Archie’s law

This wall material conductivity 2W is also an effective conductivity since the wall is porous and it is determined from the wall porosity and intrinsic material thermal conductivity, usually through a power law relation known as Archie s Law. 

It is generally believed that the pores are interconnected and the formation factor can be related with porosity by Archie s law (59) ... 

Fig. 2.7 Formation factor versus porosity for six gravity cores retrieved from different sedimentation provinces in the South Atlantic. Porosities were determined on discrete samples by wet and dry weights and volumes, formation factors by resistivity measurements. The dashed lines indicate Archie s law for a = 1 and cementation exponents (m) between 1 and 5. For a description of the sedimentation provinces, core numbers, coring locations, sediment compositions, water depths and constants (a) and (m) derived from linear least square fits please refer to Table 2.1. Unpublished data from M. Richter, University Bremen, Germany.

Fig. 2.9 Model based computation of a wet bulk density log from resistivity measurements on ODP core 690C. (a) Porosity log derived from formation factors having used Boyce s (1968) values for (a) and (m) in Archie s law. (b) Carbonate content (O Conell 1990). (c) Wet bulk density log analyzed from gamma ray attenuation measurements onboard of JOIDES Resolution (gray curve). Superimposed is the wet bulk density log computed from electrical resistivity measurements on archive halves of the core (black curve) having used the grain density model shown in (d). Unpublished data from B. Laser and V. SpieB, University Bremen, Germany.

If the tortuosity is not known on account of electric conductivity measurements and the formation factor (F), its value may be estimated, a bit less accurately, by its empirical relationship to the porosity. About a dozen different empirical equations are known from literature. The most frequently used is Archie s Law (Archie 1942) ... 

Balberg I (1986) Excluded-volume explanation of Archie s law, Phys Rev 5 33 3618-3620. Florian R and Neda Z (2001) Improved percolation thresholds for rods in three-dimensional boxes, ArXiv Condensed Matter e-prints arXiv cond-mat/0110067vl. 

Balberg 1 (1986) Excluded-volume explanatitm of Archie s law. Phys Rev B 33 3618 Balberg 1 (1987) Tunneling and nonuniversal ctmdnctivily in composite matraials. Phys Rev Lett... 

As mentioned previously, the bed-sediment of aquatic systems is a saturated porous medium and is similar to soil in many respects. The basic process of chemical transport by diffusion is comparable in these two systems. Section 12.5 contains equations to estimate the effective diffusion coefficients for several chemicals in bed sediment. See specifically Equation 12.18, Archie s law, as well as Eigures 12.2 and 12.3. Information on typical porosities in natural sedimentary materials appears in Table 12.7 and Figure 12.4. This section also contains information on pore structure and particle distributions. Tables 12.9 and 12.10 contain compilations of data on this transport parameter in porewaters. Generally speaking, the porous material types, particle sizes, porosities, and structures are comparable to surface soils so that the information and data provided is very relevant to the user. 

Lerman (1979) proposes m = 2 for the porewaters of sediments, for lack of a better model. Grathwohl (1998) found that the experimental data show that solute transport through porous samples follows the diffusion models reasonably well and that the porosity-based Archie s law estimates of effective diffusion coefficients are in good... 

FIGURE 12.3 Normalized diffusion coefficients (Zy> versus et (e corrected based on water adsorbed onto dry samples at a relative humidity of 98.8%). The lines represent Archie s law. (Reproduced from Grathwohl, R 1998. Diffusion in Natural Porous Media. Kluwer Academic Publisher, MA, p. 92. With permission). 

D is the Brownian diffusion coefficient of the particles. Within the porous media, an Archie s law type expression (Equation 12.18) is appropriate for estimation purposes. 

In the absence of hydraulic or wind forces, the water becomes quiescenf but natural or free convection processes remains operative. Driven by bottom residing thermal or concentration gradients. Equations 12.14 and 12.15 may be used for estimating these low-end MTCs. The chemical diffusion coefficient in the porewaters of the upper sediment layer is the key to quantifying the sediment-side MTC. Use Archie s law. Equation 12.18, to correct the aqueous chemical molecular diffusivity for the presence of the bed material. Bed porosity is the key independent variable that determines the magnitude of the correction factor. See Table 12.7 for typical porosity values in sedimentary materials. Eor colloids in porewaters. Equation 12.18 applies as well. The Stokes-Einstein equation (Equation 12.19) is recommended and some reported particle Brownian diffusion coefficients appear in Tables 12.9 and 12.10. Under quasisteady-state conditions, Equation 12.23 is appropriate for estimating the bed-side MTCs. 

Archie s law, Equation 12.18, with m = 2 will be used. For estimating the overall MTC on the bed-side Equation 12.23 will be used. Substituting the appropriate parameters... 

Transforming the results. The end result of a well logging measurement will be values of parameters such as electrical resistivity. This is not the information that is actually required. Instead, porosity, permeability, elastic constants and so on are needed. Some of these can be inferred by combining the results of several different logs and using a law, such as Archie s law (see [45, 138] for an explanation) to infer the property of direct interest. 

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