The "Perfect - Cleaning" Theory of Rotary Drilling
Authors: Maurer W.C. 
Published: December 2007

MAURER, W.C., JERSEY PRODUCTION RESEARCH CO., TULSA, OKLA.

Abstract
A drilling-rate formula for roller-cone bits is derived from rock cratering mechanisms. This formula holds for "perfect cleaning", which is defined as the condition where all of the rock debris is removed between tooth impacts. Under these conditions, the drilling rate of is directly proportional to the rotary speed and to the bit weight squared, and inversely proportional to the bit diameter squared and to the rock strength squared. Under imperfect cleaning conditions, such as those usually present in field drilling, regrinding of the cuttings occurs under the bit, and the drilling rates fall below those for perfect cleaning.

Introduction
Drilling with a roller-cone bit consists of two fundamental operations: the formation of craters under the bit teeth, and the removal of the broken rock from the craters. To delineate the mechanisms of drilling, it is first necessary to understand the mechanisms involved in the formation of individual craters and then to relate these mechanisms to the over-all drilling operation.

CRATER FORMATION
When a bit tooth impacts a rock, the rock is elastically deformed until the crushing strength of the rock is exceeded, at which time a wedge of crushed rock is formed below the tooth as shown in Fig. 1. As additional force is applied to the tooth, the crushed material is compressed and exerts high lateral forces on the solid material surrounding the crushed wedge. When these forces become sufficiently high, fractures are initiated below the tooth and propagate to the free surface of the rock. The trajectories of these fractures intersect the principal stresses at a constant angle, as predicted by both Mohr's and Griffith's theories of failure. It has been experimentally shown that the volume of an impact crater, Vc, varies as
.........(1)
where Ec is the total energy imparted to the rock during the formation of the crater, and Eo is the threshold energy required before cratering is initiated (Fig. 2). Indexing experiments have shown that, when there is a second free face to which the crater can fracture, a larger volume of material is removed and that above a transition zone there is a linear relationship between Vc and Ec, which has a slope equal to the relationship for the single crater. The beneficial effect of indexing tends to offset the threshold energy requirement to a large extent, as shown in Fig. 2, and the relationship is closely represented by
......(2)
Rate-of-loading effects have been used to explain the nonproportional relationships usually obtained between drilling rate and rotary speed. The data in Fig. 3 show that the Ec/Vc parameter remained constant for craters produced with impact velocities ranging from 10 to 8,000 ft/sec. Since no rate-of-loading effects were detected over this wide range of loading rates, it seems improbable that there would, be detectable rate-of-loading effects present in the small changes in impact velocities produced by varying the rotary speed. The writer has shown (Fig. 4) that, for craters produced in rock by impacting spheres,
..............(3)
where X is the depth of penetration. Simon has shown that this relationship also holds for craters produced in rock by penetrating wedge-shaped chisels.
JPT
P. 1270^