Spatial weighting functions: Transient hydraulic tests and heterogeneous media

Abstract

To improve understanding of property measurements in heterogeneous media, an energy-based weighting function concept is developed. In (assumed) homogeneous media, the instrument spatial weighting function (ISWF) depends only on the energy dissipation distribution set up by the measurement procedure and it reduces to simply inverse sample volume (uniform weighting) for 1-D parallel flow case (ideal permeameter). For 1-D transient flow in homogeneous media, such as with slug tests, the ISWF varies with position and time, with 95% of the total weighting contained within 115 well radii, even late in the test. In the heterogeneous case, the determination of the ISWF is connected to the problem of determining an equivalent hydraulic conductivity (K), where the criterion for equivalence is based on equal energy dissipation rate rather than equal volume discharge. The discharge-based equivalent K (K(E)) and the energy-based equivalent K in heterogeneous media (K(eh)) are not equal in general, with K(eh) typically above the nodal arithmetic mean K. The possibly more fundamental problem is that as one makes K measurements in heterogeneous media at different locations or on different cores of heterogeneous materials, the ISWF will be heterogeneity dependent, implying that the averaging process resulting in the equivalent K value also varies with position. If the testing procedure is transient, then the averaging process varies with time. This suggests a fundamental ambiguity in the interpretation of hydraulic conductivity measurements in heterogeneous media that may impact how we approach modeling and prediction in a practical sense (Molz 2003). Further research is suggested.