Numerical Simulation of Pressure Behavior In a Fractured Reservoir

A simplified model was employed to develop mathematically equations that describe the unsteady-state behavior of naturally fractured reservoirs. The analysis resulted in an equation of flow of radial symmetry whose solution, for the infinite case, is identical in form and function to that describing the unsteady-state behavior of homogeneous reservoirs. Accepting the assumed model, for all practical purposes one cannot distinguish between fractured and homogeneous reservoirs from pressure build-up and/or drawdown plots. Introduction The bulk of reservoir engineering research and techniques has been directed toward homogeneous reservoirs, whose physical characteristics, such as porosity and permeability, are considered, on the average, to be constant. However, many prolific reservoirs, especially in the Middle East, are naturally fractured. These reservoirs consist of two distinct elements, namely fractures and matrix, each of which contains its characteristic porosity and permeability. Because of this, the extension of conventional methods of reservoir engineering analysis to fractured reservoirs without mathematical justification could lead to results of uncertain value. The early reported work on artificially and naturally fractured reservoirs consists mainly of papers by Pollard, Freeman and Natanson, and Samara. The most familiar method is that of Pollard. A more recent paper by Warren and Root showed how the Pollard method could lead to erroneous results. Warren and Root analyzed a plausible two-dimensional model of fractured reservoirs. They concluded that a Horner-type pressure build-up plot of a well producing from a factured reservoir may be characterized by two parallel linear segments. These segments form the early and the late portions of the build-up plot and are connected by a transitional curve. In our analysis of pressure build-up and drawdown data obtained on several wells from various fractured reservoirs, two parallel straight lines were not observed. In fact, the build-up and drawdown plots were similar in shape to those obtained on homogeneous reservoirs. Fractured reservoirs, due to their complexity, could be represented by various mathematical models, none of which may be completely descriptive and satisfactory for all systems. This is so because the fractures and matrix blocks can be diverse in pattern, size, and geometry not only between one reservoir and another but also within a single reservoir. Therefore, one mathematical model may lead to a satisfactory solution in one case and fail in another. To understand the behavior of the pressure build-up and drawdown data that were studied, and to explain the shape of the resulting plots, a fractured reservoir model was employed and analyzed mathematically. The model is based on the following assumptions:1. The matrix blocks act like sources which feed the fractures with fluid;2. The net fluid movement toward the wellbore obtains only in the fractures; and3. The fractures' flow capacity and the degree of fracturing of the reservoir are uniform. By the degree of fracturing is meant the fractures' bulk volume per unit reservoir bulk volume. Assumption 3 does not stipulate that either the fractures or the matrix blocks should possess certain size, uniformity, geometric pattern, spacing, or direction. Moreover, this assumption of uniform flow capacity and degree of fracturing should be taken in the same general sense as one accepts uniform permeability and porosity assumptions in a homogeneous reservoir when deriving the unsteady-state fluid flow equation. Thus, the assumption may not be unreasonable, especially if one considers the evidence obtained from examining samples of fractured outcrops and reservoirs. Such samples show that the matrix usually consists of numerous blocks, all of which are small compared to the reservoir dimensions and well spacings. Therefore, the model could be described to represent a "homogeneously" fractured reservoir. SPEJ P. 60ˆ

This article presents a comparative analysis of various filtering methods for synthetic measurements that simulate data from well test analysis (WTA). The main objective of this work is to identify the most effective filtering methods for noisy WTA data, with the aim of preserving useful information and facilitating the subsequent interpretation of the results. The initial dataset consisted of 200 synthetic pressure drawdown (PDD) and pressure buildup (PBU) curves with varying levels of artificially introduced noise. Both classical filtering methods (Kalman filter, Savitzky–Golay filter, one-dimensional Gaussian filtering) and numerical methods based on neural networks (autoencoders) and machine learning (support vector machines) were considered for data filtering. The comparative analysis demonstrated that the performance of different filtering methods depends on the type of curve (PDD or PBU) and the well characteristics. The best results in terms of signal-to-noise ratio (SNR) and root mean square error (RMSE) were achieved using modern autoencoder-based methods. The conclusion is that the choice of an optimal filtering method requires a detailed analysis of the specific problem and the characteristics of the input data. A combination of different filtering methods is proposed to improve the quality of processing and interpretation of WTA data for complex well designs. The obtained results have practical significance, as they can simplify the segmentation of PDD and PBU curves, which is necessary for the correct identification of various operating periods of the well during the investigation process.

The transient pressure behavior of a well which produces a single compressible fluid through a single-plane vertical fracture has been investigated mathematically. The fracture is assumed to possess infinite flow capacity, to be of limited radial extent, and to penetrate the producing formation completely in the vertical direction. Previous studies of vertically fractured wells have been concerned primarily with production rate performance or semisteady-state pressure behavior. This study was undertaken to ascertain the influence of vertical fractures on transient pressure tests such as pressure build-ups and flow tests. In a vertically fractured system, flow in the region nearest the fracture is practically linear, whereas farther away from the fracture essentially radial flow prevails. Thus, transient pressure analyses based on radial flow theory are sometime inaccurate. As fracture penetration increases radially, kh values calculated from pressure build-up and flow test curves become increasingly larger than true values. Failure to consider the effect of fracture penetration also introduces inaccuracies into the calculation of fracture length from the apparent skin factor and into the determination of average reservoir pressure. If the total length of the fracture is 20 per cent, or greater, of the drainage radius of the well, corrections must be made to pressure build-up and flow test results. Methods for correcting such results are discussed in this paper. For wells with prefracturing pressure build-up or flow test data, it is possible to estimate fracture length by comparison with postfracturing build-up or flow test results. In new wells or wells without prefracturing build-up or flow test data, fracture length must be estimated to correct the values obtained from analysis of pressure tests after fracturing. Fracturing efficiency calculations should be made whenever possible to provide an estimate of fracture length. Tables of the dimension less pressure drop as a function of time and fracture penetration are included in this paper. Using these values should permit analysis of other types of transient pressure behavior in vertically fractured wells. Introduction Hydraulic fracturing has been used quite successfully for over a decade as a completion and stimulation technique in oil and gas wells completed in low-permeability reservoirs. During this period a considerable amount of theory has evolved on the performance of hydraulically fractured reservoirs and on more efficient means of artificial fracturing. Although theory has been developed, no rigorous investigation has been made of pressure build-up and flow test behavior in such wells. Prats et al. first discussed the performance of vertically fractured reservoirs for the case of a compressible fluid. Their work was primarily concerned with production performance at constant flowing pressure. These authors also considered large-time (semisteady-state) constant production rate behavior for vertically fractured wells: however, transient pressure behavior at constant rate was not investigated. McGuire and Sikora and Dyes, Kemp, and Caudle employed an electrical analog to investigate the influence of artificial vertical fractures on well productivity and pressure build-up. They found that fractures which extend beyond 15 per cent of the drainage radius away from the well alter the position and slope of the straight-line portion of the build-up curve. They concluded that these effects must be considered both in the determination of the effective permeability of the formation and in any calculations of final build-up pressure. Although these authors did not undertake an exhaustive study of the influence of vertical fractures on pressure build-up performance, their limited results were quite interesting from the standpoint of the effects they demonstrated. In a more recent paper, Scott reported the results of an investigation of the effect of vertical fractures on pressure behavior, which was conducted with a heat flow model. Scott's results appear to be consistent with those reported in Refs. 1 and 2. However, the effects of different fracture lengths on performance were not investigated. Pressure build-ups and transient flow tests are among the most diagnostic tools available to the reservoir engineer or production engineer. Since a very high percentage of present-day well completions incorporate the hydraulic fracturing technique, a definite need exists for information on the effect of fractures on transient pressure performance. For these reasons we have undertaken a rigorous study of pressure build-up and flow test behavior in vertically fractured reservoirs. The objectives of this study were to obtain synthetic pressure build-up and flow test curves to assess the effects of a vertical fracture, and to determine the modifications which need to be made to conventional pressure build-up and flow test analysis theory for the case of a vertically fractured well. JPT P. 1159ˆ

This paper presents two examples to illustrate the uses of "depth of investigation" for hydraulically fractured wells in oil and gas reservoirs. Both high and low conductivity fractures are represented. We present calculations of depth of investigation for hydraulically fractured wells. We examine reservoir pressure distributions at different flow times.

Regarding hydraulic fractured horizontal well with stimulated reservoir volume well test analysis, most of the current research on pressure drawdown curve, and pressure buildup test is a common test method, so it is necessary to carry out test analysis of transient pressure buildup of multi-fractured horizontal well (MFHW) with seam network. Aiming at the characteristics of the formation after multiple hydraulic fracturing of the tight oil reservoir, the stimulated reservoir volume (SRV) is described as the dual medium system consisting of matrix and micro-fracture, and the unstimulated reservoir volume (USRV) is described as the matrix system with poor permeability. The mathematical model of transient flow in hydraulic fractured horizontal well with SRV is established. The finite element method is used to solve the problem. The typical test curve of pressure buildup for MFHW with seam network is given, the characteristics of typical curve and the sensitivities are analyzed. The results show that the pressure buildup test curve is divided into six flow stages, the linear flow around the hydraulic fracturing, pseudo-steady state in the transient region, the inter-porosity flow stage, the linear flow in USRV, pseudo-radial flow in USRV and boundary effect stage. The storability ratio and inter-porosity flow coefficient mainly affect the inter-porosity flow stage. The smaller the storability ratio is, the deeper the “groove” is; the larger the inter-porosity flow coefficient is, the earlier the “groove” is formed; the number of fracture increases, the volume of the SRV is larger and the radial flow of USRV is formed later; the worse the fluid flow capacity in USRV, the closer the slope of the pressure drawdown derivative curve is to 1 in pseudo-steady state in the transient region, and the slope of the pressure buildup curve deviates from 1. Finally, an example of pressure buildup test interpretation of a well is given. This study can provide a scientific basis for the interpretation of pressure buildup test data for hydraulic fractured horizontal well with stimulated reservoir volume.

Previously, if a multiple-boundary situation was Previously, if a multiple-boundary situation was suspected in a hydrocarbon reservoir, about the best one could hope to do was to obtain an estimate of the distance to the nearest boundary. Also, although a well in a closed rectangular drainage area presents a very important flow problem in conventional reservoir engineering, no comprehensive method existed in the literature to determine the location of the well relative to each of the sealing boundaries of the drainage area. This study presents a type-curve matching technique, based on the time rate of change of dimensionless pressure, for interpreting the pressure transient behavior of a well located in pressure transient behavior of a well located in various multiple-sealing-fault systems and inside closed rectangular reservoirs. Type-curve plots generated in this manner may be used to match drawdown curves, based on the time rate of change of field pressure data, to determine several essential reservoir parameters such as the kh and phi c products, extent of drainage area, and distance to surrounding seating boundaries. In multiple-sealing-fault systems, buildup curves are similar to drawdown curves for long producing times. Introduction Because the presence of a fault in a reservoir is of great importance, a considerable number of pressure analysis techniques dealing with this situation have been proposed in the literature. However, very little attention has been given to the case of a multiple-boundary situation. The first application of the image method to multiple-sealing-fault systems was made by Jones. He considered a flowing gas well in an areally extensive quadrant formed by two linear no-flow boundaries that intersect at 90 deg. and showed that, after a sufficient number of producing days, the slope of the drawdown curve ultimately would be four times the initial value. Later van Poollen showed how drawdown curves from a well located between two intersecting faults can be used to find the angle between the two boundaries. Both Jones and van Poollen implied that the rate of change of pressure with time might be useful for analyzing the pressure with time might be useful for analyzing the pressure behavior of a well in a two-fault block. pressure behavior of a well in a two-fault block. Prasad presented an analytical solution for Prasad presented an analytical solution for calculating the well pressure distribution in a wedge reservoir system. Tiab and Kumar demonstrated that for a well between two parallel sealing faults, the time rate of change of pressure provides a unique behavior to detect and determine the distance to each fault. Pressure transient testing also has been applied extensively to study the case of a well in a closed drainage area. In 1937, Muskat developed a method to determine the eventual static pressure of a well in a closed circular reservoir. In the late 1940's, van Everdingen and Hurst published a fundamental study of the unsteady pressure distribution for both finite and infinite reservoirs. These two publications laid the foundation for two major reports by Horner and Miller et al. Horner presented a method of analysis of pressure buildup data obtained from a well in an infinite reservoir. He also reported the influence of a sealing fault on pressure buildup curves and the behavior of a well at the center of a finite circular reservoir. About the same time, Miller et al. published the results of a study in which they compared the effects of no-flow and constant-pressure conditions existing at the external boundary of a circular reservoir. In 1954, Matthews et al. presented a technique for estimating both the average reservoir pressure and pressure distributions within a large variety of pressure distributions within a large variety of bounded geometric shapes. SPEJ P. 378

Hydraulic fracturing has been used extensively over the past fifteen years to stimulate low permeability oil and gas wells. A considerable permeability oil and gas wells. A considerable amount of fractured well performance theory has accumulated during this period. Transient drawdown solutions for vertically fractured liquid wells based on numerical simulation were published in 1964. These solutions established the influence of vertical fractures on transient pressure buildup and drawdown testing. Others have investigated well tests of vertically fractured gas wells using both analytical and numerical models. Recent studies have provided new information for type-curve matching of pressure data obtained from fractured (vertical and horizontal) wells. The objective of this paper is to illustrate the application of numerical simulation in evaluation of fracture stimulation of gas wells. The previously published interpretation methods, such as pressure buildup and drawdown analyses and type-curve pressure buildup and drawdown analyses and type-curve matching, form an extremely important part of the complete analysis. Better and more comprehensive well test interpretation can often be obtained by using the so-called conventional methods and numerical modeling together. Introduction Prats, et.al., originally developed analytical solutions for the performance of vertically fractured reservoirs for the compressible fluid case. They considered both the constant terminal pressure and constant terminal rate cases. In the case of constant rate, however, the early-time pressure transient solutions were not investigated. In 1964, Russell and Truitt published transient pressure solutions for vertically-fractured oil or water wells based on numerical simulation. From their solutions they developed methods of analyzing pressure buildup and drawdown tests with conventional plotting techniques. Clark later applied the Russell-Truitt results in analysis of water-injection well falloff data. Analytical solutions and example applications for vertically fractured wells which produce slightly compressible fluids also were presented by van Everdingen and Meyer. More recently Gringarten, et.al., have reviewed the work of previous authors and published new solutions especially useful for published new solutions especially useful for type-curve analysis. They illustrated the use of their results (for wells with either vertical or horizontal fractures) in a companion papers.

One of the main concerns regarding flexible pipe integrity is its annulus condition, as a flooded annulus can lead to excessive corrosion and reduce fatigue life of the armor layers. The current approach to address this is to periodically perform a vacuum or pressure test to check the annulus integrity and to measure its gas-filled volume, in order to detect an accumulation of condensation water, or the ingress of sea water (Bondevik, 2004). These measurements are sometimes complemented by a continuous measurement of the flow rate of gas escaping the flexible riser's vent ports (MCS International, October 2002). The vacuum or pressure test is a costly operation, performed intermittently, while the conventional vent-gas monitoring does not provide reliable information on gas diffusion rates or water vapor emissions. To address these issues, TOTAL and Schlumberger have developed the subC-racs* riser annulus condition surveillance system for continuous monitoring of flexible riser integrity, which eliminates the need for vacuum tests. The gas that permeates the riser pressure sheath is depressurized while measuring its pressure, temperature, and flow rate. As in a well production test, the pressure drawdown and buildup curves are analyzed to give detailed information about fluid content and connectivity. The instrument's resolution and accuracy allow frequent calculations of gas diffusion rate and of the volume of liquid that may have entered the annulus, weekly, daily, or more frequently, depending on gas diffusion rate and riser parameters. In this paper we describe the measurement principle and hardware, modeling of the gas diffusion in the annulus compared with experimental results, and field test results on various risers in a West Africa field. Emphasis is placed on the measurement results, but the implementation in hardware and real-time software for alarms and remote monitoring is also shown. Introduction Monitoring of flexible pipe integrity is a main concern for all offshore fields. It has become more significant as the number of flexible risers increases, and as they age. The main issues for flexible risers are the status of outer sheath and the presence of water in the annulus due either to condensation or by damage to the outer sheath (Figure 1).

A down-the-hole device has been designed for gas dynamics analysis in coal. The device is manufactured based on the layout of a straddle packer with an adjustable interval. The device design is suitable for hydrofracturing and gas dynamics researches using the methods of indicator diagrams and pressure buildup and drawdown curves in package with relaxation of coal and rock mass by means of radially symmetric loading of hole walls in the hydrofracture interval.

Najurieta, Humberto L.,* SPE, Inst. Mexicano del Petroleo A method to calculate the unsteady-state pressure behavior within the fractures of a homogeneously fractured reservoir is presented. The technique allows the analysis of pressure buildup and drawdown tests for layer- like and block-shaped fractured reservoirs. The fractures pressure is shown to depend on four parameters, and a method for its pressure is shown to depend on four parameters, and a method for its determination is proposed. Introduction The importance of the pressure behavior as a source of information on reservoir characteristics is evidenced by the extensive theoretical work developed in this field. Homogeneous reservoirs can be described theoretically for a great number of boundary and production conditions.The basic theory for the analysis of homogeneous isotropic reservoirs is based on the line-source solution to the radial diffusivity equation: (1) The solution is (2) Eq. 2 shows that homogeneous isotropic reservoirs can be described using at least two parameters: the reservoir transmissivity T=k h/mu and the reservoir diffusivity eta=k/phi muc. Several techniques are used widely to estimate these parameters; from them, we can calculated two unknowns (e.g., k and phi provided we know the remaining (h, mu, and c). The provided we know the remaining (h, mu, and c). The definition of the skin effect and the use of the superposition principle and image wells widen the practical applications of Eq. 2. practical applications of Eq. 2. Several authors have dealt with nonhomogeneous and nonisotropic reservoirs. The solutions proposed are complex and generally require computer handling.Among the heterogeneities in a reservoir, there is an important one that is due to the presence of natural matrix fractures. In this case the productive pay is fragmented by a spatial fracture network as a pay is fragmented by a spatial fracture network as a result of the natural geologic factors. Few authors have suggested theories to aid in calculating the characteristics of naturally fractured reservoirs. A detailed theory, developed by Warren and Root, based on the theoretical work by Barenblatt and Zheltov, assumes a network of orthogonal, equally spaced features. Thus, the reservoir is made up of blocks that are able to exchange fluids with the fractures.Barenblatt and Zheltov suggested that the flow from the matrix could be considered as a first approach in a semisteady-state regime. With this assumption Warren and Root developed the differential equations and obtained analytical solutions for well test analysis. When these solutions are plotted in a conventional way (e.g., a Horner plot), plotted in a conventional way (e.g., a Horner plot), they show two parallel straight lines connected by a transition zone of variable slope. The vertical distance between them is related to the relative storage capacity of the fractures and the slopes with the flow capacity of the reservoir. JPT p. 1241

Methods of using pressure build-up or flow tests to estimate formation permeability, formation pressure and well damage are reviewed. A number of phenomena which cause actual pressure build-up behavior to differ from the idealized case, such as the effects of boundaries, skins, inhomogeneities, partial penetration and two-phase flow, are discussed. It is concluded that the same method of analysis of build-up curves can be applied with slight modification to oil reservoirs, gas reservoirs and reservoirs producing both oil and gas. The calculations can be carried out on a form sheet, a copy of which is included in the paper. Examples are worked out for four different types of reservoirs. Introduction Although the basic theory of pressure build-up behavior in wells was developed many years ago, important contributions since that time have extended the original applicability to a much wider variety of situations. The purpose of the present paper is to summarize the present status of pressure build-up theory and of its applicability. The approach in this paper will be to start with the simplest type of pressure build-up curve and to show how reservoir rock properties, reservoir fluid properties and wellbore conditions tend to distort the idealized picture. Methods for taking these distortions into account and for determining values of reservoir formation properties from build-up curves will then be considered.

This work investigates the effects of initial reservoir pressure on the computation of effective permeabilities for homogeneous and heterogeneous solution-gas-drive reservoirs which are initially above the bubble-point pressure. The heterogeneous system considered is that of a two-zone composite reservoir with different absolute permeability and/or relative permeability curves in the two regions. It is shown that the initial effective oil permeability may be estimated from drawdown and buildup pressure data. It is also shown that one can obtain accurate estimates of computed effective permeabilities as pointwise functions of wellbore pressure from drawdown data during the time period that the wellbore pressure is less than the initial bubble-point pressure. From the estimates of effective permeabilities obtained, it is shown that for the case of homogeneous solution-gas-drive systems one can construct approximate effective (or relative) permeability curves by nonlinear regression analysis techniques. For the case of heterogeneous solution-gas-drive reservoirs it is shown that if the initial reservoir pressure is above the reservoir bubble-point pressure then the effective oil permeability of the inner or the outer zone might be obtained from the analysis of drawdown and/or buildup pressure data.

Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 49–54. Abstract This paper presents results of a study to determine to what extent errors in estimated free gas saturation affect the results of static pressure calculations from build-up curves in two-phase systems. Use is made of the method of pressure build-up curve analysis developed by Miller, Dyes and Hutchinson, with modifications for two-phase flow advanced by Perrine. Combined fluid compressibility of gas-oil mixtures is included, with special attention given to the discontinuity that occurs in fluid compressibility at the saturation pressure and its effect on static pressure calculations from build-up curves. The sensitivity of static pressure calculations to errors in gas saturation is considered over wide ranges of formation and fluid properties and reservoir pressure conditions. It is concluded from this study that for analyses of two-phase pressure build-up curves (1) the use of undersaturated crude compressibility data, even when producing GOR's are at or near solution values, will result in very significant negative errors in calculated static pressure, and (2) except for large values of buildup curve slope and/or very low crude gravities, rather large errors in calculated gas saturation can be tolerated without significant effect on calculated static pressures. The conclusions are qualified as referring only to the effects of gas saturation and exclude possible errors in the numerous other factors that enter into the analysis of pressure build-up curves. Introduction As part of the analysis of basic data for several major Eocene reservoirs in the Bolivar Coastal field (Western Venezuela), it was necessary to analyze a large number of pressure build-up curves for oil permeability and static reservoir pressure. From performance data it became apparent that gravity segregation is very active in these reservoirs. As a result of this performance it was recognized that the free gas saturation within the oil leg of the reservoirs would be small but would be subject to certain errors in calculation due to the inaccuracy of relative permeability data in the low gas saturation region. Since all of the various methods of build-up curve analysis were originally derived for single-phase, constant compressibility systems, and since the presence of a small but rather indeterminate gas saturation would greatly affect the reservoir fluid compressibility, a study was undertaken to investigate the effect of gas saturation on the results of two-phase pressure build-up analyses.

RUSSELL, D.G.,* MEMBER AIME, SHELL DEVELOPMENT CO., HOUSTON, TEX. Abstract Two techniques have been developed with which the applicability of pressure build-up analyses can be extended to include pressure data which previously have been considered virtually unusable. One of the interpretation methods makes possible the analysis of pressure build-up performance during the wellbore fill-up or after production period which occurs soon after a well is closed in. The other technique is an extension of a method for analyzing pressure build-up performance during the late-time portion of the pressure build-up which occurs after boundary effects first begin to alter the shape of a conventional pressure build-up curve. With both of these methods it is possible to obtain estimates of the kh product, the skin factor and the reservoir pressure. In addition, with the late-time analysis technique it is possible to obtain an estimate of the contributory drainage volume of the well being tested. This means that in some cases a check on reservoir limit test and (or) material-balance calculations can now be obtained from pressure build-ups. Both methods are slightly more time-consuming than conventional pressure build-up analysis methods because trial-and-error plots of pressure data must be made. The late-time method for analysis of pressure build-ups is in principle applicable to the late-time portion of a two-rate flow test or a pressure drawdown test. The interpretation formulas and procedures for these types of tests are also outlined. In these cases, as with pressure build-ups, it is significant that an estimate of the contributory pore volume is also obtained. Oil the basis of limited experience with the new techniques, it appears that satisfactory estimates of the kh product, skin factor, reservoir pressure and, for late-time analysis, contributory drainage volume can be obtained. Introduction The analysis of bottom-hole pressure build-up behavior in closed-in wells has been a subject of interest in petroleum engineering circles for many years. In fact, few other subjects have received as much attention as pressure buildup analysis methods have. The cause for this interest is essentially twofold in nature. First, the pressure behavior of a well can normally be measured with a reasonably high degree of accuracy so that good data for analysis can be obtained. Secondly, over a fairly wide range of operating conditions, valuable information as to the quality of the reservoir rock and completion efficiency of the well can be obtained at a nominal cost. In recent years, numerous papers have been prepared on the effects of various operating conditions and reservoir heterogeneities on pressure buildup behavior. Very little work has been done, however, on extension of pressure build-up analysis methods to those pressure data which are not amenable to analysis by the present methods. The theory upon which the analysis of shut-in bottomhole pressure build-up data is based is derived from the solution of the radial flow equation for a slightly compressible fluid for constant-rate conditions. It requires that the well be closed in for a sufficient period of time to obtain a clearly defined linear portion on the plot of observed bottom-hole pressure vs log (t + t)/ t (where t is shutin time, and t is producing time to the instant of shut-in). From the slope of the plot and other normally obtainable data, the formation permeability, the well damage or skin factor, and the reservoir pressure at infinite shut-in time (if the reservoir were infinite) can be estimated. The successful application of this procedure depends on being able to recognize the straight-line section on the basic pressure build-up plot. The presently used pressure build-up interpretation theory also assumes that a well is closed in at the sand face and that no production into the well occurs after shut-in. In practice, of course, the well is closed in at the surface, and inflow into the well continues until the well fills sufficiently to transmit the effect of closing-in to the formation. This adjustment period is commonly referred to as the "after production" or "fill-tip" portion of the pressure build-up. During the period that the well fillup effect is most pronounced, the basic pressure build-up plot is nonlinear. At later shut-in times after the effects of a drainage boundary have been felt at the well, deviation from the straight-line behavior of the pressure build-up plot also results. In many cases either of these effects or a combination of both can make the straight-line portion on the pressure build-up plot difficult to recognize. Obviously, an extension of pressure build-up analysis methods to include the after production period and the period in which boundary effects are being felt would be desirable and might render valuable pressure data which for years have been considered virtually unusable. The principal reference of note concerning pressure buildup analysis during the after production period is a paper by Gladfelter, Tracy and Wilsey. In the approach of these authors it is necessary to measure the rate of influx into the well during the after production period. JPT P. 1624ˆ

Pressure buildup, interference, and pulse tests in a naturally fractured Pressure buildup, interference, and pulse tests in a naturally fractured dry gas reservoir are influenced by reservoir limits. Type curves are matched to test data to estimate drainage area and to compute porosity and permeability. Calculated porosity and permeability values compare well permeability. Calculated porosity and permeability values compare well with published data for natural fracture systems. Introduction The case studied is a dry gas reservoir in which three wells are completed. The wells are spaced 2 and 8 miles apart in a 10-mile line along the crest of an anticline with about 100 sq miles of closure (Fig. 1). The dashed contour in Fig. 1 is the drainage boundary that was initially estimated from geologic and production test data assuming a uniform gas-water contact. This drainage area is about 18 miles long and 3 miles wide. Only one productive stratigraphic unit is common to all three productive stratigraphic unit is common to all three wells. This is a naturally fractured zone of thinly bedded, clean orthoquartzites that accounts for 90 percent of deliverability at Well 1, 95 percent at Well 2, and 100 percent at Well 3. Type of completion, fractured zone thickness, and other reservoir data are presented in Table 1. No cores were taken directly from the naturally fractured orthoquartzite zone, but cores from other orthoquartzites had 2.5-percent average porosity and less. than 0.1-md permeability to air. Test data studied in this field case history have two chronological groupings:data recorded when Well 2 was completed, consisting of one pressure drawdown and four pressure buildup tests at Well 2; anddata obtained 4 years later, consisting of pressure interference at Wells 3 and 1 caused by flowing Well 2 for 450 hours, pressure buildup at Well 2 immediately following the interference test, and pulse response at Well 3 caused by pulsing Well 2. The field was never on production except to conduct pressure transient and production except to conduct pressure transient and deliverability tests. Analysis of the field tests data is organized into four sections:(1)general discussion of the pressure drawdown and buildup behavior in light of recently published well-test theory;(2)computation of porosity and published well-test theory;(2)computation of porosity and estimation of drainage area by matching the buildup data to type curves;(3)computation of porosity, permeability, and drainage area by matching the permeability, and drainage area by matching the interference data to type curves; and(4)analysis of pulse behavior in the presence of reservoir limits. General Pressure-Buildup Behavior Buildup Tests 1 through 4, recorded at completion of Well 2, are presented in Tables 2 through 5. The pressure drawdown corresponding to Buildup Test 4 is also pressure drawdown corresponding to Buildup Test 4 is also shown in Table 5. Fig. 2 is a graph of pressure as a function of the logarithm of time for the drawdown test. All four buildup tests are plotted in Fig. 3, using the technique of Horner. Pressure buildup during Test 1 becomes a linear function of the logarithm of the Horner time ratio, and extrapolates to initial pressure at infinite shut-in time. Each of the other tests plotted in Fig. 3 has an early period in which pressure is a linear function of the period in which pressure is a linear function of the logarithm of the Horner time ratio and a late period in which pressure bends upward. JPT P. 1097