Patent Application: US-201414288677-A

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 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:
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 . the mode - i stress intensity factor k i is given for 2 - d plane - strain crack geometries by or its mathematical equivalent , in which σ is the driving stress or pore - fluid pressure and a is crack half - length . 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 . 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 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 . 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 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 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 . 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 ) 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 : 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 . the volume of a 3 - d fracture is given by v = dlh , so that using ( 10 ), at propagation 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 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 : 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 : 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 : whereas for ductile rocks ( by substituting ( 13 ) into ( 14 )) we have : 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 : in which the appropriate values for e and k are used in ( 18 ) to calculate j . the various parameters are plotted in fig4 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 . 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 . fig1 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 . 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 . fig2 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 fig2 . the elastic index ( ei ) parameter is then calculated . in this example : the script then checks which order ( n ) of polynomial curve best fits each individual section of the p ( 1 )* x ̂ n + p ( 2 )* x ̂( n − 1 )+ . . . + p ( n )* x + p ( n + 1 ) curve and defined the coefficients for each order . 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 . 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 . 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 . 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 . 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 . 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 . 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( 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 . 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( 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 .