8.1 Flow and Transport in Multiple Dimensions

Flow and transport processes, including advection, dispersion and diffusion are described in the lesson Flow and Transport Processes in 1D System. In natural system we often need to consider these processes in multiple dimensions and in heterogeneous systems where the physical and geochemical properties of the subsurface are not evenly distributed. As an example, in Figure 1, we show that the distribution of permeability dictates the spatial distribution of injected tracer from injection wells during a biostimulation experiments at Old Rifle, Colorado (Li et al., 2010).

Figure 1. A. The spatial distribution of permeability in an biostimulation field experiment plot at Old Rifle, Colorado. Circles are injection wells and squares are monitoring well. B. The spatial distribution of injected tracer (bromide). The injected tracer tends to bypass the low permeability zone (modified from (Li et al., 2010))

The general Advection-Dispersion Equation (ADE) in multiple dimensions is as follows:

Here C is the tracer concentration (mol/m³ pore volume), t is time (s), D is the combined dispersion–diffusion tensor (m²/s), v (m/s) is the flow velocity vector and can be decomposed into v_L and v_T in the directions longitudinal and transverse to the main flow in a 2D system. The dispersion-diffusion tensor D is defined as the sum of the mechanical dispersion coefficient and the effective diffusion coefficient in porous media D^e(m²/s). At any particular location (grid block), their corresponding diffusion / dispersion coefficients D_L (m²/s) and D_T (m²/s) are calculated as follows:

Here $\alpha_L$ and $\alpha_T$ are the longitudinal and transverse dispersivity (m). The dispersion coefficients vary spatially due to the non-uniform permeability distribution. Values of $\alpha_T$ are typically at least one order of magnitude smaller than $\alpha_L$ (Gelhar et al., 1992; Olsson and Grathwohl, 2007).