Calculation of Vertical Fracture Containment in Layered Formations

Abstract

Summary An analytic procedure for calculating vertical fracture extent in symmetrical trilayered formations was extended to multilayered, asymmetrical formations using a semianalytic technique. The fracture extents computed by this method were compared with those calculated with the finite-element method. It was found that even for modulus variations between layers as large as a factor of 5, the semianalytic procedure gave exactly the same results as the finite-element solution in a fraction of the computation time and with significantly less manual data manipulation. It is recommended that the analytic and numerical procedures be used in a complementary manner to calculate fracture-width profiles in layered formations. Introduction Massive hydraulic fracturing (MHF) has become an important technique to improve productivity of tight gas sands prevalent in Canada and the U.S. In such extremely low-permeability formations, very long propped fractures are desirable to achieve acceptable production rates. Such MHF treatments can transform an uneconomical field into a viable producer. A key uncertainty that has been identified as an important consideration in MHF operations is fracture containment (see Refs. 1 through 3). It is essential that fractures be contained mainly to the pay zone because fracture breakout into overlying or underlying formations can have serious consequences on the effectiveness of MHF operations, including the following.If only a small portion of the fracture surface is in contact with the pay zone, the result may be an uneconomical well that could be economical if properly treated.The fracture could penetrate from the tight zone identified for stimulation into an adjacent high-permeability gas zone. In this case, production tests may lead to overestimating the resources and productivity of the tight zone. Typically, when an MHF operation is designed, it is assumed that the fracture will be contained from above and below by horizontal inhomogeneities in the strata. Several studies have demonstrated by theory3,4 and experiment5–7 that contrasts in critical stress intensity factor do not act as effective barriers to vertical fracture extension. Rather, it has been demonstrated that the primary mechanism of fracture containment is vertical variation in tectonic stress. Vertical fracture extension tends to be arrested in zones with high horizontal tectonic stress. Interfaces that are weak in shear can also be effective barriers to fracture penetration.5,7 Contrast in Young's modulus between layers has also been suggested to have some effect on containment,4,8 although the effect is not as dramatic as classic theory9 would predict. Experimental evidence5,7 has been unable to demonstrate an effect of modulus variations on containment. Several authors have attempted to predict quantitatively the vertical extent of fractures in layered formations. Simonson et al.2 presented an analytic expression for the extent of fracture penetration across a stress discontinuity in an otherwise homogeneous reservoir. Advani and Lee8 used the finite-element method for the same problem and included the effects of modulus contrast. Both of these studies dealt with vertically static fractures subject to the vertical, plane-strain approximation. van Eekelen4 presented an approximate, but simple, dynamic analysis of fracture growth in which the fracture was assumed to be growing at a significant rate in both the vertical and axial directions. Estimates for the fracture shape were obtained from the approximate solution of the coupled plane-strain-elasticity/fluid-flow equations. More recently, Settari and Cleary10 discussed a very elaborate fracture growth model that apparently has the capability to calculate vertical fracture containment. However, the procedure for calculating vertical fracture extent was not explained. The major limitation of previous studies on fracture containment is the assumption of reservoir property and stress symmetry about the midheight of the reservoir. Generally, reservoir properties and tectonic stresses are not symmetric. The problem of calculating vertical fracture extent in an asymmetric reservoir is considerably more complex than the symmetric case because an extra degree of freedom is introduced by the asymmetry, While in the symmetric case, the midheight of the fracture is known to be the line of vertical symmetry, in the asymmetrical case, the fracture midheight is unknown and varies as the fracture grows. In this paper, the analytic fracture penetration formula of Simonson et al.2 is generalized into a straightforward, semianalytic procedure for computing vertical fracture extent in homogeneous reservoirs with arbitrary horizontal-stress distribution. It is also shown how the finite-element method can be used to generate accurate estimates of vertical fracture extent in inhomogeneous reservoirs composed of layers of differing Young's modulus and tectonic stress. It is demonstrated by comparison of the numerical and semianalytic results that in most realistic situations the homogeneous, semianalytic approach gives adequate estimates of vertical fracture extent, even for large contrast in moduli between layers. Theoretical Framework As was done in most of the previous studies on fracture containment, the present analysis is performed for a vertical plane-strain fracture. This approximation is valid if the fracture is much longer than it is high, as should be the case for MHF. The plane-strain fracture problem under consideration is shown in Fig. 1, which depicts a vertical section perpendicular to the fracture plane at an arbitrary horizontal position along one of the fracture wings. The principal direction of fracture growth is orthogonal to the plane of the figure, and growth in this direction is limited by the finite rate at which fracturing fluid can flow to the advancing tip. Propagation of the fracture in this direction is reasonably well understood and will not be addressed in detail in this paper. For a fracture that has been vertically "contained," movement of the upper and lower tips in Fig. 1 will be slow and the vertical pressure profile will be approximately hydrostatic. Hence, for the purposes of our analysis, it will be assumed that the fracture is at equilibrium with respect to the vertical tip movement and that the vertical pressure profile is hydrostatic within the fracture, These assumptions are consistent with previous analyses of vertical fracture containment.2,8