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Relationship among porosity, permeability, electrical and elastic properties
Zakir Hossain
Alan J Cohen
RSI, 2600 South Gessner Road, Houston, TX 77063, USA
Summary
Electrical resisivity is usually easier to measure in the
laboratory and in-situ than permeability. Therefore, a
method of combination between permeability and electrical
resistivity might be used to define the fluid flow of
reservoir rocks from resistivity data. However, estimating
permeability from resistivity has been a problem examined
by different authors. Furthermore, neither electrical nor
elastic data seldom allow us to accurately quantify the
hydrocarbon saturation. Hence, a combination of elastic
and electrical properties could offer a powerful means of
solving the problem of hydrocarbon saturation production.
The objective of this study is to experimentally and
theoretically revise the relations among the electrical
properties, porosity, permeability, and elastic wave
velocity. A data set of laboratory measured petrophysical
properties, electrical properties and elastic properties of
glauconitic greensand from the North Sea Nini Field was
used for this study. A linear relationship between
laboratory measured electrical properties and permeability
could be established if the diagenesis of greensand is know.
By combing Archie’s relation and Kozeny’s equation, the
greensand diagenesis may be described by the specific
surface area of pores. A linear relationship between
laboratory measured electrical and elastic properties could
be established if the effect of micro structure of greensand
is known. Rock physics modeling results show that quartz
cementation has a larger effect on elastic properties than
electrical properties, while berthierine cementation has a
similar effect on elastic and electrical properties. Selfconsistent modeling results show that pore aspect ratios are
more sensitive for electrical properties than elastic
properties.
Introduction
Electrical resistivity is commonly used types of to define
the hydrocarbon saturation of reservior rocks. A method of
combination between permeability and electrical resistivity
may be used to define the fluid flow of reservoir rocks.
Even though both resistivity and permeability strongly
depend on porosity, no rigorous relationship between
permeability and resistivity has yet found (Gomez, 2009).
Estimating permeability from resistivity has been a
problem examined by different authors, including Archie
(1942), who showed an average trend of formation factor
versus permeability for sandstones, but recognized that the
scatter was too large to establish a definite relation between
the two properties.
Like electrical resistivity, sonic velocities is also one of the
most common collected types of geophysical well logging
data used in hydrocarbon investigations. However, neither
electrical nor elastic data seldom allow us to accurately
quantify the hydrocarbon saturation. Therefore, a
combination of elastic and electrical properties could offer
a powerful means of solving the problem. In the case of
common reservoir rocks resistivity strongly depends on
Figure 1: (a) BSE image of a cemented greensand (Hossain et al. 2011). (b) Cemented greensand model shows micro crystalline quartz cement
(QC) on quartz (Q) grains and berthierine (B) cementation within large pores. (c) Glauconite (G) grain of greensand with complex pore structure
(Hossain et al. 2009).
© 2012 SEG
SEG Las Vegas 2012 Annual Meeting
DOI http://dx.doi.org/10.1190/segam2012-1496.1
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porosity, pore geometry and saturation (Archie 1942) while
the elastic properties depend on porosity (Mavko 1980,
Murphy 1984), saturation history (Mavko and Mukerji
1995), pore geometry (Mavko 1980; Mavko and Nur 1978),
mineralogy and fluids types (Mavko et al. 2009). However,
elastic and electric methods can contribute in different
ways to characterizing rock properties.
The objective of this study is to experimentally and
theoretically revise the relations among the electrical
properties, porosity, permeability, and elastic wave
velocity. Laboratory measured data from the North Sea
greensand was used. Greensand is composed of a mixture
of quartz and micro-porous glauconite grains. (Figure 1).
Diagenesis of greensand can be described by micro
crystalline quartz cement and pore-filling berthierine
cement. Petrophysical models and rock physics models
were used to describe the effect of micro structure of
greensand on elastic and electrical properties.
Method
A laboratory measured core data set of 16 greensand
samples from the Nini field of the the North Sea was used
for this study. Helium porosity and Klinkenberg
permeability data were obtained from Hossain et at. 2011
while resistivity and elastic wave velocity data were
obtained from Hossain et al. (2012).
A physical relationship between permeability and
resistivity may be explained by combining Archie’s
equation (Archie, 1942) and Kozeny’s equation (Kozeny
1927). The ratio of the pore fluid resistivity of, Rw to bulk
resistivity of the fully saturated rock, Ro is known as 1 over
the formation factor, F (Archie 1942). Archie’s law is an
empirical relation relating the formation factor and
cementation factor, m to the porosity, and a factor
correcting for conducting minerals, a in brine saturated
reservoir rock:
F
a
m
(1)
The relationship among porosity (), permeability (k) and
specific surface area of bulk volume (S) may be written by
using Kozeny’s equation (Kozeny 1927) as:
k c
3
S2
© 2012 SEG
SEG Las Vegas 2012 Annual Meeting
(2)
where, c is Kozeny’s factor and the relationship between
permeability and formation factor can be expressed as:
a
k  c 
F
3/ m
1
S2
(3)
Worthington (1997) revisited the relationship between
formation factor and permeability by Archie and showed
how formation factor F decreases as permeability increases
according to the following relation:
b
k  
F
1/ c
(4)
Results and discussion
Electrical properties of greensands are higher than those for
consolidated sandstone, unconsolidated sandstone, average
sands, shaley sands and clear granular rock (Figure 2).
These higher electrical properties of greensand are related
to the micro-porosity within glauconite and pore-filling
berthierine cementation of greensand.
10
Formation factor
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Relationship among porosity, permeability, electrical and elastic properties
10
10
10
3
Consolidated sandstone
Unconsolidated sandstone
Average sands
Shaley sands
Clean granular rock
Greensand
Greensand lab data
2
1
0
0
0.2
0.4
0.6
Porosity
0.8
1
Figure 2: (a) Comparison of greensand formation factor with
different types of rocks.
A linear relationship can be established between
permeability and formation factor (Figure 3a). Any scatter
between these properties can be described due to greensand
diagenesis particularly at low resistivity and low
permeability bearing samples (Figure 3a). Equation (3) was
used to describe this diagenesis.
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Relationship among porosity, permeability, electrical and elastic properties
For this data set, I defined m is equal to 1.9, a is equal to
1.67 and c is close to 0.21 for porosity 0.27 to 0.42 so in
equation (3) only specific surface area of pore is the
controlling factor for relationship between permeability and
formation factor (Figure 3c).
The relationship between resistivity and elastic wave
velocity is not linear; indeed, the data exhibit an
approximate quadric trend (Figure 4a). Note that the
expression could be linear if I the three highest elastic
velocity bearing greensand from the lower Ty Formation
are omitted. By combining a rock physics soft-sand and
stiff-sand model (Mavko et al. 2009) with the Archie
equation (Archie 1942) the scatter could be described. The
modeling shows that micro crystalline quartz cement has a
larger effect on elastic properties and a smaller effect on
electrical properties. In contrast berthierine cementation has
a simultaneous effect on elastic and electrical properties
and berthierine cementation is mainly responsible for
higher elastic and electrical properties (Figure 5b).
Using self-consistent modeling with grain aspect ratio 1,
and pore aspect ratio between 0.2 and 0.1, the laboratory
measured resistivity data fall into this theoretical range
(Figure 5a). Whereas, using self-consistent modeling with
grain aspect ratio 1, and pore aspect ratio between 0.05 and
0.3, the laboratory measured elastic velocity data fall into
this theoretical range (Figure 5b). Self-consistent modeling
Figure 3: (a) Relation between permeability and electrical resistivity, (b) relationship among porosity, permeability and specific surface area.
The reference lines represent equal specific surface area with respect to pores (Sp) as calculated from Kozeny’s equation (Kozeny 1927) , (c)
Permeability and resistivity relationship are superimposed by combining Kozeny’s equation (Kozeny 1927) and Archie equation (Archie 1942),
(d) Statistical significant of permeability prediction by using equation (3).
© 2012 SEG
SEG Las Vegas 2012 Annual Meeting
DOI http://dx.doi.org/10.1190/segam2012-1496.1
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Relationship among porosity, permeability, electrical and elastic properties
Figure 4: Relationship between electrical and elastic properties. (a) Lab measured data, (b) lab measured data are superimposed by combing
rock physics models with Archie equation (Archie 1942).
Figure 5: Self-consistent modeling of greensand samples (a) lab measured electrical properties by using grain aspect ratio of 1, and pore aspect
ratio from 0.2 to 1. (b) Lab measured elastic properties by using grain aspect ratio of 1 and pore aspect ratio from 0.05 to 0.3.
results show that pore aspect ratios are more sensitive to
electrical properties than elastic properties.
Conclusions
A linear relationship between laboratory measured
electrical properties and permeability could be established
if the diagenesis of greensand is known. By combing
Archie’s relation and Kozeny’s equation, this greensand
© 2012 SEG
SEG Las Vegas 2012 Annual Meeting
diagenesis may be described by the specific surface area of
pores.
A linear relationship between laboratory measured
electrical and elastic properties could be established if the
effect of micro structure of greensand is known. Rock
physics modeling results show that quartz cementation has
a larger effect on elastic properties than on electrical
properties, while berthierine cementation has a
simultaneous effect on elastic and electrical properties.
DOI http://dx.doi.org/10.1190/segam2012-1496.1
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EDITED REFERENCES
Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012
SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for
each paper will achieve a high degree of linking to cited sources that appear on the Web.
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© 2012 SEG
SEG Las Vegas 2012 Annual Meeting
DOI http://dx.doi.org/10.1190/segam2012-1496.1
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