The approach to modeling diffusivity in soils discussed in  Chemical Property Estimation, Chapter 12, differs slightly from that used to develop some important diffusion models such as that of  P. C. Johnson and R. A. Ettinger  (ES&T, 25, 1445-1452, 1991) widely used to predict the intrusion rate of chemical vapors into buildings.  For instance, the effective diffusivity of a chemical in soil used in the Johnson and Ettinger model is estimated from soil vapor concentrations of chemical while the apparent diffusivity, defined in Section 12.2.2, is estimated from the chemical’s bulk concentration in soil.  The Johnson and Ettinger model is discussed here. 

Johnson and Ettinger present an analytical solution for combined convective and diffusive transport of vapor-phase contaminants in a soil column.  The transport equation is:

where:

                e_i = volume fraction of phase i, dimensionless (phase i is vapor (g), sorbed (s), condensed, and soil moisture(w))

                C_i = chemical concentration in phase i, g/cm³

                t = time, s

                u_i = Darcy velocity vector in phase i, cm/s

                D_i^eff = effective diffusion coefficient in phase i, cm²/s

                Ri = formation rate constant in phase i, g/cm³ s

 Assuming  (1) that chemical concentrations are low enough that there is no condensed chemical in the soil column, (2) that the chemical in soil vapor is in equilibrium with sorbed chemical and chemical dissolved in soil moisture, and (3) that only transport in soil vapor and soil moisture is important, Henry’s law and the Millington-Quirk model can be used to derive a simple transport model.  It is further assumed that convective transport occurs only in the vapor phase, and vapor flow is described by Darcy’s law.  The mass balance solution to the transport equation is obtained for a simple model system in which the soil column consists of a discrete number of  isotropic soil layers in which  the effective diffusion coefficients  are constant, the chemical concentration at the soil-air interface is zero,  the chemical is not leached from the soil (no transport through the lower soil boundary), and the chemical is stable and does not degrade.  With these assumptions, the mass transport rate through soil is approximated by:

where:

                E = one-dimension mass transport rate, g/s

                A = cross section area through which chemical is transported. cm²

                C = concentration, g/cm³

                D_Teff = “overall” effective diffusion coefficient based on vapor concentration, cm²/s

                L_T = distance from source to receptor, cm.

 The value of D_T^eff/ L_T is obtained by summing the values of  D^eff/ L for the individual isotropic soil layers.  For each layer,

and

.