Abstract
In two recent papers, Nœtinger and Estébenet, 2000; Nœtinger et al., submitted, we set-up a method allowing to compute both the transient and steady-state exchange terms between the matrix and fractured regions of a naturally fractured porous medium using continuous time random walk methods (CTRW). The goal of the present paper is to show that a new version of the CTRW algorithm provides a direct determination of the so called transient exchange function f(t) (or its Laplace transform f(s)) widely used in well test interpretation. It is shown that this function is directly linked with the probability density of the first escape time in the fractured region of a Brownian particle launched initially in the matrix region. This new interpretation allows relating directly the exchange coefficient α∞ with the mean escape time of brownian particles in the matrix. From a practical point of view, these new results allow to derive a simpler version of the CTRW method. In addition, we obtain a considerable speed up of the CTRW method for up-scaling fractured reservoirs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arbogast, T., Douglas, J. and Hornung, U.: 1990, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21, 823-836.
Barenblatt, G. I. and Zheltov, Yu. P.: 1960, Fundamental equations of homogeneous liquids in fissured rocks, Dokl. Akad. Nauk SSSR. 132, 545-548.
Berkowitz, B. and Balberg, I.: 1993, Percolation theory and its application to grounwater hydrology Water Resour. Res. 29, 775-794.
Bour, O. and Davy, P.: 1997, Connectivity of random fault networks following a power law fault length distribution, Water Resour. Res. 33, 1567-1583.
Cacas, M. C., Ledoux, E., Marsily, G. de, Barbreau, A., Calmels, P., Gaillard, B. and Margritta, R.: 1990, Modeling fracture flow with a stochastic discrete network: calibration and validation, Water Resour. Res. 26, 491-500.
Dautray, R. and Lions, J.-L.: 1995, Analyse mathematique et Calcul Numerique Pour Les Sciences et Les Techniques, Vol 2, Masson.
Daviau, F.: 1986, Interprétation des essais de puits, les méthodes nouvelles, publications de l'institut français du pétrole, ed. Technip, Paris.
Dershowitz, W. and Einstein H. H.: 1988, Characterizing rock joint geometry with joint system models, Rock Mech. Rock Engng 1, 21-51.
Dershowitz W., Hurley N. and Been K.: 1992, Stochastic Discrete Fracture Modelling of Heterogeneous Fractured Reservoirs, Proceedings, Third European Conference on the Mathematics of Oil Recovery, Delft.
Dershowitz W. and Miller, I.: 1995, Dual porosity fracture flow and transport, Geophys. Res. Letters 22, 1441-1444.
Deruyck, B. Bourdet, D. Da Prat, G. and Ramey H. J. Jr.: 1982, Interpretation of Interference Tests in Reservoirs with Double Porosity Behavior; Theory and Field Examples, SPE 11025.
Gerke, H. H. and VanGenuchten, M. T.: 1993, Evaluation of a first-order water transfer term for variably-saturated dual-porosity models, Water Resour. Res. 29, 1225-1238.
Gilman, J. R.: 1986, An Efficient Finite-Difference Method for Simulating Phase Segregation in the Matrix Blocks in Double Porosity Reservoirs SPE Res. Eng. 403-413, July 1986.
Hornung, U. and Showalter, R. E.: 1990, Diffusion Models for Fractured Media, J. of Math. Anal. and Appl. 147, 69-80.
Huseby, O., Thovert, J. F., and Adler, P. M.: 1997, Geometry and Topology of Fracture Systems J. Phys. A: Math. Gen. 30, 1-30.
Kazemi, H., Merril, L. S. Porterfield, K. L. and P. R. Zeman.; 1976, Numerical Simulation of Water-Oil Flow in Naturally Fractured Reservoirs, SPE J., 317.
Landerau, P., Noetinger, B. and Quintard, M.: submitted for publication in Transport in Porous Media.
Lim, K. T. and Aziz, K.: 1995, Matrix-fracture shape factors for dual-porosity simulators, J. Pet. Sci. Eng. 13, 169.
Long, J. C. S., and Witherspoon, P. A.: 1985, The relationship of the degree of interconnection to permeability in fracture networks, J. Geophys. Res. 90, 2087-2098.
Long, J. C. S, Gilmour, P. and Witherspoon, P. A.: 1985, A model for steady fluid flow in random three-dimensional network of disc-shaped fractures, Water Resour. Res. 21, 1105-1115.
Mc Carthy, J. F.: 1993a, Reservoir Characterization: Efficient Random-Walk Methods for Upscaling and Image Selection, SPE 25334.
Mc Carthy, J. F.: 1993b, Continuous-time random walks on random media, J. Phys. A: Math. Gen. 26, 2495-2503.
Moyne, C.: 1997, Two-equation model for a diffusive process in porous media using the volume averaging method with an unsteady-state closure, Advances in Water Resour. 20, 63-76.
Noetinger, B. and Estébenet, T.: 2000, Up scaling flow in fractured media using continuous-time random walks methods, Transport in Porous Media 39, 315-337.
Noetinger, B., Estébenet, T. and Quintard, M.: Up scaling of fractured media: Equivalence between the large scale averaging theory and the continuous time random walk method, submitted in Transport in Porous Media.
Odeh, A. S.: 1965, Unsteady-State Behavior of Naturally Fractured Reservoirs, SPE J., 60-66.
Quintard, M. and Whitaker, S.: 1993, One-and two-equation models for transient diffusion processes in two-phase systems, Advances in Heat Transfer 23, 369-464.
Quintard, M. and Whitaker, S.: 1996a, Transport in chemically and mechanically heterogeneous porous media I: Theoretical development of region averaged equations for slightly compressible single-phase flow, Advances in Water Resour. 19, 29-47.
Quintard, M. and Whitaker, S.: 1996b, Transport in chemically and mechanically heterogeneous porous media II: Comparison with numerical experiments for slightly compressible single-phase flow, Advances in Water Resour. 19, 49-60.
Sahimi, M.: 1995, Flow and Transport in Porous Media and Fractured Rocks, VCH.
Showalter R. E.: 1991, Diffusion models with microstructure, Transport in Porous Media 6, 567-580.
de Swann, A.: 1976, Analytic solutions for determining naturally fractured reservoir properties by well testing, SPE J., 117-122.
Warren, J. E. and Root, P. J.: 1963, The behavior of naturally fractured reservoirs, The Soc. Petrol. Engs. J. 3, 245-255.
Zimmerman, R. W. Chen, G. Hadgu, T. and Bodvarsonn G. S.: 1993, A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow, Water Resour. Res. 29, 2127-2137.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nœtinger, B., Estebenet, T. & Landereau, P. A Direct Determination of the Transient Exchange Term of Fractured Media Using a Continuous Time Random Walk Method. Transport in Porous Media 44, 539–557 (2001). https://doi.org/10.1023/A:1010647108341
Issue Date:
DOI: https://doi.org/10.1023/A:1010647108341