Summary. An algorithm for determining pressure-derivative functions from measured well-test data is presented. The pressure data are fit with a spline function that can account for curve shape constraints. The first and second derivatives of the resulting spline function are calculated analytically and can be used for typecurve matching. Two examples illustrate the performance of the algorithm. Introduction Use of the pressure-derivative function to interpret well-test data has gained wide acceptance over the past several years. Many analysts believe that pressure-derivative type curves display more character than conventional type curves, which makes it easier to identify various flow regimes and well-test-interpretation models and to obtain unique typecurve matches from well-test data. The main disadvantage of using derivative type curves, however, is that the rate of pressure change cannot be measured directly during a well test but instead must be inferred from discrete pressure measurements that may contain significant noise. It is well known that measurement noise tends to be amplified during the differentiation process; thus, some analysts recognize that derivative process; thus, some analysts recognize that derivative type-curve-matching techniques are limited by both the quality of the measured pressure data and, perhaps more important, the algorithm used to calculate the pressure derivative. Most pressure-derivative type curves evolved from the ideas presented by Bourdet et al., who produced derivative type curves presented by Bourdet et al., who produced derivative type curves for wells with wellbore storage and skin. These curves require that the pressure derivative with respect to the natural logarithm of time be plotted on a log-log scale. Other derivative type curves that use this same plotting function have since been published for a variety of reservoir descriptions. More recently, type curves that use a pressure/pressure-derivative-ratio plotting function have been suggested, and second-derivative type curves have also been proposed. To use any of these type curves, however, pressure derivatives must be evaluated from inexact, discrete, well-test pressure measurements. Bourdet et al. later presented an algorithm that has perhaps become the standard method for calculating pressure derivatives from measured well-test data. This algorithm uses a weighted central-difference approximation to calculate the pressure derivative at any Point i: (1) where X is the natural logarithm of the appropriate time function and Points i - 1 and i + 1 are the points preceding and following Point i, respectively. Bourdet et al. noted that when consecutive Point i, respectively. Bourdet et al. noted that when consecutive points are used, the noise in the calculated derivative often makes points are used, the noise in the calculated derivative often makes analysis impossible. They suggested that noise be reduced by choosing points farther away from Point i: (2) where L may range from L=0 (for consecutive points) to L=0.5 in extreme cases. While Eq. 1 is simple to use, this algorithm has several shortcomings:estimates for the derivative are not obtained throughout the entire interval (i.e., they are available only at those points with abscissa values that correspond to measured data, which may points with abscissa values that correspond to measured data, which may be widely spaced);calculated derivative values may still contain high levels of noise, even after smoothing with a large value of L;truncation errors associated with Eq. 1 will increase as L increases, causing the approximation for the derivative to deteriorate so that the curve may begin to lose its characteristic shape; andso-called end effects may arise when data corresponding to early and late times are differentiated because Eq. 2 can be satisfied only on one side of those data. To alleviate the end effects for late-time data, Bourdet et al. suggested that a pseudoright limit be used for p and delta X. For early-time data, they indicate that a forward-difference approximation for the derivative is usually suitable because the rate of pressure change in this region is typically large enough to mask any noise effects. To avoid the noise problems associated with the differentiation of measured data, Blasingame et al. and Onur et al. proposed that pressure data be integrated before the various derivatives of the integral pressure function are determined. The cited advantage of this approach is that the integral of noisy data yields a much smoother function than the derivative of noisy data, thus making typecurve matching easier. Note, however, that the integral of a function appears to have less character than the original function while the derivative appears to have more character. Thus, while integrating may smooth noisy pressure data, it also will result in a loss of the original character of the data, which should make finding a unique type-curve match more difficult. We believe that any algorithm that calculates discrete derivatives from inexact measurements will meet with only limited success. In addition to the noise problems mentioned above, the lack of continuity in the discrete data (i.e., large time gaps may exist between consecutive points) may make determination of the derivative curve shape highly subjective and sometimes impossible. With these disadvantages, it is easy to see why the use of second derivatives have not been examined in more detail. In this paper, we present an algorithm for fitting a set of measured pressure data with a spline function that effectively filters out the measurement noise so that the true underlying pressure function can be approximated. The first and second derivatives of the resulting spline function can then be calculated analytically to produce smooth, continuous functions that are free of the noise produce smooth, continuous functions that are free of the noise typically associated with discrete pressure derivatives. These smooth derivatives may greatly enhance our ability to identity flow regimes from the derivative curve shape and should make both manual and automatic typecurve matching easier and more accurate. The algorithm allows the user to impose shape constraints on the spline function and its derivatives to help evaluate whether a proper well-test-interpretation model was chosen. The performance of the new algorithm is illustrated by both simulated and actual well-test data. Spline Approximation A spline is a collection of piecewise polynomial segments with every two consecutive pieces joining smoothly at the knots (segment join points). The spline function and some of its derivatives are required points). The spline function and some of its derivatives are required to be continuous across the knots. SPEFE P. 493

Permanent ocean-bottom seismic cables have been installed in the Valhall Field, offshore Norway.Five seismic surveys were acquired from the permanent system between October 2003 and April 2005, in addition to two towed-streamer surveys acquired in 1992 and 2002.The various 3D seismic data sets show strong time-lapse or 4D effects resulting from primary production.During the same time period an extensive drilling program has provided high-quality data on reservoir pressure, pressure gradients, and reservoir fluid distribution in the field. This paper outlines how changes in seismic attributes were generated synthetically from a reservoir simulation model and compared against observed 4D seismic data in one area of the field.An objective function describing the misfit between the simulated and measured data was defined using both seismic and conventional production data.BP's Top-Down Reservoir Modelling (TDRM) technology[1] was used to generate multiple reservoir descriptions that would minimize the objective function.This involved computer-assisted history matching with a genetic algorithm that varied more than 60 parameters in the reservoir simulator to improve the overall history match. Introduction The Valhall field is located in the North Sea approximately 290 km offshore Norway in Blocks 2/8 and 2/11 (Fig. 1).Valhall is a double-plunging NNW-SSE trending anticline (Fig. 2), covering approximately 65 km[2].The reservoir is a high porosity, overpressured, undersaturated Upper Cretaceous chalk located at depths of 2400–2650 m.The main reservoir is the Tor Formation, deposited during Danian, Maastrichtian and Campanian periods.A secondary reservoir is the Hod Formation (Santonian, Coniacian and Turonian).A stratigrahpic section is shown in Fig. 3.Seismic imaging is challenging in the crestal part of the field due to gas charge in the overburden. Valhall was discovered in 1975 and began producing in 1982.The field was highly overpressured with an initial pressure of approximately 6500 psi.The reservoir chalk is extremely soft with significant rock compaction, which has resulted in high intrinsic reservoir energy.This has allowed the field to be produced on primary depletion since 1982, with approximately 500 MMstb produced to date.The oil density is light (36 API) and viscosity is roughly 0.4 cp.A waterflood program has recently begun, which is expected to extend the production plateau significantly. The high rock compaction at Valhall has resulted in significant seabed and platform subsidence and presented many challenges associated with wellbore failures.Compaction has also influenced the dynamic reservoir properties, closing natural fractures with depletion and causing normally "static" parameters such as reservoir thickness and porosity to vary significantly over time.This introduces several challenges in modeling the reservoir properties and simulating field behaviour. More than 100 wells have been drilled at Valhall, with a standard logging suite providing data on porosity, thickness, water saturation, and static reservoir pressures.Since 1990, most wells have been horizontal, providing excellent data on lateral pressure gradients across the field.Twenty-three years of production history provide good quality data on production performance (oil rate, GOR, water cut). In 2003 a wellhead platform was installed in the southern part of the field, and a corresponding infill drilling campaign targeting flank areas was carried out.During this infill campaign, several seismic surveys were acquired with the permanent seismic array.

Summary. The estimation of reservoir properties from production or pressure data measured during production or well tests is an important process for reservoir characterization and performance prediction. A key step in that process is the selection of a reservoir model for use in the interpretation of the data. We develop a procedure for selecting the most appropriate model from a pool of candidates. A parameter estimation algorithm is used to evaluate parameters within candidate models. Statistical measures are then used to select the most appropriate model. We demonstrate our procedure for model selection with actual well test data and production data from the Devonian shale. We show how our methodology can be used to evaluate whether certain reservoir features can be identified from measured production or pressure data. We also present a test for evaluating whether independent estimates of reservoir properties are consistent with measured pressure and production data. Introduction Production or pressure data measured at wells during reservoir production or well tests are significant sources of information for estimating reservoir properties to predict future reservoir performance. Two key steps in the process for estimating reservoir properties from pressure/production data are selection of an appropriate reservoir model and estimation of parameters within that model. The latter step has received much attention in the literature. A number of automatic history-matching (or parameter-estimation) algorithms have been proposed for that purpose. Type curve and other graphic methods are also available for estimating parameters for selected reservoir models. Perhaps more difficult, but no less important, is the selection of the most appropriate reservoir model. Generally, this requires first selecting the appropriate set of material and energy balances for the physical processes involved, as well as the fluid properties and reservoir geometry. Then, the reservoir properties must be chosen. These properties may vary significantly with location in the reservoir. However, we cannot expect to estimate those properties accurately throughout the reservoir on the basis of the limited data usually available. Instead, we usually parameterize the unknown properties in a way that is consistent with geologic data and other data from the reservoir so that fewer unknown parameters need be established on the basis of the pressure/production data. A common method of parameterization is zonation, for which reservoir properties are assumed to be uniform throughout various regions of the reservoir. As we use "model" to refer to the set of equations that relate the reservoir parameters to the dependent, or simulated, quantities, different parameterizations correspond to different reservoir models. Here we consider situations for which the measured pressure/production data may be explained adequately with reservoir models containing relatively few unknown parameters. Specifically, we deal with reservoir models in which the unknown properties are assumed to be uniform; our method for model selection is not limited, however, to such situations. We do not deal specifically with selection of appropriate material and energy balances or fluid properties; these rather broad topics are addressed in a number of texts (e.g., see Refs. 1 through 3). We develop a systematic, quantitative method for choosing from among candidate reservoir models the most appropriate model for a given set of pressure/production data. We illustrate the importance of model selection for the accuracy of estimates of the reservoir properties. We also demonstrate how our method for model selection can be used to establish whether a given set of measured data is sufficient for identifying certain reservoir features. For example, it can be used to ascertain whether sufficient data have been measured to determine boundary effects, skin, or the presence of more than a single-porosity system. Thus, our method can be used to quantify the information about reservoir properties that is "contained" in the measured data. Model Selection A key difficulty in choosing the most appropriate reservoir model is that several different reservoir descriptions (or models) may apparently satisfy the available information about the reservoir. That is, the models may be consistent with available geological and petrophysical information and seem to provide more or less equivalent matches of the measured pressure/production data. The issue may be further complicated when type curve or other graphic methods are used to evaluate the parameters within a given model because different sets of parameters may provide matches of different precision corresponding to different regions of the data. That is, no "perfect" matches can be expected because of the finite accuracy of the measurements and simplified nature of the models, so the best compromise in matching the data is not at all apparent. This latter consideration can often be alleviated when parameter estimation methods are used. Proper formulation of the performance index can lead to a unique set of estimates that may be called best (in a certain sense discussed in more detail later) provided that the proper reservoir model is selected. Thus, we believe a method for selecting the most appropriate reservoir model for use in conjunction with parameter estimation methods is a particularly attractive method for analyzing reservoir production or well test data. We have developed a method to choose, on the basis of measured pressure/production data, the most appropriate reservoir model (or reservoir description) from among a set of candidate reservoir models. A pool of candidate models that are consistent with all available information about the reservoir is to be determined. We then use statistical measures to choose the most appropriate model from among these candidates. As discussed later, the pool of candidates may be formed as a hierarchy of models, as determined by the most complete model (in the sense of the greatest number of unknown parameters) that the engineer believes is necessary for describing the reservoir. The hierarchy is characterized by increasing numbers of unknown parameters. It is important that at least one model satisfactorily describes (or "explains") the data (it is sufficient that the most complete model does so). This can generally be established satisfactorily by examination of a plot of the differences between the observed data and the corresponding quantities simulated with parameters obtained from a history match of the data. SPEFE P. 215^

No abstract

Summary Production data from more than 800 Devonian shale wells have been analyzed. Permeability-thickness product and gas in place estimated from production data have been found to correlate with well performance . Empirical performance equations, production type curves, and an analytical dual-porosity model with automatic history-matching scheme were developed for the Devonian shale. Introduction Thousands of wells have been completed in the Devonian gas shales Of the Appalachian basin. Although a wealth of historical production data exists for these wells, these data have never been studied systematically on a large scale. The purpose of our work was to gather and to analyze as many of these data as possible with the following objectives in mind:to analyze Devonian shale production data to reservoir characteristics,to identify those reservoir characteristics that correlate with superior well quality, andto develop analytical tools for the practicing engineer to use in analyzing and predicting production in the Devonian shales. Production data and other pertinent well information have been gathered for more than 1,500 Devonian shale wells in Kentucky, Ohio, and West Virginia. We first conducted a pilot data-analysis study on wells located. in southwestern West Virginia and eastern Kentucky (Area 1 in Fig. 1) to ensure that Devonian shale production data could be analyzed logically and systematically. Upon successful completion of the pilot study, we selected four areas (Fig. 1) for study. During the course of this work, we developed an analytical dualporosity reservoir model for analyzing and predicting Devonian shale production; analyzed flow test and/or production data on more than 1,000 wells; correlated permeability thickness product and gas in place with observed well performance; developed a family of type curves for analyzing and producing Devonian shale production; and developed empirical equations for predicting Devonian shale production. In the next section, we discuss the data gathered for this study. We then discuss the analytical dual-porosity model and history-matching scheme used to analyze the available production data. The analysis of initial-open-flow (IOF) data and the development of type curves for analyzing production data are also briefly outlined. We then present a statistical analysis of the results of our production data analysis. Finally, we discuss the application of our results to predicting future gas-well performance. Data Acquisition Data have been gathered from four areas in Kentucky, Ohio, and West Virginia (Fig. 1). Area 1, often referred to as the Big Sandy area, contains a large number of wells with long producing This area includes most of the best shale gas wells and produces chiefly dry gas. Area 2, in central West Virginia, contains many wells of average quality that produce mainly dry gas. Area 3 is in southeastern Ohio and generally has poorer-quality wells that produce dry gas, except near Area 4. Area 4, or the Burning Springs area, includes a large number of recently completed wells that produce gas and liquids.