Introduction During the last few years, there has been an explostion of information in the field of well-test analysis. Because of increased physical understanding of transient fluid flow, it is possible to analyze the entire pressure history of a well test, not just long-time data as in conventional analysis.1 It is now often possible to specify the time of beginning of the correct semilog straight line and determine whether the correct straight lie has been properly identified. It is also possible to identify wellbore storage effects, and the nature of wellbore stimulation as to permeability improvement, or fracturing, and to quantitatively analyze those effects. Such accomplishments have been augmented by attempts to understand the short-time pressure data from well testing - data that were often classified as too complex for analysis. One recent study of short-time pressure behavior2 showed that it was important to specify the physical nature of the stimulation in considering the behavior of a stimulated well. That is, stating that the van Everdingen-Hurst infinitesimal skin effect was negative was not sufficient to define short-time well behavior. For instance, acidized (but not acid-fractured) and hydraulically fractured wells might not necessarily exhibit the same behavior at early times, even though they could possess the same value of negative skin effect. In the same manner, hydraulic fracturing leading to horizontal or vertical fractures could produce the same skin effect, but with possibly different short-time pressure data. This could then provide a way to determine the orientation of fractures created by this type of well stimulation. In fact, it is generally agreed that hydraulic fracturing usually results in one vertical fracture, the plane of which includes the wellbore. Most studies of the flow behavior for a fractured well consider vertical fractures only.3–11 Yet it is also agreed that horizontal fractures could occur in shallow formations. Furthermore, it would appear that notch-fracturing would lead to horizontal fractures. Surprisingly, no detailed study of the horizontal fracture case had been performed until recently.12 A solution to this problem was presented by Gringarten and Ramey.13 In the course of their study, it was found that a large variety of new transient pressure behavior solutions useful in well and reservoir analysis could be constructed from instantaneous Green's functions.14 Possibilities included a well with a single vertical fracture in an infinite reservoir, or at any location in a rectangle.

Although it is an old method, the use of Green's functions to solve unsteady flow problems in reservoir engineering is not widely practiced. The reason is that it is difficult to find the appropriate Green's junction. In this study, tables of instantaneous Green's and source functions were prepared that can be used with the Newman's prepared that can be used with the Newman's product method to generate solutions for a wide product method to generate solutions for a wide variety of reservoir flow problems. New solutions for infinite conductivity sources were also prepared. Introduction The transient flow of a slightly compressible fluid in a homogeneous and anisotropic porous medium D, bounded by a surface Se (Fig. 1), is described by the diffusivity equational derived from the continuity equation and Darcy's law? Assuming constant permeabilities, porosity, and fluid viscosity and small pressure gradients everywhere, and neglecting the effect of gravity, the diffusivity equation can be written as ..(1) where x, y and z are the principal axes of permeability, and the coefficients Nx, Ny, Nz are permeability, and the coefficients Nx, Ny, Nz are the principal diffusivities. When Nx = Ny = Nr (cylindrical systems), the diffusivity equation can be written as ........................(2) The diffusivity constants are given by ...........(3) Many techniques have been used for solving Eqs. 1 and 2. Most of these were first used for solving heat conduction problems and have since been applied by different authors to petroleum engineering. In the literature, most problems were solved either with Laplace transforms or Fourier transforms. One useful method employs Lord Kelvin's instantaneous point source solution. Another method that is of value although very rarely used is the Green's function method. This study presents the point source solution as part of a more general theory of Green's functions. part of a more general theory of Green's functions. This theory is applied in combination with other techniques to yield immediate solutions to difficult flow problems, some of which either have not been published or have been solved by long analytical published or have been solved by long analytical methods or sophisticated numerical techniques only. SPEJ P. 285

A theory of heat extraction from fractured hot dry rock is presented, based on an infinite series of parallel vertical fractures of uniform aperture. Fractures are uniformly spaced and drain heat from blocks of homogeneous and isotropic impermeable rock. Cold water enters at the bottom of each fracture, and solutions are given in terms of dimensionless parameters from which the exiting water temperatures at the top of the fractures can be determined. An example of the application of the theory demonstrates how a multiply fractured system provides a more efficient mechanism for heat extraction than a single fracture in hot dry rock.

Summary We present a new time-domain method for the deconvolution of well test data which is characterized by three novel features:Instead of the rate-normalized pressure derivative itself, we estimate its logarithm, which makes explicit sign constraints necessary;the formulation accounts for errors in both rate and pressure data, and thus amounts to a Total Least Squares (TLS) problem; andregularization is based on a measure of the overall curvature of its graph. The resulting separable nonlinear TLS problem is solved using the Variable Projection algorithm. A comprehensive error analysis is given. The paper also includes tests with a simulated example and an application to a large field example. Introduction With current trends towards permanent downhole instrumentation, continuous bottomhole well pressure monitoring is becoming the norm in new field developments. The resulting well-test data sets, recorded mainly during production, can consist of hundreds of flow periods and millions of pressure data points stretched over thousands of hours of recording time. Such data sets contain information about the reservoir at distances from the well which can be several orders of magnitude larger than the radius of investigation of a single flow period. Conventional derivative analysis is thus ill equipped to access the full potential information content. What is required is an analysis method which can extract the response which the reservoir would exhibit when subjected to a single drawdown at constant rate over any period of time up to the entire production period. In mathematical terms, this is a deconvolution problem. Since its first formulation by Hutchinson and Sikora in 1959,1 it has received sporadic, but recurring attention.2-17 This paper presents a new approach which is based on a regularized, nonlinear TLS formulation. It is an update on earlier versions which were presented at the SPE Annual Meetings in 200118 and 200219 (henceforth referred to as "Paper I" and "Paper II"). More recently, our approach was taken up by Levitan,20 who subjected it to a critical evaluation and suggested some modifications. In terms of the usual classification into time-domain and spectral approaches, ours is a time-domain approach. It differs from earlier approaches in this category in three important ways:The solution is encoded in terms of the logarithm of the rate-normalized pressure derivative, which automatically ensures strict positivity of the derivative itself at the expense of rendering the problem nonlinear. However, we are thus able to avoid explicit constraints on the solution space which made previous constrained approaches so difficult, yet still cannot prevent zeros in the deconvolved derivative.A new error measure accounts for uncertainties not only in the pressure, but also in the rate data, which are usually much less accurately known. Thus, provided sufficient data are available, our method can provide a joint estimate of initial pressure, rates, and response parameters; the time-consuming manual correction of rate errors is rendered obsolete. The mathematical formulation is an instance of what is known as a TLS problem in the numerical analysis literature and as an "Errors-In-Variables" problem in statistics. TLS has become a standard approach in parameter estimation problems, but its application to well-test analysis seems to be new.Regularization is based on a measure of the total curvature of the deconvolved pressure derivative, instead of its average slope, as in an earlier approach 15 and Paper I. Here, the motivation is that slopes provide important information about the flow regime and should therefore be preserved as much as possible. The paper is organized as follows: The first two sections are introductory and give a summary of the deconvolution problem in well-test analysis and a concise survey of its treatment in the petroleum engineering and hydrology literature. Based on the mathematical framework developed in these sections, we then give a comprehensive account of our own approach. We also derive analytic expressions for bias and variance of the estimated parameter set based on simple Gaussian models for the measurement errors in pressure and rate signals. We illustrate our method with a small simulated data set, demonstrating the effect of varying levels of regularization on the confidence intervals. The final section presents an application to a large field example which allows a direct comparison of our method with conventional derivative analysis.

A mathematical model is presented for investigating the non‐steady state temperature behavior of a pumped aquifer during reinjection of a fluid at a temperature different from that of the native water. Results are presented in terms of dimensionless parameters and should be helpful in the design of geothermal space‐heating projects. Applications to practical cases are also included.