Determining hydraulic fracture profiles in multi-layered subsurface formations

Abstract

Techniques for fracturing a formation include receiving a data representing depth-dependent horizontal in-situ stress and a fracture toughness for one or more layers of the formation, predicting a hydraulic fracture profile within the formation, and pumping a fluid into a wellbore to fracture the formation at a perforation location based on the predicted hydraulic fracture profile. Predicting the hydraulic fracture profile can include determining a location of an upper portion of the fracture profile when an upper fracture tip stress intensity factor satisfies an upper fracture tip propagation condition, and determining a location of a lower portion of the fracture profile when a lower fracture tip stress intensity factor satisfies a lower fracture tip propagation condition. Predicting the hydraulic fracture profile can include determining a depth-dependent width profile of the hydraulic fracture based on the determined locations of the upper and lower portions.

What is claimed is: 
     
         1 . A method for fracturing a formation, the method comprising:
 receiving, at a processor, data representing a depth-dependent horizontal in-situ stress for at least two layers of the formation and a fracture toughness for the at least two layers of the formation;   predicting, by the processor, a hydraulic fracture profile within the formation for at least one perforation pressure by:
 determining an upper fracture tip stress intensity factor within the formation for the at least one perforation pressure, the determining being based on the depth-dependent horizontal in-situ stress of one or more layers of the at least two layers of the formation; 
 determining a first vertical depth within the wellbore at which the upper fracture tip stress intensity factor satisfies an upper fracture tip propagation condition based on the fracture toughness of one or more layers of the at least two layers of the formation; 
 determining a lower fracture tip stress intensity factor within the formation for the at least one perforation pressure, the determining being based on the depth-dependent horizontal in-situ stress of one or more layers of the at least two layers of the formation; 
 determining a second vertical depth within the wellbore at which the lower fracture tip stress intensity factor satisfies a lower fracture tip propagation condition based on the fracture toughness of one or more layers of the at least two layers of the formation; 
 determining a height of the hydraulic fracture profile for the at least one perforation pressure, the height extending from the first vertical depth to the second vertical depth; and 
 determining a depth-dependent width of the hydraulic fracture profile at the at least one perforation pressure, the depth-dependent width extending along the height of the hydraulic fracture profile, the predicted hydraulic fracture profile for the at least one perforation pressure being at least partially defined by the determined height and the determined depth-dependent width for the at least one perforation pressure; 
   determining, by the processor, a perforation pressure value based on at least one of the predicted hydraulic fracture profiles of the formation; and   injecting, by a pump, a fluid into the wellbore at the determined perforation pressure to hydraulically fracture the formation.   
     
     
         2 . The method of  claim 1 , wherein the upper fracture tip stress intensity factor, the lower fracture tip stress intensity factor, and the depth-dependent width of the hydraulic fracture profile are determined using Gauss quadrature. 
     
     
         3 . The method of  claim 1 , comprising measuring, by a logging device, data representing a Young's modulus, a Poisson's ratio, and a layer thickness for each layer of at least two layers of the formation, wherein predicting the hydraulic fracture profile is based on the Young's modulus, the Poisson's ratio, and the layer thickness for each layer of the at least two layers of the formation. 
     
     
         4 . The method of  claim 1 , wherein predicting the hydraulic fracture profile of the formation comprises determining, by the processor on a layer-by-layer basis, whether the upper fracture tip stress intensity factor satisfies the upper fracture tip propagation condition and whether the lower fracture tip stress intensity factor satisfies the lower fracture tip propagation condition. 
     
     
         5 . The method of  claim 4 , wherein predicting the hydraulic fracture profile of the formation comprises dynamically discretizing, by the processor on the layer-by-layer basis, each layer of the at least two layers of the formation into a plurality of one dimensional finite elements having an order greater than one. 
     
     
         6 . The method of  claim 1 , wherein the depth-dependent width of the hydraulic fracture profile along the height of the hydraulic fracture profile is determined using a plane strain Young's modulus. 
     
     
         7 . The method of  claim 1 , wherein injecting the fluid into the wellbore causes a hydraulic fracture to propagate until a height of the hydraulic fracture reaches the determined height of the hydraulic fracture profile. 
     
     
         8 . The method of  claim 1 , wherein the upper and lower fracture tip stress intensity factors are determined based on a depth-dependent hydrostatic pressure within the formation. 
     
     
         9 . The method of  claim 1 , comprising plotting, by the processor, a fracture height map on a first plot representing a progression of the height of the predicted hydraulic fracture profile as a function of the at least one perforation pressure. 
     
     
         10 . A method for fracturing a formation, the method comprising:
 receiving, at a processor, data representing a depth-dependent horizontal in-situ stress and a fracture toughness for one or more layers of the formation;   predicting, by the processor, a hydraulic fracture profile of the formation by:
 determining a location of an upper portion of the hydraulic fracture profile when an upper fracture tip stress intensity factor satisfies an upper fracture tip propagation condition based on the depth-dependent horizontal in-situ stress and the fracture toughness for the one or more layers of the formation, the upper portion being above a perforation location of a wellbore; 
 determining a location of a lower portion of the hydraulic fracture profile when a lower fracture tip stress intensity factor satisfies a lower fracture tip propagation condition based on the depth-dependent horizontal in-situ stress and the fracture toughness for the one or more layers of the formation, the lower portion being below the perforation location of the wellbore; and 
 determining a depth-dependent width of the hydraulic fracture profile based on the determined location of the upper portion and the determined location of the lower portion; and 
   injecting, by a pump, a fluid into the wellbore to hydraulically fracture the formation at the perforation location based on the predicted hydraulic fracture profile.   
     
     
         11 . The method of  claim 10 , wherein the depth-dependent width of the hydraulic fracture profile is determined using a plane strain Young's modulus. 
     
     
         12 . The method of  claim 10 , wherein predicting the hydraulic fracture profile of the formation comprises using one-dimensional finite elements to discretize each layer of the one or more layers of the formation. 
     
     
         13 . The method of  claim 12 , wherein the one-dimensional finite elements are at least second order elements. 
     
     
         14 . The method of  claim 10 , wherein predicting the hydraulic fracture profile within the formation comprises accounting for a depth-dependent hydrostatic pressure acting on one or more surfaces of the hydraulic fracture profile. 
     
     
         15 . The method of  claim 10 , wherein the location of the lower portion of the predicted hydraulic fracture profile is in a first layer of the one or more layers, and the location of the upper portion of the predicted hydraulic fracture profile is in a second layer of the one or more layers, the first layer being below the second layer. 
     
     
         16 . The method of  claim 10 , wherein the upper portion of the predicted hydraulic fracture profile is a upper-most tip of the hydraulic fracture profile, and the lower portion of the predicted hydraulic fracture profile is a lower most-tip of the hydraulic fracture profile. 
     
     
         17 . The method of  claim 10 , comprising determining, by the processor, a pump schedule for the pump based on the predicted hydraulic fracture profile, wherein the fluid is injected into the wellbore according to the pump schedule. 
     
     
         18 . A method for fracturing a formation, the method comprising:
 receiving, by a processor, data representing (i) fracture toughness properties for a plurality of layers of the formation, and (ii) depth-dependent horizontal in-situ stresses exerted on the plurality of layers of the formation;   automatically generating, by the processor, a plurality of one-dimensional elements along a vertical direction of a first layer of the formation;   numerically predicting, by the processor, whether a hydraulic fracture is expected to occur within the first layer based on (i) a perforation pressure, (ii) a fracture toughness of the layer based on the received fracture toughness properties, and (iii) a depth-dependent horizontal in-situ stress of the layer based on the received depth-dependent horizontal in-situ stresses; and   when the hydraulic fracture is expected to occur in the first layer, injecting, by a pump, a fluid into a wellbore at the perforation pressure to hydraulically fracture the first layer of the formation such that a hydraulic fracture propagates through at least a portion of the first layer of the formation and stops within the first layer of the formation.   
     
     
         19 . The method of  claim 18 , comprising:
 when the hydraulic fracture is not expected to occur in the first layer, automatically generating, by the processor, a plurality of one-dimensional elements along the vertical direction of a second layer of the formation and numerically predicting whether a hydraulic fracture is expected to occur within the second layer; and   when the hydraulic fracture is expected to occur in the second layer, injecting, by the pump, the fluid into the wellbore at a perforation pressure at which the hydraulic fracture is expected to occur in the second layer to hydraulically fracture the second layer of the formation such that a hydraulic fracture propagates through the first layer and stops within the second layer of the formation.   
     
     
         20 . The method of  claim 19 , comprising:
 when the hydraulic fracture is not expected to occur in the second layer, automatically generating, by the processor, a plurality of one-dimensional elements along the vertical direction of a third layer of the formation and numerically predicting whether a hydraulic fracture is expected to occur within the third layer; and   when the hydraulic fracture is expected to occur in the third layer, injecting, by the pump, the fluid into the wellbore at a perforation pressure at which the hydraulic fracture is expected to occur in the third layer to hydraulically fracture the third layer of the formation such that a hydraulic fracture propagates through the first and second layers and stops within the third layer of the formation.