Radially convergent flow in heterogeneous media

Abstract

We present an analytical solution for apparent effective transmissivity under radially convergent steady state flow conditions, produced by constant pumping from a single well of finite radius, $r_{w}$. Apparent effective transmissivity, $T_{e}$, is defined as the value that relates the expected values of flow and head gradient at a certain location. The domain is two‐dimensional, of annular shape, and the size of the pumping well is explicitly taken into account. The solution for the steady state heads is obtained by solving the perturbed flow equation and substituting it into Darcy's law to obtain a consistent second‐ order expansion for $T_{e}$. We show that apparent effective transmissivity is a scalar for any choice of isotropic covariance model, with an expression given in integral form. Our main result is that $T_{e}$ in a heterogeneous, statistically isotropic random field, under radial steady state flow conditions, is a monotonie increasing function of $r$(distance from the well) that rises from the harmonic mean of the point transmissivity values (close to the well) and tends asymptotically towards the geometric mean (far from the well). The asymptotic value is reached at a distance of a few integral scales (1.5–2 for the Gaussian model and 3–5 for the exponential one). The apparent effective transmissivity versus normalized r curves are in excellent agreement with previously published numerical work carried out using Monte Carlo method.

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