Proppant Concentration in and Final Shape of Fractures Generated by Viscous Gels
Authors: van Domselaar H.R. , Visser W. 
Published: December 2007

Abstract
A mathematical procedure is given for calculating proppant concentration and final fracture shape for proppant concentration and final fracture shape for a fracture generated by injection of a viscous gel in which the propping material does not settle. To prevent bridging in the fracture, a decreasing pad prevent bridging in the fracture, a decreasing pad volume is present ahead of the proppant slurry. If combined with a criterion for proppant admittance - expressing the minimum width required for nonbridging particle transport - the developed procedure will result in a realistic design of fracturing treatments.

Introduction
Hydraulic fracturing is a well known technique for improving the productivity of wells by creating a highly conductive path in the reservoir. This path is made by fracturing the formation through the injection of fluid into a well at pressures above the breakdown pressure. To keep the fracture open after the treatment, propping material is injected with the fracturing fluid. Settling of the proppant can be reduced or even prevented by using viscous oil- and water-based gels as fracturing fluids.From theoretical considerations it follows that the cross-section of a propagating hydraulic fracture is approximately elliptical. The dimensions of the ellipse are determined by the injection rate, the injected volume of fracturing fluid, and fracturing-fluid properties-taking into account the volume of fluid loss to the permeable formation. The fluid loss depends partly on the fluid potential gradients at the fracture walls, resulting in a time-dependent fluid-loss coefficient, which is proportional to the reciprocal of the square root of the exposure time.Since the propping material can cause early screenout, a relatively large sand-free pad of fracturing fluid is injected to initiate the fracture. This pad volume moves ahead of the fluid (gel) containing the propping material. Owing to spurt and filtration losses - highest at the fracture tip but decreasing gradually toward the well - the pad length in the fracture will decrease. The proppant-laden fracturing fluid is also subject to fluid loss, which causes the proppant concentration to increase with distance from the well. The propped fracture width obtained after the fracturing treatment depends on the balance between pad volume and proppant concentration in the fracture. A treatment design should therefore aim at optimization of pad volume, fracturing-fluid volume, and proppant concentration. A design program should deliver practical pumping schedules, which generate fractures of required penetration and conductivity. penetration and conductivity. The present study is a new step toward a more realistic design of fracturing treatments. Differential equations describing the proppant distribution in fractures created by very viscous fluids (no settling) are derived and solved.

DEFINITION OF THE MATHEMATICAL MODEL
The derivation of the differential equations describing the proppant distribution is based on the following model premises.1. Vertical fractures are of rectilinear shape.2. Two symmetric fracture wings move diametrically from the well.3. Fracture dimensions follow the relations established by Geertsma and de Klerk.4. Gel and proppant move with the same velocity in a piston-like manner.5. Rheological properties of the gel prevent settling of the proppant.6. Fluid-loss is proportional to the square root of the exposure time.7. A decreasing proppant-free pad moves ahead of the proppant suspension.Based on these conditions, a set of differential equations subject to the boundary conditions has been derived (see Appendix A). In Appendix B the applied finite-difference scheme, and in Appendix C the solution procedure are discussed. To describe radial fractures, a simple coordinate transformation has been given in Appendix D.
SPEJ
P. 531