New well test interpretation methods are presented that eliminate wellbore storage (afterflow) effects. These new methods use simultaneously measured sandface flow rate and well bore pressure data. It is shown that formation behavior without storage effects (unit response or influence function) can be obtained from deconvolution of sandface flow rate and well bore pressure data. The storage-free formation behavior can be analyzed to identify the system (reservoir flow pattern) that is under testing and to estimate its parameters. Convolution (radial multirate) methods for reservoir parameter estimation and a few synthetic examples for deconvolution and convolution also are presented.
Summary This paper presents new testing and analysis techniques to obtain individual-layer permeabilities and skin factors for layered reservoirs. The new multilayer testing technique consists of a number of sequential flow tests with a production logging tool that simultaneously measures the wellbore pressure and flow rate at the top of each layer. Two different analysis techniques are presented to estimate layer parameters. The first technique, which is the logarithmic convolution parameters. The first technique, which is the logarithmic convolution method, estimates the approximate values of parameters. The second technique, which is the nonlinear least-squares estimation method, improves the first estimates. It is shown that layer permeabilities and skin factors can be estimated uniquely from simultaneously measured wellbore pressure and flow-rate data that are acquired from all layers sequentially. It is also shown that these individual-layer parameters cannot be estimated from the conventional drawdown- or buildup-test wellbore pressure data. Several synthetic examples are presented to illustrate pressure data. Several synthetic examples are presented to illustrate the application of multilayer testing and analysis techniques. Introduction Most oil and gas reservoirs are layered (stratified) to various degrees because of sedimentation processes over long geological times. Layered reservoirs are composed of two or more layers that may have different formation and fluid characteristics. These reservoirs are usually divided into two groups:layered reservoirs without crossflow (commingled systems), where layers communicate only through the wellbore, andlayered reservoirs with crossflow, where layers communicate at the contact planes throughout the reservoir. Accurate determination of permeability, skin factor, and pressure for each layer is permeability, skin factor, and pressure for each layer is necessary to understand the reservoir performance. For example, unbalanced depletion of layers with different parameters creates many problems, such as high GOR in parameters creates many problems, such as high GOR in high-permeability layers. Conventional buildup tests from layered reservoirs usually suffer from crossflow between layers, particularly if the layers communicate only through the wellbore and/or the permeability contrast between layers is high. The crossflow problem becomes more severe if the pressure and/or the drainage radius of each zone is different. The wellbore crossflow may continue during the entire period of the buildup test. A false straight line on a period of the buildup test. A false straight line on a semi-log plot may even be observed. In many instances, the pressure data alone may not reveal any information about pressure data alone may not reveal any information about the wellbore or formation crossflow. Even if crossflow is not a complicating factor, the major problem for layered systems is still the estimation of problem for layered systems is still the estimation of individual-layer permeabilities and skin factors from conventional well tests. The conventional drawdown and buildup tests usually reveal only the behavior of the total system. Furthermore, the behavior of a multi layer formation may not be distinguished from the behavior of a single-layer formation even though a multilayer reservoir may have a distinct behavior without wellbore storage effects. There are, however, a few special cases where the conventional tests may work. A detailed study of the behavior of two-layer reservoirs with crossflow was done by Prijambodo et al. Unlike many earlier researchers, they investigated the effect of each layer's skin on the semilog straight-line behavior of the two-layer systems without the wellbore storage effect. They also examined limitations of the semilog methods for two-layer systems with many different combinations of vertical and horizontal permeabilities and skin factor with and without crossflow. permeabilities and skin factor with and without crossflow. A different approach for estimating layer parameters by means of optimum test design was investigated by Dogru and Seinfeld. They used a numerical model, but did not include skin and wellbore storage effects. Nevertheless, they show that there are serious problems with the ability to observe and question how well-posed parameter estimation is for layered reservoirs with a parameter estimation is for layered reservoirs with a single transient test. Most work on layered reservoirs has been the derivation of solutions for boundary-value and initial-value problems, the investigation of sensitivity of solutions to layer parameters (forward problem), and the estimation of the parameters (forward problem), and the estimation of the average flow capacity and skin factor of the total formation from a single transient well test. We present a new testing technique for layered reservoirs to estimate individual layer permeabilities and skin factors uniquely. This new test will be called a "multilayer test" hereafter. The multilayer testing technique consists of a number of sequential flow tests, with a production logging tool measuring the wellbore pressure and flow rate at the top of each different layer. SPEFE P. 342
Summary Determination of the influence function of a well/reservoir system from thedeconvolution of wellbore flow rate and pressure is presented. Deconvolution isfundamental and is particularly applicable to system identification. A varietyof different deconvolution algorithms are presented. The simplest algorithm isa direct method that works well for presented. The simplest algorithm is adirect method that works well for data without measurement noise but that failsin the presence of even small amounts of noise. We show, however, that amodified algorithm that imposes constraints on the solution set works verywell, even with significant measurement errors. Introduction In reservoir testing, we generally know the characteristic features of thesystem from its constant-flowrate and constant-pressure behavior. Thus it isimportant to determine the constant-rate or -pressure behavior of the systemfor the identification of its characteristic features. For instance, identification of a one-half on a log-log plot of the pressure data mayindicate a vertically fractured well, as two parallel straight lines on a Homergraph may indicate a fractured reservoir. The presence of eitherwellbore-storage or flow-rate variations, however, usually masks characteristicsystem behavior, particularly at early times. For many systems, it is desirableto have a wellbore pressure that is free of wellbore-storage and/orvariable-flowrate effects to obtain information about the well/reservoirgeometry and its parameters. For example, the effects of partial penetration, hydraulic fractures, solution gas within the vicinity of the wellbore, gas cap, etc., on the wellbore pressure can be masked entirely by wellbore storage, flowrate variations, or both. Although deconvolution of pressure and flow rate has not been commonly usedfor reservoir engineering problems, one can still find a few works ondeconvolution (computing influence function) in the petroleum engineeringliterature. Hutchinson and Sikora, Jones et al., and Coats et al. presentedmethods for determining the influence function directly from field data. Jargon and van Poollen were perhaps first to use the deconvolution ofwellbore-flowrate and pressure data to compute the constant-rate behavior (theinfluence function) of the formation in well testing. Bostic et al. used adeconvolution technique to obtain a constant-rate solution from a variable-rateand -pressure history. They also extended the deconvolution technique tocombine production and buildup data as a single test. Pascals also usedproduction and buildup data as a single test. Pascals also used deconvolutiontechniques to obtain a constant-rate solution from variable-rate (measured atthe surface) and -pressure measurementof a drawdown test. Kucuk and Ayestaranpresented several deconvolution methods including the Laplace transform andcurve fit. Thompson et al. and Thompson and Reynolds presented a stableintegration procedure for deconvolution. This paper focuses on deconvolution methods. Mathematically, thedeconvolution operation can be defined as obtaining solutions forconvolution-type, linear, Volterra integral equations. In reservoir testing, itis defined as determining the pressure behavior (in-fluence function orunit-response behavior) of a system fro simultaneously measured downholepressure and flow rate. In other words, deconvolution computes the pressurebehavior of a well/reservoir system as if the well were producing at a constantrate. We call the computed pressure behavior of the system "deconvolvedpressure." Convolution Integral The convolution integral, which is a special case of the Volterra integralequations, is widely known for providing techniques for solving time-dependentboundary-value problems. It is also known as the superposition theorem (i.e., Duhamel's principle) and has played an important part in transient well-testanalysis. In recent played an important part in transient well-test analysis. In recent years, there has been more interest in the solution of theconvolution integral in connection with analysis of simultaneously measuredwellbore pressure and flow rate. Although we restrict our discussion mostly todetermining influence functions for the constant-rate case, we do treat theconvolution integral in a general manner. In other words, the influencefunction can also be the solution of the constant-pressure case. In a linear causal system (reservoir), the relationship between input (thetime-dependent boundary condition that can be either the flow rate or pressure)and output (the system response measured as either the flow rate or pressure)at the wellbore can be described as a convolution operation. We let thequantities measured at the wellbore, above the sandface, be p =wellborepressure, Q =cumulative wellbore production, and q =wellbore flow rate. The convolution integral is (1) where The functions delta p (t) and Q (t) or q (t) are the solutions of thediffusivity equation for the constant-flowrate or -pressure case with orwithout wellbore-storage and skin effects. Although it is usually small, thedeconvolved pressure will always be affected by the wellbore volume between themeasurement point and the sandface because the sandface flow rate is differentfrom the wellbore flow rate, q, when it is measured by the flowmeter at somewellbore location above the perforations. As shown by Coats et al., the general solutions of the diffusivity equationwith the first and second kind of internal boundary conditions and nonperiodicinitial and outer-boundary conditions, satisfy the constraints (2) (3) (4) and (5) when the real time is greater than 1 second and if the diffusivity constant, k/phi mu c, is not very small. Coats et al. used linear programming with theabove constraints to compute K(t) from measured g(t) programming with the aboveconstraints to compute K(t) from measured g(t) and f(t). Here we use theseconstraints to compute the system influence function. SPEFE P. 53
introduced the log-log type curves to find out when AB:TllAcT wellbore stor~ge effects are negligible f~r a test. Later McKinley iind Earlougher and Kersch presented A general method of analysis for pressure transient type curves for the interpretation of bo?.h pressu~= tests 1s presented. This i.echnique is based on the drawdown and build up tests under the influence of pressure response of an instantaneous source and it wellbore storage and skin damage. rovides a mean to compute the first and second derivatives of the influence function (unit flow rate Gringarten et al.4 improved Ramey's type curves to response) of the well-reservoir system. This reduce the uniqueness problem in analysls. During the information is basic in identifying the flow regimes Lime wellbore storage and skin type curves were occurring during the test. This method eliminates developed, similar studies were conducted for the the effect of producing time on pressure buildup analysis of pressure data in hydraulically fractured data. wells; for this cg:f5the use cf type curve matching was also proposed . These type curves (log PD vs An explicit and stable procedure is discussed to log T ) were presented ! for both reservoir and compute both the derivatives of the influence frd~tll e Darameter
Summary. This paper addresses the problem of estimating horizontal and vertical permeabilities, skin factor, and average fluid pressure within the drainage area in each layer in a multilayered reservoir in the presence of formation crossflow between the layers and commingling through the wellbore. A multistep testing procedure is proposed that involves downhole measurement of the pressure and fluid flow rate. The measurements are made successively above the individual layers as transients are induced in the reservoir. A history-matching procedure is used to analyze the data from the entire multistep test simultaneously. The performance of this approach is compared with that of separate, conventional interpretation of each step in five examples with synthetic data. The simultaneous analysis benefits from the synergy between the steps and consistently yields better estimates. A linearized sensitivity analysis is developed to determine the probable errors in the parameter estimates. It indicates whether a given set of measurements with a given level of measurement noise contains sufficient information to determine the individual parameters uniquely. Introduction Many of the fields discovered during the last two decades in the U.S. gulf coast, Alaska, North Sea, Middle East, and Far East areas are thick reservoirs. Because such reservoirs are the result of changing sedimentary processes over geologic times. they often consist of distinct layers. The characterization of such layered reservoirs is important because of the influence of the layers on primary and secondary oil recovery. The difference in layer permeabilities may lead to unbalanced depletion. Waterflooding of such systems is in-efficient because of the unswept oil in the tighter zones. The major problems concerning such reservoirs are geological characterization of the layers and evaluation from the reservoir en-gineering standpoint. Because these problems are closely related, it is important to integrate all geological, geophysical, and open-hole log information in the analysis of well test data. The interpretation of well test data may be aided by information derived from the other sources. For example, they may provide ... initial estimates" for the reservoir engineering parameters, which may then be refined by analysis of the well test data. The problem addressed here is the estimation of layer permeabilities, skin factors, and formation pressures from well test data. Conventional tests (drawdown and/or buildup) reveal the behavior of the total system. Consequently, a two-layer formation often cannot be distinguished from a single-layer formation, especially in the presence of the wellbore storage effect. There are, of course, a few special cases where conventional well tests work. These are cited by Tariq and Ramey and Raghavan et al. Furthermore, the interpretation of the conventional tests is usually limited by crossflow between the layers and commingling through the wellbore. In many instances, the pressure data do not reveal commingling at all and a false straight line is obtained in a Homer plot. Many authors have investigated the behavior of layered reservoirs and have analyzed the well test data from such systems a survey may be found in Ref. 3. The idea of using production logging measurements in conjunction with transient pressure response of the reservoir to estimate layer permeabilities and skin factors has been discussed by Stewart et al. They proposed that measurements of pseudosteady-state flow profile and transient pressure during the shut-in period be used to determine the layer parameters by a history match using surface flow rates and a single-well simulator. As suspected by Stewart et al. and demonstrated in this work, however, it is only possible to estimate either layer permeabilities or skin factors, but not both, with these measurements. Kuchuk et al. suggested that for a layered reservoir, the parameters of the individual layers can be estimated by a sequential drawdown test in which wellbore pressure and layer flow rates are recorded simultaneously. in that work, they used a two-layer model without crossflow effects. In this paper, their testing technique is extended to the general situation of arbitrary number of layers with full crossflow effects. A new interpretation approach is presented that uses a layered reservoir model with crossflow and commingling and that simultaneously processes the full set of measurements. Ehlig-Economides and Joseph, contemporaneously with the development presented here, proposed a single transient testing procedure in multilayered reservoirs where the transient flow rates from all layers were monitored simultaneously. They interpreted the individual-layer flow contributions with the aid of an analytical solution of multilayer systems that could include formation crossflow. Their method is often not applicable in practice because of the inability of the currently available production monitoring equipment to measure satisfactorily the downhole flow rates simultaneously at multiple and arbitrary depth locations. Ref. 5 includes an excellent review of the published work until 1985 relating to this problem. Instead of the numerical solution, analytical solutions may be used when available in the interpretation procedure presented here. The test proposed here includes measurement of wellbore pressures and downhole flow rates at different depths in the wellbore as transients are introduced by step-wise changes in the surface flow rates. The philosophical approach taken here consists of viewing all the data as a single unit and determining the reservoir parameters such that, when they are used in a comprehensive, realistic, single-well reservoir model, they cause the response of the model to match all the data simultaneously. Thus the interpretation problem is cast in terms of history matching of the model response to the data over the entire duration of the test. The history-matching problem is solved by the standard minimization techniques. The result of the conventional single-layer analysis including convolution used on individual pieces of data may be used as a starting point for the history-matching procedure. SPEFE P. 555^
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