Hollow proppants and a process for their manufacture

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Reference is hereby made to the drawings that illustrate that which is presently regarded as the best mode for carrying out the invention.
As a developmental step, a theoretical model was formulated to show that a packed bed of hollow proppants will provide sufficient support strength for a deep well fracture.
As shown in FIG.
1, assuming a typical -20, +40 mesh proppant (with an average diameter of 0.
07 cm), in a non-Newtonian fluid (n=0.
35 and K=9.
0 poises of density 1.
1 g/cc) for use in a 0.
250 cm wide fracture, such hollow proppants will provide better fracture conrol because they settle slower due to their lower density.
A three-dimensional close-packed array of spherical particles was assumed for the theoretical model.
In practice, it was shown that due to the much larger compressive strength of ceramic materials as compared to tensile strength, fracture will initiate due to tensile stresses at the edge of the contact zone between particles.
For solid spheres, a critically stressed volume will initiate fracture due to these tensile stresses.
For hollow spheres, failure may be due to the bending of the wall during loading.
In the model, these stresses were calculated using an elastic solution for a concentrated The stress at the inner surface is critical.
This stress is calculated by: ##EQU1## The fracture pressure, P, can be predicted by the following equation: ##EQU2## where .
epsilon.
is the ratio of the inner central void to the radius of the proppant (r), .
sigma.
.
sub.
o is the bend strength of the material, and .
beta.
is an empirical constant.
From the above equation for hollow proppant and a similar approach for solid proppants, the failure pressure can be predicted based on literature values for bend strength and Young's modulus of a specific material.
These theoretical values have shown in practice to be in excellent agreement with experimental fracture conductivity values that were determined from the obvious assumption that both proppant strength and modulus will be reduced as porosity of the proppant shpere is increased