Summary Evaluation of reservoir parameters through well-test and decline-curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely, but it can be found in the distribution of fractures within a naturally fractured reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected. However, evidence from outcrops, well logging, production-behavior studies, and the dynamic behavior observed in these systems, in general, indicate the above assumptions are not representative of these systems. Thus, the fractal theory can contribute to explain the above. The objective of this paper is to investigate the production-decline behavior in an NFR exhibiting single and double porosity with fractal networks of fractures. The diffusion equations used in this work are a fractal-continuity expression presented in previous studies in the literature and a more recent generalization of this equation, which includes a temporal fractional derivative. The second objective is to present a combined analysis methodology, which uses transient-well-test and boundary-dominated-decline production data to characterize an NFR exhibiting fractures, depending on scale. Several analytical solutions for different diffusion equations in fractal systems are presented in Laplace space for both constant-wellbore-pressure and pressure-variable-rate inner-boundary conditions. Both single- and dual-porosity systems are considered. For the case of single-porosity reservoirs, analytical solutions for different diffusion equations in fractal systems are presented. For the dual-porosity case, an approximate analytical solution, which uses a pseudosteady-state matrix-to-fractal fracture-transfer function, is introduced. This solution is compared with a finite-difference solution, and good agreement is found for both rate and cumulative production. Short- and long-time approximations are used to obtain practical procedures in time for determining some fractal parameters. Thus, this paper demonstrates the importance of analyzing both transient and boundary-dominated flow-rate data for a single-well situation to fully characterize an NFR exhibiting fractal geometry. Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation consistently explains the peculiar behavior observed in some real production-decline curves. Introduction Evaluation of reservoir parameters through decline-curve analysis has become a common current practice (Fetkovich 1980; Fetkovich et al. 1987). The main objectives of the application of decline analysis are to estimate formation parameters and to forecast production decline by identifying different flow regimes. Different solutions have been proposed during both transient (Ehlig-Economides and Ramey 1981; Uraiet and Raghavan 1980) and boundary-dominated (Fetkovich 1980; Fetkovich et al. 1987; Ehlig-Economides and Ramey 1981; Arps 1945) flow periods. Both single- and double-porosity (Da Prat et al. 1981; Sageev et al. 1985) systems have been addressed. During the boundary-dominated-flow period, in homogeneous systems, there is a single production decline, but for NFRs in which the matrix participates, there are two decline periods, with an intermediate constant-flow period (Da Prat et al. 1981; Sageev et al. 1985). Carbonate reservoirs contain more than 60% of the world's remaining oil. Yet, the very nature of the rock makes these reservoirs unpredictable. Formations are heterogeneous, with irregular flow paths and circulation traps. In spite of this complexity, at present, all studies on constant-bottomhole-pressure tests found in petroleum literature assume Euclidean or standard geometry is applicable to both single-porosity reservoirs and NFRs (Fetkovich 1980; Fetkovich et al. 1987; Ehlig-Economides and Ramey 1981; Uraiet and Raghavan 1980; Arps 1945; Da Prat et al. 1981; Sageev et al. 1985), even though real reservoirs exhibit a higher level of complexity. Specifically, natural fractures are heterogeneities that are present in carbonate reservoirs on a wide range of spatial scales. It is well known that flow distribution within the reservoir is controlled mostly by the distribution of fractures (i.e., geometrical complexity). There could be regions in the reservoir with clusters of fractures and others without the presence of fractures. The presence of fractures at different scales represents a relevant element of uncertainty in the construction of a reservoir model. Thus, highly heterogeneous media constitute the basic components of an NFR, so Euclidean flow models have appeared powerless in some of these cases. Alternatively, fractal theory provides a method to describe the complex network of fractures (Sahimi and Yortsos 1970). The power-law behavior of fracture-size distributions, characteristic of fractal systems, has been found by Laubach and Gale (2006) and Ortega et al. (2006). Distributions of attributes such as length, height, or aperture can frequently be expressed as power laws. Scaling analysis is important because it enables us to infer fracture attributes such as fracture strike, number of fracture sets, and fracture intensity for larger fractures from the analysis of microfractures found in oriented sidewall cores. This approach offers a method to overcome fracture-sampling limitations, with microfractures as proxies for related macrofractures in the same rock volume (Laubach and Gale 2006; Ortega et al. 2006). The first fractal model applied to pressure-transient analysis was presented by Chang and Yortsos (1990). Their model describes an NFR that has, at different scales, poor fracture connectivity and disorderly spatial distribution in a proper fashion. Acuña et al. (1995) applied this model and found the wellbore pressure is a power-law function of time. Flamenco-Lopez and Camacho-Velazquez (2003) demonstrated that to characterize a NFR fully with a fractal geometry, it is necessary to analyze both transient- and pseudosteady-state-flow well pressure tests or to determine the fractal-model parameters from porosity well logs or another type of source. Regarding the generation of fracture networks, Acuña et al. (1995) used a mathematical method for this purpose, while Philip et al. (2005) used a fracture-mechanics-based crack-growth simulator, instead of a purely stochastic method, for the same objective. In spite of all the work done on decline-curve analysis, the problem of fully characterizing an NFR exhibiting fractal geometry by means of production data has not been addressed in the literature. Thus, the purpose of this work is to present analytical solutions during both transient- and boundary-dominated-flow periods and to show that it is possible to characterize an NFR having a fractal network of fractures with production-decline data.

This study presents a new way to model high secondary porosity, mainly vuggy porosity, in naturally fractured reservoirs. New solutions are presented for two cases, one in which there is no primary flow through the vugs (which is an extension of the Warren and Root model) and one in which the dissolution process of pore throats has created an interconnected system of vugs and caves. In both cases, there is an interaction between matrix, vug, and fracture systems. New insights are provided.Both pressure and production responses during transient and boundary-dominated flow periods are explored. In transient well tests, for the case in which there is no primary flow through the vugs, a change of slope could be present during the transition period. Thus, this study shows that slope ratios of 2:1 of an earlyor late-time segment vs. a transition segment do not necessarily imply transient interaction between matrix and fractures. It is also shown that the presence of vugs and caves may have a definitive influence on decline-curve and cumulative production behaviors; therefore, it is necessary to incorporate vuggy porosity in the process of type-curve match.Finally, the use of the methodology obtained in this work is illustrated with synthetic and field examples.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractEvaluation of reservoir parameters through well test and decline curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely but it can be found in the distribution of fractures within a Naturally Fractured Reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected.However, evidences from outcrops, well logging, and production behavior studies, and in general, the dynamic behavior observed in these systems, indicate that the above assumptions are not representative for these systems. Thus, the fractal theory can contribute to explain the above.Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation explains consistently the peculiar behavior observed in some real production decline curves.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractEvaluation of reservoir parameters through well test and decline curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely but it can be found in the distribution of fractures within a Naturally Fractured Reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected.However, evidences from outcrops, well logging, and production behavior studies, and in general, the dynamic behavior observed in these systems, indicate that the above assumptions are not representative for these systems. Thus, the fractal theory can contribute to explain the above.Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation explains consistently the peculiar behavior observed in some real production decline curves.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractDuring the last years, several efforts have been done focused on developing the automatic computer-aided well test analysis.Traditionally, conventional gradient-based algorithms such as Levenberg-Marquardt have been used for parameters estimations. However, recently the use of new approaches based on global optimization methods has been introduced in the literature, mainly attempting to eliminate the influence of the initial guess on the overall regression, and overcoming the uniqueness problems.The aim of this work is to study the characterization of naturally fractured vuggy reservoirs via the automatic well test analysis by using a triple porosity model recently developed. Thus, an Evolutionary Algorithm (EA) was designed, taking into account the important fact that the triple porosity case may yield multiple solutions. Therefore, several aspects such as the parameters estimations with high precision and the necessity of developing analysis techniques computationally efficient are taken into account in the design of algorithms.According to the above, it is demonstrated that the algorithm developed in this work allows finding the multiple optimal solutions, with the desired precision. Advantages of the EA over the classical Levenberg-Marquardt method for this specific well test analysis problem are also discussed.