Some Mathematical Problems Arising From the Oil Service Industry

Summary

INTRODUCTION

An outline is given of the mathematical modelling of some problems which occur in various situations in the oil service industry. In each case the objective is to use a mathematical model as a means of interpreting properties of the formation either during drilling or by well tests after drilling has been completed.

We begin with a brief outline of oil well testing. The main aim of such tests is the recovery of a fluid sample and an estimation of the flow capacity of the formation. We start with a description of a standard procedure for single phase flow (Homer's method (1)) and then discuss the complications introduced when the flow equations are coupled with the temperature. A description is also given of some two phase flow problems.

A second example is that of a mathematical model to be used for interpreting measurements while drilling. This consists of studying the axial vibrations generated by a roller cone bit during the drilling process. Through the model the axial force and displacement histories are calculated at the bit and related to the force and acceleration measured at an MWD (measurements while drilling) site just above the drill bit (possibly as far away as 60 ft.) or at the surface. It is, of course, important to identify the relation between the measured quantities and the values of these quantities at the bit. The objective of this interpretation is to determine formation and bit properties as drilling proceeds so as to facilitate the drilling process.