This study presents analytical models for naturally fractured tectonic reservoirs (NFTRs), which essentially correspond to type I fractured reservoirs, including the effects of the nonlinear gradient term for radial flow, single phase (oil), for constant rate in an infinite reservoir. Using an exact solution of Navier-Stokes equation and Cole-Hopf transform, NFTRs have been modeled. Our models are applied for fissured formations with extensive fractures. Smooth and rough extension fractures were analyzed using single and slab flow geometries. The motivation for this study was to develop a real and representative model of a NFTR, with extension fractures to describe its pressure behavior. A discussion is also presented with field examples, regarding the effect of a quadratic gradient term and the difference between the nonlinear and linear pressure solutions, comparing the Darcy laminar flow equation, with the exact solution of the Navier-Stokes equation applied to the diffusion equation and boundary conditions in wellbore. Keywords
New ideas are presented for the interpretation of pressure transient tests for wells in naturally fractured reservoirs. This work is based on the transient matrix flow model formulated by de Swaan. The differences between this model and the Warren and Root model occur during the transition flow period. It is demonstrated that the behavior of a naturally fractured reservoir can be correlated by using three dimensionless parameters. (i.e.). It is established that regardless of matrix geometry the transition period might exhibit a straight line whose slope is equal to half the slope of the classical parallel semilog straight lines, provided the transient matrix linear flow is present. In addition, information is provided on the estimation of fracture area per unit matrix volume or matrix parameters from the transition period semilog straight line. it is shown, that matrix geometry might be identified when pressure data are smooth. Field examples are included to illustrate the application and the validity of the theoretical results of this study. Introduction One of the rock heterogeneities that has deserved the attention of many investigators is the presence of fractures crossing the producing formations. Many of the important producing fields around the world are found in fractured formations in spite of this, during the last decades most of the reservoir engineering studies have been oriented towards homogeneous systems. It appears that the first study on the performance of a fractured reservoir was published in 1932. Regarding pressure transient analysis on fractured formations, commonly called naturally fractured, a review of the literature shows that it was initially discussed by Pollard. He was interested on the determination of fracture volume from pressure build up tests. His method was advanced by Pirson and Pirson' to include a calculation method for the matrix volume of a fractured reservoir. The first to present a detailed discussion of the radial flow of a slightly compressible fluid through a naturally fractured reservoir were Barenblatt and Zheltov and Barenblatt et al. (Fig 1). These authors assume that the flow occurs only in the fracture medium and that the matrix blocks are a source that delivers flow to the fracture system and that this flow could be considered under pseudosteady-state flow conditions.
This paper presents solutions for the continuous, finite step and spike injection of radioactive tracers in naturally fractured reservoirs. Solutions are presented for linear flow-vertical fractures, and for the radial flow cases of horizontal fractures and cubic block matrix-fracture geometry. The three derived solutions consider as particular cases the flow of a chemical tracer. The reservoir is treated as being composed of two regions: a mobile (fractures) where dispersion and convection take place and a. stagnant (matrix) where only diffusion and adsorption are allowed. Radioactive decay is considered in both regions. The solution that considers vertical fractures is analytical, thus avoiding the double Laplace space numerical inversion used in previous studies. Another main advantage is that the important numerical dispersion reported by previous investigators when using the Stehfest Laplace transform inversion algorithm is avoided. The radial coupled matrix to fracture solutions are presented in Laplace space, and are accurately inverted by means of the Crump algorithm. The influence of the main dimensionless parameters that enter into the solutions was carefully investigated. A comparison of results for the three different naturally fractured systems investigated, indicates that a uniqueness problem may arise in the interpretation of a test, especially to distinguish between the radial cases. As expected, this problem alleviates for the finite step injection cases. The results of this study can be applied to interpret tracer tests in naturally fractured reservoirs, allowing the estimation of fracture and matrix practical parameters of interest.
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