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Anomalous Transport in "Classical" Soil and Sand Columns

Dep. of Environmental Sciences and Energy Research, Weizmann Institute of Science, 76100 Rehovot, Israel

View larger version (50K): [in a new window]   Fig. 1. Breakthrough curves, (1 – j) vs. , at x = 1 for the truncated power law () in Eq. [4], where = t/t1 and the characteristic time t1 = 1. The effect of the ß and 2(t2/t1) parameters is analyzed. The velocity v = 1 and the dispersion D = 0.05. The boundary conditions are = 1/u, at x = 0 and /x = 0 at x = 1. The advection-dispersion equation (ADE) solution (dashed line) is plotted for reference.

View larger version (26K): [in a new window]   Fig. 2. Measured breakthrough curve (BTC) with continuous time random walk/advection-dispersion equation (CTRW/ADE) fits for a Berea sandstone core; (dimensionless) flux-averaged concentration j vs. time t. Upper figure shows the complete BTC; lower figure focuses on the region identified by the bold-framed rectangle in the upper figure. Note the difference in scale units between the figures. Column length equals 0.762 m. Porosity n = 0.204. Flux q = 1.73 cm3 min–1 (after Scheidegger's [1959] Fig. 2). Dashed line: best ADE model fit. Solid line: best CTRW fit.

View larger version (29K): [in a new window]   Fig. 3. Measured breakthrough curve (BTC) with continuous time random walk/advection-dispersion equation (CTRW/ADE) fits for Aiken clay loam 0.23- to 0.50-mm aggregates; (dimensionless) flux-averaged concentration j vs. pore volumes. Upper figure shows the complete BTC; lower figure focuses on the region identified by the bold-framed rectangle in the upper figure. Note the difference in scale units between the figures. Column length equals 30 cm. Velocity v = 3.40 cm h–1 (After Nielsen and Biggar's [1962] Fig. 9, open circles). Dashed line: best ADE model fit. Solid line: best CTRW fit.

View larger version (27K): [in a new window]   Fig. 4. Measured breakthrough curve (BTC) with continuous time random walk/advection-dispersion equation (CTRW/ADE) fits for Aiken clay loam 0.23- to 0.50-mm aggregates; (dimensionless) flux-averaged concentration j vs. pore volumes. Upper figure shows the complete BTC; lower figure focuses on the region identified by the bold-framed rectangle in the upper figure. Note the difference in scale units between the figures. Column length equals 30 cm. Velocity v = 0.058 cm h–1 (After Nielsen and Biggar's [1962], Fig. 9, filled circles). Dashed line: best ADE model fit. Solid line: best CTRW fit.

View larger version (28K): [in a new window]   Fig. 5. Measured breakthrough curve (BTC) with continuous time random walk/advection-dispersion equation (CTRW/ADE) fits for an unsaturated Oakley sand at a volumetric water content = 0.27; (dimensionless) flux-averaged concentration j vs. pore volumes. Upper figure shows the complete BTC; lower figure focuses on the region identified by the bold-framed rectangle in the upper figure. Note the difference in scale units between the figures. Column length equals 30 cm. Velocity v = 1.03 cm h–1 (After Nielsen and Biggar's [1962], Fig. 10, filled circles). Dashed line: best ADE model fit. Solid line: best CTRW fit.

View larger version (27K): [in a new window]   Fig. 6. Measured Br– breakthrough curve (BTC) with continuous time random walk/advection-dispersion equation (CTRW/ADE) fits for a soil column; (dimensionless) flux-averaged concentration j vs. time t. Upper figure shows the complete BTC; lower figure focuses on the region identified by the bold-framed rectangle in the upper figure. Note the difference in scale units between the figures. Pressure head h = 0 cm (Fully saturated) (After Jardine et al.'s [1993] Fig. 2, open circles). Column length equals 24 cm. Velocity v = 14.66 cm h–1. Dashed line: best ADE model fit. Solid line: best CTRW fit.

View larger version (29K): [in a new window]   Fig. 7. Measured Br– breakthrough curve (BTC) with continuous time random walk/advection-dispersion equation (CTRW/ADE) fits for a soil column; (dimensionless) flux-averaged concentration j vs. time t. Upper figure shows the complete BTC; lower figure focuses on the region identified by the bold-framed rectangle in the upper figure. Note the difference in scale units between the figures. Pressure head h = –10 cm (After Jardine et al.'s [1993] Fig. 2, open squares). Column length equals 24 cm. Velocity v = 2.82 cm h–1. Dashed line: best ADE model fit. Solid line: best CTRW fit.

View larger version (29K): [in a new window]   Fig. 8. Measured Br– breakthrough curve (BTC) with continuous time random walk/advection-dispersion equation (CTRW/ADE) fits for a soil column; (dimensionless) flux-averaged concentration j vs. time t. Upper figure shows the complete BTC; lower figure focuses on the region identified by the bold-framed rectangle in the upper figure. Note the difference in scale units between the figures. Pressure head h = –15 cm (After Jardine et al.'s [1993] Fig. 2, open triangles). Column length equals 24 cm. Velocity v = 0.354 cm h–1. Dashed line: best ADE model fit. Solid line: best CTRW fit.