Water Flow in the Subsurface Environment - Contaminant Geochemistry - page 246

The traditional, continuum-based approach uses Darcy's law, modified for
partially saturated porous media, to quantify the flux of water:

q ¼ K ð h Þ dh ð h Þ= dx ð 9 : 1 Þ

where q denotes specific discharge (volume of water per unit cross-sectional area
of porous medium, per time), and K(h) and h(h) are the hydraulic conductivity and
hydraulic head, respectively, which are functions of the volumetric water content
h. The water content (or moisture content), h, is defined as the ratio of the volume
of pore water to the total (bulk) volume of the porous medium. The functional
dependence of K on h is based on empirical considerations, and a range of func-
tional forms have been suggested (e.g., Brooks and Corey 1966 ; Mualem 1976 ;
van Genuchten 1980 ).
Richards' equation (Richards 1931 ), based on a mass conservation balance,
together with Eq. 9.1 , can be used to describe the transient flow of water through a
partially saturated porous medium. In one dimension (vertically), Richards'

equation is given as

oz þ 1

o h

ot ¼ o

K z ð h Þ o h ð 9 : 2 Þ

oz

This equation incorporates bulk parameters and provides a continuum-level,
averaged quantification of flow.
Pore-scale network models (including percolation models), on the other hand,
are best suited for analyzing fluid and contaminant distribution and movement
within pores and clusters of pores. Such models are particularly effective for
capturing fingering processes during infiltration, drainage, and imbibition. These
models conceptualize a porous material as an arrangement of pore throats (tubes or
cones) connected by pores (spheres); extensive discussion of these networks is
given in Berkowitz and Ewing ( 1998 ). Description of flow patterns is then
achieved by solving for the fluid advance within these networks, using local scale
Darcy's law (Eq. 9.1 ), for either fully (e.g., Margolin et al. 1998 ) or partially
saturated systems (e.g., Blunt and Scher 1995 ). In both cases, flow patterns
demonstrating strong fingering and preferential flow arise naturally.

9.2 Flow Through the Capillary Fringe

The capillary fringe (CF), which is the region where saturation increases quickly,
near the water table, deserves special consideration. Flow through the vadose zone
commonly is characterized by a mean vertical flow, with substantial temporal and
spatial variability. Within this zone, water pressures are less than atmospheric, and
the focus is on issues of fingering of water fronts and characterization of the
relationships among moisture content, matric potential, and relative permeability.