Numerical Solution of Sand Transport in Hydraulic Fracturing
Authors: Daneshy A.A. 
Published: December 2007

A numerical solution is developed for the deposition of a propping agent inside a hydraulic fracture. Such parameters as fluid leak-off into the formation, increase in sand concentration caused by leak-off, non-Newtonian fracturing fluids, hindered settling velocity, and an up-to-date geometry are taken into consideration. Three examples investigate the proppant deposition for low-, medium-, and high-viscosity fracturing fluids.

Introduction
Most hydraulic fracturing treatments are performed with a slurry composed of a fracturing fluid mixed with a propping agent. The propping agent usually consists of propping agent. The propping agent usually consists of sand particles varying in size from 4 to 60 mesh. High-strength glass beads of the same sizes also are used occasionally. The dominant use of sand as a propping agent has resulted in frequent substitution of the term "sand" for propping agent. The two terms will be used interchangeably in this paper.
The reason for using a propping agent is simple. At the end of a fracturing treatment, the hydraulic fracture has to be kept open so that the reservoir fluid can flow through it. A propping agent serves this purpose and has to meet certain requirements. A proppant must be introduced in sufficient amounts, and individual particles should be strong enough so that they will not be crushed by the action of the in-situ stresses. Furthermore, the sand bed should be conductive enough to transmit the reservoir fluid flowing into it.
Although particle transport is well established in chemical engineering, most research has been conducted on flow through cylindrical pipes. The available research for particle transport through parallel plates is scarce. One of particle transport through parallel plates is scarce. One of the fast investigations of proppant transport in hydraulic fractures was conducted by Kern et al., who considered bed buildup for vertical fractures. After several studies of proppant movement in horizontal fractures, bed buildup in vertical fractures was treated by Babcock et al. In their comprehensive study, these authors conducted a theoretical and experimental investigation of the proppant movement and deposition between two parallel plates with limited vertical extent. Schols and parallel plates with limited vertical extent. Schols and Visser investigated the mechanics of bed buildup and related it to particle size. Van Domselaar and Visser considered proppant concentration in a fracture created by a viscous gel in which there is no particle settlement.
An attempt is made to consider proppant settlement in hydraulic fractures with conditions similar to those encountered in the field. Allowing for a change in either the proppant type or concentration during treatment, this proppant type or concentration during treatment, this study includesthe influence of fluid leak-off on fracture geometry and the concentration of the proppant in it, andconsideration of pseudoplastic as well as Newtonian fluids. The results were obtained by numerical computations carried out with a computer.
Proppant Deposition in Hydraulic Fractures Proppant Deposition in Hydraulic Fractures The determination of a proppant schedule usually involves the fracture geometry.
Suppose L denotes the length, omega max the wellbore width, and h the height of an induced vertical hydraulic fracture. Assuming that q, the fluid injection rate, stays constant during treatment, that the treatment fluid does not change, and that h is constant, then the variations of L and omega max with treatment time, t, can be expressed as
(1) (2)
JPT
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