Storativity Ratio Research Articles

A study on transient fluid flow of horizontal wells in dual-permeability media

To adopt horizontal wells in dual media reservoirs, a good understanding of the related fluid flows is necessary. Most of the recent studies focus on dual porosity media instead of dual permeability media. In this article, through both integral transformation and sink-source superposition, a new Laplace-domain solution is obtained for the slightly-compressible fluid flow in the 3-D dual-permeability media in which the horizontal well is operating in a constant rate of production. Major asymptotic characteristics of diagnosis curves of dimensionless downhole pressure are analyzed by the limited analysis. Effects of parameters of dual-permeability media including mobility ratio κ, storativity ratio ω and inter-porosity flow parameter κ on the downhole pressure are studied by using the Laplace numerical inversion. The new solution obtained in this article includes and improves the previous results and then can be used as a basis for either pressure transient analysis or formation behavior evaluation for the typical reservoir with horizontal wells.

Transient Pressure Behaviour Under Non- Darcy Flow, Formation Damage and Their Combined Effect for Dual Porosity Reservoirs

Abstract A semi-analytical model is presented to study transient pressure behaviour of vertical wells in a dual porosity reservoir under the influence of non-Darcy flow inside the reservoir (Forchheimer number), formation damage around the wellbore (mechanical skin factor), and their combined effect (near-wellbore non-Darcy skin factor). Unique transient pressure behaviour is presented in both semi-log pressure and log-log pressure derivative plots. Semi-log pressure plots suggest that for a vertical well producing with a constant sandface flow rate, its pressure responses exhibit two linear portions for Darcy flow, even with non-Darcy skin factor (i.e. the near-wellbore non-Darcy skin factor is larger than zero). However, the first linear portion will dismiss if non-Darcy flow presents inside the reservoir (i.e. the Forchheimer number is larger than zero). Further pressure derivative analyses in a log-log plot suggest that:both Forchheimer number and nearwellbore non-Darcy skin factor lead to a narrower and steeper transition region between pure wellbore storage and radial flow region for build up tests and a wider and gentler one for drawdown tests; andthe derivative curves under the effect of the Forchheimer number always exhibit a slight convex shape in the radial flow region, rather than a horizontal line with a constant value of 0.5 for Darcy flow. The type curves presented may be applied to estimate mechanical skin factor, Forchheimer number and near-wellbore non- Darcy skin factor from a single rate test. Synthetic cases show that ignoring the globally distributed non-Darcy flow causes errors of up to 300% in interpreting dual porosity characteristic parameters. Both synthetic and field case studies suggest that the non-Darcy flow has a significant effect on the interpretation of the inter-porosity flow coefficient, ?, and less effect on the interpretation of the storativity ratio, ?. Introduction The Forchheimer equation suggests that high velocity gas flow in porous media obeys the following quadratic equation: Equation (available in full paper) It has been recognized that Forchheimer non-Darcy flow is introduced by gas inertial effects; thus, ß?is called the inertial factor. Forchheimer non-Darcy flow is a universal phenomenon in gas reservoir development. It could affect reservoir parameter estimations, well productivity evaluation and production performance analysis. Observations in the field(1) and the laboratory(2, 3) have suggested that, in the fracture system of a naturally fractured reservoir, the pressure loss is dominated by the square term in the Forchheimer equation. Studies on modelling non-Darcy flow in naturally fractured reservoirs are limited. The first attempt was reported by Villalobos-L. et al.(4). Based on numerical results of a radial liquid flow model with the Forchheimer non-Darcy flow, these authors derived an expression of a rate-dependent pseudo-skin term similar to that in homogeneous single porosity reservoirs. Wu(5) modelled non-Darcy flow in a naturally fractured system using the Multiple Interacting Continua (MINC) method by applying the results of Darcy flow to approximate the characteristic length of non-Darcy flow distance between fractures and the matrix crossing the interface for the dual porosity formation. He observed that semi-log plots of pressure drop versus time are extremely sensitive to the values of non-Darcy coefficient.

Transient Flow in Discretely Fractured Porous Media

Summary This paper presents a semianalytical model for transient flow into multiple vertical wells producing from a porous medium containing randomly distributed discrete fractures. Both vertical openhole wells and hydraulically fractured vertical wells are considered. The semianalytical model simulates pressure and pressure-derivative characteristics of wells and flow distribution along and through both the natural and the hydraulic fractures. The study shows that single or multiple isolated natural fractures yield negative pseudoskin factors in vertical wells near isolated fractures. The negative pseudoskin factor is a function of fracture conductivity, density, length, distance from the wellbore, and azimuth. Using the model, we demonstrate that the shape of the pressure derivative is related to fracture distribution. The results of this study indicate that the conventional double-porosity analysis to predict the storativity ratio of a naturally fractured system is not reliable. Also, the displacement between two semilog straight lines is not necessarily a good indicator of the storativity ratio.

A new solution of transient confined–unconfined flow driven by a pumping well

A new solution of transient confined–unconfined flow driven by a pumping well is developed and compared to previous approximate solutions of Moench and Prickett [Moench AF, Prickett TA. Radial flow in an infinite aquifer undergoing conversion from artesian to water table conditions. Water Resour Res 1972;8:494–9] and Hu and Chen [Hu L, Chen C. Analytical methods for transient flow to a well in a confined–unconfined aquifer. Ground Water 2008;46(4):642–6]. The problem is rewritten in dimensionless form with the Boltzmann transform. The nonlinear equation for flow in the unconfined zone is solved with the Runge–Kutta method. Position of the conversion interface is determined with an iteration scheme. This study shows that the confined–unconfined flow depends on three dimensionless parameters that represent the confined–unconfined storativity ratio ( a D ), the ratio of the initial hydraulic head over the aquifer thickness ( f i ), and the dimensionless pumping rate ( q D ). The rate of expansion of the unconfined zone increases with q D , but decreases with a D and f i . Differences between the two previous approximate solutions and the new solution of this study are observable in the estimated position of the conversion interface and the drawdown–time curves. The new solution can be applied to estimate the time for confined–unconfined conversion to occur (critical conversion time), and the time when the pumping well becomes dry (critical drying time). The critical conversion time is found to be very sensitive to the initial hydraulic head. The critical drying time is often much larger than the critical conversion time and may never be observed during a finite pumping period.

Estimation of Storativity Ratio in a Layered Reservoir With Crossflow

Summary This paper presents a practical method to estimate the storativity ratio of a dual-permeability layered reservoir with crossflow from pressure-transient data. The method uses an analytical formula for the storativity ratio in terms of the separation between the two semilog straight lines on the pressure vs. log-time plot, similar to the method used for dual-porosity systems. Knowing the storativity ratio from a well test allows individual-layer properties to be estimated if the layer flow rates are available from production logs. Demonstrations of the method to estimate the storativity ratio and individual-layer properties are presented by examples. Comparison of the results with those obtained from the existing techniques is also provided to highlight the accuracy of the proposed technique.

Condensate Bank Characterization from Well Test Data and Fluid PVT Properties

Summary Published well-test analyses in gas/condensate reservoirs in which the pressure has dropped below the dewpoint are usually based on a two- or three-region radial composite well-test interpretation model to represent condensate dropout around the wellbore and initial gas in place away from the well. Gas/condensate-specific results from well-test analysis are the mobility and storativity ratios between the regions and the condensate-bank radius. For a given region, however, well-test analysis cannot uncouple the storativity ratio from the region radius, and the storativity ratio must be estimated independently to obtain the correct bank radius. In most cases, the storativity ratio is calculated incorrectly, which explains why condensate bank radii from well-test analysis often differ greatly from those obtained by numerical compositional simulation. In this study, a new method is introduced to estimate the storativity ratios between the different zones from buildup data when the saturation profile does not change during the buildup. Application of the method is illustrated with the analysis of a transient-pressure test in a gas/condensate field in the North Sea. The analysis uses single-phase pseudo pressures and two- and three-zone radial composite well-test interpretation models to yield the condensate-bank radius. The calculated condensate-bank radius is validated by verifying analytical well-test analyses with compositional simulations that include capillary number and inertia effects. Introduction and Background When the bottomhole flowing pressure falls below the dewpoint in a gas/condensate reservoir, retrograde condensation occurs, and a bank of condensate builds up around the producing well. This process creates concentric zones with different liquid saturations around the well (Fevang and Whitson 1996; Kniazeff and Nvaille 1965; Economides et al. 1987). The zone away from the well, where the reservoir pressure is still above the dewpoint, contains the original gas. The condensate bank around the wellbore contains two phases, reservoir gas and liquid condensate, and has a reduced gas mobility, except in the immediate vicinity of the well at high production rates, where the relative permeability to gas is greater than in the bank because of capillary number effects (Danesh et al. 1994; Boom et al. 1995; Henderson et al. 1998; Mott et al. 1999).

After-Closure Analysis for Naturally Fractured Reservoirs: A Field Validation Study

This article, written by Assistant Technology Editor Karen Bybee, contains highlights of paper SPE 97015, "After-Closure Analysis for Naturally Fractured Reservoirs: A Field Validation Study," by S. Chipperfield, SPE, Santos Ltd., prepared for the 2005 SPE Annual Technical Conference and Exhibition, Dallas, 9–12 October. Economic optimization of fracture stimulation designs requires accurate reservoir description. The full-length paper presents a new technique for evaluating a minifracture injection in a manner similar to a conventional pressure-transient test for identification of productive natural fractures. Appendix A in the full-length paper presents a brief theoretical background. Three examples from Australia and one from Algeria review field application of this technology. Introduction An understanding of critical reservoir parameters aids in determining whether a zone should be stimulated and the most appropriate stimulation design. Until recently, this evaluation was made by use of pressure-buildup tests. However, these tests are performed less frequently now because of the costs in terms of additional equipment and the delay in bringing the product on line. The advent of after-closure analysis (ACA) has allowed reservoir characterization to be conducted in a more cost-effective manner through merging reservoir engineering and minifracture analysis. The full-length paper presents an extension of ACA to identify and characterize naturally fractured reservoirs. This analysis approach identifies natural fractures that are open under in-situ conditions and assists the engineer in differentiating these natural fractures from those fractures that may be open only under high injection pressures. Because naturally fractured reservoirs may require alternative design strategies, this ACA approach not only provides a means of reservoir characterization but also drives stimulation-design choices. Theory Highlights Naturally Fractured vs. Matrix-Style Reservoirs. Flow regimes observable during the after-closure period for a matrix reservoir include pseudolinear and pseudoradial. These flow regimes are identified on a Nolte diagnostic plot (NDP) from a half slope and unit slope, respectively. Reservoir quality can be determined from the pseudoradial-flow period. Fracture half-length Xf can be estimated from the intercept of pseudolinear flow and pseudoradial flow (the "knee" time). Matrix reservoir response is typified by the derivative, and the pressure-difference curves maintain a degree of separation and overlay each other only in late time. The dual-porosity scenario is typified by a dual-porosity dip feature where the derivative and the pressure difference cross. The time and size of the dip during the decline period depend on the natural-fracture properties—namely the storativity ratio ω and the matrix-permeability/fracture-permeability ratio λ. The terms λ, ω, and Xf can be determined from key time markers during the decline period—namely the start of the dip, the dip base, and the end of the dip. Table 1 in the full-length paper outlines the broad range of possible outcomes for the dual-porosity naturally fractured case.

Analysis of flow behavior in fractured lithophysal reservoirs

This study develops a mathematical model for the analysis of pressure behavior in fractured lithophysal reservoirs. The lithophysal rock is described as a tri-continuum medium, consisting of fractures, rock matrices, and cavities. In the conceptual model, fractures have homogeneous properties throughout and interact with rock matrices and cavities that have different permeabilities and porosities. Global flow occurs through the fracture network only, while rock matrices and cavities contain the majority of fluid storage and provide fluid drainage to the fractures. Interporosity flows between the triple media are described using a pseudosteady-state concept and the system is characterized by interporosity transmissivity ratios and storativity ratio of each continuum. Pressure behavior is analyzed by examining the pressure drawdown curves, the derivative plots, and the effects of the characteristic parameters. Typical pressure responses from fractures, matrices, and cavities are represented by three semilog straight lines; the transitions by two troughs below the stabilization lines in the derivative plots. The analytical solution to the proposed model is further verified using a numerical simulation. The analytical model has also been applied to a published field-buildup well test and is able to match the pressure buildup data.

A Generalized Pressure Derivative Analysis For Composite Reservoirs

Abstract Pressure derivatives have been shown to be more sensitive to disturbances in the reservoir than pressure signals; resulting in more detail on derivative graphs than is apparent on pressure graphs. The semilog pressure derivative is widely used in well test analysis. One reason for its popularity is that, for radial systems, the response appears as a horizontal line during the infinite- acting radial flow period, resulting in easier identification. However, when the semilog pressure derivative is applied to flow geometries other than radial, the responses are not horizontal; making identification of flow regimes more difficult. Thus, a generalized pressure derivative is necessary to simplify the identification of flow regimes in any flow geometry. In this study, a generalized pressure derivative is defined and used to identify the various flow regimes for composite systems in radial, elliptical, linear and spherical geometries. This generalized pressure derivative is of the power law type, and is characterized by a different exponent for each of the flow geometries. Using well test data from analytical solutions for radial, elliptical, linear and spherical composite reservoirs, a graph of the generalized pressure derivative versus time, for any of the flow geometries appears as a horizontal line during the primary flow regime characteristic of that geometry. Design and analysis equations, based on the generalized pressure derivative, are presented for well testing of composite reservoirs in various flow geometries. Reservoir parameters estimated using these equations will add to the degree of confidence in the estimated parameters based on pressure analysis. The generalized pressure derivative is also used to investigate differences and similarities among the four flow systems. Results from this study confirm that for radial and elliptical systems, the long term pressure derivative behaviour is influenced only by the mobility ratio between the inner and outer regions of the composite system. For linear and spherical systems, however, long term derivative behaviour is governed by both the mobility ratio and the storativity ratio. This finding has a significant impact on the development of type curves for either manual or automated type curve matching for the various flow geometries. Introduction Pressure derivatives have been shown to be more sensitive to disturbances in the reservoir than pressure signals. This results in greater detail on a derivative graph than is apparent on a pressure graph. Pressure derivatives were first introduced by Tiab and Kumar(1), who presented the derivative of pressure with respect to time. Later, Bourdet et al.(2) introduced the semi-log pressure derivative, defined as the derivative of the well pressure with respect to the natural logarithm of time. The semilog pressure derivative response appears as a horizontal line during the infinite- acting radial flow period, resulting in an easy identification of the radial flow regime. As a result, the semilog pressure derivative is widely used in well test analysis of not only homogeneous, but also composite reservoirs. To analyse well tests for thermal recovery projects, reservoirs have been idealized as composite reservoirs.

Well Test Analysis of Naturally Fractured Gas Condensate Reservoirs

Abstract Approximate equations are presented for evaluation of naturally fractured gas condensate reservoirs represented by dualporosity models in radial systems. The reader is cautioned that this work is in progress. Additional research will help to corroborate and refine these techniques. The solutions have not been published previously in the petroleum engineering literature. The solutions are presented for drawdown and buildup tests. The model assumes flow of gas condensate from a tight matrix into permeable natural fractures. The fractures conduct the fluids to the wellbore. Preliminary work shows that a conventional crossplot of m(p) vs. time on semilogarithmic coordinates results in approximately two parallel straight lines with a separation that is related to the storativity ratio between fractures and matrix. This plot allows determination of key parameters such as absolute permeability, effective permeabilities to oil and gas, skin, storativity ratio (ω), fracture porosity (Φ2), average distance between natural fractures (hm), radius of investigation, and extrapolated pressure (Pi or p*). In addition the method permits generating a liquid saturation profile and a general composition profile around the wellbore at shut-in. Introduction Naturally fractured reservoirs have been the object of intensive research during the last few years in the geologic as well as the engineering fields. Transient pressure analysis has received particular attention. Barenblatt and Zheltov(1), and Warren and Root(2) handled naturally fractured reservoirs by assuming pseudo steady-state (restricted) interporosity flow in a model made out of cubes with spaces in between. Flow toward the wellbore was assumed to be radial via the natural fractures. Their work led to the conclusion that a conventional crossplot of pressure vs. log of time should result in two parallel straight lines with a transition period in between. The separation of the two straight lines allowed calculation of the storativity ratio omega, i.e., the fraction of the total storage within the natural fractures. Kazemi(3) used a numerical model of a finite reservoir with a horizontal fracture under the assumption of unsteady state interporosity flow and substantiated Warren and Root's conclusion with respect to the two parallel straight lines. The transition period, however, was different due to the unsteady rather than pseudo steady-state interporosity flow assumption. de Swaan(4) developed a diffusivity equation and analytical solutions to handle the first and last straight lines. His method, however, could not analyse the transition period. Najurieta(5,6) developed analytical solutions of de Swaan's radial diffusivity equation which could handle the transition period as well as the first and last straight lines. Streltsova(7) used a gradient flow model and indicated that the transition period should yield a straight line with a slope equal to 1/2 the slope of the early and late straight lines. Her examples showing the 1/2 slopes gave values of storativity ratios approximately equal to 0.37, 0.26, and 0.48. Serra et al.(8) reached the same conclusion with the use of a stratum model for the cases in which the storativity ratio, omega, was smaller that 0.0099. Various type curves have been developed to analyse naturally fractured reservoirs with transient(9, 10) and pseudo steady-state(11) interporosity flow.

Decline Curve Analysis For Composite Reservoirs

Abstract Analysis of production decline curves presents a useful tool in forecasting the future production from a well or reservoir. A knowledge of the future production is an important factor in the economic analysis of exploration and production expenditures. Decline curve analysis can be used to estimate the production performance of a stimulated well due to acidizing. A comparison of the production decline before and after acid treatment will enable a determination of the technical and economic success of the treatment. Decline curve analysis for homogeneous reservoirs has been discussed extensively in the literature. However, the conventional homogeneous decline type curves are not appropriate for composite reservoirs. While production decline curves for radial, composite reservoirs have been presented, the same is not the case for composite reservoirs in elliptical, linear and spherical flow geometries. This study presents a comparison of the production performances of two-region, composite reservoirs in radial, elliptical, linear and spherical flow geometries. Transient rate and cumulative production responses for both infinite and closed, finite reservoirs for the various flow geometries are discussed. Normalizing factors are presented to enable comparison of production rate and cumulative production responses for the various composite reservoirs. A detailed investigation of the effects of mobility and storativity ratios, and also, the reservoir size on the production rate and cumulative production responses from radial and linear, composite reservoirs are presented. Results from this study show that the mobility-storativity product, MF, is not a correlating parameter for transient rate and cumulative production responses from finite, linear, composite reservoirs, unlike the case for infinitely large, linear, composite reservoirs. Some type curves for decline curve analysis of radial and linear composite reservoirs are presented. Introduction Analysis of production decline curves presents a useful tool in forecasting the future production from a well or reservoir. A knowledge of the future production is an important factor in the economic analysis of exploration and production expenditures. Decline curve analysis can be used to estimate the production performance of a stimulated well due to acidizing. A comparison of the production decline curves before and after acid treatment will enable a determination of the technical and economic success of the treatment. For homogeneous reservoirs, Fetkovich(1) presented an extensive discussion of decline curve analysis using type curves. He showed, among other things, how to use production decline type curves to estimate the formation flow capacity (kh) and the porosity- compressibility (Φct). For composite reservoirs, however, Olarewaju and Lee(2) have argued that the conventional homogeneous decline type curves are not appropriate. Olarewaju and Lee(2–4) have presented production decline curves for the analysis of transient rate data from radial, composite reservoirs with a damaged or stimulated well. Turki et al.(5) presented rate decline and rate decline derivative type curves for constant-pressure production in a two-region, radial, composite reservoir. They conducted a parametric study of the effects of mobility ratio, storativity ratio and distance to the discontinuity on the rate decline and rate decline derivative for both infinite and finite composite reservoirs.

A Numerical Investigation of Pressure Transient Analysis For Horizontal Wells In Composite Reservoirs Mimicking Thermal Recovery Situations

Abstract Horizontal wells have become quite popular for primary and enhanced oil recovery operations due to their well-documented advantages over vertical wells. Steam injection through horizontal wells has also been attempted at several places to improve heavy oil recovery. For horizontal wells undergoing steam injection, a steam chamber containing high mobility steam is established. This steam chamber, which can be of a complex shape, is surrounded by low mobility reservoir fluids. Such reservoir situations are referred to as composite reservoirs. An analytical solution for pressure transient tests for horizontal wells under composite reservoir situations with complex swept region shapes is not available yet. This study attempts a numerical investigation of pressure transient analysis for horizontal wells in two-region, composite reservoirs mimicking thermal recovery situations. A specialized three-dimensional, single-phase simulator was developed for this purpose. A closed, box-shaped reservoir was considered with a horizontal well. A detailed sensitivity study of transient pressure responses is presented with respect to grid size, well location in different directions, swept region shape, mobility ratio and storativity ratio. This study emphasizes the effects of the aforementioned factors on the swept volume estimation using the pseudosteadystate method. The pseudosteady-state method is based on an analysis of a Cartesian graph of pressure response versus time. This study establishes that for horizontal wells, the swept (or inner) region volume in a two-region system can be accurately estimated by the pseudosteady-state method for large mobility and storativity contrasts between the two regions. Introduction Horizontal wells have become quite popular in recent years due to their well-documented advantages over vertical wells(1). Steam injection through horizontal wells has also been attempted at several places to improve heavy oil recovery. Steam injection through horizontal wells includes scenarios such as steam-assisted gravity drainage process pioneered by Butler(2). Under such situations, a steam chamber containing high mobility steam is established around a horizontal well. A fall off test conducted on a horizontal injection well can be analysed using the pseudosteady-state (PSS) method. The PSS method was first proposed by Eggenschwiler et al.(3) to analyse thermal well tests conducted on vertical injection wells to estimate the steam chamber volume. Though a number of papers(4–9) have evaluated the applicability of the pseudosteadystate method for vertical injection wells, to the best of our knowledge, Reference (10) is the only study concentrating on the applicability of the pseudosteady-state method for horizontal injection wells. Issaka and Ambastha(10) used a numerical thermal simulator to generate fall off test data for horizontal, steam-injection wells. In this study, a specialized three-dimensional, single-phase, nonthermal simulator is used to study the applicability of the pseudosteady- state method for horizontal injection wells. In the next section, a brief description of the numerical model is presented followed by a section reporting our validation efforts for the numerical model. Thereafter, a detailed sensitivity study of transient pressure responses is presented with respect to various parameters, followed by concluding remarks.

Transient Pressure Behaviour Of A Heavy Oil Reservoir Undergoing Steam Injection In The Presence Of A Bottom-Water Zone

Abstract There are numerous heavy oil reservoirs in Canada which are underlain by bottom-water zones and are undergoing steam injection. Steam may be injected in the reservoir or the bottomwater zone, depending on the injectivity considerations. Due to gravity override effects, zones swept by steam in these reservoirs may have irregular shapes. Using a new analytical solution for a multi-layer, composite reservoir model, this study investigates transient pressure responses for partially-penetrating, steam injection wells with irregularly-shaped fronts in the presence of infinite or finite bottomwater zones. Pseudosteady-state crossflow is allowed among adjacent layers to model the vertical flow of fluids. Pressure and pressure derivative responses have been studied to determine the effects of penetration ratio, crossflow parameters, mobility ratio, storativity ratio, and dimensionless front radii in different layers. Results of this study demonstrate the variety of well-test responses expected in such complex scenarios. Also, this study outlines the difficulties associated with the traditional pseudosteady-state approach to estimate the swept volume in the presence of bottom-water zone effects. However, this analytical transient-pressure model should prove helpful in automated (or automatic) type-curve matching analysis of well-test data obtained from thermal recovery operations with and/or without bottom-water zone effects. Introduction Steam injection process is widely used in heavy oil recovery operations. As a result of steam injection, at least two regions of different fluid properties are created and the reservoir resembles a composite reservoir. Because of gravity override effect, an inclined fluid front is created between the hot and the cold zones. Many times, heavy oil reservoirs are accompanied by a bottomwater zone. The purpose of this study is to investigate the transient pressure behaviour of a steam-stimulated heavy oil reservoir under bottom-water conditions. A reservoir undergoing a thermal recovery process has been idealized as a composite reservoir for a long time(l-6). But most of the studies consider-piston-like movement of the fluid front, neglecting the gravity override effect. Satman(7) used the concept of a tilted (inclined) front in pressure transient analysis of a two layer, composite reservoir. He proposed that fluid front would propagate at different rates in different layers. For steam flooding, Satman and Oskay(8) considered the discontinuity boundary as a tilted front to account for the gravity override effect and modelled the reservoir as a multi-layer, composite reservoir without crossflow. They concluded that the tilted-front model is a better representation of the actual reservoir. According to published reports on steam-drive projects(9,10), gravity override effect is a common phenomenon which results in a tilted front between the swept and the unswept zones. If gravity override effect is not taken into consideration, the predicted performance of the steam flooding project may be quite different than the actual performance. Singhal(l1) conducted some scaled physical model studies of steam-flood in a pool containing heavy oil. He presented some temperature profiles obtained from his model which showed very strong gravity override effect. Blevins et al.(9) discussed the application of steam-flooding for light oil reservoirs.

Practical Aspects of Well Test Analysis Under Composite Reservoir Situations

Abstract Well test analysis using a composite reservoir model has become popular because of the flexibility provided by a composite reservoir model in visualizing numerous reservoir situations. A composite reservoir model is useful for analysing transient pressure data from acidization and fluid injection projects, oil or gas reservoirs with rock or fluid property contrasts) and geothermal reservoirs with thermal discontinuities. In the simplest case, a composite reservoir model is helpful in all reservoir situations, where we can consider the reservoir to be composed of two regions of different properties. Therefore, well test analysis based on a composite reservoir model is aimed at obtaining estimates of reservoir properties and skin factors corresponding to bath the inner and outer region data. Introduction Though linear, spherical(3), and elliptical(1–6) composite models have been discussed in the literature a radial(7–17) composite model has been a popular model to analyse well test data from a variety of reservoir situations. Generally, a two-region, radial composite reservoir model is used in practice (see Figure 1). In a two-region, composite reservoir idealization the reservoir is considered to be composed of two regions of different properties. The well may be producing (or injecting) at a constant rate and may exhibit wellbore storage and skin effects. The variable R denotes the discontinuity (or front) radius. There may also be a skin at the discontinuity. The dimensionless pressure solution, in Laplace space, for such a situation has been presented in References (9) and (11). Title Laplace space solution is inverted to the real space using Stehfest algorithm (1–8). Well test analysis based on a radial, two-region composite reservoir model is aimed at obtaining estimates of discontinuity radius, and reservoir properties and skin factors corresponding to both the inner and outer region data. Problems associated with estimating discontinuity radius (or swept volume for fluid injection projects) have been discussed before(9.10,14.19.20). This paper addresses the issues related to data misinterpretation, and consequent misinterpretations of reservoir characteristics and skin factors, while analysing well test data using a radial, two-region composite reservoir model. All results presented in this paper have been generated using the solution presented in Reference (11). Ideal Model Response Figure 2 shows drawdown semi-log pressure derivative (PwD = d Pwd I d In tD) responses for a well located in an infinite, two-region, radial composite reservoir. The pressure derivative responses are graphed as a function of tRD with M and Fs as parameters. All variables are identified in the Nomenclature. Figure 2 applies for CD = S = sr = 0 situations. Pressure derivative responses are shown for several possible combinations of mobility and storativity ratios, which may be equal to, less than, or greater than unity. For two-region composite reservoir situations, Figure 2 shows that at early times an infinite-acting radial flow regime corresponding to the inner region mobility appears. The early time infinite-acting radial flow regime is identified as that portion of the responses where PWD has a constant value equal to one-half. Dimensionless pressure response during the early-time infinite-acting radial flow regime is expressed as:

Automated Well Test Analysis Using Robust (LAV) Nonlinear Parameter Estimation

ABSTRACT Most of the nonlinear regression algorithms used in automated well test interpretation perform a minimization of an objective function formed by the sum of the squares of the differences between the data and the model values. This constitutes a typical nonlinear least-squares problem. One of the drawbacks of the least squares approach is that it is significantly affected by outliers, which are data points that can be considered bad observations. This study proposes a robust algorithm that uses the LAV (Least Absolute Value) as the criterion for the minimization. One of the greatest advantages of the LAV approach is that it provides a smooth transition between full acceptance and total rejection of a given observation, providing a systematic way of rejecting outliers by automatically assigning them less weight in the objective function. The technique described in this work is based on the expansion of the model function in a Taylor series up to the first order terms. This poses a new problem which is solved iteratively after being transformed into a multiple linear regression parameter estimation. The procedure becomes therefore a sequence of multiple linear regression problems, which are solved using the least absolute value criterion. Because the equations of condition are solved, instead of the so-called normal equations (as in several implementations of nonlinear parameter estimation algorithms discussed in the literature), the method avoids unnecessary exacerbation of the ill-conditioning of the design matrix. Examples where no large errors in the data are present show that this method produces results similar to least-squares minimization. When outliers are introduced into the observations, however, the present technique yields much better estimates. Also, the least absolute value approach works better than the least-squares criterion when estimating parameters whose sensitivity is small, as in the case of obtaining the storativity ratio and the interporosity flow coefficient in a naturally fractured reservoir with double-porosity behavior.

Interference Testing In a Two-region Composite Reservoir With a Frontal Skin

Abstract A two-region composite reservoir model is used to analyze well-test data from a variety of enhanced oil recovery projects, geothermal reservoir, and acidization projects. Dynamic phenomena, such as phase changes and multi-phase flow effects in a region near the front, can cause a sharp pressure drop at the front. Such a sharp pressure drop can be modeled as a thin frontal skin. This study considers interference testing in a two-region composite reservoir with a thin frontal skin. Wellbore storage and skin at the active well are also included in the analytical solution to the problem. Correlating parameters are established in the presence of a frontal skin for type-curve marching analysis of interference data from such reservoirs. Our results indicate that for an observation well in the inner region surrounding the wellbore, the effects of storativity ratio (i.e. Φc, ratio) and frontal skin on interference tests are similar. Thus, frontal skin cannot be estimated independently of storativity ratio from an interference test on an observation well in the inner region. However, interference test data collected at an observation well located in the outer region beyond the front helps estimate storarivity ratio and frontal skin independently of each other. The information presented in this paper should be useful in obtaining a consistent analysis of well tests conducted in composite reservoir situations. Introduction This study considers interference pressure and pressure derivative responses in a two-region composite reservoir with an infinitesimally thin skin at the discontinuity. The observation well may be located in any region. Figure 1 shows a schematic diagram of a radial, two-region composite reservoir with an observation well located in the outer region (region 2) at a distance a from the active well. Region 1 (inner region) extends to a radius R, and a thin skin sr may occur at the discontinuity. Ambastha and Ramey(1) presented an analytical solution for transient pressure behavior at any point in a reservoir depicted by Figure 1. Their analytical solution as presented in the Appendix of this paper has been used to investigate interference test analysis in a radial, two-region composite reservoir with a frontal skin. All results presented in this study are for an infinite reservoir. FIGURE 1: Two-region composite reservoir. (Available in full paper) Correlating Parameters Satman(2) presented an study of interference responses in radial, two region infinite composite reservoir. He did not consider the effects of a thin skin at the discontinuity. He correlated transient interference pressure responses using CD/aD, CDe2s, a/R, M and F as correlating parameters. He defined dimensionless pressure and time using the properties in the outer region, and considered the interference pressure responses for the observation wells located in the outer region only. This study also considers the same correlating parameters as used by Satman(2). A thin skin at the discontinuity, Sr is an additional correlating parameter, Figure 2 through 7 verify the selected correlating parameters under different conditions. Figure 2 through 7 are for M = 10 and Fs = 100. TABLE 1: Parameters used to generate responses shown on Figures 3, 6 and 7. (Available in full paper)

Transient 2D Flow in Layered Reservoirs With Crossflow

Summary Analytical solutions of transient 2D flow through double-permeability media are presented. The pressure response of a vertically fractured well intercepting a layered reservoir with crossflow is described through the methods of integral transform and variable separation. The influences that parameters of a double-permeability model such as the permeability ratio, Fk interporosity flow parameter, λ, and storativity ratio, ω, have on the pressure response of a fractured well are calculated and analyzed by numerical inversion. Log-log diagnosis graphs of fractured well pressure are given and analyzed here.

Double Porosity Well Testing in the Fractured Carbonate Rocks of the Ozarks

AbstractThe objective of this research was to determine a well test model which could be used for determining the hydraulic parameters of T and S in the complex fractured carbonates of the Ozarks of southern Missouri. Municipal water well testing in the Ozarks is performed on deep wells completed in confined fractured carbonate aquifers where all of the flow to the wellbore is through the fractures but most of the storage is in the matrix blocks. Observation wells are not used because of cost. The methodology uses double porosity well test analysis including the inner boundary conditions of skin and wellbore storage. Using this method, S and T are obtained as well as the storativity ratio, ω, and the block to fracture flow parameter λ. Because wellbore storage, skin, and storativity are interrelated, a computer simulation to find the final S was necessary. Derivative curve analysis provided a constraint in determining the correctness of the simulation results. Failure to use this well test method proved to result in storativities which are much too large. An example is presented.

An Approximate Solution Of Linear Flow In Naturally Fractured Reservoirs

Abstract Equations are presented for evaluation of linear flow in naturally fractured reservoirs represented by dual-porosity systems. The problem has not been treated previously in the petroleum engineering literature. This situation might happen, for example, in the case of drawdowns or buildups in dual-porosity systems which are hydraulically fractured or semi-infinite dual-porosity systems which are bounded by two parallel linear discontinuities. The equations apply only for oil systems. The extension to gas wells, however, is straightforward. The model assumes flow from the matrix into the fractures. There is no flow from a matrix block into another matrix block. It is shown that a conventional cross plot of log of delta p vs log of time should result in two parallel straight lines with a slope equal to 0.5 and a transition period which depends on the type of inter-porosity flow (pseudo steady-slate, transient or gradient) and the shape of the matrix blocks. This plot allows determination of the storativity ratio, omega, fracture porosity and distance between natural fractures. A conventional cross plot of delta p vs square root of time on Cartesian coordinates should result in two distinct straight lines. The first straighr line has a slope bigger than the second straight line. The ratio of these slopes squared gives the storativity ratio omega. The analysis also permits calculation of the hydraulic fracture length. Introduction Naturally fractured reservoirs have been the object of intensive research during the last few years in the geologic as well as the engineering fields. Transient pressure analysis has received particular attention. Barenblatt and Zheltov(1) and Warren and Root(2) handled naturally fractured reservoirs by assuming pseudo steady-state inter-porosity flow in a model made out of cubes with spaces in between. Flow toward the wellbore was assumed to be radial via the natural fractures. Their work led to the conclusion that a conventional cross plot of pressure vs log of time should result in two parallel straight lines with a transition period in between. The separation of the two straight lines allowed calculation of the storativity ratio omega, i.e. the fraction of the total storage within the natural fractures. Kazemi(3) used a numerical model of a finite reservoir with a horizontal fracture under the assumption of unsteady state inter-porosity flow and substantiated Warren and Root's conclusion with respect to the two parallel straight lines. The transition period, however, was different due to the unsteady rather than pseudo steady-state inter-porosity flow assumption. de Swaan(4) developed a diffusivity equation and analytical solutions to handle the first and last straight lines. His method, however, could not analyze the transition period. Najurieta(5,6) developed analytical solutions of de Swaan's radial diffusivity equation which could handle the transition period as well as the first and last straight lines. Streltsova(7) used a gradient flow model and indicated that the transition period should yield a straight line with a slope equal to ½ the slope of the early and late straight lines.

Transient Pressure Analysis of Horizontal Wells in Anisotropic Naturally Fractured Reservoirs

Summary Analytical solutions are presented for pressure-drawdown and -buildup analyses of horizontal wells in anisotropic reser voirs with tectonic, regional, or contractional natural fractures. (The reservoirs might be limited by parallel sealed boundaries.) These solutions have led to the identification of various flow periods, including radial in a vertical plane, transitional as a result of flow from the matrix into the fractures, linear when the transient pressure reaches the upper and lower boundaries of the reservoir, pseudoradial toward the wellbore in a horizontal plane, and linear when the transient reaches the outer parallel boundaries. Recognition of these flow periods leads to the calculation of pi or p*; permeabilities in the x, y, and z directions; storativity ratio, ω; average distance be tween natural fractures; fracture porosity; and fracture aperture, skin, and pseudoskin caused by vertical and horizontal partial penetra tion. Calculations are illustrated with an example.