PRESSURE TRANSIENT FIELD DATA SHOWING FRACTAL RESERVOIR STRUCTURE

Conventional pressure transient models strictly apply to areally homogeneous reservoirs. Yet, core and log data indicate this assumption is often not justified. This paper describes a model for heterogeneous reservoirs and supporting field data from the Grayburg/ San Andres formations in southeastern New Mexico. Conventional models fail to match these field data. Instead, a model for a heterogeneous reservoir with a fractal structure provides a quantitative analysis. The fractal reservoir model reduces to the conventional solution in the case of a homogeneous reservoir.

Conventional pressure transient models strictly apply to a really homogeneous reservoirs. Yet, core and log data indicate this assumption is often not justified. This paper describes a model for heterogeneous reservoirs and supporting field data from the Grayburg/San Andres formations in southeastern New Mexico. Conventional models fail to match these field data. Instead, a model for a heterogeneous reservoir with a fractal structure provides a quantitative analysis. The fractal reservoir model reduces to the conventional solution in the case of a homogeneous reservoir. Introduction Conventional pressure transient models do not match some recent pressure transient tests on the Grayburg and San Andres Formations in southeastern New Mexico. The problem is illustrated by comparing a recent pressure buildup test in Fig. 1 with conventional type curves in Fig. 2. Since all producing zones have been sand fractured, we would expect this set of curves for a well with a vertical fracturel to apply. The type curves show the pressure change along with the usual pressure derivative. The pressure curve initially has a slope of unity if wellbore storage effects dominate the early time data. Without wellbore storage, linear flow near the vertical fracture produces an early straight pressure line of slope one half. Straight lines with other slopes do not appear even for curves that fall between these two extreme cases shown. The derivative curve eventually approaches a constant value. The field data in Fig. 1 do not follow these trends. Since the production time prior to shut-in is much larger than the shut-in time, Δt, a log Δp versus log Δt plot should match the drawdown curves in Fig. 2. Instead, the pressure data between 1 and 150 hours fall on a straight line with slope 0.60, substantially different from the lines in Fig. 2. The pressure derivative curve is parallel to the pressure curve. These parallel lines for over 2 log cycles of shut-in time cannot be matched with homogeneous reservoir models. After 150 hours, the pressure derivative flattens due to interference from neighboring wells. The shortcomings of homogeneous reservoir models become clear by looking at core photographs and well log data. The Grayburg and San Andres formations are very heterogeneous as recent outcrop and geological studies2,3 demonstrate. The present paper presents a pressure transient model for a heterogeneous reservoir that matches the field data in Fig. 1. In the model, the reservoir contains both permeable and impermeable rock. The resulting permeable areal network is assumed to have a fractal structure. The fractal reservoir model is an extension of earlier work by Chang And Yortsos.4 Chang and Yortsos proposed their model for naturally fractured reservoirs without any supporting field pressure data. With some modifications, their model can also represent a reservoir with impermeable rock embedded within the permeable pay zones. For this application, we rewrite the pressure transient equations for a fractal reservoir in a form that can use available core estimates of near wellbore porosity and permeability.

Often and for many reasons the wellbore does not completely penetrate the entire formation, yielding a unique early-time pressure behavior. Some of the main reasons for partial penetration, in both fractured and unfractured formations, are to prevent or delay the intrusion of unwanted fluids into the wellbore, i.e. water coning. A similar early-time pressure behavior may be due to the presence of plugged perforations. Drilling problems associated with high mud losses when the well encounters fractures, often prevent well penetration of the total formation thickness. Penetration in naturally fractured reservoirs is usually minimal (10 to 20%), but with the right mud, it can reach 50% and in some cases 100%. Such well completions are referred to as limited-entry, restricted-entry or partially penetrating wells. The transient flow behavior in these types of completions is different and more complex compared to that of a fully penetrating well. This paper proposes a method for identifying, on the pressure and pressure derivative curves, the unique characteristics of the different flow regimes resulting from these types of completions and to obtain various reservoir parameters, such as vertical and horizontal permeability, fracture properties, and various skin factors. Both naturally fractured and unfractured (homogeneous) reservoirs have been investigated. For unfractured and homogeneous formations, a spherical or hemispherical flow regime occurs prior to the radial flow regime whenever the penetration ratio is twenty percent or less. A half-slope line on the pressure derivative is the unique characteristic identifying the presence of the spherical flow. This straight line can be used to calculate spherical permeability and spherical skin values. These parameters are then used to estimate vertical permeability, anisotropy index and skin. For a naturally fractured formation, the type curves of the pressure and pressure derivative reveal that the combination of partial penetration and dual-porosity effects yields unique finger prints at early and transition periods. These unique characteristics are used to calculate several reservoir parameters, including the storage capacity ratio, interporosity flow coefficient, permeability and pseudo-skin. Equations have been developed for calculating the skin for three partial completion cases: top, center and bottom. The analytical solution was obtained by combining the partially penetrating well model in a homogeneous reservoir with the pseudo-steady model for a naturally fractured reservoir. The interpretation of pressure tests in both systems, i.e. fractured and unfractured reservoirs, is performed using Tiab's Direct Synthesis (TDS) technique for analyzing log-log pressure and pressure derivative plots. TDS uses analytical equations to determine reservoir and well characteristics without using type-curve matching. These characteristics are obtained from unique fingerprints, such as flow regime lines and points of intersection of these lines, that are found on the log-log plot of pressure and pressure derivative. It is applied to both drawdown and buildup tests. Several numerical examples are included to illustrate the step-by-step application of the proposed technique.

The current type-curve matching technique is essentially a trial-and-error procedure without any independent and accurate way of checking the validity of the end results. This paper introduces a new technique for interpreting pressure tests. This technique uses log-log plots of the pressure and pressure derivative versus time to calculate reservoir and well parameters WITHOUT type-curve matching. This paper concentrates on the interpretation of pressure tests in which wellbore storage and skin are present. The technique essentially consists of obtaining characteristic points of intersection of various straight line portions of the pressure and pressure derivative curve, slopes and starting times of these straight lines. These points, slopes and times are then used with appropriate equations to solve directly for permeability, wellbore storage and skin. A step-by-step procedure for calculating these parameters without type-curve matching for five different cases is included in the paper. The most important aspect of this new technique is undoubtedly its accuracy because it uses exact analytical solutions to calculate permeability, skin, and wellbore storage. The second most important feature of this new technique is that it is verifiable. Any two parameters calculated from two independent equations corresponding to two different portions of the pressure derivative curve are verified by a third equation which corresponds to a known and unique point relating the two parameters. The proposed technique is applicable to the interpretation of pressure buildup and drawdown tests. This technique is illustrated by several numerical examples.

This paper describes rate derivative curves of constant pressure tests similar to derivative analysis for constant rate tests. Four rate decline derivatives: Cartesian, log-semi, semi-log, and log-log, are constructed, Typically, the rate derivative curves have some special characters. The specific characteristics of these derivative curves are related to the different flow regimes and various boundary effects. The derivative curves and decline curves can be used for well test interpretation and diagnostic in a similar way to the constant rate tests. Therefore, in some cases nonunique problems in well test analysis are eliminated. In this paper, we consider rate declines of an arbitrarily placed well in various shapes such as a circle, a rectangle, a stringer, and an irregular shape.

Well testing provides a powerful tool to detect and to evaluate heterogeneities in naturally fractured reservoirs (NFR`s). Experience has shown that this type of reservoir may display behavior that consists of a variety of flow models. This paper presents a discussion of the applications and limitations of pressure-transient tests in the evaluation of NFR`s.

The combined plot of log pressure change and log derivative of pressure with respect to superposition time as a function of log elapsed time was first introduced by Bourdet et al. as an aid to type-curving matching. Features that are hardly visible on the Horner plot or are hard to distinguish because of similarities between one reservoir system and another was easier to recognize on the pressure-derivative plot. Once the patterns have been diagnosed on the log-log plot, specialized plots can be used to compute reservoir parameters or the data can be matched to a type curve.

Pressure communication between adjacent wells is frequently encountered in multi-stage hydraulic fractured shale gas reservoirs. An interference test is one of the most popular methods for testing the connectivity of a reservoir. Currently, there is no practical analysis model of an interference test for wells connected by large fractures. A one-dimensional equation of flow in porous media is established, and an analytical solution under the constant production rate is obtained using a similarity transformation. Based on this solution, the extremum equation of the interference test for wells connected by a large fracture is derived. The type-curve of pressure and the pressure derivative of an interference test of wells connected by a large fracture are plotted, and verified against interference test data. The extremum equation of wells connected by a large fracture differs from that for homogeneous reservoirs by a factor 2. Considering the difference of the flowing distance, it can be concluded that the pressure conductivity coefficient computed by the extremum equation of homogeneous reservoirs is accurate in the order of magnitude. On the double logarithmic type-curve, as time increases, the curves of pressure and the pressure derivative tend to be parallel straight lines with a slope of 0.5. When the crossflow of the reservoir matrix to the large fracture cannot be ignored, the slope of the parallel straight lines is 0.25. They are different from the type-curves of homogeneous and double porosity reservoirs. Therefore, the pressure derivative curve is proposed to diagnose the connection form of wells.

This paper presents a case history of characterization of a gas condensate reservoir using pressure transient analysis. Pressure transient tests from wells in this field led to test data plots with complex shapes. Specifically, the pressure derivative in a typical test flattened at intermediate shut-in times (after wellbore storage effects diminished) and then trended downward. This curve shape indicates lower mobility near the wellbore and increased mobility some distance away. Using conventional interpretation techniques, this pressure derivative response may be interpreted (erroneously) as a composite reservoir with low transmissibility in a region with radius of almost 500 feet near the well, surrounded by a region of higher transmissibility, and a positive skin factor. In this study, we modeled well tests in this field with a fully compositional reservoir simulator. We demonstrated that we can reproduce the observed test behavior in a homogenous reservoir. The decrease in pressure derivative is caused by reservoir fluid property changes with pressure, and the apparent positive skin factor is a result of liquid condensing in the formation near the wellbore. The region with reduced transmissibility (high liquid saturation) was on the order of only 20feet in radius. Our study included sensitivity analysis to determine the effect of selected variables on pressure transient test response. Production time prior to shut-in proved to be particularly important. Longer production periods prior to shut-in can modify the shape of the derivative curve plot but do not change the possible erroneous interpretations resulting from essentially perfect fits of test data with composite reservoir models. Introduction Analysis of well tests from gas condensate reservoirs is a significant challenge for engineers. If pressure drops below the dew point near the wellbore during the test, a condensate ring will accumulate immediately around the well. This can cause a significant loss in well productivity. The formation of this ring is documented by McCain and Alexander1. In this paper, we report an investigation of the effect of the condensate ring on pressure transient analysis and document the distinctive behavior of the pressure derivative caused by the ring. A drill stem test (DST) can include a series of production and shut-in periods, and thus can produce particularly "interesting" pressure derivative curves in gas condensate reservoirs. The test that we analyzed and discuss in this paper was from a multi-flow period, multi-shutin period DST. Although many papers discuss fluid flow in gas condensate reservoirs, we found none that propose adequate methodology to determine formation properties from analysis of well test data from gas condensate reservoirs. Bourbiaux2 investigated depletion behavior in gas condensate wells using a parametric modeling study. Carlson and Myer3 studied the effect of condensate drop out on the performance of fractured wells and presented some information on well test analysis of fractured gas condensate reservoirs. Afidick, et al.4 presented a case study of a gas condensate reservoir. Jones, et al.5 presented a two-phase analog that can be used for build up analysis from wells producing below the dew point pressure.

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

The present paper describes the use of pressure derivative and second derivative of integral of pressure in a fractal reservoir with matrix participation with phase redistribution in a geological environment that are not possible by conventional techniques. The analysis of this type of data in reservoir characterization is known as “inverse problem” and one can obtain information about interwell and vertical permeability distribution in a reservoir. The fractal geometry in a dynamic pressure transient tests data plays a very vital role for heterogeneity characterization. The pressure transient response is analyzed for flow in a connected fracture network and fracture with matrix participation. The computer aided matching technique for both pressure and its derivative by nonlinear regression techniques are used in estimating the reservoir properties from measured drawdown/buildup and falloff pressure data of heterogeneous reservoir. In the present paper the fractional calculus approach has been utilized to solve the diffusivity equation with phase redistribution in fractal reservoir. The pressure solution of the diffusivity is in terms of Laplace space and its analytical inversion is not possible. We have obtained numerically inversion of the problem and the pressure, pressure derivative, integral of pressure and its first and second derivative has been calculated. The permeability estimated from pressure transient test data of a well are in good agreement with the identified the geological model.

Conventional analysis method of DST buildup period uses Agwarl's equivalent time handling variable rate. Assumption of the method is semi-log radial flow for homogenous reservoir. However for heterogeneous reservoir (double porosity, double permeability) the method will cause pressure derivative curve distortion and the pressure derivation curve cannot match type curve of conventional constant rate. This paper presents a new analysis method of DST pressure history using flow rate deconvolution in Laplace space. Through wellbore storage effect computing history rate and determined pressure response by applying Laplace deconvolution method, we can estimate parameters in Laplace space. Through simulated data and oil field data, the numerical results obtained are stable and not sensitive to noise. The established method provides a new way to understand DST pressure history in homogenous and heterogeneous reservoirs. Introduction DST is widely adopted due to its fast speed, attaining a lot of information and low cost. Particular interpretation methods of DST such as flow period analysis (Ramey2,3,4 slug test analysis, Peres's5 pressure integrated method), buildup period analysis (Peres's6 pressure deconvolution, modified Horner method) and pressure history analysis are special for homogenous formation. On the other hand, the analysis theory and interpretation methods of conventional drawdown and buildup test have become mature. If DST buildup period data are converted into equivalent conventional drawdown data, conventional test analysis theory and interpretation method can be used in DST. At the present time, using variable rate superposition (Agwarl's1 equivalent time) method can convert DST buildup data into equivalent conventional drawdown data. This method comes from variable rate superposition of semi-log radial flow behavior and it is only suitable for homogenous reservoir. According to simulated data analysis of DST, obtained data through variable rate superposition processing do not match type curve of conventional constant rate for naturally fractured reservoir. Therefore, this conventional method can not correctly estimate parameters. In this paper, according to DST full pressure history characteristic and throughout wellbore storage to calculate history rate, we can get pressure response of reservoir for constant rate and directly estimate parameters in Laplace space by numerical Laplace transformation. Application of simulated and oilfield data show that the new method is more accurate than Agwarl's method and can effectively determine estimated error of initial formation pressure. Effect of Variable Rate Superposition Time on Pressure Derivative Curve DST test simulator can be used to generate "flow and buildup", history pressure data to calculate flow period and buildup period rate. According to variable rate superposition time handling buildup period data, we can attain double-log diagnose curve and pressure derivative curve of standard drawdown response. Comparing the actual analysis curve with the standard curve, it is shown that the curve match is very good between the analysis curve of variable rate superposition time and the standard curve for infinite acting homogenous reservoir. But, if using average rate superposition time Δ te = tf × Δ t / (tf + Δ t), pressure derivative of analysis curve shows up in the late period and forms assumption phenomenon with sealing boundary.

Often and for many reasons the wellbore does not completely penetrate the entire formation, yielding a unique early-time pressure behaviour. Some of the main reasons for partial penetration, in both fractured and unfractured formations, are to prevent or delay the intrusion of unwanted fluids into the wellbore, i.e., water coning. The transient flow behaviour in these types of completions is different and more complex compared to that of a fully penetrating well. This paper proposes a method for identifying, on the pressure and pressure derivative curves, the unique characteristics of the different flow regimes resulting from these types of completions and to obtain various reservoir parameters, such as vertical and horizontal permeability, fracture properties and various skin factors. Both naturally fractured and unfractured (homogeneous) reservoirs have been investigated. For a naturally fractured formation, the type curves of the pressure and pressure derivative reveal that the combination of partial penetration and dual-porosity effects yields unique finger prints at early and transition periods. These unique characteristics are used to calculate several reservoir parameters, including the storage capacity ratio, interporosity flow coefficient, permeability and pseudoskin. Equations have been developed for calculating the skin for three partial completion cases: top, centre and bottom. The analytical solution was obtained by combining the partially penetrating well model in a homogeneous reservoir with the pseudo-steady model for a naturally fractured reservoir (NFR). The interpretation of pressure tests is performed using the TDS (Tiab's Direct Synthesis) technique for analyzing log-log pressure and pressure derivative plots. The TDS technique uses analytical equations to determine reservoir and well characteristics without using type-curve matching. These characteristics are obtained from unique fingerprints, such as flow regime lines and points of intersection of these lines, which are found on the log-log plot of pressure and pressure derivative. Two numerical examples are included to illustrate the application of the. proposed technique. Introduction Consider a vertical well partially penetrating a naturally fractured reservoir, i.e., only a portion of hydrocarbon-bearing formations is perforated. The naturally fractured reservoir has an infinite radial extent. The Warren and Root(1) model is used in which the matrix blocks are replaced by a system of uniform rectangular parallelepipeds with identical properties. The fractures are assumed to be parallel with the principal axes. FIGURE 1: Different types of partially penetrating vertical wells based on the position in the perforated interval hw. Available in Full Paper The pressure solution is derived using the Laplace transformation and the separation of variables technique as proposed by Bui et al.(2). This solution is expressed as an infinite Fourier-Bessel series in Laplace domain. The theory for a partially penetrating well in a homogenous reservoir developed by Yildiz and Bassiouni(3) is used for comparison purposes. The analytical solution for constant flow rate in Laplace space was inverted into real dimensionless pressure using the Stehfest algorithm(4). Pressure Derivative Behaviour Four types of partial penetration or completion schemes are considered (as shown in Figure 1). A plot of the dimensionless pressure derivative tD * P'D versus tD is shown in Figures 2 and 3.

A simulation model of a reservoir with a symmetrical-horizontal fracture extending from the wellbore to the midpoint of the drainage radius was constructed. The mathematical equation was developed for the case of single-phase unsteady state fluid flow. A solution for an infinite reservoir was obtained numerically and used for pressure drawdown and buildup analysis. The pressure drawdown and buildup analysis. The numerical results shows that: 1. On pressure drawdown and buildup curves, two straight lines are obtained; the first straight line with lower slope yields the effective permeability of the matrix and fracture, and the second straight line with greater slope yields the permeability of matrix. 2. The time of bend between the straight lines increases with increase in fracture radius. As the fracture radius approaches infinity, only one straight line of the Odeh type is obtained. 3. Extrapolation of the first straight line portion of the buildup curve may lead to an incorrect value of the static reservoir pressure. Introduction Analysis of pressure buildup and drawdown data is recognized as a powerful tool by the production and reservoir engineer seeking to production and reservoir engineer seeking to characterize the reservoir. Most pressure analysis techniques have assumed homogeneous reservoirs, i.e. the porosity and permeability are constant. However, some prolific wells produce from fractured reservoirs. These produce from fractured reservoirs. These reservoirs contain two distinct types of porosity and permeability, namely fracture porosity and permeability, namely fracture and matrix. Since the fractured region has higher permeability, reservoir-engineering analysis based on a homogeneous reservoir may lead to erroneous results. The purpose of this study is to develop a mathematical model which will simulate the pressure drawdown and buildup curves that would be obtained from a reservoir with a symmetrical-horizontal fracture around the wellbore. The mathematical model is developed by assuming a cylindrical reservoir of drainage area of uniform thickness is penetrated by a single production well at its center. The two -dimensional diffusivity equation for single phase flow was used to obtain pressure buildup phase flow was used to obtain pressure buildup and drawdown curves. It was necessary to obtain a constant rate solution to the equation because of the mathematical complexities introduced by the fractured reservoir geometry.

Large-scale karst caves are the principal storage spaces for hydrocarbon resources in fracture–cavity carbonate reservoirs. Drilling directly into these caves is considered the ideal mode of development, but many wells do not effectively penetrate karst caves. Therefore, acid fracturing is employed to generate artificial fractures that can connect with these caves. However, there are no appropriate well test methods for fracturing wells in fracture–cavity reservoirs. This study establishes a novel pressure transient analysis model for such wells. A new mathematical model is proposed that couples linear flow in acid fracturing cracks with radial flow in the oil drainage area. The Laplace transform and Stehfest numerical inversion provided analytical solutions for the bottomhole pressure. Typical log–log well testing curves were plotted to analyze oil flow, which occurs in ten stages. During the flow stage in fracturing cracks, the pressure and pressure derivative curves are parallel lines with a slope of 0.5. In the stage of karst cave storage, the pressure derivative curve is a straight line with a slope of 1. A comparison with previous models confirmed the validity of the proposed model. The influence of key parameters on the behavior of typical curves is analyzed. A field case study of the proposed model was carried out. Parameters related to fracturing cracks and karst caves, such as the crack length and cave radius, were successfully estimated. The proposed model has great potential for determining formation parameters of fracture–cavity reservoirs.