Introduction

The study of pulse propagation in one dimensional random media arises in many applied contexts. While reflection and transmission of monochromatic waves was studied extensively some time ago [1–6 and references therein], new and perhaps surprising results emerge in the study of pulses that cannot be understood simply from the single frequency analysis by Fourier synthesis. The numerical study of Richards and Menke [7] drew our attention to these questions and led to [8] and [9]. Here we extend and simplify the analysis of [8] and give several new results. The computations are at a formal level comparable to the one in [8].

In [8] we analyzed the reflection of a pulse that is broad compared to the size of the inhomogeneities of the random medium. The random functions characterizing the medium properties were statistically homogeneous. We gave a rather complete description of the reflected signal process in a well defined asymptotic limit in which it has a canonical structure. We introduced the notion of a windowed process and showed that the canonical reflection process is windowed and Gaussian. We found a scaling law for the power spectral density but not its explicit form. All this was subjected to extensive numerical simulations in [9] where an intrinsic scaling, localization length scaling, was introduced that makes comparison to the theory much more reliable. This intrinsic scaling idea is not fully understood theoretically but seems to be very promising.

In this paper we extend the analysis to random media that are not statistically homogeneous. The incident pulse is now broad compared to the size of the inhomogeneities but short compared to the scale of variation of the mean properties.

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.

INTRODUCTION

This paper discusses certain types of stability questions that have been largely ignored in the literature, i.e. continuous dependence on geometry and continuous dependence on modeling. Although we shall consider these questions primarily in the context of ill-posed problems we shall briefly indicate some difficulties that might arise under geometric and/or modeling perturbations in well posed problems.

In setting up and analyzing a mathematical model of any physical process it is inevitable that a number of different types of errors will be introduced e.g. errors in measuring data, errors in determining coefficients, etc. There will also be errors made in characterizing the geometry and in formulating the mathematical model. In most standard problems the errors made will induce little error in the solution itself, but for ill-posed problems in partial differential equations this is no longer true.

Throughout this paper we shall assume that a “solution” to the problem under consideration exists in some accepted sense, but in the case of ill-posed problems such a “solution” will invariably fail to depend continuously on the data and geometry. We must appropriately constrain the solution in order to recover the continuous dependence (see [8]); however, appropriate restrictions are often difficult to determine. In the first place any such constraint must be both mathematically and physically realizable. At the same time a given constraint must simultaneously stabilize against all possible errors that may be made in setting up the mathematical model of the physical problem. Since a constraint restriction has the effect of making an otherwise linear problem nonlinear, one must use care in treating the various errors separately and superposing the effects.

This textbook was initially developed for the introductory course in finite element methods at the Department of Mechanical Engineering and Applied Mechanics, the University of Michigan, Ann Arbor, Michigan. It is based on four years of teaching experience of first-year graduate students and some senior undergraduate students in the engineering college. Because of the mechanical engineering environment, heat conduction problems are covered, as are standard stress analysis of solids and structures. Many small-size BASIC and FORTRAN programs are given so that readers can apply them to solve exercises using either microcomputers such as IBM PCs and compatibles or mainframes of computer networks that support FORTRAN iv. BASIC programs are available in separate volumes with a diskette for the IBM PC and compatibles by reader's request (see back of book for ordering instructions). FORTRAN programs are also available to readers who specially request them from the author. At present, BASIC compilers are available for some microcomputers in order to improve speed of computation. In the author's opinion, they are very impressive and encourage the use of microcomputers even for finite element methods. It is noted, however, that these programs are primarily designed for educational purposes – to teach the theory of finite element methods.