# On Perturbative Expansions to the Stochastic Flow Problem

Chapter

## Abstract

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, 1nK G , is assumed to have finite variance, σ f 2 . Historically, perturbation schemes have involved the assumption that σ f 2 <1. Here it is shown that of may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, a Δf 2 , is a more appropriate choice. By solving the problem with this parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, ε, defined. Since the processes f and V f can often be considered independent, further assumptions on V f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of avf in terms of o f and various length scales. The ratio of the integral scale in the main direction of flow (A,) to the total domain length (L*), px = a. x /L*, plays an important role in the convergence of the perturbation scheme. For p x smaller than a critical value p c , px < p c , the scheme’s perturbation parameter is e = o f/p x for one-dimensional flow, and s = o f/p, 2 for two-dimensional flow with mean flow in the x direction. For p x p c , the parameter e = of/p x 3 may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.

## References

1. Ash, R. B.: 1972, Real Analysis and Probability. Academic Press, London, San Diego.Google Scholar
2. Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M.: 1999, Development and application of the fractional advection—dispersion equation. Water Resour. Res. (submitted).Google Scholar
3. Bonilla, F. A. and Cushman, J. H.: 2000, Role of boundary conditions on convergence and nonlocality of solutions to stochastic flow problems in bounded domains. Water Resour. Res. 36 (4), 981–997.
4. Carrier, G. F. and Pearson, C. E.: 1988 Partial Differential Equations Theory and Technique. Academic Press, Boston, San Diego.Google Scholar
5. Cushman, J. H. (ed.): 1990, Dynamics of Fluids in Hierarchical Porous Media. Academic Press, London, San Diego.Google Scholar
6. Cushman, J. H.: 1997, The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. Kluwer Academic Publishers, NY, 1997.
7. Cushman, J. H., Hu, B. X. and Deng, F. W.: 1995, Nonlocal reactive transport with physical and chemical heterogeneity: Localization errors. Water Resour. Res. 31 (9), 2219–2237.
8. Dagan, G.: 1982, Stochastic modeling of groundwater flow by unconditional and conditional probabilities 1. Conditional simulation and the direct problem. Water Resour. Res. 18 (4), 813–833.
9. Dagan, G.: 1989, Flow and Transport in Porous Formations. Springer-Verlag, New York.
10. Dagan, G.: 1993, Higher-Order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution. Transport in Porous Media 12, 279–290.
11. Deng, F. W. and Cushman, J. H.: 1995, On higher-order corrections to the flow velocity covariance tensor. Water Resour. Res. 31 (7), 1659–1672.
12. Eckhaus, W.: 1973, Matched Asymptotic Expansions and Singular Perturbations. North-Holland Pub. Co., American Elsevier Pub. Co., Amsterdam, New York.Google Scholar
13. Gelhar, L. W.: 1993, Stochastic Subsurface Hydrology. Prentice-Hall, Englewood Cliffs, NJ. Gomez-Hernandez, J. J. and Wen, X. H.: 1998, To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology. Adv. In Water Resour. 21, 47–61.Google Scholar
14. Hassan, A., Cushman, J. H. and Delleur, J. W.: 1998, A Monte Carlo assessment of Eulerian flow and transport perturbation models. Water Resour. Res. 34 (5), 1143–1163.
15. Hinch, E. J.: 1990, Perturbation Methods. Cambridge University Press, Cambridge, New York. Hsu, K. C., Zhang, D. X. and Neuman, S. P.: 1996, Higher-order effects on flow and transport in randomly heterogeneous porous media. Water Resour. Res. 32 (3), 571–582.Google Scholar
16. Magnus, W., Oberhettinger, F. and Soni, R. P.: 1966, Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, New York.Google Scholar
17. Mizell, S. A., Gutjahr, A. L. and Gelhar, L. W.: 1982, Stochastic analysis of spatial variability in two-dimensional steady groundwater flow assuming stationary and nonstationary heads. Water Resour. Res. 18 (4), 1053–1067.
18. Mukhopadhyay S. and Cushman, J. H.: 1998, Diffusive transport of volatile pollutants in nonaqueous-phase liquid contaminated soil: A fractal model. Transport in Porous Media 30, 125–154.
19. Neuman, S. P.: 1995, On advective transport in fractal permeability and velocity fields. Water Resour. Res. 31 (6), 1455–1460.
20. Neuman, S. P. and Orr, S.: 1993, Prediction of steady state flow in nonuniform geologic media by conditional moments -exact nonlocal formalism, effective conductivities, and weak approximation. Water Resour. Res. 29 (2), 341–364.
21. Roach, G. F.: 1982, Green’s Functions. Cambridge University Press, London.Google Scholar
22. Rubin, Y. and Dagan, G.: 1992, A note on head and velocity covariances in three-dimensional flow through heterogeneous anisotropic porous media. Water Resour. Res. 28, 1463–1470.
23. SanchezVila, X., Carrera, J. and Girardi, J. P.: 1996, Scale effects in transmissivity. J. Hydrology 183 (1–2), 1–22.
24. Serrano, S. E.: 1992, Semianalytical methods in stochastic groundwater transport. Applied Mathematical Modeling 16 (4), 181–191.
25. Zhang, D. X.: 1999, Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media. Water Resour. Res. 35, 1127–1141.
26. Zhang, D. X. and Winter, C. L.: 1999, Moment-equation approach to single-phase fluid flow in heterogeneous reservoirs. Soc. Petroleum Eng. 3 (2).Google Scholar