A Three-Dimensional Hydraulic Propagation Theory Coupled With Two-Dimensional Proppant Transport

Abstract

ABSTRACT A three dimensional hydraulic fracture propagation theory, coupled with two dimensional proppant transport, has been developed. Principal features of the theory are listed below. Initial fracture is assumed to be initiated from a line source with length equal to the perforated interval, and the fracture will, propagate as an ellipsoid if the formation is homogeneous with no stress variation.Fracture shape deviates from the perfect ellipsoid as soon as the fracture reaches a layer having stress change.Pressure drops from the center of the fracture in the perforated interval to fracture tips in all directions are equal. Thus the width profile in every flow line varies according to the stress variation in that direction.Proppant is assumed to move with equal velocity as fluid along the flow line but has an additional vertical settling velocity which allows proppant pumped at earlier stages to mix with proppant pumped in latter stages. Settled proppant will continue to fall to a place where fracture is too narrow to allow further drop.Fluid losses are considered for both the pay zone and non-pay zones. With state-of-the-art improvement in microfracturing processes, fluid losses in non-pay zones are measurable and should be measured. The new theory can handle the special case of a fracture that is confined to a constant gross height either from above or below. Deficiencies of two dimensional models are removed by the new theory: Assumption that constant height is achieved instantly.Assumption that the width at any vertical plane is either constant (Christianovich and Zheltov)1 or with a fixed shape (Perkins and Kern).2 Subject paper presents development of theory and an example of selecting proppant-laden fluids for optimum proppant transport.