Abstract:
A method is disclosed including simulating a hydraulic fracture in an Earth formation, the formation including a multilayered reservoir, wherein a mesh overlays the hydraulic fracture thereby defining a plurality of fracture elements. The method further includes calculating and determining an influence coefficient matrix having spatially related indices, wherein the spatially related indices relate influence coefficient matrix elements to corresponding fracture elements. The mesh overlays the hydraulic fracture in more than one layer of the multilayered reservoir. In one embodiment, the method is executed on a program storage device readable by a machine tangibly embodying a program of instructions executable to perform the method.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation in part application of a prior application Ser. No. 09/301,961 corresponding to U.S. Pat. No. 6,876,959 filed Apr. 29, 1999, entitled “Method and Apparatus for Hydraulic Fracturing Analysis and Design”; this application is also a continuation in part of a prior pending application Ser. No. 10/831,799 corresponding to U.S. Pat. No. 7,063,147 filed Apr. 26, 2004, entitled “Method and Apparatus and Program Storage Device for Front Tracking in Hydraulic Fracturing Simulators.” This application is also a continuation application of application Ser. No. 11/095,104 filed Mar. 31, 2005, now U.S. Pat. No. 7,509 245 entitled “Method System and Program Storage Device for Simulating A Multilayer Reservoir and Partially Active Elements In a Hydraulic Fracturing Simulator.” This application claims priority to, and incorporates by reference in the entirety, each of the preceding applications. This application is related to a prior pending application corresponding to U.S. Pat No. 7,386,431 entitled “Method System and Program Storage Device for Simulating Interfacial Slip In a Hydraulic Fracturing Simulator Software” having the same inventors and the same assignee as the present invention. 
    
    
     BACKGROUND 
     The subject matter set forth in this specification relates to a Hydraulic Fracturing Simulator Software adapted for designing and monitoring and evaluating reservoir, for example petroleum reservoir, fracturing, and, in particular, to a Hydraulic Fracturing Simulator Software adapted for simulating a multilayer reservoir and partially active elements of a mesh overlaying a fracture footprint when designing and monitoring and evaluating reservoir fracturing. 
     In hydraulic fracturing, thousands of gallons of fluid are forced under high pressure underground to split open the rock in a subterranean formation, a process which, for petroleum reservoirs, is known as ‘petroleum reservoir fracturing’ associated with ‘a fracturing event’. Proppant or propping agent is carried into the fracture by a viscosified fluid, and deposited into the fracture. Proppant provides a permeable flow channel for formation fluids, such as oil and gas, to travel up the wellbore and to the Earth&#39;s surface. Fracturing involves many variables, including: viscosity of the fracturing fluid, rate of leak-off of fracturing fluid into the reservoir, proppant carrying capacity of the fluid, viscosity of the fluid as a function of temperature, time history of fluid volumes (i.e., the amount of fluid pumped over a given period of time), time history of proppant volumes, fluid physical constants, proppant properties, and the geological properties of various zones in the reservoir. 
     A Hydraulic Fracturing Simulator software is capable of modeling the ‘fracturing event’. In fact, the Hydraulic Fracturing Simulator software will design and monitor and evaluate the ‘petroleum reservoir fracturing’ associated with the ‘fracturing event’ from a time extending before, during, and after the ‘fracturing event’. However, when the Hydraulic Fracturing Simulator software designs and monitors and evaluates the ‘petroleum reservoir fracturing’ associated with the ‘fracturing event’, the Hydraulic Fracturing Simulator software should function to model a ‘multilayered reservoir’ when modeling the ‘fracturing event’. In addition, the Hydraulic Fracturing Simulator software should also function to model ‘partially active elements’ (partially enclosed by the fracture footprint, as opposed to ‘fully active elements’ fully enclosed by the fracture footprint), among a plurality of elements of a mesh overlaying a fracture footprint, when the Hydraulic Fracturing Simulator software models the ‘fracturing event’. 
     Therefore, a Hydraulic Fracturing Simulator Software is needed that is capable of modeling or simulating a ‘multilayered reservoir’ during a time when the Hydraulic Fracturing Simulator Software is designing and monitoring and evaluating ‘petroleum reservoir fracturing’ associated with the ‘fracturing event’. 
     In addition, a Hydraulic Fracturing Simulator Software is needed that is capable of modeling or simulating ‘partially active elements’, among a plurality of elements of a mesh overlaying a fracture footprint, during a time when the Hydraulic Fracturing Simulator Software is designing and monitoring and evaluating ‘petroleum reservoir fracturing’ associated with the ‘fracturing event’. 
     SUMMARY 
     One aspect of the present invention involves a program storage device readable by a machine tangibly embodying a program of instructions executable by the machine to perform method steps of simulating a hydraulic fracture in an Earth formation where the formation includes a multilayered reservoir, a mesh overlaying the fracture thereby defining a plurality of elements, the method steps comprising: calculating and determining an influence coefficient matrix, the matrix having a plurality of numbers, the plurality of numbers of the influence coefficient matrix being calculated and determined such that the method steps for simulating the hydraulic fracture will take into account an existence of the multilayered reservoir. 
     Another aspect of the present invention involves a method of simulating a hydraulic fracture in an Earth formation where the formation includes a multilayered reservoir, a mesh overlaying the fracture thereby defining a plurality of elements, comprising the step of: calculating and determining an influence coefficient matrix, the matrix having a plurality of numbers, the plurality of numbers of the influence coefficient matrix being calculated and determined such that the method steps for simulating the hydraulic fracture will take into account an existence of the multilayered reservoir. 
     Another aspect of the present invention involves a program storage device readable by a machine tangibly embodying a program of instructions executable by the machine to perform method steps of simulating a hydraulic fracture in an Earth formation where a mesh overlays the fracture, the mesh and the fracture collectively defining one or more partially active elements, the method steps comprising: calculating and determining an influence coefficient matrix, the matrix having a plurality of numbers, the plurality of numbers of the influence coefficient matrix being calculated and determined such that the method steps for simulating the hydraulic fracture will take into account an existence of the one or more partially active elements. 
     Another aspect of the present invention involves a method of simulating a hydraulic fracture in an Earth formation where a mesh overlays the fracture, the mesh and the fracture collectively defining one or more partially active elements, comprising the step of: calculating and determining an influence coefficient matrix, the matrix having a plurality of numbers, the plurality of numbers of the influence coefficient matrix being calculated and determined such that the method steps for simulating the hydraulic fracture will take into account an existence of the one or more partially active elements. 
     A further aspect of the present invention involves a system adapted for simulating a hydraulic fracture in an Earth formation where the formation includes a multilayered reservoir, a mesh overlaying the fracture thereby defining a plurality of elements, comprising: apparatus adapted for calculating and determining an influence coefficient matrix, the matrix having a plurality of numbers, the plurality of numbers of the influence coefficient matrix being calculated and determined such that the system adapted for simulating the hydraulic fracture in the formation will take into account an existence of the multilayered reservoir. 
     A further aspect of the present invention involves a system adapted for simulating a hydraulic fracture in an Earth formation where a mesh overlays the fracture, the mesh and the fracture collectively defining one or more partially active elements, comprising: apparatus adapted for calculating and determining an influence coefficient matrix, the matrix having a plurality of numbers, the plurality of numbers of the influence coefficient matrix being calculated and determined such that the system adapted for simulating the hydraulic fracture will take into account an existence of the one or more partially active elements. 
     Further scope of applicability will become apparent from the detailed description presented hereinafter. It should be understood, however, that the detailed description and the specific examples set forth below are given by way of illustration only, since various changes and modifications within the spirit and scope of the ‘Hydraulic Fracturing Simulator Software’, as described and claimed in this specification, will become obvious to one skilled in the art from a reading of the following detailed description. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A full understanding will be obtained from the detailed description presented hereinbelow, and the accompanying drawings, which are given by way of illustration only and are not intended to be limitative to any extent, and wherein: 
         FIGS. 1 through 3  illustrate a typical Hydraulic Fracturing (HF) job in a wellbore; 
         FIGS. 4 through 6  illustrate the fracture footprint created in the formation penetrated by the wellbore when the HF job is pumped; 
         FIGS. 7 through 9  illustrate how a mesh comprised of a plurality of grid cells will overlay on top of the fracture footprint of  FIGS. 4 through 6 , each grid cell of the mesh having a width and pressure, some of the grids cells called ‘tip elements’ being intersected by a perimeter of the fracture footprint, the tip elements having a width and a pressure (w, p), a portion of each tip element having fracturing fluid disposed therein; 
         FIG. 10  illustrates a mesh overlayed on top of a fracture footprint, one or more ‘fully active elements’ being enclosed by the fracture footprint, one or more ‘partially active elements’ being partially enclosed by the fracture footprint, the Hydraulic Fracturing Simulator software of  FIG. 16  modeling the ‘partially active elements’; 
         FIGS. 11 and 12  illustrate, in greater detail, the ‘partially active elements’ of  FIG. 10 , the Hydraulic Fracturing Simulator software of  FIG. 16  modeling the ‘partially active elements’; 
         FIGS. 13 and 14  illustrate a ‘multi-layered reservoir’, the Hydraulic Fracturing Simulator software of  FIG. 16  modeling the ‘multi-layered reservoir’; 
         FIG. 15  illustrates an apparatus used in connection with a Hydraulic Fracturing (HF) job adapted for fracturing a formation penetrated by a wellbore, the apparatus including a computer system for storing a software called a ‘Hydraulic Fracturing Simulator software’; 
         FIG. 16  illustrates the computer system of  FIG. 15  which stores the software called a ‘Hydraulic Fracturing Simulator software’ adapted for modeling a ‘multilayered reservoir’ and for modeling ‘partially active elements’ of a mesh overlaying a fracture footprint during a ‘petroleum reservoir fracturing’ event; 
         FIG. 17  illustrates in greater detail the ‘Other Data’ of  FIG. 16 ; 
         FIG. 18  illustrates the ‘Other Software Instructions’ of  FIG. 16 ; 
         FIG. 19  illustrates a construction of the ‘Hydraulic Fracturing Simulator software’ of  FIG. 16 ; 
         FIG. 20  illustrates the function associated with the ‘Set Up Influence Coefficient Matrix [C]’ step associated with the construction of the ‘Hydraulic Fracturing Simulator software’ which is illustrated in  FIG. 19 ; 
         FIGS. 21 and 22  illustrate a more detailed construction of the ‘Hydraulic Fracturing Simulator software’ which is illustrated in  FIG. 19 ; 
         FIGS. 23 and 24  illustrate a construction of the ‘Set Up Influence Coefficient Matrix [C]’ step  102  of  FIGS. 19 and 21 ; and 
         FIGS. 25-28  are used during a discussion of a detailed construction of the ‘Set Up Influence Coefficient Matrix [C]’ step  102  in  FIGS. 19 and 21 . 
     
    
    
     DESCRIPTION 
     This specification discloses a Hydraulic Fracturing Simulator software adapted to be stored in a memory of a program storage device of a computer system for modeling and simulating a multilayered reservoir and for modeling and simulating partially active elements of a mesh overlaying a fracture footprint when the Hydraulic Fracturing Simulator software is designing and monitoring and evaluating petroleum reservoir fracturing. The Hydraulic Fracturing Simulator software includes a first step further including the step of Setting Up an Influence Coefficient Matrix and a second step further including two iteration loops whereby, in a first iteration loop at a first time step, a second iteration loop will continuously calculate fracture width for each element of the mesh given a previously determined fluid pressure and will continuously calculate fluid pressure for each element of the mesh given a previously determined fracture width until convergence of the solution of width and pressure is reached at which time the fracture footprint is updated to a first value and ‘output data’ is generated, then, in the first iteration loop at a second time step, the second iteration loop calculates fracture width for each element of the mesh given a previously determined fluid pressure, and fluid pressure for each element of the mesh is calculated given a previously determined fracture width at which time the fracture footprint is updated to a second value and ‘output data’ is generated, and the process repeats, at which point, ‘additional output data’ is generated. The ‘output data’ includes the previously determined first value of the fracture footprint and the previously determined second value of the fracture footprint. The ‘output data’, representative of the updated fracture footprint at each of the incremented time steps, is recorded or displayed on a recorder or display device. In the first step including the step of ‘Setting Up an Influence Coefficient Matrix’, the Influence Coefficient Matrix is calculated and determined in a special way such that, when the Influence Coefficient Matrix is determined, and during any ‘fracturing event’, the Hydraulic Fracturing Simulator software will model and simulate the following: (1) a multilayered reservoir of the type illustrated in  FIGS. 13 and 14 , and (2) partially active elements of a mesh which overlays a fracture footprint of the type illustrated in  FIGS. 10 ,  11 , and  12 . 
     Referring to  FIG. 1 , a perforating gun  10  is disposed in a wellbore  12  and a packer  14  isolates a plurality of shaped charges  16  of the perforating gun  10  downhole in relation to the environment uphole. The shaped charges  16  detonate and a corresponding plurality of perforations  18  are produced in a formation  20  penetrated by the wellbore  12 . 
     Referring to  FIG. 2 , when the formation  20  is perforated, a fracturing fluid  22  is pumped downhole into the perforations  18  in accordance with a particular pumping schedule  24 . In response thereto, the formation  20  surrounding the perforations  18  is fractured. 
     Referring to  FIG. 3 , when the formation  20  surrounding the perforations  18  is fractured, oil or other hydrocarbon deposits  26  begin to flow from the fractures, into the perforations  18 , into the wellbore  12 , and uphole to the surface. The oil or other hydrocarbon deposits flow at a certain ‘production rate’  28  (e.g., in m 3 /day). 
     Referring to  FIG. 4 , when the wellbore  12  of  FIG. 2  is fractured, pump trucks  30  situated at the surface of the wellbore will pump fracturing fluid down a tubing and into the perforations  18  in the formation  20  penetrated by the wellbore, as shown in  FIG. 2 . The formation  20  includes different layers, such as the different layers  42 , one such layer being a perforated interval  40 . In response thereto, at time t 1 , a fracture footprint  32  will be formed in the perforated interval  40  (and possibly in additional adjacent intervals  42 ) of a formation  20  penetrated by the wellbore  12 . At time t 2 , the fracture footprint  32  will change to a new footprint  34  in the perforated interval  40  (and possibly in additional intervals  42 ) of a formation  20  penetrated by the wellbore  12 . At time t 3 , the fracture footprint  34  will change to a new footprint  36  in the perforated interval  40  (and possibly in additional intervals  42 ) of a formation  20  penetrated by the wellbore  12 . At time t 4 , the fracture footprint  36  will change to a new footprint  38  in the perforated interval  40  (and possibly in additional intervals  42 ) of a formation  20  penetrated by the wellbore  12 . 
     Referring to  FIGS. 5 and 6 , referring initially to  FIG. 5 , a schematic three dimensional view of a fracture footprint, such as the fracture footprints  32 - 38  of  FIG. 4 , is illustrated. In  FIG. 5 , each fracture footprint  32 - 38  has a length ‘L’ and a height ‘H’ and a width ‘w’. In  FIG. 6 , the wellbore  12  is illustrated again, and a plurality of perforations  18  are shown in the formation  20  penetrated by the wellbore  12 , as illustrated in  FIGS. 1-3 . As noted in  FIG. 4 , the formation  20  includes a plurality of different layers  42 . In  FIG. 6 , when the pump trucks  30  of  FIG. 4  pump the fracturing fluid into the perforations  18 , a ‘fracture footprint’  46  is created in the formation  20  which is identical to the fracture footprints  32 ,  34 ,  36 , and  38  shown in  FIG. 4  that are created, respectively, over the different periods of time t 1 , t 2 , t 3 , and t 4 . Note that the ‘fracture footprint’  46  in  FIG. 6  has a cross section  44 , the cross section  44  having a fracture width ‘w’ similar to the width ‘w’ of the fracture footprint  32 - 38  shown in  FIG. 5 . 
     Referring to  FIG. 7 , recalling the fracture footprint  46  shown in  FIG. 6 , in  FIG. 7 , a mesh  48  comprised of a plurality of grid elements  48   a  or grid cells  48   a  is illustrated. The fracture footprint  46  is assumed (by the model of this specification) to lie in the 2D plane, although, in principle and in reality, the fracture footprint  46  does lie in the 3D plane. In  FIG. 7 , the mesh  48  is overlayed over the top of the fracture footprint  46  of  FIG. 6 . The mesh  48  includes a plurality of active elements or active grid cells  48   a   1  and a plurality of inactive elements or inactive grid cells  48   a   2 . The active grid cells  48   a   1  will overlay the fracture footprint  46 , whereas the inactive grid cells  48   a   2  will not overlay the fracture footprint  46 . Each of the active grid elements or grid cells  48   a   1  of the mesh  48  has a width ‘w’ and a pressure ‘p’ assigned thereto, denoted by the symbol: (w, p). Therefore, each active grid cell  48   a   1  of the mesh  48  has a width/pressure value (w, p) assigned to that active grid cell. In  FIG. 6 , since the fracturing fluid propagating down the wellbore  12  enters the perforations  18  and creates the fracture footprint  46 , in  FIG. 7 , each of the active grid cells  48   a   1  in the mesh  48  has a fracturing fluid disposed therein. In  FIG. 7 , there are two types of active grid cells  48   a   1 : (1) an active grid cell  48   a   1  which is intersected by a perimeter  46   a  of the fracture footprint  46 , and (2) an active grid cell  48   a   1  which is not intersected by the perimeter  46   a  of the fracture footprint  46 . An active grid cell  48   a   1  that is intersected by the perimeter  46   a  of the fracture footprint  46  is known as a ‘tip element’. For example, in  FIG. 7 , ‘tip element’ 50 is an active grid cell  48   a   1  that is intersected by the perimeter  46   a  of the fracture footprint  46 . An active grid cell  48   a   1  which is not intersected by the perimeter  46   a  of the fracture footprint  46  has a volume which is wholly occupied by the fracturing fluid (i.e., 100% of the volume of the active grid cell is occupied by the fracturing fluid), where the fracturing ‘fluid’ may or may not include proppant. For example, in  FIG. 7 , active grid cell  52  is not intersected by the perimeter  46   a  of the fracture footprint  46  and its volume is wholly occupied by the fracturing fluid (100% of the volume of the active grid cell  52  is occupied by fracturing fluid). However, an active grid cell  48   a   1  that is intersected by the perimeter  46   a  (i.e., a ‘tip element’) has a volume that is occupied by ‘less than 100%’ of the fracturing fluid. For example, the active grid cell or ‘tip element’ 54 is intersected by the perimeter  46   a  of the fracture footprint  46 , however, only 45% of the volume of the active grid cell  54  is occupied by the fracturing fluid. In comparison, an inactive grid cell  48   a   2 , such as inactive grid cell  55 , has a volume which is wholly devoid of any fracturing fluid (0% of the volume of the inactive grid cell  55  is occupied by fracturing fluid). In prior pending application Ser. No. 10/831,799, filed Apr. 27, 2004, entitled “Method and Apparatus and Program Storage Device for Front Tracking in Hydraulic Fracturing Simulators” which is directed to the ‘VOF software’, the ‘VOF software’ of that prior pending application calculates, over a series of time steps, the ‘amount of fracturing fluid that is contained within each of the active grid cells  48   a   1  that are intersected by the perimeter  46   a  of the fracture footprint  46 ’. That is, the ‘VOF software’ of prior pending application Ser. No. 10/831,799 now U.S. Pat. No. 7,063,147 calculates, over the series of time steps, the ‘amount of fracturing fluid that is contained within each of the tip elements  50 ’. The ‘amount of fracturing fluid that is contained within each of the tip elements  50 ’ is calculated from the ‘fill fraction’, the ‘fill fraction’ being denoted by the letter ‘F’. For example, in  FIG. 7 , the ‘fill fraction’ F for the ‘tip element’  54  is 45%. Therefore, the ‘VOF software’ of prior pending application Ser. No. 10/831,799 calculates, over a series of time steps, the ‘fill fraction’ (F) for each of the ‘active grid cells  48   a   1  of the mesh  48  that are intersected by the perimeter  46   a  of the fracture footprint  46 ’. That is, the ‘VOF software’ of prior pending application Ser. No. 10/831,799 calculates, over the series of time steps, the ‘fill fraction’ (F) for each of the ‘tip elements’  50  of the mesh  48  of  FIG. 7 . As a result, by calculating the ‘fill fraction’ (F) for each of the ‘tip elements’  50  over a series of time steps, the amount or degree by which the perimeter  46   a  of the footprint  46  of the fracture is expanding (or contracting), as a result of the pumping of the fracturing fluid into the perforations  18  in the formation  20  by the pump trucks  30  of  FIG. 4 , can be determined. The ‘VOF software’ of prior pending application Ser. No. 10/831,799 is embodied in steps  138  and  96  of  FIG. 22 . However, in this specification, the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  of  FIG. 21  (and the ‘Set Up Influence Coefficient Matrix [C]’  102  of  FIG. 19 ) is discussed in detail. 
     Referring to  FIGS. 8 and 9 , two more examples of a mesh  48  similar to the mesh  48  of  FIG. 7  are illustrated. In  FIG. 8 , a mesh  48  is illustrated as overlaying the fracture footprint  46  having a perimeter  46   a . Fracturing fluid is disposed inside the perimeter  46   a , but the fracturing fluid is not disposed outside the perimeter  46   a . In  FIG. 8 , since the inactive grid cell  48   a   2  is disposed outside the perimeter  46   a , there is no fracturing fluid disposed inside the inactive grid cell  48   a   2  and, therefore, the ‘fill fraction’ F associated with the inactive grid cell  48   a   2  of  FIG. 8  is ‘zero’ (F=0). In  FIG. 8 , the active grid cells  48   a   1  are disposed wholly within the perimeter  46   a  (i.e., the perimeter  46   a  does not intersect the active grid cells  48   a   1 ); therefore, the entire volume (i.e., 100%) of the active grid cells  48   a   1  is occupied by fracturing fluid and, as a result, the ‘fill fraction’ F associated with the active grid cells  48   a   1  in  FIG. 8  is ‘1’ (F=1). However, in  FIG. 8 , let us analyze the active grid cell  56 . The active grid cell  56  is intersected by the perimeter  46   a  and, as a result, 80% of the volume of the active grid cell  56  is occupied by the fracturing fluid; therefore, the ‘fill fraction’ F for the active grid cell  56  is 0.8 (F=0.8). In  FIG. 9 , the ‘VOF software’ of prior pending application Ser. No. 10/831,799 also calculates the volume of an active grid cell occupied by a first type of fluid and the volume of that same active grid cell occupied by a second type of fluid. For example, active grid cell  58  includes a first volume of 80% occupied by a first type of fluid, and a second volume of 20% occupied by a second type of fluid. The ‘VOF software’ of prior pending application Ser. No. 10/831,799 calculates, over a series of time steps, the ‘fill fraction’ (F) for each of the ‘active grid cells  48   a   1  that are intersected by the perimeter  46   a  of the fracture footprint  46 ’ in the mesh  48 ; that is, the ‘VOF software’ of prior pending application Ser. No. 10/831,799 will calculate, over the series of time steps, the ‘fill fraction’ (F) for each of the ‘tip elements’ in the mesh  48  shown in  FIGS. 7 ,  8 , and  9 . As a result, the amount or degree by which the perimeter  46   a  of the footprint  46  is expanding (or contracting), in response to the pumping of fracturing fluid into the perforations  18 , can be determined. 
     Referring to  FIG. 10 , ‘fully active elements’ and ‘inactive elements’ and ‘partially active elements’ are illustrated. In  FIG. 10 , a wellbore  12  is illustrated, and a fracture  17  grows in a direction that is directed away from the wellbore  12 . A mesh  19  overlays the fracture  17 . In the mesh  19 , an ‘inactive element’  21  is illustrated, and a ‘fully active element’  23  is also illustrated. Note that  FIG. 10  relates to ‘fracture growth’ of the fracture  17  where the ‘growth’ of the fracture is in a direction that is either directed away from the wellbore  12  (i.e., expansion) or is directed toward the wellbore  12  (contraction). That is, the fracture  17  represents a moving boundary that is growing in a direction which is either directed away from the wellbore  12  (expansion) or is directed toward the wellbore  12  (contraction). In  FIG. 10 , ‘partially active elements’  25  and  27  and  29  and  43  are illustrated. In the ‘partially active element’  25  and  27 , only a portion  25   a  and  27   a  of the ‘partially active element’  25  and  27  is disposed inside of the boundary  17  of the fracture  17 . Since only portions  25   a  and  27   a  of ‘partially active elements’  25  and  27  are disposed inside the boundary  17  of fracture  17 , the elements  25  and  27  of mesh  19  of  FIG. 10  are each known as a ‘partially active element’. 
       FIG. 11  illustrates a close up view of the ‘partially active element’  29  of  FIG. 10 . The ‘partially active element’  29  has a fracture leading edge  31  with crossing points  33  and  35 , respectively. Straight line  37  is erected between the crossing points, and it forms the boundary for the active portion  39  of the element  29  and the inactive portion  41  of the element  29 . 
       FIG. 12  illustrates the same features for the ‘partially active element’  43  of  FIG. 10  as did  FIG. 11  with respect to the ‘partially active element’  29  of  FIG. 10 . In  FIG. 12 , a ‘partially active element’  43  has a fracture leading edge  31  with crossing points  33  and  35 , respectively. Straight line  37  is erected between the crossing points  33 ,  35 , and it forms the boundary for the active portion  39  of the element  43  and the inactive portion  41  of the element  43 . 
     The concept of ‘partially active elements’, such as the ‘partially active elements’  25 ,  27 ,  29 , and  43  shown in  FIGS. 10 ,  11 , and  12 , is set forth in the following two publications, each of which is incorporated herein by reference: (1) Ryder, J. A. and Napier, J. A. L., 1985,  Error Analysis and Design of a Large Scale Tabular Mining Stress Analyzer , Proceedings of the Fifth International Conference on Numerical Methods in Geomechanics, Nagoya, Japan, [Balkema] 1549-1555, the disclosure of which is incorporated by reference into the specification of this application; and (2) Ryder, J. A., Eds.: E. G., Beer, J. R. Booker, and J. P. Carter,  Optimal Iteration Schemes Suitable for General Non - linear Boundary Element Modeling Applications : Proceedings of the 7th International Conference on Computer Methods and Advances in Geomechanics, Cairns, Australia, [Balkema], 1991, the disclosure of which is incorporated by reference into the specification of this application. 
     Referring to  FIG. 13 , a reservoir or Earth formation  20  is shown. In  FIG. 13 , pumping truck  47  provides fluid at high pressures and flow rates to wellhead  49 , which is operably connected to the wellbore  12  at or near the ground surface  53 .  FIG. 13  illustrates the fracture boundary  55  at a particular time. Two other fracture fluid boundaries  57  and  59  also are indicated in  FIG. 13 . In  FIG. 13 , the reservoir  20  represents a ‘multi-layered reservoir’  20  because the reservoir  20  in  FIG. 13  includes the following plurality of layers  61 - 71  of Earth formation where the layers  61 - 71  represent various zones or laminations of underground geological formation: (1) a first layer  61  of ‘shale’, (2) a second layer  63  of ‘sandstone’, (3) a third layer  65  representing an ‘oil/gas pay zone’, (4) a fourth layer  67  of ‘shale’, (5) a fifth layer  69  representing an ‘oil/gas pay zone, and (6) a sixth layer  71  being a ‘water-bearing zone’. The fluid boundaries in  FIG. 13  reveal separate types or compositions of pumped fluid. In  FIG. 13 , the fracture preferably is stopped prior to the water bearing zone  71  seen at the lower portion of  FIG. 13 . 
     Referring to  FIG. 14 , a ‘multilayered’, hydraulically fractured reservoir  20  or Earth formation  20  is penetrated by a wellbore  12 . The wellbore  12  penetrates the Earth formation  20  and the hydraulic fracture  59  of  FIG. 13  is shown between layers  20   f  and  20   h . The fracture  59  has a fracture width  59   a  which was created in response to the pressurization of the formation  20  by a fracturing fluid. In  FIG. 14 , the ‘multilayered’ reservoir/Earth formation  20  includes a plurality of Earth formation layers  20   a ,  20   b ,  20   c ,  20   d ,  20   e ,  20   f ,  20   g ,  20   h ,  20   i , and  20   j . Each of the layers  20   a - 20   j  can be characterized by a Young&#39;s Modulus (E) and a Poisson&#39;s Ratio (v). The Young&#39;s Modulus (E) and the Poisson&#39;s Ratio (v) describe the elastic properties of each of the layers  20   a - 20   j  in  FIG. 14 . For example, the elastic properties of layer  20   a  can be characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E a , v a ), the elastic properties of layer  20   b  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E b , v b ), the elastic properties of layer  20   c  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E c , v c ), the elastic properties of layer  20   d  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E d , v d ), the elastic properties of layer  20   e  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E e , v e ), the elastic properties of layer  20   f  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E f , v f ), the elastic properties of layer  20   g  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E g , v g ), the elastic properties of layer  20   h  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E h , v h ), the elastic properties of layer  20   i  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E i , v i ), and the elastic properties of layer  20   j  being characterized by Young&#39;s Modulus and Poisson&#39;s Ratio (E j , v j ). 
     Referring to  FIG. 15 , the pump trucks  30  of  FIG. 4  will pump a fracturing fluid  62  (usually including proppant) down the wellbore  12  of  FIG. 4  in accordance with a pumping schedule  60  (an example used in connection with this discussion). The fracturing fluid  62  will enter the perforations  18 , and, responsive thereto, create a ‘fracture footprint’  46 , similar to the fracture footprint  46  shown in  FIG. 6 . Micro-seismic data sensor(s)  64  and tiltmeter data or other sensor(s)  66  will monitor the approximate geometry of the fracture footprint  46  and, responsive thereto, the sensor(s)  64  and  66  will generate output signals, the micro-seismic data sensor(s)  64  generating a micro-seismic data output signal  64   a , the tiltmeter data sensor(s)  66  generating a tiltmeter data output signal  66   a , and the pumping schedule  60  generating a pumping schedule output signal  60   a  representative of the pumping schedule  60 . The pumping schedule output signal  60   a , the tiltmeter data output signal  66   a , and the micro-seismic data output signal  64   a  are time line merged via a ‘time line merging’ step  68 . In this ‘time line merging’ step  68 , the pumping schedule output signal  60   a , the tiltmeter data output signal  66   a , and the micro-seismic data output signal  64   a  are ‘time synchronized’ in a particular manner such that the tiltmeter data output signal(s)  66   a  and the micro-seismic data output signal(s)  64   a  are synchronized with the times present in the pumping schedule  60 . When the ‘time line merging’ step  68  is complete, a ‘time line merged pumping schedule and tiltmeter data and micro-seismic data’ output signal  70  is generated which is provided as ‘input data’ to a ‘computer system’  72  optionally disposed within a truck  74  situated at the site of the wellbore  12 , such as a monitoring truck  74  or a ‘FracCAT vehicle’  74  (which is a vehicle with software that monitors and measures the fracture and controls the fracture treatment). 
     Referring to  FIG. 16 , the ‘computer system’  72  which is optionally disposed within the truck  74  of  FIG. 15 , such as the ‘FracCAT vehicle’  74 , is illustrated. In  FIG. 16 , recall that the ‘time line merged pumping schedule and tiltmeter data and micro-seismic data’ output signal  70  of  FIG. 15  is provided as ‘input data’ to the computer system  72  disposed within the truck  74 , the output signal  70  being comprised of a time line merged pumping schedule and tiltmeter data and micro-seismic data plus other data including downhole temperature and bottom hole pressure. The computer system  72  of  FIG. 16  includes a processor  72   a  operatively connected to a system bus, a memory or other program storage device  72   b  operatively connected to the system bus, and a recorder or display device  72   c  operatively connected to the system bus. The memory or other program storage device  72   b  stores the following software ( 76 ,  78 , and  80 ): a Planar 3D User Interface software  76 , a Planar 3D ‘engine’ or software  78 , and a Hydraulic Fracturing Simulator Software  80 . The software  76 ,  78 , and  80 , which is stored in the memory  72   b  of  FIG. 16 , can be initially stored on a CD-ROM, where that CD-ROM is also a ‘program storage device’. That CD-ROM can be inserted into the computer system  72 , and, then, the software  76 ,  78 , and  80 , which includes the Hydraulic Fracturing Simulator Software  80 , can be loaded from that CD-ROM and into the memory/program storage device  72   b  of the computer system  72  of  FIG. 16 . The Hydraulic Fracture Simulator Software  80  will be described in detail in the following paragraphs. The computer system  72  of  FIG. 16  receives Input Data  82 , including: (1) the time line merged pumping schedule, tiltmeter data, and micro-seismic data  84  (which corresponds to the ‘time line merged pumping schedule and tiltmeter data and micro-seismic data’ output signal  70  of  FIG. 15 ), and (2) Other Data  86 . The computer system  72  also receives Other Software Instructions  88 . The processor  72   a  will execute the Hydraulic Fracturing Simulator Software  80  (including the Planar 3D User Interface software  76  and the Planar 3D ‘engine’  78 ), while simultaneously using the Input Data  82  and Other Software Instructions  88 ; and, responsive thereto, the recorder or display device  72   c  will generate a set of ‘Output Data’  72   c   1  which is adapted to be recorded by or displayed on the recorder or display device  72   c . The computer system  72  may be a personal computer (PC), a workstation, or a mainframe. Examples of possible workstations include a Silicon Graphics Indigo 2 workstation or a Sun SPARC workstation or a Sun ULTRA workstation or a Sun BLADE workstation. The memory or program storage device  72   b  is a computer readable medium or a program storage device which is readable by a machine, such as the processor  72   a . The processor  72   a  may be, for example, a microprocessor, microcontroller, or a mainframe or workstation processor. The memory or program storage device  72   b , which stores the Hydraulic Fracturing Simulator Software  80 , may be, for example, a hard disk, ROM, CD-ROM, DRAM, or other RAM, flash memory, magnetic storage, optical storage, registers, or other volatile and/or non-volatile memory. 
     Referring to  FIGS. 17 and 18 , the Other Data  86  and the Other Software Instructions  88  of  FIG. 16  are illustrated. In  FIG. 17 , the Other Data  86  will include: layer confining stresses and properties, perforating interval and depth, wellbore data, fluid and proppant properties, time history of fluid volumes to be pumped, time history of proppant volumes to be pumped, and logs identifying the identity, properties, and location of geological zones. In  FIG. 18 , the Other Software Instructions  88  include instructions to calculate values representing physical dimensions of the fracture and pressures inside the fracture. 
     Referring to  FIG. 19 , a construction of the Hydraulic Fracturing Simulator Software  80  disposed within the Planar 3D software  78  and the Planar 3D User Interface  76  is illustrated. In  FIG. 19 , the Input Data  82  is provided to the Planar 3D User Interface  76 , to the Planar 3D software  78 , and to the Hydraulic Fracturing Simulator Software  80 . The Hydraulic Fracturing Simulator Software  80  includes: an initial ‘time stepping’ step  90 , a second step  92  which calculates fracture width (w) given the fluid pressure (p); a third step  94  which calculates fluid pressure (p) given the fracture width (w); and a fourth step  96  adapted to update the fracture footprint. The fracture ‘footprint’ is shown in  FIGS. 4 and 13  at times t 1 , t 2 , t 3 , . . . , t n  (for example at times t 1 , t 2 , t 3 , and t 4  in  FIG. 4  and at times corresponding to the three boundaries  59 ,  57  and  55  in  FIG. 13 ). In operation, note the time stepping loop  98  in  FIG. 19  wherein the ‘time stepping’ step  90  will increment from a first time step (t 1 ), to a second time step (t 2 ), to a third time step (t 3 ), . . . , and to an nth time step (t n ). Operating within the first time step t 1 , the second step  92  and the third step  94  will iterate on each other (as indicated by arrow  100 ) until the fracture width (w) and the fluid pressure (p) are solved at every point (i.e., within each ‘element’ or ‘grid cell’) in the fracture geometry of  FIGS. 4 and 13 . When the iteration  100  of steps  92  and  94  has converged, the fracture footprint is updated in step  96 . In step  97 , the proppant concentration is calculated for each grid cell in the updated fracture footprint. The ‘output data’  72   c   1 , associated with the last iteration loop  100 , is generated and stored. 
     The time step  90  is then incremented to the second time step t 2 , and, responsive thereto, the second step  92  and the third step  94  will then iterate again on each other (as indicated by arrow  100 ) until the fracture width (w) and the fluid pressure (p) are solved at every grid cell in the fracture geometry of  FIGS. 4 and 13 . When the iteration  100  of steps  92  and  94  has converged, the fracture footprint is updated in step  96 . In step  97 , the proppant concentration is calculated for each grid cell in the updated fracture footprint. The ‘output data’  72   c   1 , associated with the last iteration loop  100 , is generated and stored. The time step  90  is then incremented once again to the third time step t 3 , the second step  92  and the third step  94  will then iterate again on each other (as indicated by arrow  100 ) until the fracture width (w) and the fluid pressure (p) are solved at every point (i.e., within each ‘element’ or ‘grid cell’) in the fracture geometry of  FIGS. 4 and 13 . When the iteration  100  of steps  92  and  94  is complete, the fracture footprint is updated in step  96 . In step  97 , the proppant concentration is calculated for each grid cell in the updated fracture footprint. The ‘output data’  72   c   1 , associated with the last iteration loop  100 , is generated and stored. The time step  90  is then incremented once again, and the above process repeats until it has reached the end of the pumping schedule. ‘Convergence’ takes place when the ‘solution does not change from one iteration to the next’. The inner iteration loop  100  is solving for two things: (1) fracture width (w) using the ‘elasticity equation’, and (2) fluid pressure (p) using the ‘fluid flow equation’. Thus, the ‘solution does not change from one iteration to the next’ when the change in the ‘elasticity equation’ solution and when the change in the ‘fluid flow equation’ solution is below a ‘tolerance’. When the change in the ‘elasticity equation’ solution and the change in the ‘fluid flow equation’ solution is below the ‘tolerance’, we know that the inner iteration loop  100  has converged. Steps  92  and  94  can be solved in various ways, such as by iteration of two equations as shown here, or by direct substitution of the one equation into the other equation, or vice-versa. 
     The Output Data  72   c   1  is generated at the end of each time step. However, ‘additional output data’  99  is generated when the time stepping in the outer iteration loop  98  is complete, the ‘additional output data’  99  being used in subsequent calculations, such as in the generation of other graphical plots. 
     In  FIG. 19 , however, before the second step  92 , a first step  102  is practiced, the first step  102  being called ‘Set Up Influence Coefficient Matrix [C]’  102 . The ‘Set Up Influence Coefficient Matrix [C]’ step  102  of  FIG. 19  will be discussed below with reference to  FIG. 20  of the drawings. 
     Referring to  FIG. 20 , a ‘Multi-Layer Elasticity Equation’  104  is illustrated. Recall from  FIG. 19  that step  92  will calculate the fracture width (w) and step  94  will calculate the fluid pressure (p). In  FIG. 20 , the fracture width (w) of step  92  is actually calculated by using the ‘Elasticity Equation’  104 . In the ‘Elasticity Equation’  104 , the fracture width (w)  106  is calculated given the inverse of a ‘Matrix of Influence Coefficients [C]’  108  (also known as an ‘Influence Coefficient Matrix’) multiplied by (fluid pressure p  110  minus confining stress σ c    112 ), as follows:
 
{ w}=[C]   −1   {p−σ   c }, where  Elasticity Equation 104
     w is the fracture width  106 ,   [C] is the ‘matrix of influence coefficients’  108 ,   p is the fluid pressure  110 , and   σ c  is the confining stress  112 .   

     In  FIG. 20 , a generic form of the ‘Influence Coefficient Matrix [C]’  108  is shown, where the ‘Influence Coefficient Matrix [C]’  108  has ‘M’ rows and ‘M’ columns. In  FIG. 20 , the ‘Influence Coefficient Matrix [C]’  108  is fully populated with numbers  114  and is used to calculate fracture width (w) for step  92  in  FIG. 19 . 
     The ‘elasticity equation’  104  did not previously take into account the existence of a ‘multilayered reservoir’ of the type illustrated in  FIGS. 13 and 14 , and the ‘elasticity equation’  104  did not previously take into account the existence of ‘partially active elements’, such as the ‘partially active elements’  25 ,  27 ,  29 , and  43  of  FIG. 10 . However, in a real reservoir, a ‘multilayered reservoir’ does exist. Furthermore, when overlaying a mesh over a fracture footprint in the manner illustrated in  FIG. 10 , ‘partially active elements’ also do exist. 
     Therefore, the Hydraulic Fracturing Simulator Software  80  of  FIG. 16  disclosed in this specification will include the effect of, and take into account the existence of, ‘multilayered reservoirs’ and ‘partially active elements’. Consequently, in order to take into account the existence of ‘multilayered reservoirs’ and ‘partially active elements’, it is necessary to execute the ‘Set Up Influence Coefficient Matrix [C]’ step  102  of  FIG. 19 , associated with the Hydraulic Fracturing Simulator Software  80  of  FIGS. 16 and 19 . The ‘Set Up Influence Coefficient Matrix [C]’ of Step  102  of  FIG. 19  will calculate the ‘Influence Coefficient Matrix [C]’  108  in the ‘Elasticity Equation’  104  of  FIG. 20  in a special way in order to allow the Hydraulic Fracturing Simulator software  80  to include the effect of, or take into account the existence of, the ‘multilayered reservoirs’ and the ‘partially active elements’. When the ‘Influence Coefficient Matrix [C]’  108  of the ‘elasticity equation’  104  of  FIG. 20  is calculated (by the ‘Set Up Influence Coefficient Matrix [C]’ step  102  of  FIG. 19  associated with the Hydraulic Fracturing Simulator Software  80  of  FIGS. 16 and 19 ) in this special way, the numbers  114  in the ‘Influence Coefficient Matrix [C]’  108  will be changed accordingly; and, when the numbers  114  of the ‘Influence Coefficient Matrix [C]’  108  are changed accordingly, the Hydraulic Fracturing Simulator Software  80  will then include the effect of, or take into account the existence of, the ‘multilayered reservoirs’ and the ‘partially active elements’. As disclosed in this specification, there is a systematic way for determining and changing the numbers  114  in the ‘Influence Coefficient Matrix’  108  of  FIG. 20  in order to include or take into account the existence of the ‘multilayered reservoirs’ and the ‘partially active elements’. Consequently, the following pages of this specification will present a ‘method’ (along with an accompanying ‘system’ and ‘program storage device’) for determining how the numbers  114  in the ‘Influence Coefficient Matrix [C]’  108  will be changed (by the ‘Set Up Influence Coefficient Matrix’ step  102  of  FIG. 19 ) for the ultimate purpose of allowing the Hydraulic Fracturing Simulator Software  80  to include the effect of, or take into account the existence of, the ‘multilayered reservoirs’ of  FIGS. 13-14  (such as the ‘multilayered reservoir’  20  having multiple layers  61 - 71  of  FIG. 13 ) and the ‘partially active elements’ of  FIG. 10  (such as the ‘partially active elements  25 ,  27 ,  29 , and  43  of  FIG. 10 ). 
     Referring to  FIGS. 21 and 22 , a more detailed construction of the Hydraulic Fracturing Simulator Software  80  of  FIG. 19 , which is disposed within the Planar 3D software  78  and the Planar 3D User Interface  76 , is illustrated. 
     In  FIG. 21 , the input data  82  is provided from the human interface, such as the injection rate and the pumping schedule, which includes the injection rate as a function of time, proppant concentration, fluid viscosity, the geology or the properties of the elastic layers of the reservoir including the elastic constants comprising the Young&#39;s Modulus and the Poisson&#39;s Ratio (E, v) as previously described, and the leakoff behavior. In  FIG. 21 , in the ‘Generate Layer Interface Locations’ step  116 , the depths of each of the ‘interfaces’ of the ‘layers’ of  FIG. 13  are calculated, such as the end of the sandstone layer, the end of the gas layer, etc. In the ‘Assign Layer Properties (Young&#39;s Modulus, Poisson&#39;s Ratio, Toughness, Leakoff Coefficients, Stresses)’ step  118 , the Young&#39;s Modulus, Poisson&#39;s Ratio, Toughness, Leakoff Coefficients, and Stresses are assigned to each of the ‘layers’ (of  FIG. 13 ) the depths of which were calculated in step  116 . Therefore, a series of numbers (comprising the Young&#39;s Modulus, Poisson&#39;s Ratio, Toughness, Leakoff Coefficients, and Stresses) is assigned to each Earth formation ‘layer’ that is shown in  FIG. 13 . In the ‘Assign Maximum Expected Fracture Height and Extent of Fracture’ step  120  of  FIG. 21 , before any simulation is performed, the maximum possible ‘length’ to which the ‘fracture’ will propagate and the maximum possible ‘height’ to which the ‘fracture’ will propagate is assumed or introduced. Then a parent mesh is assigned to the ‘fracture surface’, where the parent mesh is divided into rectangular ‘elements’. See  FIGS. 7 and 10  for examples of the parent mesh. The parent mesh is broken down into rows and columns comprising ‘grids’ or ‘elements’ which are generally rectangular in shape. It is assumed that the ‘fracture’ will grow into the ‘elements’ of the parent mesh, but no further. In the ‘Generate Numerical Parent Mesh’ step  122 , the numerical parent mesh is generated; that is, in step  122 , the ‘coordinates’ of each ‘grid cell’ or ‘element’ of the parent mesh are generated. The ‘coordinates’ will define where each ‘grid cell’ or ‘element’ exists within an axis system, such as the (x, y) axis system. Therefore, in step  122  of  FIG. 21 , the ‘coordinates’ of each of the ‘grid cells’ (such as ‘grid cells’  23 ,  25 ,  29 , and  43  shown in  FIG. 10  or each of the ‘grid cells’  48   a   1  and  52  shown in  FIG. 7 ) are allocated. At this point, we know where our ‘layers’ exist, we know the numerical mesh which includes the rectangular mesh of elements, and we know where the fracture exists within the parent mesh including all their coordinates. In  FIG. 21 , the next step is the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  which is the ‘Set up Influence Coefficient Matrix [C]’ step  102  of  FIG. 19 . In step  102  of  FIGS. 19 and 21 , the Influence Coefficient Matrix [C] ( 108  of  FIG. 20 ) is generated. As previously mentioned, the Influence Coefficient Matrix [C] (as shown in  FIG. 20 ) comprises rows and columns of ‘numbers’  114 , the Influence Coefficient Matrix [C] being fully populated with such ‘numbers’  114 . Each of the ‘numbers’  114  of the Influence Coefficient Matrix [C] will indicate how one ‘element’ of the mesh relates to another ‘element’ of the mesh in an ‘elastic manner’. For example, if a fracture exists within one ‘element’ of the mesh, what kind of stress or strain exists within another ‘element’ of the mesh in response thereto (a phenomenon that is called an ‘elasticity behavior’). Each of the ‘numbers’  114  in the Influence Coefficient Matrix [C] describes this ‘elasticity behavior’. Therefore, the Influence Coefficient Matrix [C] ( 108  of  FIG. 20 ) is a matrix that indicates how one ‘element’ of the parent mesh (such as the mesh shown in  FIGS. 7 and 10 ) talks to another ‘element’ of the parent mesh in an ‘elastic manner’. Therefore, step  102  of  FIGS. 19 and 21  (i.e., the ‘Set Up Influence Coefficient Matrix’ step  102  of  FIG. 19  and the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  of  FIG. 21 ), which determines the Influence Coefficient Matrix  108  of  FIG. 20 , is responsible for allowing the Hydraulic Fracturing Simulator Software  80  of  FIG. 16  to simulate the ‘multilayered reservoir’ of  FIG. 13  and the ‘partially active elements’ of  FIGS. 10 ,  11 , and  12 . Step  124  of  FIG. 21  entitled the ‘Restart Option’ will allow the user to restart the simulator without having to recalculate the Influence Coefficient Matrix (it is stored on the hard drive). Such situations will occur if the reservoir properties are unchanged, but the user wants to change the injection schedule, for example. 
     In  FIG. 22 , the ‘Time=Time+A t’ step  90  is a ‘time stepping’ step. Note the loop  98  which represents a time stepping loop. The loop  98  controls how the fracture grows as a function of time during the duration of the ‘pumping schedule’, where the term ‘pumping schedule’ includes the ‘shut-in time’. In the ‘Assign Latest Fracture Extent (Active Elements)’ step  126 , at any particular time step, the fracture will have propagated to some extent in the parent mesh, which is called the ‘footprint’ of the fracture; step  126  will sweep through all the ‘grid cells’ in the entire parent mesh, and then step  126  will determine if a particular ‘element’ or ‘grid cell’, within the perimeter of the ‘fracture footprint’, is an ‘inactive element’, or an ‘active element’, or a ‘partially active element’. In step  128  entitled ‘Extract Elastic Influence Coefficient Matrix for Current Fracture Geometry’, the Influence Coefficient Matrix [C]  108 , as previously described with reference to step  102 , contains ‘all possible combinations within the parent mesh’; and, at any particular time step, we need a subset of the aforementioned ‘all possible combinations within the parent mesh’; thus, step  128  extracts, from the Influence Coefficient Matrix [C]  108 , a ‘submatrix’ which contains essential information regarding the current size of the ‘fracture footprint’. In step  130  entitled ‘Assign Special Characteristics to Coefficients for Tip Elements’, a ‘particular subset’ of the ‘grid cells’ or ‘elements’ of the parent mesh will cross a ‘perimeter of the fracture footprint’; in connection with only the ‘particular subset’ of the ‘grid cells’ which cross the ‘perimeter of the fracture footprint’, it is necessary to change some of the properties of the ‘submatrix’ (and recall that the ‘submatrix’ contains essential information regarding the current size of the ‘fracture footprint’); step  130  will make this change to the ‘submatrix’; consequently, step  130  in  FIG. 22  entitled ‘Assign Special Characteristics to Coefficients for Tip Elements’ will deal specifically with the ‘partially active elements’ of  FIGS. 10 ,  11 , and  12 ; that is, step  130  in  FIG. 22  will change the ‘partially active element submatrix’ (i.e., a submatrix which pertains specifically to the ‘partially active elements’) by ‘assigning special characteristics to the coefficients’ in the ‘partially active element submatrix’. In step  132  entitled ‘Assign Loading (Fluid Pressure) to Each Element in Current Mesh’, we now have a current time step and a current fracture footprint; in addition, each ‘active element’ has a ‘fluid pressure’; thus, step  132  will assign a ‘fluid pressure’ to each ‘element’ of the parent mesh. In step  92  entitled ‘Solve Elastic Equation System for Fracture Widths’, we use the ‘elasticity equation’  104  of  FIG. 20  (including their Fourier Transforms) to solve for the ‘fracture width’ in each ‘element’ or ‘grid cell’ of the mesh, at the current time step, given the ‘fluid pressure’ that was assigned to each ‘element’ of the mesh in step  132 . In step  134  entitled ‘Calculate Influence Matrix for Fluid Flow in Current Mesh Using Widths from Above’, now that we have calculated the ‘fracture width’ in each ‘element’ at the current time step from step  92 , we can do the same thing for the fluid flow equations in step  94 . Therefore, in step  94  entitled ‘Solve Fluid Flow Equations for Fluid Pressure in Current Mesh’, we can solve for the ‘fluid pressures’. Note the loop  100  in  FIG. 22  whereby, when the ‘fluid pressures’ are determined in step  94 , those ‘fluid pressures’ determined from step  94  are used to solve for ‘fracture widths’ in step  92 , and that loop  100  will continue to process as indicated until ‘global mass balance’ is achieved in step  136 . Thus, from step  136  to step  132 , an inner iteration will take place; in that inner iteration, we continue to iterate on fluid pressure and fracture width until convergence of the solution takes place at that time step; convergence takes place when the solution does not change from one iteration to the next. The inner iteration loop  100  is solving for two things: (1) fracture width using the ‘elasticity equation’, and (2) the fluid pressure using the ‘fluid flow equation’; when the change in each of these ‘equations’ is below a tolerance, we know that the inner iteration  100  has converged. Steps  138  and  96  involve updating for the next time step. Step  138  (‘Calculate Local Fracture Tip Velocity of Propagation’) and step  96  (‘Grow Fracture: Update New Layout’) are described in prior pending application Ser. No. 10/831,799, filed Apr. 27, 2004, directed to the ‘VOF Algorithm’, entitled “Method and Apparatus and Program Storage Device for Front Tracking in Hydraulic Fracturing Simulators”, the disclosure of which is incorporated by reference into the specification of this application. In step  97 , the proppant concentration is calculated for each grid cell in the updated fracture footprint. The ‘output data’  72   c   1 , associated with the last iteration loop  100 , is generated and stored. In step  140 , if the time is not greater than the maximum allowed, time step  90  is incremented once again, and the above process repeats until it has reached the end of the pumping schedule. In step  140 , if the ‘time’ is greater than the maximum allowed, ‘additional output data’  99  is generated. The ‘output data’  72   c   1  and the ‘additional output data’  99  is stored in a storage device, such as the memory or program storage device  72   b  of  FIG. 16 . In  FIG. 22 , the Output Data  72   c   1  includes, at each time step, the fluid pressure (p) and the fracture width (w) at each of the ‘elements’ or ‘grid cells’  23 ,  25 ,  29 , and  43  shown in  FIG. 10  or each of the ‘elements’ or ‘grid cells’  48   a   1  and  52  shown in  FIG. 7  (again, at each time step). Since the Output Data  72   c   1  includes fluid pressure (p) and fracture width (w) for each ‘grid cell’ of  FIGS. 7 and 10  at each time step, the Output Data  72   c   1  can also include a number of 2D or 3D plots representative of the fluid pressure (p) and the fracture width (w) at each of the ‘elements’ or ‘grid cells’  23 ,  25 ,  29 , and  43  shown in  FIG. 10  or each of the ‘elements’ or ‘grid cells’  48   a   1  and  52  shown in  FIG. 7  at each time step. Steps  92 ,  134 , and  94  can be solved in various ways, such as by iteration of the two equations shown here (i.e., the ‘elasticity equation’ and the ‘fluid flow equation’), or by direct substitution of one equation into the other equation, or vice-versa. 
     Recall that the ‘Set Up Influence Coefficient Matrix [C]’ step  102  in  FIG. 19  and the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  in  FIG. 21  will recalculate the Influence Coefficient Matrix  108  in  FIG. 20  in a ‘special way’ in order to allow the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16  and  19  to model, or take into account the existence of, ‘partially active elements’ (of the type illustrated in  FIGS. 10 ,  11 , and  12 ). The concept of ‘partially active elements’, such as the ‘partially active elements’  25 ,  27 ,  29 , and  43  shown in  FIGS. 10 ,  11 , and  12 , is set forth in the following two publications, each of which is incorporated herein by reference: (1) Ryder, J. A. and Napier, J. A. L. 1985,  Error Analysis and Design of a Large Scale Tabular Mining Stress Analyzer , Proceedings of the Fifth International Conference on Numerical Methods in Geomechanics, Nagoya, Japan, [Balkema] 1549-1555, the disclosure of which is incorporated by reference into the specification of this application; and (2) J. A. Ryder, Eds.: E. G. Beer, J. R. Booker, and J. P. Carter,  Optimal Iteration Schemes Suitable for General Non - linear Boundary Element Modeling Applications : Proceedings of the 7th International Conference on Computer Methods and Advances in Geomechanics, Cairns, Australia, [Balkema], 1991, the disclosure of which is incorporated by reference into the specification of this application. 
     Referring to  FIGS. 23 and 24 , a construction of the ‘Set Up Influence Coefficient Matrix [C]’ step  102  of  FIG. 19  and the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  of  FIG. 21 , which would allow the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16 and 19  to model or take into account the existence of a ‘multi-layer reservoir’ (of the type illustrated in  FIG. 13 ), is illustrated. In  FIGS. 23 and 24 , recall that the ‘Set Up Influence Coefficient Matrix [C]’ step  102  in  FIG. 19  and the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  in  FIG. 21  will also recalculate the Influence Coefficient Matrix  108  in  FIG. 20  in a ‘special way’ in order to allow the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16 and 19  to model or take into account the existence of a ‘multi-layer reservoir’ (of the type illustrated in  FIG. 13 ). 
     In  FIGS. 23 and 24 , a ‘construction of the Set Up Influence Coefficient Matrix step  102 ’ of  FIGS. 19 and 21 , for the purpose of allowing the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16 and 19  to model or take into account the existence of a ‘multi-layer reservoir’ (of the type illustrated in  FIG. 13 ), is illustrated. The ‘construction of the Set Up Influence Coefficient Matrix step  102 ’ shown in  FIGS. 23  and  24 , for the purpose of allowing the ‘Hydraulic Fracturing Simulator software’  80  to model or take into account the existence of a ‘multi-layer reservoir, includes a ‘plurality of calculation steps’  102 , the ‘plurality of calculation steps’  102  being practiced by the ‘Set Up Influence Coefficient Matrix’ step  102  in  FIG. 19  and the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  in  FIG. 21  when the Influence Coefficient Matrix  108  of  FIG. 20  is calculated. 
     In  FIG. 23 , the ‘construction of the Set Up Influence Coefficient Matrix step  102 ’ of  FIGS. 19 and 21 , for the purpose of allowing the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16 and 19  to model or take into account the existence of a ‘multi-layer reservoir’, includes ‘three basic steps’  102   a ,  102   b , and  102   c , as follows: (1) Find Spectral Coefficients in Each Layer, step  102   a , (2) Perform Exponential Approximation, step  102   b , and (3) Assemble an ‘Influence Coefficient Matrix’ (i.e., matrix  108  of  FIG. 20 ) Using Exponential Expansion Coefficients, step  102   c . In step  200 , the Influence Coefficient Matrix is sent to a storage device. 
     The above referenced ‘three basic steps’  102   a ,  102   b , and  102   c  in  FIG. 23 , adapted for constructing an Influence Coefficient Matrix  108  for the purpose of allowing the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16 and 19  to model or take into account the existence of a ‘multi-layer reservoir’, will be discussed in greater detail below with reference to  FIG. 24 . 
     In  FIG. 24 , a detailed construction of each of the ‘three basic steps’  102   a ,  102   b , and  102   c  of  FIG. 23  will be discussed. 
     In  FIG. 24 , the ‘plurality of calculation steps’  102 , which are practiced by the ‘Set Up Influence Coefficient Matrix’ step  102  in  FIG. 19  and by the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  in  FIG. 21  for the purpose of allowing the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16 and 19  to model or take into account the existence of a ‘multi-layer reservoir’, include the following steps: 
     Find Spectral Coefficients in Each Layer, step  102   a  of  FIG. 23 : 
     The ‘Find Spectral Coefficients in Each Layer’ step  102   a  of  FIG. 23  includes the following step (1), as follows:
         (1) In order to obtain the stresses and displacements due to a prescribed source [e.g., Displacement Discontinuity (DD) element], determine the spectral coefficients A j   l (k) by solving a system of algebraic equations (7) and (9) that express the continuity of tractions and displacements across layer interfaces; Equations (7) and (9) will be discussed in detail in the following paragraphs; however, equations (7) and (9) are duplicated below, as follows:       

                       u   ^     r   l     =       ∑   j     ⁢           ⁢       (       d   jr   l     +       f   jr   l     ⁢   y       )     ⁢     ⅇ       α   j   l     ⁢   ky       ⁢       A   j   I     ⁡     (   k   )                   (   7   )                   σ   ^     pq   l     =       ∑   j     ⁢           ⁢       (       s   jpq   l     +       t   jpq   l     ⁢   ky       )     ⁢     ⅇ       α   j   l     ⁢   ky       ⁢       A   j   I     ⁡     (   k   )                   (   9   )               
Perform Exponential Approximation Step  102   b  of  FIG. 23 :
 
     The ‘Perform Exponential Approximation’ step  102   b  of  FIG. 23  includes the following steps (2), (3), and (4), as follows:
         (2) Determine the (free space) spectral coefficients A j   l (∞) by solving the algebraic equations (7) and (9), set forth above and discussed below, for an infinite homogeneous medium corresponding to the high frequency components associated with the prescribed Displacement Discontinuity (DD); the explicit expressions are identified in  FIG. 24  by numeral  102   b   1 ; where λ and μ are Lame constants:       

     
       
         
           
             
               λ 
               = 
               
                 vE 
                 
                   
                     ( 
                     
                       1 
                       + 
                       v 
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       1 
                       - 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         v 
                       
                     
                     ) 
                   
                 
               
             
             , 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 μ 
               
               = 
               
                 
                   E 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         v 
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
         
         
           
             (3) Evaluate the low frequency spectral components which are defined by a further expression that is identified in  FIG. 24  by numeral  102   b   2 ; 
             (4) Approximate these low frequency components (A j   l (k)) LOW  by a series of exponential functions by solving for the unknown constants a jr   l  and b jr   l  in the expansion that is identified in  FIG. 24  by numeral  102   b   3 .
 
Assemble Influence Coefficient Matrix Using Exponential Expansion Coefficients, Step  102   c  in  FIG. 23 :
 
           
         
       
    
     The ‘Assemble Influence Coefficient Matrix Using Exponential Expansion Coefficients’ step  102   c  of  FIG. 23  includes the following steps (5), (6), and (7), as follows:
         (5) Substitute these expansions for the low frequency components into equations (7) and (9) to obtain the expressions for the ‘displacement and stress components’ that are identified in  FIG. 24  by numeral  102   c   1 ;   (6) Invert the displacements and stresses associated with the low frequency Fourier Transforms by evaluating integrals of the form that is identified in  FIG. 24  by numeral  102   c   2 , where ‘i’ is the imaginary number, and ‘e’ is the exponential operator; and   (7) Combine the low frequency displacement and stress components with the infinite space displacements and stresses in the manner which is identified in  FIG. 24  by numeral  102   c   3 .       

     Referring to  FIGS. 25 through 28 , a detailed construction of the ‘Set Up Influence Coefficient Matrix’ step  102  of  FIG. 19  and the ‘Generate Elastic Influence Coefficient Matrix for Parent Mesh’ step  102  of  FIG. 21 , for the purpose of allowing the ‘Hydraulic Fracturing Simulator software’  80  of  FIGS. 16 and 19  to model or take into account the existence of a ‘multi-layer reservoir’, is set forth in the following paragraphs with reference to  FIGS. 25 through 28 . Note that step  102  of  FIG. 19  is the same step as step  102  in  FIG. 21 . 
     The numerical algorithm employed in this invention comprises an efficient technique to determine the local width of a hydraulic fracture due to local pressure applied to the fracture faces by the injection of hydraulic fluid and proppant into the fracture. Further, a method to track the dimensions and width of said fracture as it grows as a function of time is shown. The hydraulic fracture(s) may span any number of layers in a laminated reservoir, with the restriction that all layers must be parallel to each other, as depicted for example in  FIG. 23 . Layers may be inclined at any angle to the horizontal. 
       FIG. 25  shows a section through multiple hydraulic fractures in a layered reservoir. The calculation of the fracture width due to the pressure front the injected fluids and proppant mixture is determined by taking into account, accurately and efficiently, the physical properties of each layer comprising the laminated reservoir. The technique used to calculate the relationship between the layered reservoir and the growing hydraulic fracture is based on a well-established numerical technique called the Displacement Discontinuity Boundary Element (hereafter “DD”) Method. The method is extended to enable efficient and accurate calculation of the physical effects of layering in the reservoir by the use of a Fourier Transform Method, whereby the relations between stress and strain in the layered reservoir are determined. The numerical method assumes that each hydraulic fracture is divided into a regular mesh of rectangular elements, as depicted in  FIG. 26 , wherein each numerical element contains its own unique properties. Such properties include applied fluid and proppant pressure, fluid and proppant propagation direction and velocity, local reservoir properties, stress-strain relations, and fracture width. 
       FIG. 26  shows a numerical mesh of elements subdividing the fracture surface for purposes of calculation. In cases where the numerical element coincides with the fracture edge or tip (see  FIG. 27 ), certain additional information is uniquely defined for such elements. For example, such information may include the local velocity of propagation of the fracture tip, the special relationship between the fluid in the fracture and the surrounding layered reservoir, and how the fluid and reservoir physically interact with each other. This interaction is accounted for by means of special properties assigned to the tip elements, comprising the interaction between a fluid-filled fracture and the host material it is fracturing. 
       FIG. 27  reveals a fracture outline on a numerical mesh. Each numerical element depicted in  FIG. 26  or  27  relates to every other element in the numerical mesh by means of special mathematical relationships. We refer to elements as: (1) sending or source elements, and (2) receiver elements. A source element sends a signal representing a mathematical relationship to a receiver element. The signal is the net pressure, which is equal to the fluid pressure (p)  110  minus the confining stress (σ c )  112  as indicated by the multi-layer elasticity equation  104  in  FIG. 20 , in that portion of the fracture. The receiver signal comprises the stress and strain experienced at the receiver location due to the applied pressure at the source element location. Many of these signals between source and receiver element are duplicated in the numerical mesh, and in these cases, special algorithms are designed to dramatically minimize the volume of storage required, so that only unique signals between different elements need to be stored. 
     The signals between each unique pair of receiver and source elements are stored in the computer memory or on a physical storage device in a matrix. The hydraulic fracture propagation numerical method is designed so that the fracture propagates in a finite number of time steps. At each time step, the signal matrix is invoked, extracting those signals which are active over the part of the numerical mesh that is covered by the current configuration of the hydraulic fracture. This matrix is then used to build a system of numerical equations that are solved for the fracture width at the current time—at each active element location. 
     During each fracture propagation step, another matrix of signals is constructed, the matrix comprising the physical behavior of the fluid in the hydraulic fracture, which relates the local fluid pressure to the local fracture width. This system of equations is also solved iteratively for local fluid pressures at each time step. 
     The combined system of equations must be coupled together in an efficient manner, so that they feed off each other until a balanced solution of fluid pressure and fracture width is obtained at each time step. This coupling between the two equation systems is accomplished by means of a special numerical algorithm that efficiently and accurately ensures that the correct solution is obtained. The entire system is designed to ensure that no fluid or proppant is unaccounted for in any time step. 
     The above process is repeated at each time step, thereby allowing the calculation of the way in which the fracture grows as a function of time. At each time step, the algorithm predicts which elements are active (i.e., filled with fluid and proppant), and the fracture width and fracture pressure on each active element. A complete description of the process of the propagation of a hydraulic fracture is thus obtained. 
     Solutions of the multi-layer equilibrium equations are provided. 
     In the following, we assume a three-dimensional body, so subscripts range from 1 to 3. The theory also applies to the two-dimensional cases (plane strain, plane stress, antiplane strain). The method provides an efficient way of determining the solution to the equilibrium equations:
 
σ ij,j   l   +b   i   l =0  (1)
 
for an, in general, transversely isotropic elastic layered medium, where superscript l indicates the layer number, with a stress-strain relationship given by:
 
σ ij   l =C ijkr   l ∈ kr   l   (2)
 
In the case of a transversely isotropic three-dimensional elastic medium, there are five independent material constants. The strain components in (2) are given by
 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     kr 
                     l 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           u 
                           
                             k 
                             , 
                             r 
                           
                           l 
                         
                         + 
                         
                           u 
                           
                             r 
                             , 
                             k 
                           
                           l 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     For a medium comprising multiple parallel layers each of which is homogeneous (see  FIG. 28 ), it is possible to obtain a solution to the governing equations (1)-(3) by means of the Fourier Transform. 
     Refer now to  FIG. 28  representing a schematic showing multiple parallel layers in a three dimensional case. 
     By substituting (3) and (2) into (1) and by taking the two-dimensional Fourier Transform with respect to x and z (where subscripts 1=x, 2=y, 3=z): 
     
       
         
           
             
               
                 
                   
                     
                       
                         u 
                         ^ 
                       
                       j 
                       l 
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         , 
                         n 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             ⅈ 
                             ⁡ 
                             
                               ( 
                               
                                 mx 
                                 + 
                                 nz 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           
                             u 
                             j 
                             l 
                           
                           ⁡ 
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           z 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     of the resulting equilibrium equations in terms of the displacements, we obtain a system of ordinary differential equations in the independent variable y. This system of ordinary differential equations determines the Fourier Transforms of displacement components u x   l , u y   l , and u z   l : 
     
       
         
           
             
               
                 
                   
                     
                       
                         L 
                         ⁡ 
                         
                           ( 
                           
                             C 
                             ijkr 
                             l 
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   u 
                                   ^ 
                                 
                                 x 
                               
                             
                           
                           
                             
                               
                                 
                                   u 
                                   ^ 
                                 
                                 y 
                               
                             
                           
                           
                             
                               
                                 
                                   u 
                                   ^ 
                                 
                                 z 
                               
                             
                           
                         
                         ] 
                       
                     
                     l 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 b 
                                 ^ 
                               
                               x 
                             
                           
                         
                         
                           
                             
                               
                                 b 
                                 ^ 
                               
                               y 
                             
                           
                         
                         
                           
                             
                               
                                 b 
                                 ^ 
                               
                               z 
                             
                           
                         
                       
                       ] 
                     
                     l 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     For a layered material, there is a system of differential equations of the form (5) for each layer, each of whose coefficients is determined by the material properties of the layer. It is possible to solve the system of differential equations for a typical layer l to obtain the general solution to the r th displacement components in the form: 
                       u   ^     r   l     =       ∑   j     ⁢           ⁢       d   jr   l     ⁢     ⅇ       α   j   i     ⁢   ky       ⁢       A   j   l     ⁡     (   k   )                   (   6   )               
where k=√{square root over (m 2 +n 2 )}
 
     In the case of repeated roots of the characteristic equation associated with (5), which occurs for the important case of isotropic layers, the system (5) has the general solution: 
     
       
         
           
             
               
                 
                   
                     
                       u 
                       ^ 
                     
                     r 
                     l 
                   
                   = 
                   
                     
                       ∑ 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             d 
                             jr 
                             l 
                           
                           + 
                           
                             
                               f 
                               jr 
                               l 
                             
                             ⁢ 
                             y 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             α 
                             j 
                             i 
                           
                           ⁢ 
                           ky 
                         
                       
                       ⁢ 
                       
                         
                           A 
                           j 
                           l 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Here d jr   l  and f jr   l  are constants that depend on the material constants of the layer, the α j   l  are the roots of the characteristic equation for the system of ordinary differential equations, and the A j   l (k) are free parameters of the solution that are determined by the forcing terms b i   l  in (1) and the interface conditions prescribed at the boundary between each of the layers (e.g., bonded, frictionless, etc.). 
     Substituting these displacement components into the stress strain law (2), we can obtain the corresponding stress components: {circumflex over (σ)} xx   l , {circumflex over (σ)} yy   l , {circumflex over (σ)} zz   l , {circumflex over (σ)} xy   l , {circumflex over (σ)} xz   l , and {circumflex over (σ)} yz   l , which can be expressed in the form: 
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       ^ 
                     
                     pq 
                     l 
                   
                   = 
                   
                     
                       ∑ 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           s 
                           jpq 
                           l 
                         
                         ) 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             α 
                             j 
                             i 
                           
                           ⁢ 
                           ky 
                         
                       
                       ⁢ 
                       
                         
                           A 
                           j 
                           l 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     In the case of repeated roots the stress components assume the form: 
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       ^ 
                     
                     pq 
                     l 
                   
                   = 
                   
                     
                       ∑ 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             s 
                             jpq 
                             l 
                           
                           + 
                           
                             
                               t 
                               jpq 
                               l 
                             
                             ⁢ 
                             ky 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             α 
                             j 
                             i 
                           
                           ⁢ 
                           ky 
                         
                       
                       ⁢ 
                       
                         
                           A 
                           j 
                           l 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     For each layer and for each sending DD element located at a particularly coordinate, there are a set of six parameters A j   l (k) that need to be determined from a system of algebraic equations that express the required body forces and boundary conditions in the model. Once the A j   l (k) have been determined, it is possible to calculate the influences of any DD having the same y component on any receiving point in any layer by taking the inverse Fourier Transform: 
                     u   j   l     =       1       (     2   ⁢           ⁢   π     )     2       ⁢       ∫     -   ∞     ∞     ⁢       ∫     -   ∞     ∞     ⁢       ⅇ     -     ⅈ   ⁡     (     mx   +   nz     )           ⁢       u   ^     j   l     ⁢           ⁢     ⅆ   m     ⁢           ⁢     ⅆ   n                     (   10   )               
of equations (6)-(9).
 
     One of the major computational bottlenecks in the procedure outlined above is the inversion process represented by (10), which involves the numerical inversion of a two-dimensional Fourier Transform for each sending-receiving pair of DD elements. The scheme we propose uses an exponential approximation of the spectral solution coefficients A j   l (k) (see for example reference [11], which was constructed only for point sources in layered dielectric materials, or reference [12]—only for horizontal conducting elements in layered dielectric materials) of the form: 
     
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         j 
                         l 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     - 
                     
                       
                         A 
                         j 
                         l 
                       
                       ⁡ 
                       
                         ( 
                         ∞ 
                         ) 
                       
                     
                   
                   ≈ 
                   
                     
                       ∑ 
                       r 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         a 
                         jr 
                         l 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             b 
                             jr 
                             l 
                           
                           ⁢ 
                           k 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Here A j   l (∞) represents the “high frequency” components of the spectrum of the solution, which represents the singular part of the solution in real space. 
     The inversion process can be achieved by evaluating integrals of the form: 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       j 
                       pl 
                     
                     = 
                     
                       
                         1 
                         
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                             ) 
                           
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             ∞ 
                           
                           ⁢ 
                           
                             
                               [ 
                               
                                 
                                   
                                     A 
                                     j 
                                     l 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     A 
                                     j 
                                     l 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     ∞ 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               k 
                               p 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   α 
                                   j 
                                   i 
                                 
                                 ⁢ 
                                 ky 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 - 
                                 
                                   ⅈ 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       mx 
                                       + 
                                       nz 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               m 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               n 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   or 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       I 
                       j 
                       pl 
                     
                     ≈ 
                     
                       
                         1 
                         
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                             ) 
                           
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           r 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             a 
                             jr 
                             l 
                           
                           ⁢ 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 ∞ 
                               
                               ⁢ 
                               
                                 
                                   k 
                                   p 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     k 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             α 
                                             j 
                                             i 
                                           
                                           ⁢ 
                                           y 
                                         
                                         + 
                                         
                                           b 
                                           jr 
                                           i 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     - 
                                     
                                       ⅈ 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           mx 
                                           + 
                                           nz 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   m 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   n 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     which can be evaluated in a closed form. The shifted components α j   l y+b jr   l  in (12) represent a finite number of complex images that approximate what would be an L-fold infinite Fourier Series (for L layers) that would be required to represent the Green&#39;s function in a closed form using the method of images. Typically three or four complex images suffice to give a high order of accuracy. 
     The expressions of the form (12) are not much more complicated than the pair-wise DD influences that apply for a homogeneous elastic medium. The only difference in this case is that for each sending DD element, the expansion coefficients a jr   l  and b jr   l  for each layer need to be determined by solving the appropriate set of algebraic equations and performing the exponential fit (11) of these coefficients. Once these coefficients have been determined we have a very efficient procedure to determine the influences between DD elements. 
     Refer now to  FIG. 26  representing a Schematic showing a duplicity of influence coefficients for a multiple layer problem. 
     In  FIG. 26 , for a regular array of DD elements there is an additional saving that can be exploited. In this case only the influence of a single sending DD element at each horizon (i.e., y level) needs to be determined in order to determine the whole influence matrix. For example, the DD elements in layer N denoted by the solid circle  202 , the shaded circle  204 , the unshaded circle  206 , and the arrow  208 , in  FIG. 26 , each have the same set of exponential expansion coefficients a jr   l  and b jr   l , in layer N−1, where the arrow  208  indicates where the expansion coefficients a jr   l  and b jr   l  are identical. 
     A DD influence at a specified point within a given layer is constructed by constructing a pseudo interface parallel to the layering across which there can be a jump in the displacement field. To construct a normal DD a jump in u y  is specified, whereas to construct a shear or ride DD a jump in u x  or u z  is specified. This technique limits the orientation of DD components to be aligned with the pseudo interface that is parallel to the layering. 
     However, it is desirable to have DD components that specify jumps in the displacement field which are across interfaces that are not parallel to the layering in the material. In particular, for hydraulic fracturing problems in the petroleum industry, it is important to be able to model vertical fracture planes that are perpendicular to the horizontally layered material. In this case, and for arbitrarily oriented DDs, it is possible to construct a DD field of a desired orientation, by utilizing the duality relationship between the stresses due to a force discontinuity (or point force) and the displacements due to a displacement discontinuity. The solution to a force discontinuity in the r th direction can be constructed by taking b r =δ(x, y, z)F r , where δ(x, y, z) represents the Dirac delta function. 
     Having obtained the stresses due to a force discontinuity:
 
{circumflex over (σ)} ij ={circumflex over (γ)} ijr F r   (13)
 
     it is possible to determine the displacements due to a DD according to the following duality relation:
 
û r ={circumflex over (γ)} ijr D ij   (14)
 
     The key idea here is to construct a planar Green&#39;s function or influence matrix, which represents the influences of all the DDs that lie in a vertical fracture plane. The influence matrix will only represent the mutual influences on one another of DDs that lie in the fracture plane. However, it will implicitly contain the information about the variations in material properties due to the layering. 
     A reduced influence matrix can be constructed by any numerical method, including that proposed above, which can represent rigorously the changes in material properties between the layers, for example the finite element method or a boundary integral method in which elements are placed on the interfaces between material layers. The in-plane influence coefficients would be calculated by means of the following procedure. 
     To calculate the influence of the ij th in-plane DD on the kl th DD anywhere else on the fracture plane, the value of the ij th DD would be assigned a value of unity and all the other fracture plane DDs would be set equal to zero. The boundary integral or finite element solution on the interfaces between the material layers would then be determined so as to ensure that there is compatibility in the displacements between the material layers as well as a balance in the forces between the layers at the interface. Once numerical solution has been calculated for the whole problem, the corresponding stresses on each of the in-plane DD elements can be evaluated to determine the in-plane stress influence of that unit DD on all the other DDs in the fracture plane. By repeating this process for each of the DDs that lie in the fracture plane, it is possible to determine the in-plane influence matrix which represents the effect that each DD in the plane has on any of the other in-plane DDs. By allowing the interface solution values to react to the sending DD element, the effect of the layers has been incorporated implicitly into this abbreviated set of influence coefficients. 
     The numerical procedure outlined to construct the desired in-plane influence matrix would take a considerable amount of time to compute. Indeed, such a process would likely exclude the possibility of real time processing with current personal computers or workstations, but could conceivably be performed in a batch mode prior to the desired simulation being performed. The semi-analytic method outlined above would be much more efficient, as the fully three-dimensional (or two-dimensional, in the case of plane strain or plane stress), problem that needs to be solved to calculate the influence of each DD element has been effectively reduced to a one-dimensional one. 
     Numerical models for multi-layered materials require that the interface between each material type is numerically “stitched” together by means of elements. For example, a boundary element method implementation would require that each interface between different material types is discretized into elements. A finite element or finite difference method implementation would require that the entire domain is discretized into elements. In the inventive method, the material interfaces are indirectly taken into account without the requirement of explicitly including elements away from the surface of the crack or fracture. The implication thereof, is a dramatic reduction in the number of equations to be solved, with a commensurate dramatic decrease in computer processing time. In addition, accuracy of the solution is maintained. One aspect of this invention which distinguishes it from previous work is that it is capable of solving problems in multi-layered elastic materials with arbitrarily inclined multiple cracks or fractures in two or three-dimensional space. 
     In this specification, note that ‘elements’ can intersect layers. This is accomplished by taking special care of the mathematical stress/strain relationships across interfaces in such a way as to obtain the correct physical response for the element which is located across the interface(s). 
     References 1-3 below are classic papers that establish the Fourier scheme to solve elastic multi-layer problems, but do not utilize the inversion scheme proposed here. In references 1 and 2, a propagator matrix approach is suggested to solve the system of algebraic equations necessary for the Fourier scheme, but this particular scheme will become unstable for problems with many layers. 
     References 4 and 5 use exponential approximation for inversion. The methods in those references do not give rise to the complex images generated by the algorithm presented in this invention that so efficiently represent the effect of many layers. Reference 5 extends the propagator approach used in references 1 and 2 to solve the algebraic equations of the Fourier method. Reference 5 discloses an inversion scheme that is an integral part of the propagator method. This method involves an exponential approximation, similar to that proposed in this patent, but it is applied to only one part of the propagator equations. As a result, a least squares fit of many terms (more than 50) is required to yield reasonable results using the teachings of this reference. Apart from stability issues involved with exponential fitting, the large number of terms would probably be less efficient than using direct numerical integration for inversion. The exponential fit of the spectral coefficients that we propose involves less than five terms. 
     References 6 through 10 extend the Fourier method to transversely isotropic media. References 7-10 use the propagator matrix for solving the algebraic equations, while reference 6 proposes direct solution. All these methods of solution would be numerically unstable for problems with many thick layers. While reference 10 proposes numerical inversion using continued fractions, little mention is made of the inversion process. 
     References 11 and 12 describe methodologies for multi-layer dielectric materials containing point electrical charges, or line charge distributions aligned parallel to interfaces (i.e., with different Green&#39;s functions to those used in elasticity). 
     Reference 13 describes a so-called “sweeping” algorithm to solve layered systems. The method disclosed in reference 13 is essentially the classic block LU decomposition for a block tri-diagonal system. In this specification, we use this algorithm to obtain a stable solution of the algebraic equations that determine the Fourier spectral coefficients in each of the layers. This method is particularly effective for problems in which the layers are thick or the wave-numbers are large. 
     It is recognized that other mathematical relationships may be used in the invention to achieve the same commercial or physical purpose. While not employing exactly the same equations, such methods are within the scope of the invention set forth in this specification. 
     The following references (i.e., references 1 through 15) are incorporated herein by reference:
     1. Ben-Menahem, A. and Singh, S. J. 1968.  Multipolar elastic fields in a layered half space . Bull. Seism. Soc. Am. 58(5). 1,519-72.   2. Singh, S. J. 1970.  Static deformation of a multi - layered half - space by internal sources . J. Geophys. Res. 75(17). 3,257-63.   3. Chen, W. T. 1971.  Computation of stresses and displacements in a layered elastic medium . Int. J. Engng. Sci. vol. 9. 775-800.   4. Sato, R. and Matsu&#39;ura, M. 1973.  Static deformations due to the fault spreading over several layers in a multi - layered medium Part I: Displacement . J. Phys. Earth. 21. 227-249.   5. Jovanovich, D. B., Husseini, M. I. and Chinnery, M. A. 1974.  Elastic dislocations in a layered half - space—I. Basic theory and numerical methods . Geophys. Jour. Roy. Astro. Soc. 39. 205-217.   6. Wardle, L. J. 1980.  Stress analysis of multilayered anisotropic elastic systems subject to rectangular loads . CSIRO Aust. Div. Appl. Geomech. Tech. paper no. 33. 1-37.   7. Singh, S. J. 1986.  Static deformation of a transversely isotropic multilayered half - space by surface loads . Physics of the Earth and Planetary Interiors. 42. 263-273.   8. Pan, E. 1989.  Static response of a transversely isotropic and layered half - space to general surface loads . Physics of the Earth and Planetary Interiors. 54. 353-363.   9. Pan, E. 1989.  Static response of a transversely isotropic and layered half - space to general dislocation sources . Physics of the Earth and Planetary Interiors. 58. 103-117.   10. Pan, E. 1997.  Static Green&#39;s functions in multilayered half spaces . Appl. Math. Modeling. 21. 509-521.   11. Chow, Y. L., Yang, J. J., and Howard, G. E. 1991.  Complex images for electrostatic field computation in multilayered media . IEEE Trans. on Microwave Theory and Techniques. vol. 39. no. 7. 1120-25.   12. Crampagne, R., Ahmadpanah, M. and Guiraud, J.-L. 1978.  A simple method for determining the Green&#39;s function for a class of MIC lines having multilayered dielectric structures . IEEE Trans. on Microwave Theory and Techniques. vol. MTT-26. No. 2. 82-87.   13. Linkov, A. M., Linkova, A. A., and Savitski, A. A. 1994.  An effective method for multi - layered media with cracks and cavities . Int. J. of Damage Mech. 3. 338-35.   14. Ryder, J. A., and Napier, J. A. L. 1985.  Error Analysis and Design of a Large Scale Tabular Mining Stress Analyzer . Proceedings of the Fifth International Conference on Numerical Methods in Geomechanics, Nagoya, Japan, [Balkema] 1549-1555.   15. J. A. Ryder, Eds.: E. G. Beer, J. R. Booker, and J. P. Carter,  Optimal Iteration Schemes Suitable for General Non - linear Boundary Element Modeling Applications : Proceedings of the 7th International Conference on Computer Methods and Advances in Geomechanics, Cairns, Australia [Balkema], 1991.   

     The above description of the ‘Hydraulic Fracturing Simulator Software’ being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the claimed method or apparatus or program storage device, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims.