Marine and Petroleum Geology 27 (2010) 1692–1697
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A permeability–porosity relationship for mudstones Yunlai Yang a b , Andrew C. Aplin a ,
a b
,
*
School of Civil Engineering and Geosciences, Drummond Building, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK PetroQuant Consultants, Newcastle upon Tyne, UK
a r t i c l e
i n f o
Article history: Received 9 February 2009 Accepted Accepted 8 July 2009 Available online 18 July 2009 Keywords: Fine-grained clastic sediments Mudstone Shale Permeability–porosity relationship
a b s t r a c t
The relationship between permeability and porosity for fine-grained clastic sediments (‘‘mudstones’’) is a key constitutive equation for modelling subsurface fluid flow and is fundamental to the quantification of a range of geological processes. For a given porosity, mudstone permeability varies over a range of 2–5 orders of magnitude. We show here that much of the range can be explained by variations in lithology, which we define simply and pragmatically by clay content (mass fraction of particles less than 2 microns in diameter). Using clay content as the quantitative lithology descriptor, we have used a dataset (clay content content range range of 12–97%; porosity range of 0.04–0.78; six orders of magnitude magnitude permeability range) range) comprising 376 data points to derive a new bedding perpendicular permeability (K, m 2) – void ratio (e ¼ porosity/(1-porosity)) relationship as a function of clay content (CF): lnðK Þ ¼ 69: 69:59 26: 26:79 CF þ 44: 44:07 CF 0:5 þ ð53: 53:61 80: 80:03 CF þ 132: 132:78 CF 0:5 Þ e 0 5 0 5 : : þð86: 86:61 þ 81: 81:91 CF 163: 163:61 CF Þ e ,
,
,
,
,
,
,
,
The coefficient of regression (r 2) ¼ 0.93. At a given porosity, the inclusion of the quantitative lithological logical descript descriptor or,, clay content content reduce reducess the predic predicted ted range of permeab permeabilit ility y from from 2 to 5 orders orders of magnitude to one order. 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Fine-grained clastic sediments, mud and its lithified counterparts mudstone mudstone and shale, shale, fill around 70% of the world’s world’s sedimentary mentary basins. basins. Since Since the permeabil permeability ity of mudstone mudstoness is several several orders of magnitude lower than that of coarser grained lithologies such as sand, mudstones control the rate at which fluids, including water, petroleum and CO 2, move through or are retained within sedimentary basins, and the rate of sediment compaction. The permeability of fine-grained, clastic sediments and its relationship with porosity are thus fundamental to the quantification quantification of a range of geological processes and geo-engineering applications, such as (a) basin evolution and the development of high pore fluid pressures (Smith, (Smith, 197 1971; 1; Bethke, 1985; Luo and Vasseur, Vasseur,1992; 1992; Schneider Schneider et al., 1993; Dugan and Flemings, 2000); 2000 ); (b) formation of mud diapirs (Graue, (Graue, 2000; Milkov, 2000); 2000); (c) destabilisation of continental nental slopes slopes (Dug Dugan an and Fle Flemin mings, gs, 2000 2000); ); (d) the long-t long-term erm subsurfac subsurface e retentio retention n of petroleu petroleum m (Engla England nd et al., 1987; 1987; Wat Watts, ts,
137, * Corresponding author. Saudi ARAMCO, Advanced Research Centre, Building 137, Dhahran 31311, Saudi Arabia. Tel.: þ 966 3 873 0205; fax: þ 966 3 873 0572. E-mail address: yunlai.yang@ara
[email protected] mco.com.sa (Y. Yang). 0264-8172/$ – see front matter 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.marpetgeo.2009.07.001
¨ mer 1987; 19 87; Schl Schlo o¨ mer and Kro Krooss, oss, 1997 1997), ), radio radioact activ ive e waste waste and CO2 (Hollow Holloway, ay, 2001 2001;; Hilde Hildenbr nbrand and et al., al., 2004 2004;; Marty Marty et al. al.,, 2003 2003;; Huysmans and Dassargues, 2006; Bickle et al., 2007); 2007 ); (e) surface geological process, such as erosion ( Jacobs ( Jacobs et al., 2007 2007)) and flowslides (Wang ( Wang and Shibata, 2007); 2007); (f) shale gas production (Luffel ( Luffel et al. al.,, 19 1993 93); ); and (g) evalu evaluati ation on of founda foundatio tion n settle settlemen mentt and landfill liner design. Permeability is a constant for a given porous medium and fluid. To separate the influence of fluid from that of the porous medium, the absolute permeability K is defined which only describes the permea permeabil bility ity of the porous porous medium medium (see (see Leonards, 1962 for a review): h
K ¼ k
r
(1)
where, K absolute absolute permeability, permeability, L 2; k permeability, permeability, LT1; 2 h coefficient of dynamic viscosity of the fluid, TFL ; r unit weight of the fluid, FL 3. For clay-rich media, equation (1) strictly only applies to non-polar fluids, since permeability is also affected by the valence and concentration of dissolved cations (Mesri ( Mesri and Olson, 197 1971; 1; Mishra et al., 2005). 2005). Mudstone permeabilities are direction dependent, with higher values parallel to bedding. This phenomenon reflects two factors:
Y. Yang, A.C. Aplin / Marine and Petroleum Geology 27 (2010) 1692–1697
particle alignment and material heterogeneity. Material heterogeneity relates primarily to sediment deposition and is most obviously linked to conditions leading to the formation of silt clay lamination (e.g. Bennett et al., 1991). Particle alignment is linked to the subsequent mechanical compaction and clay mineral recrystallisation (e.g. Ho et al.,1999; Bolton et al., 2000; Aplin et al., 2006; Day-Stirrat et al., 2008). Published permeability data for fine-grained clastic sediments, especially well characterised fine-grained clastic sediments, are sparse, but suggest a range of around six orders of magnitude, with a three orders of magnitude range at a single porosity ( Mesri and Olson, 1971; Coyner et al., 1993; Neuzil, 1994; Nagaraj et al., 1994; Yang and Aplin, 1998, 2007; Dewhurst et al., 1998, 1999a, 1999b; Hildenbrand et al., 2004; Kwon et al., 2004; Mallon et al., 2005). Current permeability–porosity relationships (e.g. Nagaraj et al., 1994) for mudstones are inadequate and only applicable over the often narrow range of porosities and lithologies used to define the relationship. The wide permeability range and the lack of a reliable permeability–porosity relationship hinders quantitatively useful flow modelling to a range of geological and geo-engineering applications. The aim of this paper is to construct a practically useful relationship between bedding perpendicular, absolute permeability and porosity for homogeneous, fine-grained clastic sediments. We are dealing with absolute permeability since it is an intrinsic property of sediments. Further, we do not consider the effect of solution chemistry on permeability. Following convention, absolute permeability is simply termed permeability hereafter. The constructed relationship can be used in two ways. First, it can be directly applied as a constitutive equation in the ‘‘dynamic’’ modelling of geological processes involving fluid flow, such as basin evolution (Smith, 1971; Bethke, 1985; Dugan and Flemings, 2000). Second, it can also be used to quickly and easily evaluate permeability from porosity, which can be much more easily measured or derived from geophysical data. The derived permeabilities can then be applied in ‘‘static’’ applications, in which the permeability, instead of permeability–porosity relationship is required. 2. Lithological control of permeability–porosity relationship
The development of a practical permeability–porosity relationship for mudstones requires an understanding of the main factors resulting in the wide permeability range observed at a single porosity. Poiseuille’s Law states that for a porous media composed of parallel tubes of equal diameter, the permeability is related to the square of the pore radius. A combination of theory and experimental data show that at a given porosity, pore size distributions of both granular media such as sandstones and mudstones are controlled in the first order by grain size ( Mesri and Olson, 1971; Bryant et al., 1993; Cade et al., 1994; Yang and Aplin, 1998, 2007; Dewhurst et al., 1998). At a given porosity, pore radii, and thus permeability decrease as grain radii decrease. We therefore anticipate that the wide range of mudstone permeabilities can be more tightly constrained by adding a lithological, or grain size descriptor to a relationship which predicts permeability from porosity. It is impractical to use particle size distribution as a lithological descriptor for two reasons; firstly, it cannot be readily employed in a simple equation and secondly, it must be measured and cannot be assessed from geophysical data such as the downhole well logs. We suggest that clay content, defined here as the mass fraction of particles less than 2 micron in diameter is a reasonable lithological descriptor for the following four important reasons: (a) it is a single number which can be easily incorporated into a permeability– porosity equation, (b) it is strongly correlated with the overall grain size distribution of mudstones (Yang and Aplin, 2004), (c) it
1693
100
) % ( 80 t n e t n 60 o c e l c i t r 40 a p y a 20 l C
This study Skempton, 1944
0 0
50
100
150
Liquid limit (%) Fig. 1. Correlation between clay content and liquid limit for fine-grained clastic sediments. Number of samples 18. ¼
correlates well with liquid limit ( Skempton, 1944; Fig. 1), a quantitative lithology descriptor of fine-grained sediments used in soil mechanics (Skempton, 1944; Burland, 1990) and (d) clay content has been employed successfully as the quantitative lithology descriptor in the definition of porosity – effective stress relationship of fine-grained clastic sediments (Yang and Aplin, 2004). For these reasons, we explore the extent to which clay content can be used as a pragmatic lithology descriptor which will help constrain permeability–porosity relationship of fine-grained clastic sediments. 3. Dataset
The dataset is summarised in Table 1 and is shown as a permeability–porosity plot in Fig. 2. All together 376 data points were collected from 303 samples. Samples from sources 1–4 and one sample from source 5 (Table 1) are marine sediments, accounting for 299 out of 303 samples. Ninety three of the 376 data points represent permeabilities measured using constant head, constant flow and transient pulse decay (e.g. Hsieh et al., 1981) techniques, or consolidation tests (Terzaghi, 1943). The remaining data are permeabilities derived from pore size distribution data using a model (Yang and Aplin, 1998) which has been calibrated using measured permeability data and which predicts permeability to factor of 3 of the measured values over a wide permeability range (Yang and Aplin, 2007). Clay contents, porosities, pore size distributions and total organic carbon contents for the samples from sources 1, 2 and 4 (Table 1) were determined using previously published techniques (Yang and Aplin, 1997, 1998, 2007; BS 1377: 1990). In order to achieve the widest possible range of clay contents, we have included five permeability data points of pure, Na-exchanged smectite for which the clay content is 97% (Mesri and Olson, 1971). Although these data are not from natural sediments, ultra-fine-grained clastic sediments can be smectiterich, so that the smectite sample approximates an end member fine-grained sediment. Clay contents were measured in most samples (95% in terms of samples, 84% in terms of data points). For the rest of the samples (sources1 and 5, Table 1), clay contents were not available and were estimated from a relationshipbetween clay content and liquidlimit derived from the data in Fig. 1:
CF ¼ 0:761 0:00438 LL þ 0:1872 LL0:5 ,
,
(2)
where CF is clay content in fraction and LL is liquid limit (%). The coefficient of regression, r 2 ¼ 0.95. The dataset used in the derivation of the above equation (equation(2)) coversa clay content
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Table 1 Dataset used in the construction of the permeability – porosity/void ratio relationship.
Data source
(1) Yang and Aplin, 2007
(2) This work
(3) Long and Hobbs, 1979 and Long, 1979
(4) Dewhurst et al., 1998, 1999b
(5) Nagaraj et al., 1994
(6) Mesri and Olson, 1971
No. of data No. of samples Sample type
22 22 Cores from North Sea and Gulf of Mexico oil wells
261 261 Cuttings and cores from North Sea and Gulf of Mexico oil wells
26 6 Shallow buried London clay
24 4 Reconstitut ed residual sediments (India)
5 1 Smectite in concentrated NaCl solution
Method of evaluation of K
Transient pulse decay technique
Direct measurement; Terzaghi’s model
Direct measurement
Terzaghi’s model from 1D consolidation
Range of clay content (%) Clay content evaluation Range of porosity Range of K (m 2)
1241
Modelled from pore size data (Yang and Aplin, 1998, 2007) 2492
38 9 Shallow (1.2130 m) marine sediments from North Sea boreholes Terzaghi’s model from 1D consolidation 3240
3369
3470
97
Measured
Measured
From liquid limit
Measured
From liquid limit
Measured
0.0550.22
0.040.45
0.260.55
0.220.43
0.36 0.7
0.560.78
2.4 1022 1.6 1019 30,000 mg/L NaCl solution
2.5 1022 3.0 1019 N/A
3.6 1019 1.7 1016 In situ pore fluid
1.1 1018 1.6 1016 Not available (distilled water?)
3.7 1021 3.5 1020 0.1 N NaCl solution
0.12.4
0.24.8
–
2.7 1021 2.1 1018 In situ pore fluid (500 mg/L NaCl and 50 mg/L CaCl2 solution) 0.20.9
–
–
Permeant
Range of TOC (% of weight)
range of 12%–80% (Fig. 1), encompassing essentially the whole range of clastic fine-grained sediment lithologies. The permeants used in the measurement of permeability in Sources 1, 3 and 4 (Table 1) are either in situ pore fluids or fluids very similar to the in situ pore fluids.Permeabilities estimated from pore size distributions (source 2 in Table 1) were derived using a model (Yang and Aplin, 1998) which was calibrated using a dataset (source 1, Table 1) in which the permeants were similar to the in situ pore fluids of marine sediments; the ‘‘virtual permeant’’ of these data points is therefore similar to the pore fluids of marine sediments. All samples from sources 1–4 are marine sediments, so that 92% of the permeability data points were measured or modelled using pore fluids close to those of marine sediments. Permeabilities from sources 4, 5 and 6 are permeabilities relative to the permeants (k) and were converted to absolute permeabilities using equation (1) from calculated viscosity data based on experimental temperatures. In one case (Mesri and Olson, 1971), 22 C was assumed. 1E-15
12-30 30-40 40-50 50-60 60-70 70-80 80-90 97
1E-16 1E-17
) 1E-18 m (
2
K1E-19 1E-20 1E-21 1E-22 0
0.2
0.4
0.6
0.8
Porosity Fig. 2. Permeability – porosity dataset. Legend shows the range of clay content for each band. Open circles and crosses are data points of measured or modelled permeability respectively.
In summary, 376 data points were collected from 303 samples, of which nearly 99% are marine sediments. The samples have the following ranges of key properties (Table 1 and Fig. 2): clay content: 12–97%; porosity: 0.04–0.78; permeability: 2.4 1022– 1.7 1016 m2; age: recent (see bed sediments) to late Jurassic (sources 1–4, Table 1) and total organic carbon content: 0.1–4.8%. The total organic carbon contents from sources 3, 5 and 6 ( Table 1) were not available, but are very likely within the range of the other samples. The dataset thus covers close to the full spectrum of finegrained clastic sediments in terms both of lithology and compaction state. 4. Porosity–permeability relationship
The permeability data in Fig. 2 range over six orders of magnitude, from about 2 1022 m2 to about 2 1016 m2. The range of permeabilities at a single porosity decreases with porosity, from over 4 orders of magnitude at porosity of 0.4 to 2 orders of magnitude at porosity of 0.15. The convergence of permeabilities at lower porosities reflects the fact that mudstones appear to compact by preferential collapse of those large pores which contribute most ofthefluid flow (Borst, 1982; Yang and Aplin, 1998, 2007; Dewhurst et al., 1998, 1999a, b). As porosities decrease, pore size distributions for lithologically diverse mudstones become increasingly similar (Borst, 1982; Yang and Aplin, 1998, 2007; Dewhurst et al., 1998, 1999a, b). Fig. 2 indicates firstly that there is no single, simple relationship between permeability and porosity for mudstones and secondly that the variability is mainly controlled by clay content; the lower the clay content, the higher the permeability at the same porosity. The influence of clay content on permeability–porosity relationship is explored explicitly in Fig. 3, which shows data for 58 samples from an 18 m long core from the Gulf of Mexico. These samples have a narrow porosity range (0.04) (0.20 to 0.24) but diverse clay contents (37%–70%). The minor variations in porosity are controlled by clay content (Fig. 3a), as predicted by mechanical compaction models for muds (Skempton, 1944; Burland, 1990; Yang and Aplin, 2004). Permeabilities vary by an order of magnitude and correlate strongly with clay content (Fig. 3b and 3c), generating confidence in
Y. Yang, A.C. Aplin / Marine and Petroleum Geology 27 (2010) 1692–1697
a
b
0.24
1695
1E-19
0.23
y t 0.22 i s o r o 0.21 P 0.2
) m1E-20 (
2
K
0.19
1E-21 30
40
50
60
70
0.19
80
0.2
Clay content (%)
c
0.21
0.22
0.23
0.24
Porosity
1E-19
) m1E-20 (
2
K
1E-21 30
40
50
60
70
80
Clay content (%) Fig. 3. Porosity, clay content and modelled permeability data for 58 samples from an 18 m core from the Gulf of Mexico. (a) porosity vs. clay content, showing the dependence of porosity on clay content; (b) permeability vs. porosity; (c) strong dependence of permeability on clay content and the good fit of equation (3) to the data.
the use of clay content as a key constraint on mudstone permeability porosity relationships. Previous studies have shown fine-grained clastic sediment permeability is roughly related to porosity ( 4) or void ratio (e ¼ 4/ (14)) by log-linear functions (Mesri and Olson, 1971; Lapierre et al., 1990; Nagaraj et al., 1994; Yang and Aplin, 1998, 2007; Dewhurst et al., 1998, 1999a, b). Our dataset suggests that whilst a linear relationship between log permeability and porosity is reasonable over a restricted porosity range, over the full porosity range the relationship between the logarithmic permeability and porosity or void ratio is betterdescribed by a slightly more complex form. The following form fits our dataset best:
(3)
,
where CF is the clay content in fraction, e is the void ratio and aK , bK , c K, c 0, c 1 and c 2 are coefficients (m2). The influence of clay content 1.E-15
1.E-17 ) 1.E-18 m (
2
,
,
,
,
e
,
(4)
for which the coefficient of regression ( r 2) ¼ 0.93. Fig. 4 shows the fit of equation (4) to the dataset. 5. Discussion
1.E-15
) 1.E-16 m ( K1.E-17 d e l l e d 1.E-18 o m r 1.E-19 o d e r 1.E-20 u s a e M1.E-21
2
K1.E-19
1.E-20 1.E-21 1.E-22 0
,
Fig. 5 shows that the constructed relationship between permeability and porosity has the ability to predict the permeability of
12-30 30-40 40-50 50-60 60-70 70-80 80-90 97
1.E-16
,
þ 53:61 80:03 CF þ 132:78 CF 0:5 e 0:5 0:5 ,
,
,
lnðK Þ ¼ 69:59 26:79 CF þ 44:07 CF 0:5
þ 86:61 þ 81:91 CF 163:61 CF
lnðK Þ ¼ aK þ bK e þ c K e0:5 aK ðor b K or c K Þ ¼ c 0 þ c 1 CF þ c 2 CF 0:5 ,
on the permeability–porosity relationship is incorporated in the coefficients. At a fixed porosity or void ratio, the permeability – clay content relationship thus takes the form of the relationship between the coefficients and clay content. Fig. 3c shows that the form between clay content and coefficients is robust. Fitting equation (3) to our dataset gives the relationship between permeability and porosity for mudstones as a function of clay content:
0.2
0.4
0.6
0.8
Porosity Fig. 4. Comparison between measured/modeled permeabilities and our constructed relationship (equation (4)). Curves are from the clay content constrained porosity permeability relationship. Legend shows the range of clay content for each band. Open circles and crosses are data points of measured or modelled permeability respectively. Each curve represents the relationship at the middle value of clay contents of the band with the same colour.
1.E-22 1E -2 2
Modelled Measured
1E -2 1
1 E- 20
1 E- 19
1E -18
1E -1 7
1 E- 16
1 E- 15
2
Calculated K (m )
Fig. 5. Comparison between permeabilities calculated using equation (4) and permeabilities which are either measured or modelled from pore size data (Yang and Aplin, 1998; 2007). Dashed lines represent a range of factor of 3 from the 1:1 line.
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Clay (%) 0
50
100
Porosity
K (m2)
0.0 0.1 0.2 0.3 0.4
1.E-21 1.E-20 1.E-19 1.E-18
6000
6500
7000
) t 7500 f ( l f s D V T 8000
8500
9000
9500 Fig. 6. Evaluated clay content, porosity and permeability for a mud-rich section of a North Sea well. Clay contents and porosities were evaluated from well logs using the approach developed by Yang et al. (2004). Permeabilities were evaluated from clay contents and porosities from equation (4).
almost all the samples to within a factor of 3 over a six orders of magnitude range of permeability. By including clay content as a discriminator, an uncertainty of 2–5 orders of magnitude in permeability at a given porosity is collapsed to 1 order of magnitude. This strikingly improved correlation reflects the basic control that grain size exerts on pore size in fine-grained clastic sediments. The remaining one order of magnitude uncertainty in the predicted permeability partly reflects measurement uncertainty, but probably primarily the fact that a simple descriptor such as clay content can only approximate the complexities of mudstone lithology and grain size distribution. The constructed relationship between permeability and porosity can be easily applied in both ‘‘dynamic’’ and ‘‘static’’ geological fluid flow modelling. For dynamic modelling, the specific constitutive equation between permeability and porosity for a given mudstone can be derived from the mudstone’s clay content. For static modelling, mudstone permeability can be evaluated from clay content and porosity, both of which can be measured in the laboratory or can be estimated from geophysicalwell logs (e.g. Yang et al., 2004). If geophysical logs are available, continuous profiles of both permeabilities and permeability–porosity relationships can be derived and incorporated into a range of flow models designed to simulate, for example, basin scale fluid flow and leakage rates of petroleum or stored CO 2. Fig. 6, as an example, demonstrates the way in which geophysical log data can be used to assess the permeability structure of mud-rich sedimentary sequences. In Fig. 6, clay contents, porosities and permeabilities were evaluated for a mud-rich section of a North Sea well from well logs and our constructed permeability–porosity relationship. Clay contents and grain densities were evaluated fromwell logs using Artificial Neural Networks models constructed by Yang et al. (2004). Porosities were calculated from grain densities and log-derived bulk densities, and permeabilities were then calculated from the evaluated clay contents and porosities using equation (4). Apart from the demonstration of the approach, this example also demonstrates two important points. In this case, the section shallower than 8475 ft has a higher porosity but an order of magnitude lower
permeability than the more heterogeneous, coarser grained deeper section. Note also that even within the relative homogeneous section above 8475 ft, there are some relatively coarse layers with much higher permeabilities. Since the permeability–porosity relationship constructed here is based mainly on marine mudstones, care should be taken when applying it to non-marine mudstones, or mudstones containing hypersaline pore fluids, or pore fluids in which the dominant cation is Ca 2þ rather than Naþ. In these cases, the thickness of the double diffuse layer around phyllosilicate minerals will differ to that typical of marine pore fluids, leading to a relative decrease (low salinity) or increase (high salinity) in permeability (Mesri and Olson, 1971; Mishra et al., 2005). The effect is related to specific surface area or particle sizes (Mesri and Olson, 1971; Mishra et al., 2005), which is correlated to clay mineral content. This effect will be greatest in mudstones with high clay mineral contents.
Acknowledgements
The work presented in this paper was supported by the GeoPOP consortium, comprising Amerada Hess, BG, BP, ChevronTexaco, ConocoPhillips, ExxonMobil, JNOC, Norsk Hydro, Shell, Statoil and Total, and precursors of those companies. Further support was provided by the UK Natural Environment Research Council. Norsk Hydro and BP kindly provided log data and samples. Gareth Yardley and Steve Larter are thanked for their very useful reviews.
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