Interpretation Of The Enhancement Of Field-scale Effective Matrix Diffusion Coefficient In A Single Fracture Using A Semi-analytical Power Series Solution

摘要

A power series solution for convergent radial transport in a single fracture (PCRTSF) is developed. Transport processes considered in PCRTSF include advection and hydrodynamic dispersion in the fracture, molecular diffusion in the matrix, diffusive mass exchange across the fracture-matrix interface, and mixing effects in the injection and the extraction boreholes. An analytical solution in terms of a power series in Laplace domain is developed first, which is then numerically inverted by de-Hoog et al.'s algorithm. Four dimensionless parameters determine the behaviour of a breakthrough curve (BTC) calculated by PCRTSF, which are, in the order of decreasing sensitivity, the matrix diffusion factor, two mixing factors, and the Peclet number. The first parameter is lumped from matrix porosity, effective matrix diffusion coefficient, fracture aperture, and retardation factors. Its value increases as the matrix diffusion effect becomes significant. A non-zero matrix diffusion factor results in a -3/2 slope of the tail of a log-log BTC, a common property for tracer diffusion into an infinite matrix. Both mixing factors have equal effects on BTC characteristics. However, the Peclet number has virtually no effect on BTC tail. PCRTSF is applied to re-analyse two published test results that were obtained from convergent radial tracer tests in a discrete, horizontal fracture in Silurian dolomite. PCRTSF is able to fit the field BTCs better than the original channel model does if a large matrix diffusion coefficient is used. Noticeably, the ratio of field-scale to lab-scale matrix diffusion coefficients can be as large as 378. This enhancement of the field-scale matrix diffusion coefficient may be ascribed to the presence of a degraded zone at the fracture-matrix interface because of karstic effects, or to flow channeling as a result of aperture heterogeneity.