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.

Finding a good algorithm for the deconvolution of pressure and flow rate data is one of the long-standing problems in well test analysis. In this paper we give a survey of methods which have been suggested in the past 40 years, and develop a new formulation in terms of the logarithm of the response function. The main advantage of this nonlinear encoding over prior methods is that it does not require explicit sign constraints. Moreover we introduce a new error model which accounts for errors in both pressure and rate data; here the rates can be cumulative or continously measured. In this formulation, deconvolution is equivalent to a separable nonlinear Total Least Squares problem for which standard algorithms exist. Preliminary numerical results with both simulated and field data suggest that the method is capable of producing smooth, interpretable reservoir response estimates from data contaminated with errors of up to 10% in rates, provided a careful choice of weight and regularization parameters is made.

Harmonic testing for obtaining dynamic reservoir information was first proposed some thirty years ago. Although not much used in the oil industry, interest in the method is revived periodically, mostly for the determination of skin effect and near-wellbore permeability. This paper looks at the practical aspects of using periodic rate variations for testing oil wells. It is shown that such tests can provide the same information as conventional well tests and can be interpreted in the same way. Their main advantage is that they do not require fluids to be brought to surface in exploration or early appraisal testing, or wells to be shut-in in production testing. They also provide data that are less affected by measurement errors and wellbore effects such as multiphase flow or phase redistribution. The main limitation is that, for the same radius of investigation, harmonic tests are significantly longer than conventional tests. Consequently, they cannot be used for reservoir characterization in exploration and appraisal wells. They appear well suited, however, for monitoring reservoir changes from production wells. Introduction The concept of harmonic testing was first proposed by Kuo1 in the early 70's, as an extension of pulse testing2.Pulse tests aim at generating interference data between two wells through a sequence of alternating production and shut-in periods in order to obtain intra-well reservoir properties and were believed to be more practical than interference tests. Kuo's suggestion was to use a periodic production history in a single well to determine the well near-wellbore properties. If the well is produced at a sinusoidal (or at least periodic) rate, the resulting pressure drop is also periodic, after early transients have died out and a pseudo-steady regime is established. As in pulse tests, the amplitude and phase lag of the pressure relative to the flow-rate can be measured and matched with the response of an interpretation model to obtain the corresponding reservoir parameters. The analysis is performed in the frequency domain instead of the time domain. Rosa and Horne offered a comprehensive review of previous publications on the subject. Many aspects of harmonic testing were studied in the late 70's and early 80's by Jouanna and co-workers3–7 at the university of Montpellier, France. They developed a number of well test interpretation models in the frequency domain and identified some of the difficulties inherent to the method. They also designed several testing devices and performed experiments in the laboratory and in shallow water wells. They concluded that harmonic testing could provide reservoir parameters such as skin factor, damaged zone depth, permeability and the group fct. They pointed out, however, that inertia effects and fluid-solid coupling should be included to fully understand the experimental results. Pan9 recently investigated harmonic testing in the medium frequency range (0.01Hz-10Hz), where more than just Darcy flow is involved. Her work focused on the identification of pore structure properties and the upscaling of these properties to infer the macroscopic behavior of the porous medium. She also suggested that wells could be stimulated by increasing effective reservoir permeability and porosity with elastic wave periodic excitation, thus providing higher flow-rates. Another example of frequency-domain analysis of pressure behavior has been presented by Firoozabadi and Chang10, who showed that reservoir compressibility and permeability can be estimated from the analysis of pressure data influenced by tidal effects. The input signal is a fixed gravitational potential variation instead of an imposed flow-rate variation and their analyses use two frequencies only (diurnal and semi-diurnal). An important development in harmonic testing came from Mercier11, who showed that the derivative of the pressure modulus in the frequency domain had a behavior similar to that of the pressure derivative in conventional well test analysis11. She therefore concluded that the interpretation methodology developed for conventional tests12 could be applied to harmonic tests.

fax 01-972-952-9435. AbstractCurrent trends towards permanent downhole instrumentation allow the acquisition of large sets of well test data ranging over much longer periods of time than previously imaginable. Such data sets can contain information about the reservoir at a substantially larger radius of investigation than that accessible to conventional derivative analysis, which is limited to the interpretation of single flow periods at constant rate. By contrast, deconvolution methods do not suffer from this constraint as they are designed to perform well test analysis at variable flow rate.Recently we presented a new method for the deconvolution of well test data in which the problem is reformulated as a separable nonlinear Total Least Squares problem which accounts for uncertainties in the measurement of both rate and pressure data. 8 In this paper we report a number of improvements to our algorithm, and derive error bounds for rate and response estimates in the presence of uncertainties in the data, for which we assume simple Gaussian models. We illustrate our method by applying it to a small simulated example and two large sets of field data with up to 6000 hours of pressure data and up to 450 flow periods.

Despite the tremendous progress achieved in well test analysis in the last twenty years, many important and practical questions that are frequently asked by practicing engineers have received little attention in the literature. Three of these are addressed in this paper. The first one is whether it is possible to obtain more information from build-up data than from drawdown data in a testing sequence. By introducing a new definition of the radius of investigation - as opposed to the radius of drainage - based on the ability to interpret data using pressure derivatives, we show that the information obtainable from a long build-up following a short drawdown is actually limited by the gauge accuracy and noises as the shut-in time increases. The second frequently asked question (FAQ) relates to the minimum amount of detailed rate history which is required in order to obtain correct pressure derivatives, as a truncated or averaged rate histories may modify the derivative shapes at late times and therefore induce erroneous interpretations. A new rule of thumb is proposed, which combines the most recent flow-rate history with an effective time based on a fraction of the cumulative well production. The last FAQ concerns the possibility to distinguish a non-uniform mechanical skin effect from a uniform one from well test data. Current interpretations with fully penetrating wells yield a single value for the mechanical skin effect, which implicitly assumes that damage is uniform over the well surface. Various skin distributions are investigated with a multilayered model to determine how a non-uniform skin distribution around the wellbore affects the interpretation. It is shown that non-uniform skin distribution can be identified during specific flow-regimes and therefore must be taken into account in the interpretation. Introduction Very significant advances have been made in well test analysis over the last twenty years with the introduction of a systematic interpretation methodology, pressure derivative analysis and new interpretation models. As a results, well test analysis has not only become more powerful, but also easier and faster to perform. There remains, however, a number of issues which are faced by practicing engineers in their day-to-day work, but have not received much attention in the literature. This paper aims at giving some answers to these frequently asked questions (FAQ) in well test analysis. FAQ # 1: Can we see more in a build-up than in a drawdown? This has been a subject a disagreement for many years. Some defend the concept that once you start producing a well, the pressure disturbance is felt everywhere instantaneously, and therefore, by shuting the well in indefinitely, you should be able to obtain all possible information on the reservoir, even that which has not been obtained in the drawdown. Others believe that "if you don't see it in the drawdown, you don't see it in the build-up". The answer to the question can be obtained from the concept of the radius of investigation of a well test, which is the distance at which a given feature of the reservoir can be interpreted, taking into account the influence of noises in the data and the rate history of the test. Radius of drainage The radius of investigation, as defined in this paper, is different from the radius of drainage which is routinely used in well test analysis. The radius of drainage has been the subject of many publications, mainly in the 1950's and early 1960's (summarized by Van Poolen1) with a few more recent ones2–4. They all propose similar expressions, of the form reD=1+D.tD0.5 in dimensionless parameters4 In most cases, the use of the line-source solution with an infinitely small wellbore radius simplifies this equation to reD=D.tD0.5. A summary of the various definitions is given in Appendix A. Radius of drainage The radius of investigation, as defined in this paper, is different from the radius of drainage which is routinely used in well test analysis. The radius of drainage has been the subject of many publications, mainly in the 1950's and early 1960's (summarized by Van Poolen1) with a few more recent ones2–4. They all propose similar expressions, of the form reD=1+D.tD0.5 in dimensionless parameters4 In most cases, the use of the line-source solution with an infinitely small wellbore radius simplifies this equation to reD=D.tD0.5. A summary of the various definitions is given in Appendix A.