2\frac{{\left(1-e^2\right)}^2}{3}-{\left(\frac{1·3·5}{2·4·6}\right)}^2\frac{{\left(1-e^2\right)}^2}{5}-\dots ]& \left(4\right)\end{array}$

The average fluid velocity v_e may be approximated as

$\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{v}}_e\approx \frac{1}{2}\left[v_{\mathrm{ex}}\left(x,y=0\right)+v_{\mathrm{ey}}\left(x=0,y=\mathrm{ex}\right)\right]\\ \approx \frac{1}{2}\left(1+e\right)v_{\mathrm{ex}}\left(x,y=0\right)\\ \approx \frac{1}{2}\left(1+1/e\right)v_{\mathrm{ey}}\left(x=0,y=\mathrm{ex}\right)\end{array}\mathrm{with}& \left(5\right)\\ v_{\mathrm{ex}}\left(x,y=0\right)=-{\left[\frac{k_x}{\mu }\frac{\partial p}{\partial x}\right]}_{\left(x,y=0\right)},& \left(6a\right)\\ v_{\mathrm{ey}}\left(x=0,y=\mathrm{ex}\right)=-{\left[\frac{k_y}{\mu }\frac{\partial p}{\partial y}\right]}_{\left(x=0,y=\mathrm{ex}\right)},& \left(6b\right)\end{array}$

[0067] where p is fluid pressure,

• • [0068] μ is fluid viscosity, and
    • [0069] k_x and k_y are permeability factors for the formation along the x-direction and the y-direction, respectively.
      For the sake of mathematical simplicity, equations below are presented for an incompressible fluid as an example, with the understanding that fluid compressibility may be accounted for by using a corresponding equation of state for the injected fluid.

[0070] Using equations (5) and (6), governing equations (3a,3b) can be written as

$\begin{array}{cc}2\pi \mathrm{ex}\frac{\partial \phi }{\partial t}-2\frac{\partial }{\partial x}\left(\frac{B\left(1+e\right){\mathrm{xk}}_x}{\mu }\frac{\partial p}{\partial x}\right)=0,\mathrm{or}& \left(7a\right)\\ \frac{2\pi y}{e}\frac{\partial \phi }{\partial t}-2\frac{\partial }{\partial y}\left(\frac{B\left(1+e\right){\mathrm{yk}}_y}{e^2\mu }\frac{\partial p}{\partial y}\right)=0.& \left(7b\right)\end{array}$

[0071] The width w of a hydraulic fracture may be calculated as

$\begin{array}{cc}w=\frac{2l}{E}\left(p-{\sigma }_c\right)H\left(p-{\sigma }_c\right),H\left(p-{\sigma }_c\right)=\{\begin{array}{cc}0& p\le {\sigma }_c\\ 1& p>{\sigma }_c\end{array}& \left(8\right)\end{array}$

where H is the Heaviside step function, σ_c is the confining stress perpendicular to the fracture, E is the plane strain modulus of the formation, and l is the characteristic length scale of the fracture segment and given by the expression

l=d+(h−d)H(d−h)  (9)

where h and d are the height and the length, respectively, of the fracture segment.

[0072] When mechanical interaction between adjacent fractures is accounted for, assuming that the size of stimulated formation is much larger than either the height of the ellipse or the averaged length of fractures, the width of fractures parallel to the x-axis and the y-axis, respectively, can be expressed as

$\begin{array}{cc}{w}_x=\frac{2d_x}{A_{\mathrm{Ex}}E}\left(p-{\sigma }_{\mathrm{cy}}\right)H\left(p-{\sigma }_{\mathrm{cy}}\right),& \left(10a\right)\\ w_y=\frac{2d_y}{A_{\mathrm{Ey}}E}\left(p-{\sigma }_{\mathrm{cx}}\right)H\left(p-{\sigma }_{\mathrm{cx}}\right)& \left(10b\right)\end{array}$

where σ_cx and σ_cy are the confining stresses, respectively, along the x-direction and the y-direction, respectively, and A_Ex and A_Ey are the coefficients for defining the effective plane strain modulus along the x-axis and y-axis, respectively.
represented by the following expressions

$\begin{array}{cc}{A}_{\mathrm{Ex}}=\frac{d_x\left[2l_x+\left(d_y-2l_x\right)H\left(d_y-2l_x\right)\right]}{d_yl_x},& \left(11a\right)\\ A_{\mathrm{Ey}}=\frac{d_y\left[2l_y+\left(d_x-2l_y\right)H\left(d_x-2l_y\right)\right]}{d_xl_y}.& \left(11b\right)\end{array}$

where l_x and l_y are the characteristic length scale along the x-axis and the y-axis, respectively.
The value of the coefficient (A_Ex) for the effective plane strain modulus along the x-axis can be simplified for different cases of d_x, d_y, and h by any one of Tables 1-2 listed below. The value of the coefficient (A_Ey) for the effective plane strain modulus along the y-axis can be simplified for different cases of d_x, d_y, and h by any one of Tables 3-5 listed below.

TABLE 1
Coefficient AEx for different cases of dx, dy, h
AEx
dx ≧ dy	dx < dy
	dx > h	dx ≦ h	dx > h
dx ≦ h	dy ≦ 2h	dy > 2h	dy ≦ 2dx	dy > 2dx	dy ≦ 2h	dy > 2h
2  d x d y	2  d x d y	d x h	2  d x d y	1	2  d x d y	d x h

TABLE 2
Coefficient AEx for different cases of dx, dy, h
AEx
dx ≧ dy	dx < dy
	dx > h	dy ≦ h	dy > h
dx ≦ h	dy ≦ 2h	dy > 2h	dy ≦ 2dx	dy > 2dx	dy ≦ 2h	dy > 2h
2  d x d y	2  d x d y	d x h	2  d x d y	1	2  d x d y	d x h

TABLE 3
Coefficient AEy for different cases of dx, dy, h
AEy
dy ≧ dx	dy < dx
	dy > h	dy ≦ h	dy > h
dy ≦ h	dx ≦ 2h	dx > 2h	dx ≦ 2dy	dx > 2dy	dx ≦ 2h	dx > 2h
2  d y d x	2  d y d x	d y h	2  d y d x	1	2  d y d x	d y h

TABLE 4a
Coefficient AEy for different cases of dx, dy, h
AEy
dx ≧ dy
		dx > h
	dx ≦ h	dy ≦ h	dy > h
	dx ≦ 2dy	dx > 2dy	dx ≦ 2dy	dx > 2dy	dx ≦ 2h	dx > 2h
	2  d y d x	1	2  d y d x	1	2  d y d x	d y h

TABLE 4b
Coefficient AEy for different cases of dx, dy, h
AEy
dx < dy
	dx ≦ h
	dy ≦ h	dy > h	dx > h
	dx ≦ 2dy	dx > 2dy	dx ≦ 2h	dx > 2h	dx ≦ 2h	dx > 2h
	2  d y d x	1	2  d y d x	d y h	2  d y d x	d y h

TABLE 5
Coefficient AEy for different cases of dx, dy, h
AEy
dx ≧ dy
	dx > h	dx < dy
dx ≦ h	dy ≦ h	dy > h		dx > h
dx ≦	dx >	dx ≦	dx >		dx >			dx >
2dy	2dy	2dy	2dy	dx ≦ 2h	2h	dx ≦ h	dx ≦ 2h	2h
2  d y d x	1	2  d y d x	1	2  d y d x	d y h	2  d y d x	2  d y d x	d y h

[0073] The increase in porosity of the fractured formation (Δφ) can be calculated as

$\begin{array}{cc}\mathrm{\Delta \phi }=n_xw_x+n_yw_y-n_xn_yw_xw_y\approx \frac{2d_x}{d_yA_{\mathrm{Ex}}E}\left(p-{\sigma }_{\mathrm{cy}}\right)H\left(p-{\sigma }_{\mathrm{cy}}\right)+\frac{2d_y}{d_xA_{\mathrm{Ey}}E}\left(p-{\sigma }_{\mathrm{cx}}\right)H\left(p-{\sigma }_{\mathrm{cx}}\right)& \left(12\right)\end{array}$

The fracture permeability along the x-axis (k_x) and the fracture permeability along the y-axis (k_y) can be determined as

$\begin{array}{cc}\begin{array}{c}{k}_x=\frac{n_xw_x^3}{12}\\ =\frac{2d_x^3}{3E^3d_yA_{\mathrm{Ex}}^3}{\left(p-{\sigma }_{\mathrm{cy}}\right)}^3H\left(p-{\sigma }_{\mathrm{cy}}\right),\end{array}\mathrm{and}& \left(13a\right)\\ \begin{array}{c}{k}_y=\frac{n_yw_y^3}{12}\\ =\frac{2d_y^3}{3E^3d_xA_{\mathrm{Ey}}^3}{\left(p-{\sigma }_{\mathrm{cx}}\right)}^3H\left(p-{\sigma }_{\mathrm{cx}}\right),\end{array}& \left(13b\right)\end{array}$

along the x-axis and y-axis, respectively.

[0074] For p>σ_cy and a negligible virgin formation permeability as compared to the fracture permeability along the x-axis, the governing equation (7a) can be integrated from x_w to x using equation (13a) for the permeability (k_x) to yield

$\begin{array}{cc}4{\left(p-{\sigma }_{\mathrm{cy}}\right)}^3\frac{p}{x}=\frac{3A_{\mathrm{Ex}}^3d_yE^3\mu }{\left(1+e\right){\mathrm{Bd}}_x^3x}\left(2\pi {\int }_{x_w}^x\frac{\partial \phi }{\partial t}\mathrm{es}s-q\right).& \left(14a\right)\end{array}$

Similarly for p>σ_cx, the governing equation (7b) can be integrated from x_w to y using equation (12b) for the permeability (k_y) to yield

$\begin{array}{cc}4{\left(p-{\sigma }_{\mathrm{cx}}\right)}^3\frac{p}{y}=\frac{3e^2A_{\mathrm{Ey}}^3d_xE^3\mu }{\left(1+e\right){\mathrm{Bd}}_y^3y}\left(2\pi {\int }_{x_w}^y\frac{\partial \phi }{\partial t}\frac{s}{e}s-q\right).& \left(14b\right)\end{array}$

In equations (13a) and (13b), x_w is the radius of the wellbore and q is the rate of fluid injection into the formation via the wellbore. The inject rate q is treated as a constant and quantified in volume per unit time per unit length of the wellbore.

[0075] Equation (14a) can be integrated from x to a and yields a solution for the net pressure inside the fracture along the x-axis as

$\begin{array}{cc}p-{\sigma }_{\mathrm{cy}}={\left[\frac{3}{\left(1+e\right)B}{\int }_x^a\frac{A_{\mathrm{Ex}}^3d_yE^3\mu }{d_x^3r}\left(q-2\pi {\int }_{x_w}^r\frac{\partial \phi }{\partial t}\mathrm{es}s\right)r\right]}^{1/4}.& \left(15a\right)\end{array}$

Equation (14b) can be integrated from y to b yields a solution for the net pressure inside the fractures along the y-axis as

$\begin{array}{cc}p-{\sigma }_{\mathrm{cx}}={\left[\frac{3e^2}{\left(1+e\right)B}{\int }_y^b\frac{A_{\mathrm{Ey}}^3d_xE^3\mu }{d_y^3r}\left(q-2\pi {\int }_{x_w}^r\frac{\partial \phi }{\partial t}\frac{s}{e}s\right)r\right]}^{1/4}.& \left(15b\right)\end{array}$

[0076] For uniform σ_c, E, μ, n and d, equation (15a) reduces to

$\begin{array}{cc}p-{\sigma }_{\mathrm{cy}}={A_{\mathrm{px}}\left[q\ln \left(\frac{a}{x}\right)-2\pi e{\int }_x^a\left({\int }_{x_w}^r\frac{\partial \phi }{\partial t}ss\right)\frac{1}{r}r\right]}^{1/4}A_{\mathrm{px}}={\left(\frac{3A_{\mathrm{Ex}}^3d_yE^3\mu }{\left(1+e\right){\mathrm{Bd}}_x^3}\right)}^{1/4}.& \left(16a\right)\end{array}$

Similarly, equation (15b) reduces to

$\begin{array}{cc}p-{\sigma }_{\mathrm{cx}}=e^{1/2}{A_{\mathrm{py}}\left[q\ln \left(\frac{b}{y}\right)-\frac{2\pi }{e}{\int }_y^b\left({\int }_{x_w}^r\frac{\partial \phi }{\partial t}ss\right)\frac{1}{r}r\right]}^{1/4}A_{\mathrm{py}}={\left(\frac{3A_{\mathrm{Ey}}^3d_xE^3\mu }{\left(1+e\right){\mathrm{Bd}}_y^3}\right)}^{1/4}.& \left(16b\right)\end{array}$

[0077] The wellbore pressure P_w is given by the following expressions:

$\begin{array}{cc}p_w={\sigma }_{\mathrm{cy}}+{A_{\mathrm{px}}\left[q\ln \left(\frac{a}{x_w}\right)-2\pi e{\int }_{x_w}^a\left({\int }_{x_w}^r\frac{\partial \phi }{\partial t}ss\right)\frac{1}{r}r\right]}^{1/4},& \left(17a\right)\\ p_w={\sigma }_{\mathrm{cx}}+e^{1/2}{A_{\mathrm{py}}\left[q\ln \left(\frac{b}{x_w}\right)-\frac{2\pi }{e}{\int }_{x_w}^b\left({\int }_{x_w}^r\frac{\partial \phi }{\partial t}ss\right)\frac{1}{r}r\right]}^{1/4}.& \left(17b\right)\end{array}$

By requiring the two expressions (17a, 17b) for the wellbore pressure p_w, to be equal, one obtains the difference between confining stresses (Δσ_c), which is also referred herein to as stress contrast Δσ_c, as

$\begin{array}{cc}\begin{array}{c}{\mathrm{\Delta \sigma }}_c={\sigma }_{\mathrm{cx}}-{\sigma }_{\mathrm{cy}}\\ ={A_{\mathrm{px}}\left[q\ln \left(\frac{a}{x_w}\right)-2\pi e{\int }_{x_w}^a\left({\int }_{x_w}^r\frac{\partial \phi }{\partial t}ss\right)\frac{1}{r}r\right]}^{1/4}-\\ e^{1/2}{A_{\mathrm{py}}\left[q\ln \left(\frac{\mathrm{ea}}{x_w}\right)-\frac{2\pi }{e}{\int }_{x_w}^{\mathrm{ea}}\left({\int }_{x_w}^r\frac{\partial \phi }{\partial t}ss\right)\frac{1}{r}r\right]}^{1/4}.\end{array}& \left(18\right)\end{array}$

[0078] Assuming negligible leakoff and incompressible fluid, the time t_p for the ellipse edge propagating from x_w to a along the x-axis and x_w to b along the y-axis is determined as

$\begin{array}{cc}\begin{array}{c}\frac{{\mathrm{qt}}_p}{\pi }=e{\int }_{x_w}^a{\mathrm{\Delta \phi }}_xxx+\frac{1}{e}{\int }_{x_w}^b{\mathrm{\Delta \phi }}_yyy\\ =e{\int }_{x_w}^a\frac{2d_x\left(p_x-{\sigma }_{\mathrm{cy}}\right)}{d_yA_{\mathrm{Ex}}E}xx+e{\int }_{x_w}^{x_{\sigma }}\frac{2d_y\left(p_x-{\sigma }_{\mathrm{cx}}\right)}{d_xA_{\mathrm{Ey}}E}xx+\\ \frac{1}{e}{\int }_{x_{\sigma }}^b\left[\frac{2d_x\left(p_y-{\sigma }_{\mathrm{cy}}\right)}{d_yA_{\mathrm{Ex}}E}+\frac{2d_y\left(p_y-{\sigma }_{\mathrm{cx}}\right)}{d_xA_{\mathrm{Ey}}E}\right]yy,\end{array}\mathrm{or}& \left(19a\right)\\ \begin{array}{c}\frac{{\mathrm{qt}}_p}{\pi e}={\int }_{x_w}^a\left[{\mathrm{\Delta \phi }}_x\left(x\right)+{\mathrm{\Delta \phi }}_y\left(y=\mathrm{ex}\right)\right]xx\\ =\frac{2}{E}[{\int }_{x_w}^{x_{\sigma }}\left(\frac{d_x}{d_yA_{\mathrm{Ex}}}+\frac{d_y}{d_xA_{\mathrm{Ey}}}\right)\left(p_x-{\sigma }_{\mathrm{cy}}\right)xx+\\ {\int }_{x_{\sigma }}^a\frac{d_x}{d_yA_{\mathrm{Ex}}}\left(p_x-{\sigma }_{\mathrm{cy}}\right)xx]+\\ \frac{2}{E}{\int }_{x_w}^a\left(\frac{d_x}{d_yA_{\mathrm{Ex}}}+\frac{d_y}{d_xA_{\mathrm{Ey}}}\right)\left(p_x-{\sigma }_{\mathrm{cx}}\right)xx+\\ \frac{2{\mathrm{\Delta \sigma }}_c}{E}\left({\int }_{x_w}^a\frac{d_x}{d_yA_{\mathrm{Ex}}}xx-{\int }_{x_w}^{x_{\sigma }}\frac{d_y}{d_xA_{\mathrm{Ey}}}xx\right),\end{array}& \left(19b\right)\end{array}$

where x_σ is defined as x_w≦xσ<a where

p≦σ_cx if x≦x_σ

p>σ_cx if x>x_σ

p=σ_cx if x=x_σ

[0079] Equation (15a) can be rewritten for the case p=σ_cx at x=x_σ as follows

$\begin{array}{cc}{\mathrm{\Delta \sigma }}_c={\left[\frac{3}{\left(1+e\right)B}{\int }_{x_{\sigma }}^a\frac{A_{\mathrm{Ex}}^3d_yE^3\mu }{d_x^3r}\left(q-2\pi {\int }_{x_w}^r\frac{\partial \phi }{\partial t}\mathrm{es}s\right)r\right]}^{1/4}.& \left(20\right)\end{array}$

[0080] The surface area of the open fractures may be calculated as follows

$\begin{array}{cc}\begin{array}{c}S\approx \pi \mathrm{ab}\times 2{\mathrm{hn}}_x+\pi x_{\sigma }b\times 2{\mathrm{hn}}_y,\\ =2\pi \mathrm{eah}\left(\frac{a}{d_y}+\frac{x_{\sigma }}{d_x}\right).\end{array}& \left(21\right)\end{array}$

[0081] For a quasi-steady state, governing equations (7a) and (7b) reduce to

$\begin{array}{cc}-2B\left(1+e\right)\frac{{\mathrm{xk}}_x}{\mu }\frac{p}{x}=q,& \left(22a\right)\\ -\frac{2B\left(1+e\right)}{e^2}\frac{{\mathrm{yk}}_y}{\mu }\frac{p}{y}=q.& \left(22b\right)\end{array}$

Moreover, for the quasi-steady state, the pressure equations (15a) and (15b) reduce to

$\begin{array}{cc}p-{\sigma }_{\mathrm{cy}}={\left[\frac{3}{\left(1+e\right)B}{\int }_x^a\frac{A_{\mathrm{Ex}}^3d_yE^3q\mu }{d_x^3r}r\right]}^{1/4},& \left(23a\right)\\ p-{\sigma }_{\mathrm{cx}}={\left[\frac{3e^2}{\left(1+e\right)B}{\int }_y^b\frac{A_{\mathrm{Ey}}^3d_xE^3q\mu }{d_y^3r}r\right]}^{1/4}.& \left(23b\right)\end{array}$

For the quasi-steady state and uniform properties of σ_c, E, ,μ, n and d, equations (16a) and (16b) reduce to

$\begin{array}{cc}p-{\sigma }_{\mathrm{cy}}={A_{\mathrm{px}}\left(q\ln \frac{a}{x}\right)}^{1/4},& \left(24a\right)\\ p-{\sigma }_{\mathrm{cx}}=e^{1/2}{A_{\mathrm{py}}\left(q\ln \frac{b}{y}\right)}^{1/4}.& \left(24b\right)\end{array}$

Correspondingly, for the quasi-steady state, the wellbore pressure equations (17a) and (17b) reduce to

$\begin{array}{cc}{p}_w={\sigma }_{\mathrm{cy}}+{A_{\mathrm{px}}\left(q\ln \frac{a}{x_w}\right)}^{1/4},& \left(25a\right)\\ p_w={\sigma }_{\mathrm{cx}}+e^{1/2}{A_{\mathrm{py}}\left(q\ln \frac{\mathrm{ea}}{x_w}\right)}^{1/4},& \left(25b\right)\end{array}$

By requiring the two expressions (25a, 25b) for the wellbore pressure p_w to be equal, one obtains

$\begin{array}{cc}\left[1-e^{1/2}A_{\mathrm{ea}}\frac{d_x}{d_y}{\left(\frac{A_{\mathrm{Ey}}}{A_{\mathrm{Ex}}}\right)}^{3/4}\right]\left(p_w-{\sigma }_{\mathrm{cy}}\right)={\mathrm{\Delta \sigma }}_c,A_{\mathrm{ea}}={\left[\frac{\ln \left(\mathrm{ea}/x_w\right)}{\ln \left(a/x_w\right)}\right]}^{1/4}.& \left(26\right)\end{array}$

[0082] For the quasi-steady state and uniform properties of σ_c, E, μ, n and d, equations (19a) and (19b), respectively, reduce to

$\begin{array}{cc}\frac{{\mathrm{qt}}_p}{\pi }=\frac{{\mathrm{eA}}_{\phi }d_y^{1/4}A_{\mathrm{Ex}}^{3/4}}{d_x^{3/4}}[\left(\frac{d_x}{d_yA_{\mathrm{Ex}}}+\frac{d_y}{d_xA_{\mathrm{Ey}}}\right){\int }_{x_w}^{x_{\sigma }}{\left(\ln \frac{a}{x}\right)}^{1/4}xx+\frac{d_x}{d_yA_{\mathrm{Ex}}}{\int }_{x\sigma }^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx]+\frac{A_{\phi }d_x^{1/4}A_{\mathrm{Ey}}^{3/4}}{e^{1/2}d_y^{3/4}}\left(\frac{d_x}{d_yA_{\mathrm{Ex}}}+\frac{d_y}{d_xA_{\mathrm{Ey}}}\right){\int }_{x_w}^b{\left(\ln \frac{b}{y}\right)}^{1/4}yy+\frac{{\mathrm{\Delta \sigma }}_c}{E}\left[\frac{d_x}{{\mathrm{ed}}_yA_{\mathrm{Ex}}}\left(b^2-x_w^2\right)-\frac{{\mathrm{ed}}_y}{d_xA_{\mathrm{Ey}}}\left(x_{\sigma }^2-x_w^2\right)\right],A_{\phi }={\left[\frac{48}{\left(1+e\right)\mathrm{BE}}\right]}^{1/4},\mathrm{and}& \left(27a\right)\\ \frac{{\mathrm{qt}}_p}{\pi e}={A_{\phi }\left(\frac{d_yA_{\mathrm{Ex}}^3}{d_x^3}\right)}^{1/4}[\left(\frac{d_x}{d_yA_{\mathrm{Ex}}}+\frac{d_y}{d_xA_{\mathrm{Ey}}}\right){\int }_{x_w}^{x_{\sigma }}{\left(\ln \frac{a}{x}\right)}^{1/4}xx+\frac{d_x}{d_yA_{\mathrm{Ex}}}{\int }_{x\sigma }^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx]+e^{1/2}{A_{\phi }\left(\frac{d_xA_{\mathrm{Ey}}^3}{d_y^3}\right)}^{1/4}\left(\frac{d_x}{d_yA_{\mathrm{Ex}}}+\frac{d_y}{d_xA_{\mathrm{Ey}}}\right){\int }_{x_w}^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx+\frac{{\mathrm{\Delta \sigma }}_c}{E}\left[\frac{d_x}{d_yA_{\mathrm{Ex}}}\left(a^2-x_w^2\right)-\frac{d_y}{d_xA_{\mathrm{Ey}}}\left(x_{\sigma }^2-x_w^2\right)\right],A_{\phi }={\left[\frac{48q\mu }{\left(1+e\right)\mathrm{BE}}\right]}^{1/4}.& \left(27b\right)\end{array}$

Correspondingly, equation (20) can be solved to yield

$\begin{array}{cc}{x}_{\sigma }=a\exp \left[-\frac{1}{q}{\left(\frac{{\mathrm{\Delta \sigma }}_c}{A_{\mathrm{px}}}\right)}^4\right].& \left(28\right)\end{array}$

[0083] The integrations in equation (27) can be numerically evaluated rather easily for a given x_σ.

[0084] 1. Constraints on the Parameters of the Model Using Field Data

[0085] In general, given the rest of the equations, equations (25a), (26) and (27) can be solved to obtain any three of the model parameters. Certain geometric and geomechanical parameters of the model as described above can be constrained using field data from a fracturing treatment and associated microseismic events. In one embodiment, the geometric properties (dx and dy) and the stress contrast (Δσ_c) are constrained given wellbore radius xw and wellbore net pressure pw−σc, fluid injection rate q and duration tp, matrix plane strain modulus E, fluid viscosity μ, and fracture network sizes h, a, e, as follows. Note that since xσ in equation (27) is calculated using equation (28) as a function of Δσ_c, the solution procedure is necessarily of an iterative nature.

[0086] Given these values, the value of d_x³/(A_Ex³,d_y) is determined according to equation (25a) by

$\begin{array}{cc}\frac{d_x^3}{A_{\mathrm{Ex}}^3d_y}=d_0^2d_0={\left[\frac{3E^3q\mu \ln \left(a/x_w\right)}{{\left(p_w-{\sigma }_{\mathrm{cy}}\right)}^4\left(1+e\right)B}\right]}^{1/2},& \left(29\right)\end{array}$

[0087] If (2d_y≧d_x≧d_y) and (d_x≦h), equation (29) leads to

d_y=√{square root over (8)}d₀.  (30)

Equations (26) and (27) become, respectively,

$\begin{array}{cc}\left[1-{A_{\mathrm{ea}}\left(\frac{{\mathrm{ed}}_y}{d_x}\right)}^{1/2}\right]\left(p_w-{\sigma }_{\mathrm{cy}}\right)={\mathrm{\Delta \sigma }}_c,\mathrm{and}& \left(31\right)\\ \frac{{\mathrm{qt}}_a}{\pi }=\frac{{\mathrm{eA}}_{\phi }}{2^{1/4}d_y^{1/2}}\left[2{\int }_{x_w}^{x_{\sigma }}{\left(\ln \frac{a}{x}\right)}^{1/4}xx+{\int }_{x_{\sigma }}^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx\right]+\frac{2^{3/4}A_{\phi }}{e^{1/2}d_x^{1/2}}{\int }_{x_w}^b{\left(\ln \frac{b}{y}\right)}^{1/4}yy+\frac{{\mathrm{\Delta \sigma }}_c}{2E}\left[\frac{b^2-x_w^2}{e}-e\left(x_{\sigma }^2-x_w^2\right)\right].& \left(32\right)\end{array}$

Using solution (30), equations (31) and (32) can be solved to obtain

$\begin{array}{cc}{\mathrm{\Delta \sigma }}_c=\left\{\frac{{\mathrm{qt}}_a}{\pi }-\frac{{\mathrm{eA}}_{\phi }}{2^{1/4}d_y^{1/2}}\left[2{\int }_{x_w}^{x_{\sigma }}{\left(\ln \frac{a}{x}\right)}^{1/4}xx+{\int }_{x_{\sigma }}^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx\right]-\frac{2^{3/4}A_{\phi }}{e^{1/2}d_x^{1/2}}{\int }_{x_w}^b{\left(\ln \frac{b}{y}\right)}^{1/4}yy\right\}\frac{2\mathrm{eE}}{b^2-x_w^2-e^2\left(x_{\sigma }^2-x_w^2\right)},\mathrm{and}& \left(33\right)\\ d_x=\sqrt{8}d_0{{\mathrm{eA}}_{\mathrm{ea}}^2\left(\frac{p_w-{\sigma }_{\mathrm{cy}}}{p_w-{\sigma }_{\mathrm{cy}}-{\mathrm{\Delta \sigma }}_c}\right)}^2.& \left(34\right)\end{array}$

[0088] If (h≧d_x>2d_y), equations (26) and (27) become, respectively,

$\begin{array}{cc}\left[1-\frac{e^{1/2}}{2^{3/4}}{A_{\mathrm{ea}}\left(\frac{d_x}{d_y}\right)}^{1/4}\right]\left(p_w-{\sigma }_{\mathrm{cy}}\right)={\mathrm{\Delta \sigma }}_c,\mathrm{and}& \left(35\right)\\ \frac{{\mathrm{qt}}_a}{\pi }=\frac{2^{3/4}{\mathrm{eA}}_{\phi }}{d_y^{1/2}}\left[\left(\frac{1}{2}+\frac{d_y}{d_x}\right){\int }_{x_w}^{x_{\sigma }}{\left(\ln \frac{a}{x}\right)}^{1/4}xx+\frac{1}{2}{\int }_{x_{\sigma }}^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx\right]+\frac{A_{\phi }d_x^{1/4}}{e^{1/2}d_y^{3/4}}\left(\frac{1}{2}+\frac{d_y}{d_x}\right){\int }_{x_w}^b{\left(\ln \frac{b}{y}\right)}^{1/4}yy+\frac{{\mathrm{\Delta \sigma }}_c}{E}\left[\frac{1}{2e}\left(b^2-x_w^2\right)-\frac{{\mathrm{ed}}_y}{d_x}\left(x_{\sigma }^2-x_w^2\right)\right].& \left(36\right)\end{array}$

Combined with solution (30) and replacing Δσ_c with equation (35), equation (36) can be solved for d_x, Δσ_c can then be calculated using equation (35).

[0089] If (d_x>h≧d_y), equation (29) leads to solution (30). Furthermore, if (d_x≦2d_y), equations (26) and (27) lead to solutions (33) and (34). On the other hand, if (d_x>2d_y), equations (26) and (27) lead to equations (35) and (36).

[0090] If (d_x≧d_y) and (h<d_y≦2h), equation (29) leads to solution (30). Furthermore, if (d_x≦2h), equations (26) and (27) lead to solutions (33) and (34). On the other hand, if (d_x>2h), equations (26) and (27) become, respectively,

$\begin{array}{cc}\left[1-{A_{\mathrm{ea}}\left(\frac{8e^2d_0^2d_x}{h^3}\right)}^4\right]\left(p_w-{\sigma }_{\mathrm{cy}}\right)={\mathrm{\Delta \sigma }}_c,\mathrm{and}& \left(37\right)\\ \frac{{\mathrm{qt}}_a}{2\pi e}=\frac{A_{\phi }}{2d_0^{1/2}}\left[\left(1+\frac{2h}{d_x}\right){\int }_{x_w}^{x_{\sigma }}{\left(\ln \frac{a}{x}\right)}^{1/4}xx+{\int }_{x_{\sigma }}^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx\right]-\frac{h\left(x_{\sigma }^2-x_w^2\right)\left(p_w-{\sigma }_{\mathrm{cy}}\right)}{{\mathrm{Ed}}_x}\left[1-{\left(\frac{8e^2d_0^2d_x}{h^3}\right)}^4\right].& \left(38\right)\end{array}$

Equation (38) can be solved for d_x and then Δσ_c can be calculated by equation (37).

[0091] If (d_x≧d_y>2h), equation (29) leads to

$\begin{array}{cc}{d}_y=\frac{h^3}{d_0^2}.& \left(39\right)\end{array}$

Equations (26) and (27) becomes, respectively,

$\begin{array}{cc}\left[1-e^{1/2}{A_{\mathrm{ea}}\left(\frac{d_0^2d_x}{h^3}\right)}^{7/4}\right]\left(p_w-{\sigma }_{\mathrm{cy}}\right)={\mathrm{\Delta \sigma }}_c,\mathrm{and}& \left(40\right)\\ \frac{{\mathrm{qt}}_a}{2\pi e}=\frac{A_{\phi }d_0^{3/2}}{h^2}\left[\left(1+\frac{h^3}{d_0^2d_x}\right){\int }_{x_w}^{x_{\sigma }}{\left(\ln \frac{a}{x}\right)}^{1/4}xx+{\int }_{x_{\sigma }}^a{\left(\ln \frac{a}{x}\right)}^{1/4}xx\right]-\frac{h\left(x_{\sigma }^2-x_w^2\right)\left(p_w-{\sigma }_{\mathrm{cy}}\right)}{{\mathrm{Ed}}_x}\left[1-{e^{1/2}\left(\frac{d_0^2d_x}{h^3}\right)}^{7/4}\right].& \left(41\right)\end{array}$