Characteristic-Time Concept in Hydraulic Fracture Configuration Evolution and Optimization (includes associated papers 23551 and 23552 )

Abstract

Summary. This paper introduces the concept of characteristic time as ameasure to separate and to optimize the roles of Griffith surface energyand fracture-fluid dissipation energy during hydraulic fracture evolutions. The analysis reveals two separate sets of solutions that correspond to twodistinct time regimes of hydraulic fracture propagation: a dominant fluiddissipation energy and a dominant fracture surface energy regime. Eachinfluence domain is governed by a characteristic time and demarcated by atransition region, where the energy contributions of each are of the sameorder of magnitude. Verification of the theoretical response trends andthe use of the characteristic-time concept as a design tool are also presented. Introduction Several methods with special emphasis on energy formulation havebeen adopted to predict hydraulic fracture configurations. BenNaceur reviewed governing equations and general modelingconsiderations. The use of generalized coordinates in the energyformulation to derive closed-form solutions for constant-height hydraulicfracture models was initiated by Blot et al. This technique hasbeen extended successfully to prediction of penny-shaped andelliptical fracture configurations including the effect of fluid leakoff. Advanced numerical investigations using finite-element, finite-difference, or boundary-integral solution techniques havealso been based on the variational methodologies associated with theminimization of the energy potential of the formation structuralresponse. The structure/fluid-pressure coupling is introduced by theviscous fracture-fluid dissipation function derived from the momentum, fluid-constitutive, and mass-conversation equations. The contributions of the governing energy components vary duringthe injection interval. These energy components include the potentialof the fracturing-fluid pressure, formation strain energy, viscousfracture-fluid dissipation, and Griffith fracture energy. Among these, the relative magnitudes of the Griffith fracture energy (associated withrock fracture toughness) and fracture-fluid dissipation during fractureconfiguration evolution have been a subject of considerable study andcontroversy. For example, Thiereclin et al. examined the role offracture toughness for penny-shaped cracks and contrasted their findingswith the results of Cleary, and Wong for the case of zero fracturetoughness. In this paper, explicit time-dependent solutions corresponding toregimes dominated by fluid dissipation energy and Griffith fractureenergy are deduced from the generalized equations for apenny-shaped fracture in the absence of leakoff effects. Characteristic times delineating the valid time domains for these solutions areidentified in terms of reservoir and fracturing-fluid parameters. Thefeasibility of the presented characteristic-time concept is alsodemonstrated by deriving supplementary results from allied rectangularfracture models. The objective of this paper is to quantify the role of the criticalhydraulic fracture parameters that govern each fracture regime. Forexample, in the aforementioned time regimes of fracturepropagation, the relative contributions of the critical Griffith surfaceenergy and fracture-fluid dissipation energy are evaluated. The transitionbetween these regimes is found in terms of the fracture parametersand formation properties. Values of characteristic time for theselected examples are determined. The solution behavior in each identifiedregime is verified by comparison with numerical results from ageneral-purpose, 3D, finite-element simulator. Also, we discuss potential applications of the characteristic-time concept to parametric sensitivity investigation, multilayered model analysis, bottomhole-treatment-pressure(BHTP) interpretation, and fracture process design optimization. Theoretical Development Energy Considerations. The pertinent energy-related contributionsfor an induced hydraulic fracture include the following. Potential of the applied traction to open the fracture. Up= A,..............................(1) where w (x, t) = crack-opening width with × = (x1, x2) on the fracturesurface. Pe (x, t) is the resultant fluid pressure defined by Pe (x, t) = P (x, t) -sigma 0, where p (x, t) = fluid pressure andsigma 0 = minimum in-situ stress of the formation. Formation strain energy corresponding to the crack openingwidth. Us = 1/2 A,..................(2) where the kernel function K (x, ) is defined by Pe(x, t)= A.....................(3) Fracture surface energy for Griffith crack propagation. Uf= A,.................................(4) where gamma cr, is the critical energy release rate of the reservoirrock. Viscous fluid dissipation energy rate. D = (i=1,2),................(5) where q (x, t) = 2D flow-rate vector components. The flow ratesatisfies the mass-conservation equation for an incompressible fluidgiven by 0,.....................(6) where qL (x, t) = fluid leakoff rate and i (x, t) = source (injection)rate. For a non-Newtonian fluid with power-law characteristics, thepressure-gradient/flow-rate relation is x, t),........................(7) where f (x, t) = -K' and qj (x, t) = [q1 (x, t) + q2 (x, t)]. Introducing Eq. 7 into Eq. 5 yields D = A. .........................................(8) P. 323^ SPEPE