A Multiple Regression Approach to Optimal Drilling and Abnormal Pressure Detection
Authors: Bourgoyne A.T. , Young F.S. 
Published: December 2007

Abstract
Over the past decade, a number of drilling models have been proposed for the optimization of The rotary drilling process and the detection of abnormal pressure while drilling. These techniques have pressure while drilling. These techniques have been largely based Upon limited held and laboratory data and often yield inaccurate results. Recent developments in onsite well monitoring systems have made possible the routine determination of the best mathematical model for drilling optimization and pore pressure detection. This modeling is accomplished through a multiple regression analysis of detailed drilling data taken over short intervals. Included in the analysis are the effects of formation strength, formation depth, formation compaction, pressure differential across the hole bottom, bit diameter and bit weight, rotary speed, bit wear, and bit hydraulics.This paper presents procedures for using the regressed drilling model for selecting bit weight rotary speed, and bit hydraulics, and calculating formation pressure from drilling data. The application of the procedure is illustrated using field data.

Introduction
Operators engaged in the search for hydrocarbon reserves are facing much higher drilling costs as more wells are drilled in hostile environments and to greater depths. A study by Young and Tanner has indicated that the average well cost per foot drilled is increasing at approximately 7.5 percent/ year. Recently, more emphasis has been placed on the collection of detailed drilling data to aid in the selection of improved drilling practices.At present, many people are using one drilling model for optimizing bit weight and rotary speed, a different drilling model for optimizing jet bit hydraulics, and yet another model for detecting abnormal pressure from drilling data. Each model has been based on meager laboratory and field data. We have tried here to combine what is known about the rotary drilling process into a single model, develop equations for calculating formation pore pressure and optimum bit weight, rotary speed, and jet bit hydraulics that are consistent with that model, and provide a method for systematically "calibrating" the drilling model using field data.

DRILLING MODEL
The drilling model selected for predicting be effect of the various drilling parameters, xj, on penetration rate, dD/dt, is given by penetration rate, dD/dt, is given by(1)
when Exp (z) is used to indicate the exponential function ez. The modeling of drilling behavior in a given formation type is accomplished by selecting the constants a, through a 8 in Eq. 1. Since Eq. 1 is linear, those constants can be determined from a multiple regression analysis of field data.

EFFECT OF FORMATION STRENGTH
The constant a, primarily represents the effect of formation strength on penetration rate. It is inversely proportional to the natural logarithm of the square proportional to the natural logarithm of the square of the drillability strength parameter discussed by Maurer. It also includes the effect on penetration rate of drilling parameters that have not yet been mathematically modeled; for example, the effect of drilled solids.

EFFECT OF COMPACTION
The terms a2x2 and a3x3 model the effect of compaction on penetration rate. x2 is defined by(2)
and thus assumes an exponential decrease in penetration rate with depth in a normally compacted penetration rate with depth in a normally compacted formation. The exponential nature of the normal compaction trend is indicated by the published microbit and field data of Murray, and also by the field data of Combs (see Fig. 1).
SPEJ
P. 371