Groundwater Geophysics Phần 10 pptx

494 Reinhard Kirsch, Ugur Yaramanci

Fig. 17.2. Hydraulic conductivity of sands in relation to porosity, grain size, and

sorting (after Beard and Weyl 1973, with permission from AAPG)

Based on the Kozeny-Carman relation, Georgi and Menger (1994) de￾veloped the formulation

2

3

2

2

hy

f T (1 )

r

k − φ

φ ⋅ ⋅ = (17.8)

The Kozeny-Carman relation was further modified by Pape et al. (1998)

to the following form:

8 T

r k

2

eff

⋅

φ⋅ = (17.9)

reff = effective radius of pore channel.

An outline of porosity– hydraulic conductivity relations based on fractal

pore models for sandstone is given by Pape (2003).

Marotz (1968) relates effective porosity (drainable pore space, see

Chap. 14) to hydraulic conductivity and found the following relation at

sandstone samples (Fig. 17.3):

25.5 4.5ln k φeff = + (17.10)

17 Geophysical characterisation of aquifers 495

Fig. 17.3. Porosity and effective porosity of unconsolidated sediments (after Mat￾thess and Ubell 2003) and hydraulic conductivity related to effective porosity (af￾ter Marotz 1968)

Porosity and effective porosity are linked by the content of undrainable

pore water Swirr (irreversible water saturation, Swirr = φ - φeff). Timur (1968)

found a relation between hydraulic conductivity, porosity and Swirr (in mD)

1

K

S 3.5 0.35

1.26

wirr − φ = ⋅ (17.11)

17.3 Geophysical assessment of hydraulic conductivity

As shown before, hydraulic conductivity is not easily linked to porosity as

geophysical parameters are. Therefore, no straight hydraulic conductivity -

resistivity or hydraulic conductivity - seismic velocity relations can be ex￾pected. However, an attempt to enable a geophysical way for interpolation

of hydraulic conductivities valid at least for a limited project area should

be made.

17.3.1 Resistivity

The relation between complex resistivity and hydraulic conductivity is dis￾cussed in details in Chap. 4 (see also Lesmes and Friedman 2005). In the

following, only the real part of resistivity which can be determined by

electrical soundings is taken into account.

The close relation of electrical formation factor F to porosity (Archies

law) and tortuosity (see Eq. 1.8, Chap. 1) makes an attempt to find relations

496 Reinhard Kirsch, Ugur Yaramanci

between hydraulic conductivity and resistivity or hydraulic conductivity

and formation factor reasonable. Field and laboratory results are reported

by many authors with puzzling results. So, e.g., one group of authors like

Urish (1981), Frohlich and Kelly (1985), Huntley (1986), and Leibundgut

et al. (1992) found positive correlation between hydraulic conductivity and

formation factor, while other authors like Worthington (1975), Heigold et

al. (1979), and Biella et al. (1983) reported negative correlation (Fig. 17.4).

Fig. 17.4. Negative and positive correlation between electrical formation factor

and hydraulic conductivity after Biella et al. (1983) and Urish (1981)

A compilation of resistivity – hydraulic conductivity relations is given

by Mazác et al. (1985, 1990), Fig. 17.5. Within one sediment group

(gravel, coarse sand, etc) resistivity and hydraulic conductivity are in￾versely correlated. As porosity and resistivity (or formation factor) are in￾versely correlated too, a positive correlation exists between porosity and

hydraulic conductivity, as it is indicated by, e.g., the Kozeny-Carman rela￾tion (Eq. 17.6). However, if the sediment groups are compared, then posi￾tive correlation between resistivity and hydraulic conductivity is observed

leading to negative correlation between porosity and hydraulic conductiv￾ity. This is in accordance with Fig. 17.3 which shows that well sorted

coarse sediments like gravel have smaller porosities than well sorted fine

sediments, although effective porosity and hydraulic conductivity of

coarser sediments is higher.

This is backed by laboratory experiments of Biella et al. (1983). They

used artificial sediments of increasing uniform grain sizes from 0.2 to 8

mm which were used to produce 2-component sediment mixtures, e.g.,

consisting of material with grainsize 1 mm and 8 mm. Different percentage

of fine and coarser material lead to different porosities. For all mixtures of

17 Geophysical characterisation of aquifers 497

grain compositions electrical formation factor was linear related to poros￾ity (Fig. 17.6). However, different correlations of hydraulic conductivity

and porosity as well as of hydraulic conductivity and formation factor were

obtained for the different mixtures (Fig. 17.7). Samples taken arbitrarily

from the different mixtures would show no correlation.

Fig. 17.5. Correlation of hydraulic conductivity and resistivity for sediment

groups (after Mazác et al. 1985, 1990, with permission from SEG)

Fig. 17.6. Correlation of porosity and formation factor for artificial sediment sam￾ples (after Biella et al. 1983), best fit of data was by 1.42 F 1.15 − = ⋅ φ or 1.54 F − = φ

498 Reinhard Kirsch, Ugur Yaramanci

Fig. 17.7. Correlation of porosity and hydraulic conductivity (left) and formation

factor and hydraulic conductivity (right), although no general correlation is obvi￾ous, clear correlation is obtained within the groups (after Biella et al. 1983)

17.3.2 Seismic velocities

Seismic velocities, as shown in Chap. 1, are strongly related to porosity.

After Gassmann (1950), porosity is linked to seismic velocities by the po￾rosity dependence of bulk modulus

(K K )

K

K K

K

K K

K

m fl

fl

m us

us

m sat

sat

φ⋅ − + − = −

with Ksat = bulk modulus of saturated material

Kus = bulk modulus of unsaturated material

Km = bulk modulus of rock matrix

Kfl = bulk modulus of pore fluid

(17.12)

Bulk modulus of saturated and unsaturated material can be obtained

from p- and s-velocities and density ρ by

v ) 3

4 K (v

2

s

2

sat,usat = ρ ⋅ psat,usat − (17.13)