Effective Reactive Surface Area: an Anisotropic Property of Physically and Chemically Heterogeneous Porous Media
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by poor sorting with respect to size and shape and commonly consist of a fair amount of clay and silt, resulting in low k. Beach processes are more effective in removing silt to clay-sized grains and in grain sorting, resulting in comparatively high k. Variations of these properties permit recognition of sediments from various parts of the depositional environment in which different processes predominated during accumulation. An example of a theoretical correlation between physical and geochemical heterogeneities for idealized porous media is discussed below. Lerman [6] reviewed numerous relationships between k, 0, and grain size derived from earlier literature sources. Additional evaluations have also been published recently [7-12]. Thompson and others [13] reviewed the physics and relationships between k, 0, and the microgeometry of sedimentary rocks. Bear [14] described general relationships for k based on the concept of hydraulic radius (defined as the ratio of porosity to internal surface area) for an isotropic porous medium given by: >2
k = fl (s)f2 (fi) -0 as,
(1)
wheref (s) is a factor describing the grain shape, f2(0) is a factor including porosity, 0 is the porosity, and ahs(Afrace Vold) is the grain surface area per unit volume of the solid. Sperry and Peirce [7] investigated and emphasized the importance of the shape and porosity factor in the application of Equation 1 to granular porous media. Equation I can be rearranged to yield:
k
S
(2)
To obtain a closed-form relationship between k and as, the shape functions must be known or assumed. Lake [9] has derived an expression relating ot, to k, resulting in an expression of the form: f (S)f2(0)
2r (I
6
0)3
where -cis the tortuosity. Substituting Equation 3 into Equation 2 results in: 0
0
I -6_ 2Tk
(4)
Equation 4 indicates that the surface area of a compositionally homogeneous porous medium is inversely proportional to the square root of k. Interfacial biogeochemical processes are typically formulated in terms of surface area per unit volume of water rather than in terms of surface area per unit volume of solid as in Equation 4. For a fully saturated porous medium, this leads to an expression of the form: As =
1-0 0=
0 2zk
(5)
where As is the surface area per unit volume of saturated porosity (As..rfacelVpores) Equation 5 links the reactive surface area to the medium k and provides a theoretical basis for the expected
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correlation between heterogeneity in reactive properties and the heterogeneity in hydraulic properties for saturated aquifers. Taking the logarithm of Equation 5, InAx =O.5In 0-.51n2T-0.5Ink (6) shows that In A, and In k for unconsolidated sediments should be correlated with a slope of -0.5. In addition, both 0 and T should influence the relationship between A, and k. In the simple case where both 0 and -care constant, Equation 6 reduces to a straight line. In the case where r is constant, variations in 0 should not significantly affect the expected relationship between A, and k. However, the assumption of constant
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