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.

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.

Permanent downhole pressure gauges are increasingly being installed in new wells in the North Sea and in other new developments around the world. Their reliability has greatly improved and they now can operate for several years. They provide a record of everything that is happening to the well and, in the long term, they will replace production tests for well and reservoir monitoring. The main difference with production tests, however, is that rate variations are not controlled, which can make the interpretation difficult. The paper illustrates how the availability of three years' worth of pressure data from a permanent downhole pressure gauge was key to understanding and explaining a tenfold loss of productivity in a North Sea horizontal gas well. Four million pressure measurements were processed and analyzed both in the conventional way, one flow period at a time, and by deconvolution, using increasing durations of pressure records from the start of production. Interpretation identified progressive changes in gas relative permeability followed by decreases in the well length, suggesting water invasion of the well zone. This points out to the need for continuous monitoring of the interpretation of permanent gauge data, which is possible by deconvolution, to identify changes in well-reservoir behavior as soon as they occur and thus avoid potentially irreversible productivity problems. Introduction As they become more reliable, downhole permanent pressure gauges are used more and more in new field developments. The advantage is obvious: permanent gauges allow the field operator to monitor the reservoir behavior in real time and in theory, to react to problems as soon as they appear. In practice, the need to interpret the raw information provided by permanent gauges and the lack of manpower for doing so prevents real time intervention. Instead, permanent gauge records are archived and only examined after the existence of a problem has been recognised, to identify its cause and evaluate possible solutions. Unfortunately, the corresponding delay often makes the damage irreversible. Clearly, some sort of automatic interpretation and alarm system is required to benefit from the full potential of downhole permanent gauges. This is discussed in this paper with an example of a horizontal well in the North Sea. Field example The well presented here (well A) is one of 20 wells in a North Sea dry gas reservoir. The field is 700–800 ft thick with a large degree of vertical (Zones A, B and C in Figure 1) and lateral (Figure 2) heterogeneity and, for the main part, was put on production in 1988. In 1996, a separate structure was discovered by vertical well B. This structure was developed by well A, which was completed in January 1999. General fault orientation is NNW-SSE, with major E-W faults bounding the field to the north and south (Figure 2). It is thought that wells A and B both intersected a NNW-SSE fault. Throw on this fault is approximately 30ft. Well A was drilled as an extended reach horizontal well with 4" sand screens in an 8 1/2" hole. The reservoir was entered at 13794' MD and the well was terminated at a total depth of 16000' MD, giving a horizontal reservoir section of 2206 ft. The horizontal section was drilled 50' TVD above the Gas-Water contact to target the better quality zone C depositional unit. Sudden mud losses were experienced at about 15537' MD and 15835' MD while the horizontal section was being drilled. 4" excluder screens were run over the entire open-hole section to 15979' MD. Of the 2206 ft total reservoir length, a total of 2071' was logged fully, with 1220' of net pay. Average net pay porosity was 15%, and the average Sw was calculated at 44%. Log permeability estimates suggest zone C exhibits permeabilities greater than 100mD. Probable intra-reservoir faults were interpreted on the sonic log at 15533' and 15794' MD. A 3-stage acid stimulation was performed, and the well cleaned up with filtered brine, with losses of 6–8 bbls/hr.

Summary Based on a linearized model for the isothermal flow of a single, compressible phase through a reservoir of arbitrary shape with impermeable orconstant-pressure boundaries and spatially varying, anisotropic rockproperties, we develop a multiwell extension of the superposition principle andre-examine the question of reciprocity between wells that may be modeled aspoint sinks or as extended sinks. In the latter case, we find that the answerdepends on the wellbore boundary conditions: Reciprocity holds for infiniteconductivity wells but fails to hold for spatially uniform sink strength. Wealso derive a multiwell generalization of the fractional transformation in theLaplace domain, which adds skin and wellbore storage to a reservoir model, andfind that its impact on reciprocity is neutral: It preserves reciprocity if itholds for the reservoir model. Introduction Data from interfering wells have been a long-standing challenge for welltest analysis. The challenge is more acute than ever as multiple active wellsper reservoir compartment are now the norm in optimized production plans; it iscompounded by the trend toward more complex well trajectories (see Fig. 1 foran example). The central signal processing task remains to estimate therate-normalized pressure drop and its time derivative (Bourdet et al. 1983,1989) for each well in response to its own production as well as to that of theother wells. As in the case of a single well (van Everdingen and Hurst 1949), this is a deconvolution problem. A recent study (Levitan 2007) showed how the same principles that provedsuccessful in single-well deconvolution (von Schroeter et al. 2004; Levitan2005) can be extended to multiple wells. This study assumed reciprocity betweenwells (i.e., that the rate-normalized pressure drop at one well in response toproduction at another is the same as vice versa), which halves the number ofinterference signals to be estimated. For wells modeled as point sinks (such asfully penetrating vertical wells in a conducting layer of constant thickness), reciprocity follows from the symmetry of Green's function in its spatialarguments, a fact established by several authors (McKinley et al. 1968; Dengand Horne 1993) under various physical assumptions. However, for extended sinks (such as horizontal, inclined, curved, andfractured wells) the picture is more complicated, as we show in this paper. Based on a linearized model for the flow of a single, compressible phasethrough a reservoir of arbitrary shape with spatially varying permeabilitytensor, we derive multiwell extensions of the superposition principle anddeduce the symmetry of Green's function, which establishes reciprocity at thelevel of point sinks for a wider class of reservoir models than hithertoconsidered, and by a simplified mathematical route. For extended wells, we findthat reciprocity depends on the boundary conditions: It holds for infiniteconductivity wells but fails to hold for spatially uniform sink strength. Moreover, the fractional transformations applied in the Laplace domain to addskin and wellbore storage to a reservoir model preserve reciprocity if it holdsfor the reservoir model. As our investigation relies heavily on Green's functions and relatedmathematical concepts, we illustrate the methodology with a simple yetinstructive analytic example.