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A Neumann expansion approach to flow through heterogeneous formations

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Abstract

A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well.

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Abbreviations

c :

compressibility

G(,):

Green's function

k :

permeability

\(\hat k\) :

fluctuated component ofk

\(\bar p\) :

Laplace transform ofp

r w :

radius of well

α i :

coefficient for mixed boundary conditions

ϕ:

trend component of ϕ

\(\bar \rho \) :

autocorrelation function

σ v :

standard deviation governing the variation of\(\hat k\)

σ A :

standard deviation governing the variation of\(\hat \phi \)

E{}:

Expectation operator

h :

thickness of aquifer

k :

trend component ofk

p :

pressure

r 0 :

autocorrelation distance

t :

time

ϕ:

porosity

\(\hat \phi \) :

fluctuated component of ϕ

μ:

Dynamic viscosity

Γ i :

portion of boundary

∇:

nabla operator

D :

dimensionless parameter

w :

well

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Authors and Affiliations

  1. Dept. of Civil Engineering, Technion, 3200, Haifa, Israel

    D. G. Zeitoun & C. Braester

  2. Dept. of Mechanical and Civil Engineering, University of Vermont, 05401, Burlington, Vermont, USA

    D. G. Zeitoun

Authors
  1. D. G. Zeitoun
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  2. C. Braester
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Zeitoun, D.G., Braester, C. A Neumann expansion approach to flow through heterogeneous formations. Stochastic Hydrol Hydraul 5, 207–226 (1991). https://doi.org/10.1007/BF01544058

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