A note on well boundary condition for flow through heterogeneous formations

In modeling flow toward wells it is mathematically convenient to replace the well by a singularity line along its axis. In the case of homogeneous aquifers, confined flow, and fully penetrating wells the strength of the sources is constant, and the common condition of constant head on the well is satisfied exactly. It also leads to constant flux on the well envelope. In the case of aquifers of spatially variable conductivity the constant head boundary condition can be satisfied accurately, if the well radius is much smaller than the log conductivity horizontal integral scale, by selecting a variable source strength that is proportional to the local conductivity along the well [Indelman et al., 1996; Fiori et al., 1998; Indelman, 2003a, 2003b]. In this case the equivalent conductivity tends to the arithmetic mean near the well. It changes with distance, approaching from above the effective conductivity in uniform horizontal mean flow far from the well. Selecting inadvertently a constant source strength violates the condition of constant well head. The equivalent conductivity tends to the harmonic mean near the well and approaches from below the effective one in uniform mean flow. The equivalent conductivity is derived for both types of sources for stratified formations as well as for aquifers of a three-dimensional heterogeneous structure. The implications on interpretation of head measurements by the two models is illustrated by a few examples.