An Empirical Approach For Relating Drilling Parameters
Authors: Cunningham R.A. 
Published: December 2007

This paper proposes using the results of one laboratory test for estimating the effects of bit weight, rotational speed, drilling strength, and differential pressures on penetration rate. The concept of using the rock bit as a fundamental testing tool for determining drilling strength also is proposed. Drilling strength then becomes a property that can be measured proposed. Drilling strength then becomes a property that can be measured and used mathematically.

Introduction
Attempts to relate drilling rate of rock bits having rolling cutters with various drilling parameters have been made before. Theoretical approaches have been based on stress and failure of rock beneath individually loaded rock bit teeth. It was hoped that this information could predict rock failure and, in turn, drilling rates could be derived. This is the basic, desirable approach for incorporating all drilling factors in a classic analysis. However, the investigator quickly is overwhelmed by complexities of fracture theories, bit geometry, formation heterogeneity, mud properties, bit hydraulics, weight, rpm, and other significant factors that are known or unknown.
The investigator is forced to retreat and accept a less scientific but more workable approach when trying to provide a useful tool for certain engineering estimates. provide a useful tool for certain engineering estimates. Perhaps the simplest and most practical approach Perhaps the simplest and most practical approach assumes that the rock bit itself is a valid testing tool. By performing drilling-rate tests on a variety of formations, a consistent and orderly relationship has been found that relates drilling rate to bit weight (Fig. 1).
Results of these tests show a consistent response of drilling rate to weight for various formations. Using these data as a basis, drilling rate for a given formation appears proportional to weight per inch of bit diameter raised to a proportional to weight per inch of bit diameter raised to a value equal to the slope of the curve for that particular formation. Since experience has shown that drilling rate is proportional to rotary speed for a wide range of rotary speeds at atmospheric pressures, drilling rate can be expressed as
(1)
To determine K and a in Eq. 1, we must conduct drilling tests in a given formation at two different bit weights and at equal rpm's. By substituting DR, W, and N from two separate tests into Eq. 1, we have two equations with two unknowns from which K and a can be determined. Once K and a are found, drilling rate can be calculated for any other weight or rotary speed desired.
Fig. 1 further indicates that, in general, the higher the compressive strength, the more weight is required to drill a formation at a specified rate. However, a close correlation between compressive strength and drilling rate does not exist.

Drilling Strength
These relationships have been known and used for some years. They are reviewed here because they are the basis for additional developments given in this study.
Let us accept the general relationship shown in Fig. 1 and expressed by Eq. 1. We now can construct a family of lines relating DR and W (Fig. 2). By spacing these systematically, it is possible to locate approximately a scale that corresponds with the compressive strength of Fig. 1. We arbitrarily called these lines, lines of equal "drilling strength" (sigma d in thousands of psi). Several formations shown in Fig. 1 are repeated as dashed lines in Fig. 2 for reference.
JPT
P. 987