Abstract:
The invention relates to the calculation of parameters to inform hydraulic stimulation of non-conventional hydrocarbon-bearing rock formations, such as shales. Unlike conventional formations, non-conventional formations tend to display elastic-plastic behavior and have stress-strain characteristics which with substantial non-linear regions. A parameter which has been termed Elastic Index (EI) is proposed, together with a demonstration of how this parameter, when coupled with principles of fracture mechanics, may be used to extract meaningful calculated or estimated values for e.g.; total required volume of fracturing fluid; treating pressure; fracturing fluid viscosity; proppant size; and proppant concentration.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a non-provisional application which claims benefit under 35 USC §119(e) of and priority to U.S. Provisional Application Ser. No. 61/828,368 filed 29 May 2013, entitled “INTEGRATING ROCK DUCTILITY WITH FRACTURE PROPAGATION MECHANICS FOR HYDRAULIC FRACTURE DESIGN,” which is incorporated by reference herein in its entirety. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0002]    None. 
       FIELD OF THE INVENTION 
       [0003]    This invention relates to hydraulic stimulation (“fracking”) in shale and other unconventional subterranean hydrocarbon reservoirs. 
       BACKGROUND OF THE INVENTION 
       [0004]    Shale and other non-conventional formations such as tight gas, mudstone, siltstone and marl systems are becoming an increasingly important source of hydrocarbon resources. Such unconventional resources, however, present challenges not only in their extraction but also in the analysis of their properties in order to design strategies for drilling and treatment such as hydraulic stimulation (“fracking”). 
         [0005]    Once a well has been drilled, it is normally necessary to design the completion of the well, including e.g. hydraulic stimulation to create artificial fractures in the rock. In order to design hydraulic stimulation, one or more of a number of design parameters need to be calculated, estimated, determined by trial and error, or determined by a mixture of these approaches. The parameters may include (i) the pressure at which hydraulic fluid must be pumped to produce the correct degree of fracturing, (ii) the viscosity of the fracturing fluid, (iii) the size of proppant particles, (iv) the concentration of proppant in the fluid, or (v) the total fluid volume. 
         [0006]    Calculation of the above parameters has proven problematic in shale and other non-conventional rock, which appears to have geomechanical properties which differ substantially from conventional rock. Conventional rock is normally assumed to behave perfectly elastically until failure and this assumption or model has proved reliable over many years in the process of calculating or estimating fracking parameters. Shale and other non-conventional rock, however, do not always behave in this way and often show a considerable degree of plastic behavior. 
         [0007]    Current approaches continue to rely on the assumed elastic properties of rock (Mullen, 2007). This will generally produce incorrect results and often a large amount of trial and error is also involved. Once a well in a given field has been completed successfully, the parameters used may provide guidance for the completion of other wells in the same or neighboring fields. 
         [0008]    Another approach has been proposed which is to include an assessment of the so-called “brittleness” of the rock in the evaluation and design of drilling and treatment. Brittleness has been defined by a cross-plot of the elastic Young&#39;s modulus and Poisson&#39;s ratio of the rock, determined from core samples (Rickman et al., 2008). However, the brittleness approach has been shown to provide inaccurate predictions for shale properties and the “frackability” of non-conventional reservoirs. Full references are given at the end of this specification. 
         [0009]    Some theoretical work on fracture growth in ductile rocks has been done in the past. An effective stress intensity factor, or apparent fracture toughness, can be calculated which on a theoretical basis may predict fracture growth in semi-brittle or ductile rocks (e.g., Heald et al., 1972; Khazan and Fialko, 1995). Similarly, Kanninen and Popelar (1985) show how the J-integral (Rice, 1968) can theoretically be used to model the growth of elastic-plastic (or non-brittle) fractures. 
         [0010]    Heald et al. (1972) showed that the apparent fracture toughness of rock having a non-linear stress strain curve exceeded the standard plane-strain brittle fracture toughness due to toughening of the rock. The strengthening in ductile rocks may be related to the additional work above K Ic  (critical stress intensity factor) required to deform the rock inelastically beyond the fracture tip (e.g., Li and Liang, 1986; Rubin, 1993). This may lead to a dependence of apparent fracture toughness on confining pressure for mode-I cracks (Perkins and Krech, 1966; Abou-Sayed, 1977; Schmidt and Huddle, 1977; Atkinson and Meredith, 1987; Thallak et al., 1993). 
         [0011]    However, there is to date no straightforward way of predicting the required fracture pressure, density and volume of fluid or size and concentration of proppant for an unconventional, e.g. shale, reservoir. Getting these parameters wrong can have serious consequences for field operations. A common issue encountered in hydraulic fracturing is called a ‘screen out’, which occurs when the formation does not break-down at the expected pumping pressure. The frac sand collects in the wellbore causing serious logistics issues; the problems arise because the sand is never properly placed in the formation i.e the hydraulic fracture operation fails. The entire field operation must cease and additional field operation services must be called in to clean out the well. This causes delays and extra expense and, in addition, the complete hydraulic fracture product (fluid, sand, manpower and horsepower) is wasted on an unsuccessful attempt. Hydraulic fracture ‘screen outs’ can be avoided if the completion design factors in the appropriate breakdown pressure based on the mechanical response of the near wellbore rock (as outlined in this document.) 
         [0012]    Also, it is well documented that not all hydraulic frac intervals in a horizontal well in non-conventional reservoirs contribute to the hydrocarbon production of the well. By understanding the fracture mechanical response and the ductility of the rock, we may have a better chance of predicting which rock types will permit the formation of continuous hydraulic fractures that stay propped open and contribute to flow versus those that will not. 
         [0013]    Despite all these issues, today hydraulic fracturing in unconventional rocks tends to be conducted using design parameters determined more from experience in similar rocks and by simple trial and error than from any systematic approach. What is required is a hydraulic fracturing system for non-conventional reservoirs that accounts for variations in elasticity, non-homogeneous behavior, and ductility of the rock in non-conventional reservoirs. 
       BRIEF SUMMARY OF THE DISCLOSURE 
       [0014]    The inventors have investigated the stress-strain curve for a number of different types of non-conventional hydrocarbon-bearing rock. They have found that the stress-strain is likely to comprise an initial non-linear portion as the rock is initially loaded. This region, known as a poro-elastic region, is not always present, though. The poro-elastic region, if present, is followed by a linear (elastic) region up to the yield point, and then normally a substantial non-linear plastic region leading up to the failure point. The non-linear region beyond the yield point may have very different characteristics depending on the particular non-conventional rock tested, but it will generally have significant plasticity or ductility. The properties of the different regions of the stress-strain curve will differ widely between different reservoirs or even between samples from different places in the same reservoir. 
         [0015]    The invention includes a method of calculating one or more parameters for performing hydraulic fracturing of a well in a non-conventional subterranean hydrocarbon reservoir, the method comprising:
   taking a core sample from the reservoir;   performing a tri-axial compressive test on said core sample;   determining from the tri-axial test:   the axial poro-elastic stress, if a poro-elastic region is present;   the axial yield stress; and   the axial failure stress;   (a) taking a core sample from the reservoir;   (b) performing a tri-axial compressive test on said core sample;   (c) determining from the tri-axial test the axial yield stress and axial failure stress;   (d) calculating an elastic index value (EI), defined as:   
 
         [0000]    
       
         
           
             
               
                 
                   EI 
                   = 
                   
                     
                       ( 
                       
                         axial 
                          
                         
                             
                         
                          
                         yield 
                          
                         
                             
                         
                          
                         stress 
                       
                       ) 
                     
                     
                       ( 
                       
                         failure 
                          
                         
                             
                         
                          
                         stress 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent; 
           
         
         (e) calculating an average fracture width D ductile  for a propagating crack using the equation: 
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     
                       D 
                       ductile 
                     
                     
                       D 
                       brittle 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         EI 
                       
                       ) 
                     
                      
                     
                       
                         E 
                         ductile 
                       
                       
                         E 
                         brittle 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent, 
             where E ductile  is the secant modulus (E s ) and E brittle  is the tangent (Young&#39;s) modulus (E T ), and where D brittle  is an average fracture width value derived from the tangent modulus, Poisson&#39;s ratio and assumed crack geometry factors; 
           
         
         (f) using said calculated value of D ductile  to derive or estimate one or more of:
       (i) fracturing fluid viscosity;   (ii) total required volume of fracturing fluid;   (iii) treating pressure;   (iv) proppant size; and   (v) proppant concentration.   
     
       
     
         [0036]    It will be understood that, once the width of a propagating crack is determined, a well informed estimate of crack volume can be made based on reasonable assumptions about crack length and geometry. 
         [0037]    An alternative method is provided, for example for use when the poro-elastic strain or poro-elastic stress is relatively small or is insignificant in that its effect on the result of the pressure calculation would be lower than the precision with which the applied pressure can be controlled. This method comprises:
       (a) taking a core sample from the reservoir;   (b) performing a tri-axial compressive test on said core sample;   (c) determining from the tri-axial test the axial yield stress and axial failure stress;       (g) calculating an elastic index value (EI), defined as:   
 
         [0000]    
       
         
           
             
               
                 
                   EI 
                   = 
                   
                     
                       ( 
                       
                         axial 
                          
                         
                             
                         
                          
                         yield 
                          
                         
                             
                         
                          
                         stress 
                       
                       ) 
                     
                     
                       ( 
                       
                         failure 
                          
                         
                             
                         
                          
                         stress 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent; 
           
         
         (h) calculating an average fracture width D ductile  for a propagating crack using the equation: 
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     
                       D 
                       ductile 
                     
                     
                       D 
                       brittle 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         EI 
                       
                       ) 
                     
                      
                     
                       
                         E 
                         ductile 
                       
                       
                         E 
                         brittle 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent, 
             where E ductile  is the secant modulus (E s ) and E brittle  is the tangent (Young&#39;s) modulus (E T ), and where D brittle  is an average fracture width value derived from the tangent modulus, Poisson&#39;s ratio and assumed crack geometry factors; 
           
         
         (i) using said calculated value of D ductile  to derive or estimate one or more of:
       (i) fracturing fluid viscosity;   (ii) total required volume of fracturing fluid;   (iii) treating pressure;   (iv) proppant size; and   (v) proppant concentration.   
     
       
     
         [0000]    
       
         
           
             EI 
             = 
             
               [ 
               
                 
                   ( 
                   
                     
                       
                         axial 
                          
                         
                             
                         
                          
                         yield 
                          
                         
                             
                         
                          
                         stress 
                       
                       - 
                       
                         poro 
                          
                         
                           - 
                         
                          
                         elastic 
                          
                         
                             
                         
                          
                         stress 
                       
                     
                     , 
                     
                       if 
                        
                       
                           
                       
                        
                       present 
                     
                   
                   ) 
                 
                 
                   ( 
                   
                     
                       
                         failure 
                          
                         
                             
                         
                          
                         stress 
                       
                       - 
                       
                         poro 
                          
                         
                           - 
                         
                          
                         elastic 
                          
                         
                             
                         
                          
                         stress 
                       
                     
                     , 
                     
                       if 
                        
                       
                           
                       
                        
                       present 
                     
                   
                   ) 
                 
               
               ] 
             
           
         
       
     
         [0000]    An alternative calculation for elastic index includes:
       (3)       
 
         [0053]    One way of deriving a value for treating pressure in either method described above is to apply the equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     
                       prop 
                        
                       
                           
                       
                        
                       
                         ( 
                         ductile 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             υ 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         K 
                         Ic 
                         2 
                       
                     
                     
                       
                         D 
                         max 
                       
                        
                       
                         E 
                         s 
                       
                        
                       Ω 
                        
                       
                           
                       
                        
                       
                         π 
                          
                         
                           ( 
                           EI 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent,
 
where ν is Poisson&#39;s ratio, K Ic  is the mode-I critical stress intensity factor for brittle rock, as determined from the triaxial testing, D max  is the fracture width given by the above equation for the ductile case (D ductile ), E s  is the ductile (secant) modulus and Ω is a crack geometry factor. K Ic  can be expressed as σ√(πa) where a is half the estimated crack length and σ is the yield stress.
 
           
         
       
     
         [0055]    One way of estimating a value for total required volume of fracturing liquid is to estimate the fracture volume using the equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     prob 
                   
                   = 
                   
                     κ 
                      
                     
                         
                     
                      
                     
                       K 
                       Ic 
                     
                      
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             υ 
                             2 
                           
                         
                         ) 
                       
                       E 
                     
                      
                     
                       
                         8 
                         π 
                       
                     
                      
                     
                       
                         
                           L 
                           
                             3 
                              
                             
                               / 
                             
                              
                             2 
                           
                         
                          
                         H 
                       
                       Ω 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent. 
           
         
       
     
         [0057]    In this expression, E is the ductile (secant) modulus and L and H are fracture length and height. κ in this expression converts the maximum opening displacement to the average opening displacement. For a typical elliptical crack opening distribution, κ=Davg/Dmax=0.78. The value for κ depends on the shape of the aperture distribution along the crack, but this value is reasonable for an elliptical opening distribution, which is typical for strong rock. A range of possible values for κ in this context is from 0.75 to 0.79. See Schultz et al., Planetary Tectonics, pp 457-510 (full reference given below) for a full discussion of κ. In fact, κ can go as low as 0.5 or thereabouts and also higher, to perhaps 1.0 or thereabouts, due to interaction with stratigraphy and/or other fractures. 
         [0058]    Although not associated with specific equations, it will immediately be understood how a better understanding of fracture width will lead to improved estimates of fracturing fluid viscosity and of proppant size and concentration. 
         [0059]    In another embodiment of the invention, a method of designing a hydraulic fracturing procedure for a well in a non-conventional subterranean hydrocarbon reservoir includes calculating one or more parameters derived or estimated by the process described in the summary above and in the detailed description below. 
         [0060]    In a further embodiment of the invention, method of performing a hydraulic fracturing procedure on a well in a non-conventional subterranean hydrocarbon reservoir includes designing said procedure with reference to one or more parameters derived or estimated by the process described in the summary above and in the detailed description below. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0061]    A more complete understanding of the present invention and benefits thereof may be acquired by referring to the follow description taken in conjunction with the accompanying drawings in which: 
           [0062]      FIG. 1  is an exemplary triaxial stress-strain plot for a shale core; 
           [0063]      FIG. 2  is a triaxial stress-strain plot after data cleanup; 
           [0064]      FIG. 3  is a plot similar to  FIG. 2  for a different core; 
           [0065]      FIG. 4  is a plot showing calculated crack opening displacement for both brittle and ductile cases as a function of elastic index; 
           [0066]      FIG. 5  is a plot showing crack or fluid volume as a function of elastic index; and 
           [0067]      FIG. 6  is a plot showing excess fluid pressure as a function of elastic index. 
       
    
    
     DETAILED DESCRIPTION 
       [0068]    Turning now to the detailed description of the preferred arrangement or arrangements of the present invention, it should be understood that the inventive features and concepts may be manifested in other arrangements and that the scope of the invention is not limited to the embodiments described or illustrated. The scope of the invention is intended only to be limited by the scope of the claims that follow. 
         [0069]    The mode-I stress intensity factor K I  is given for 2-D plane-strain crack geometries by 
         [0000]      K I =σ√{square root over (πa)}  (6)
       or its mathematical equivalent,   in which σ is the driving stress or pore-fluid pressure and a is crack half-length.       
 
         [0072]    Although this expression can be modified for non-plane crack geometries (e.g., three-dimensional (3-D) crack shapes) by either a multiplicative factor or polynomial expression or by a more complete stress analysis (e.g., Kanninen and Popelar, 1985), it is not necessary since that will be incorporated directly into the calculation of crack opening displacements (widths) below. 
         [0073]    The stress intensity factor for a perfectly brittle rock is given by (6) with σ being the yield strength (corresponding to the rock&#39;s tensile strength), and EI=1.0. Substituting the value of peak or ultimate stress into (6) provides an estimate of the apparent stress intensity factor for the ductile rock, K ductile . This approach is justified because it implicitly incorporates the inelastic deformation mechanisms that control the value of ultimate failure strength in a ductile rock. Combining (3) and (6) leads to 
         [0000]    
       
         
           
             
               
                 
                   
                     K 
                     ductile 
                   
                   = 
                   
                     ( 
                     
                       
                         K 
                         I 
                       
                       EI 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which illustrates how the ductile apparent stress intensity factor can be estimated from the brittle stress intensity factor and the elastic index. Combining (1) and (6) gives essentially the same result, but allows for an initial poro-elastic region of the stress-strain curve. In practice, the brittle stress intensity factor can be measured in the laboratory, in which case the apparent fracture toughness of the ductile rock would be given as the brittle fracture toughness divided by its elastic index. 
         [0074]    The case of a crack in 3-D is straightforward to consider by calculating the displacements on a crack having an elliptical tipline and subjected to a crack-normal stress (e.g., Irwin, 1962; Kassir and Sih, 1966). The maximum opening displacement (width) at the center of a symmetrically loaded mode-I crack is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     max 
                   
                   = 
                   
                     2 
                      
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             υ 
                             2 
                           
                         
                         ) 
                       
                       E 
                     
                      
                     σ 
                      
                     
                       L 
                       Ω 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent, 
           
         
       
     
         [0000]    
       
         
           
             Ω 
             = 
             
               
                 1 
                 + 
                 
                   1.464 
                    
                   
                     
                       ( 
                       
                         a 
                         b 
                       
                       ) 
                     
                     1.65 
                   
                 
               
             
           
         
       
     
         [0000]    in which ν is Poisson&#39;s ratio, E is Young&#39;s modulus, and Ω is the flaw shape parameter (Anderson, 1995, pp. 115-116) that is within 5% (Schultz and Fossen, 2002) of the complete elliptic integral of the second kind (Irwin, 1962; Kanninen and Popelar, 1985, p. 153; Lawn, 1993, p. 33) which is given by 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       ? 
                     
                      
                     
                       
 
                     
                      
                     
                       
                         ? 
                       
                        
                       
                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent,
 
in which b is crack half-height. For a tall 2-D fracture (a&lt;&lt;b), Ω=1.0; for a circular “penny-shaped” fracture (a=b), Ω=π/2; for a long “blade-like” fracture (a=10b), Ω˜8.
 
           
         
       
     
         [0077]    The stress intensity factors and maximum opening or shear displacements (below) for an elliptical, three-dimensional fracture thus depend explicitly on both its horizontal (length L=2a) and vertical (height H=2b) dimensions (e.g., Irwin, 1962; Willemse et al., 1996; Gudmundsson, 2000; Schultz and Fossen, 2002). The average opening displacement for a crack that is propagating (i.e., K I =K Ic ) is limited by the rock&#39;s fracture toughness K Ic . The maximum opening displacement (width) for a propagating crack is given by (see Olson, 2003) 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       max 
                        
                       
                           
                       
                        
                       prop 
                     
                   
                   = 
                   
                     
                       K 
                       Ic 
                     
                      
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             υ 
                             2 
                           
                         
                         ) 
                       
                       E 
                     
                      
                     
                       
                         8 
                         π 
                       
                     
                      
                     
                       
                         L 
                       
                       Ω 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent. 
           
         
       
     
         [0079]    The value of E used in (8) or (10) would be the tangent Young&#39;s modulus for the brittle case (i.e., E T =σ yield /ε) but the secant Young&#39;s modulus for the ductile case (i.e., E s =σ ultimate /ε). The crack opening displacement (width) thus depends on the values of stress and rock stiffness, which differ for brittle and ductile cases. Rewriting (8) or (10) with (3) gives: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       D 
                       ductile 
                     
                     
                       D 
                       brittle 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         EI 
                       
                       ) 
                     
                      
                     
                       
                         E 
                         ductile 
                       
                       
                         E 
                         brittle 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which shows that the opening displacement (width) is described by both the elastic index and the ratio of the ductile to brittle Young&#39;s modulus. Given specification of crack lengths and geometries, (2) implies that fluid volumes can be calculated for crack growth in brittle and ductile rocks, as given below. Using (1) instead of (3) gives essentially the same result, but allows for an initial poro-elastic region of the stress-strain curve. 
         [0080]    The volume of a 3-D fracture is given by V=DLH, so that using (10), at propagation 
         [0000]    
       
         
           
             
               
                 
                   
                     V 
                     prob 
                   
                   = 
                   
                     κ 
                      
                     
                         
                     
                      
                     
                       K 
                       Ic 
                     
                      
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             υ 
                             2 
                           
                         
                         ) 
                       
                       E 
                     
                      
                     
                       
                         8 
                         π 
                       
                     
                      
                     
                       
                         
                           L 
                           
                             3 
                              
                             
                               / 
                             
                              
                             2 
                           
                         
                          
                         H 
                       
                       Ω 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent, 
           
         
       
     
         [0082]    Similarly, (8) can be combined with (6) and solved for the driving stress, which for a hydrofracture would be the excess pore-fluid pressure above the minimum compressive principal stress σ h . The excess fluid pressure necessary to propagate a hydrofracture is then given by 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     prop 
                   
                   = 
                   
                     
                       
                         4 
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               υ 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           K 
                           Ic 
                           2 
                         
                       
                       
                         
                           D 
                           max 
                         
                          
                         E 
                          
                         
                             
                         
                          
                         Ω 
                          
                         
                             
                         
                          
                         π 
                       
                     
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent. 
           
         
       
     
         [0084]    Equation (12) shows that the injection pressure depends on the maximum crack opening displacement (width), the crack geometry (through the flaw shape parameter), the rock stiffness, and the fracture toughness. Equation (12) can be expressed for ductile rocks by using elastic index by using (3) and specifying that E=E s , giving: 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     
                       prop 
                        
                       
                           
                       
                        
                       
                         ( 
                         ductile 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             υ 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         K 
                         Ic 
                         2 
                       
                     
                     
                       
                         D 
                         max 
                       
                        
                       
                         E 
                         s 
                       
                        
                       Ω 
                        
                       
                           
                       
                        
                       
                         π 
                          
                         
                           ( 
                           EI 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent. 
           
         
       
     
         [0086]    The J-integral (Kanninen and Popelar, 1985) provides a convenient method of comparing the deformation in brittle and ductile rocks. For crack growth, J may be given as: 
         [0000]      J=σD max    (14)
 
         [0000]    with D being the maximum crack-opening displacement (width), given approximately by (4) or (6). J can be calculated for crack growth in brittle or ductile rocks given the corresponding values for stress (yield or ultimate) and opening displacement. For example, by substituting the expression for the excess fluid pressure necessary to propagate a hydrofracture (12) into (14), for brittle rocks we have: 
         [0000]    
       
         
           
             
               
                 
                   
                     J 
                     
                       ( 
                       brittle 
                       ) 
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             υ 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         K 
                         Ic 
                         2 
                       
                     
                     
                       
                         E 
                         T 
                       
                        
                       Ω 
                        
                       
                           
                       
                        
                       π 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent. 
           
         
       
     
         [0088]    Whereas for ductile rocks (by substituting (13) into (14)) we have: 
         [0000]    
       
         
           
             
               
                 
                   
                     J 
                     
                       ( 
                       ductile 
                       ) 
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             v 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         K 
                         Ic 
                         2 
                       
                     
                     
                       
                         E 
                         s 
                       
                        
                       
                         
                           Ωπ 
                            
                           
                             ( 
                             EI 
                             ) 
                           
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent. 
           
         
       
     
         [0000]    
       
         
           
             G 
             = 
             
               
                 
                   ( 
                   
                     1 
                     - 
                     
                       v 
                       2 
                     
                   
                   ) 
                 
                  
                 
                   K 
                   Ic 
                   2 
                 
               
               E 
             
           
         
       
     
         [0000]    is interpreted as the strain energy release rate (for brittle conditions) or more generally, the crack propagation energy, both having units of kJ/m 2 . Because the strain energy release rate for brittle and low-ductility rocks is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       ? 
                     
                      
                     
                         
                     
                      
                     
                       
                         ? 
                       
                        
                       
                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent. 
           
         
       
     
         [0091]    The expressions for J simplify to become 
         [0000]    
       
         
           
             
               
                 
                   J 
                   = 
                   
                     
                       4 
                        
                       G 
                     
                     Ωπ 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
       
         
           
             or its mathematical equivalent,
 
in which the appropriate values for E and K are used in (18) to calculate J.
 
           
         
       
     
         [0093]    The various parameters are plotted in  FIGS. 4 through 6 . Although the magnitude of increase with increasing ductility varies for each, the form is the same for all cases. It can be seen that the ductility of the host rock leads to systematic increases in the crack opening displacement (i.e., wider cracks in a ductile rock), crack or fluid volume, and excess fluid pressure. 
         [0094]    To summarize, the measured properties of a stress-strain curve can be used to calculate the resistance to crack growth in brittle or ductile rocks, which controls the injection pressure needed for hydrofracture propagation. The fluid volume increases with ductility as does the energy release rate consumed by crack growth. Crack interaction problems (e.g., a hydrofracture interacting mechanically with nearby joints or faults) can be treated by using conventional means with the substitution of the apparent fracture toughness calculated by using elastic index with fracture toughness obtained from core. 
         [0095]      FIG. 1  shows a typical triaxial stress-strain plot for a shale system, the failure point, yield point and poro-elastic limit are shown. The yield point and failure point are well defined stress-strain analysis parameters. The poro-elastic limit is defined as the early departure point from the tangential modulus defining the Young&#39;s modulus (or the tangent elastic modulus) of the stress deformation curve. The secant modulus is also shown, which is the slope of the line connecting the start of the elastic region (poro-elastic limit) with the ultimate yield point. 
         [0096]    A script has been developed to analyze the non-linear behavior of the rock from a laboratory strain-stress curve. Laboratory data may have erroneous points and inconsistent sampling, which are a hindrance to consistent and automatic analysis of this data. This script does the following steps for analyzing the data:
       1. Statistical analysis to clean the erroneous points and inconsistent sampling of the data and quality checks that the data does not lose its original behavior;   2. Defining the poro-elastic limit, yield point, and maximum compressive strength points on the data.   3. Defining the elastic index parameter.       
 
       EXAMPLE 
       [0100]      FIGS. 2 and 3  show cleaned and re-sampled data with the original data from laboratory triaxial testing from couple of core samples. As is usual with this type of testing, the data has erroneous points and inconsistent sampling. The script performs statistical analysis to clean the erroneous points and do quality checks (step 1 above). Once the data is cleaned, the script defines the poro-elastic limit, yield point and failure point (step 2) and these points are shown on  FIG. 2 . The elastic index (EI) parameter is then calculated. In this example: 
         [0000]      EI=0.9098 
         [0101]    The script then checks which order (N) of polynomial curve best fits each individual section of the P(1)*X̂N+P(2)*X̂(N−1)+ . . . +P(N)*X+P(N+1) curve and defined the coefficients for each order. 
         [0000]    
       
         
           
             
               
                 
                   
                     ? 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ? 
                     
                      
                     
                       indicates text missing or illegible when filed 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
       
         Poro Elastic= 
         1.0e+010* 
         1.5537 −0.0021 0.0001 0.0000 
         Elastic= 
         1.0e+006* 
         3.3805 −0.0131 
         Yield= 
         1.0e+09* 
         3.6139 −0.6628 0.0173 −0.0001 
         Failure= 
         1.0e+008* 
         −6.5061 0.1935 −0.0011 
       
     
         [0114]    P(N+1) is the offset from zero or a permanent strain, which can be ignored for individual analysis. Between the poro-elastic and yield point, the polynomial fit is first order, which is consistent with the theory that stress=Young&#39;s modulus*strain, where P(N) indicates the Young&#39;s modulus. For the other regions, polynomial fit is of higher order, indicating that rock behaves non-linearly in these regions defined by different strain limits. Therefore, stress values have to be corrected from a pure elastic calculation (plain-strain type) in these regions. 
         [0115]    The tangent and secant moduli (also called E  brittle  and E ductile ) can then be calculated. These values can be used in equation (2), together with a value for EI calculated from the yield and failure stress (and, optionally, poro-elastic stress), to estimate an average fracture width D ductile . Any of the following parameters may then be derived or estimated using D ductile  and the methodology discussed above:
       (i) fracturing fluid viscosity;   (ii) total required volume of fracturing fluid;   (iii) treating pressure;   (iv) proppant size; and   (v) proppant concentration.       
 
         [0121]    Using one or more of these estimated or derived values, hydraulic fracturing of a non-conventional formation may be planned, designed and carried out with enhanced safety and efficacy. 
         [0122]    In closing, it should be noted that the discussion of any reference is not an admission that it is prior art to the present invention, especially any reference that may have a publication date after the priority date of this application. At the same time, each and every claim below is hereby incorporated into this detailed description or specification as additional embodiments of the present invention. 
         [0123]    Although the systems and processes described herein have been described in detail, it should be understood that various changes, substitutions, and alterations can be made without departing from the spirit and scope of the invention as defined by the following claims. Those skilled in the art may be able to study the preferred embodiments and identify other ways to practice the invention that are not exactly as described herein. It is the intent of the inventors that variations and equivalents of the invention are within the scope of the claims while the description, abstract and drawings are not to be used to limit the scope of the invention. The invention is specifically intended to be as broad as the claims below and their equivalents. 
       REFERENCES 
       [0124]    All of the references cited herein are expressly incorporated by reference. The discussion of any reference is not an admission that it is prior art to the present invention, especially any reference that may have a publication data after the priority date of this application. Incorporated references are listed again here for convenience:
   Abou-Sayed, A. S. (1977). Fracture toughness, K Ic , of triaxially-loaded Indiana limestone.  Proc. U.S. Symp. Rock Mech.  18:2A3/1-2A3/8.   Anderson, T. L. (1995).  Fracture mechanics: fundamentals and applications  (2nd ed.). CRC Press, Boca Raton, Fla.   Atkinson, B. K. and Meredith, P. G. (1987). Experimental fracture mechanics data for rocks and minerals. In  Fracture mechanics of rock  (ed. B. K. Atkinson) pp. 477-525, Academic Press, New York.   Gudmundsson, A. (2000). Fracture dimensions, displacements and fluid transport.  J. Struct. Geol.  22:1221-1231.   Heald, P. T., Spink, G. M. and Worthington, P. J. (1972). Post yield fracture mechanics.  Mat. Sci. Engng.  10:129-138.   Irwin, G. R. (1962). The crack extension force for a part-through crack in a plate.  J. Appl. Mech.  29:651-654.   Kanninen, M. F. and Popelar, C. H. (1985).  Advanced fracture mechanics.  Oxford University Press, New York, 563 pp.   Kassir, M. K. and Sih, G. C. (1966). Three-dimensional stress distribution around an elliptical crack under arbitrary loadings.  J. Appl. Mech.  33:601-611.   Khazan, Y. M. and Fialko, Y. A. (1995). Fracture criteria at the tip of fluid-driven cracks in the earth.  Geophys. Res. Lett.  22:2541-2544.   Lawn, B. (1993).  Fracture of brittle solids—second edition.  Cambridge University Press, 378 pp.   Li, V. C. and Liang, E. (1986). Fracture processes in concrete and fiber reinforced cementitious composites.  J. Eng. Mech.  112:566-586.   Mullen, M., R. Roundtree, and B. Barree. 2007. A composite determination of rock properties for stimulation design . . .  SPE  08139;  Proc. Rocky Mountain Oil  &amp;  Gas Techn. Symp.,  Denver, Colo., USA 16-18 Apr. 2007.   Olson, J. E. (2003). Sublinear scaling of fracture aperture versus length: An exception or the rule?  J. Geophys. Res.  108:2413, doi:10.1029/2001JB000419.   Perkins, T. K. and Krech, W. W. (1966). Effect of cleavage rate and stress level on apparent surface energies of rocks.  Soc. Petrol. Eng. J.  6:308-314.   Rice, J. R. (1968). A path-independent integral and the approximate analysis of strain concentration by notches and cracks.  J. Appl. Mech.  35:379-386.   Rickman, R., Mullen, M., Petre, E., Grieser, B. and Kundert, D. (2008). A practical use of shale petrophysics for stimulation design optimization: all shale plays are not clones of the Barnett Shale. Presented at SPE 2008 Annual Technical Conference and Exhibition, Sep. 21-24, 2008, Denver, Colo.; paper SPE 115258.   Rubin, A. M. (1993). Tensile fracture of rock at high confining pressure: Implications for dike propagation.  J. Geophys. Res.  98:15,919-15,935.   Schmidt, R. A. and Huddle, C. W. (1977). Effect of confining pressure on fracture toughness of Indiana limestone.  Int. J. Rock Mech. Min. Sci. Geomech. Abstr.  14:289-293.   Schultz, R. A. and Fossen, H. (2002). Displacement-length scaling in three dimensions: The importance of aspect ratio and application to deformation bands.  J. Struct. Geol.  24:1389-1411   Schultz, R. A., R. Soliva, C. H. Okubo, and D. Mege, 2010, Fault populations, in Planetary Tectonics, edited by T. R. Watters and R. A. Schultz, Cambridge University Press, Cambridge, pp. 457-510   Thallak, S., Holder, J. and Gray, K. E. (1993). The pressure dependence of apparent hydrofracture toughness.  Int. J. Rock Mech. Min. Sci. Geomech. Abstr.  30:831-835.   Willemse, E. J. M., Pollard, D. D. and Aydin, A. (1996). Three-dimensional analyses of slip distributions on normal fault arrays with consequences for fault scaling.  J. Struct. Geol.  18:295-309.