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A modified Swanson method to determine permeability from mercury intrusion data in marine muds
Journal Pre-proof A modified Swanson method to determine permeability from mercury intrusion data in marine muds Hugh Daigle, Julia S. Reece, Peter B. Flemings PII:
S0264-8172(19)30607-5
DOI:
https://doi.org/10.1016/j.marpetgeo.2019.104155
Reference:
JMPG 104155
To appear in:
Marine and Petroleum Geology
Received Date: 2 October 2019 Revised Date:
22 November 2019
Accepted Date: 28 November 2019
Please cite this article as: Daigle, H., Reece, J.S., Flemings, P.B., A modified Swanson method to determine permeability from mercury intrusion data in marine muds, Marine and Petroleum Geology (2019), doi: https://doi.org/10.1016/j.marpetgeo.2019.104155. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
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A modified Swanson method to determine permeability from mercury intrusion data in
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marine muds
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Hugh Daigle1*, Julia S. Reece2, Peter B. Flemings3
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Austin, Austin, Texas, USA
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2
Department of Geology and Geophysics, Texas A&M University, College Station, Texas, USA
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Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas, USA
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*Corresponding author. Email: [email protected] Tel: +1-512-471-3775
Hildebrand Department of Petroleum and Geosystems Engineering, The University of Texas at
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Abstract
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The permeability of shallow marine sediments is an extremely important parameter to
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constrain, as it affects fluid and nutrient transport near the sediment-water interface, mediates
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mass exchange between igneous basement and oceans, and plays a role in seismicity along
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convergent margins. Determining the permeability of these sediments in the laboratory is
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difficult because existing methods typically require fully saturated, intact samples of large
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volume (tens of cm3), which are usually not collected with high spatial resolution in scientific
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ocean drilling operations. We demonstrate how mercury injection capillary pressure (MICP) data
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may be used to predict the permeability of marine muds using a modification of the widely used
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Swanson method. Our results show that MICP measurements performed on small, irregular, and,
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most importantly, unpreserved samples can yield important permeability information. This will
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improve the spatial resolution of permeability data in the shallow marine subsurface and allow
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analyses to be performed on the significant quantities of existing legacy core.
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Key words: permeability, marine sediments, ocean drilling
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1. Introduction The prevalence of muds in sedimentary sequences – comprising 70% of sediments in
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most basins (Dewhurst et al., 1998) – mean that their permeability controls fluid and chemical
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fluxes at the basin scale (Neuzil, 1994; Bethke, 1989; Person et al., 1996). This in turn affects
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processes on both active and passive margins, including fault localization, sediment accretion,
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and seismic slip on active margins (Davis et al., 1983; Moore and Saffer, 2001), and
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overpressure generation, seafloor fluid expulsion, and submarine slope failure on passive
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margins (Dugan and Sheahan, 2012). The importance of marine mud permeability has led to
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extensive characterization efforts based on samples from scientific ocean drilling (e.g., Daigle
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and Screaton, 2015, and references therein).
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Permeability of marine sediments is usually determined in the laboratory. Measurement
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techniques include steady-state flow-through tests, uniaxial consolidation tests, and transient
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pulse decay measurements (Daigle and Screaton, 2015). All three laboratory methods typically
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require several days to perform (e.g., Daigle, 2011; Reuschle, 2011). An additional complication
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in the laboratory is the need for fully saturated, intact, regularly shaped samples several tens of
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cm3 in volume. In the scientific ocean drilling community, these requirements have limited the
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number of permeability measurements conducted, as significant amounts of core must be
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dedicated to such efforts, and analysis of older cores, particularly archived, unpreserved material,
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is not possible.
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Various methods have been developed to determine permeability on samples that are
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small, irregularly shaped, and, in the case of samples from oil and gas wells, initially saturated
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with two or more fluids. These include pressure-pulse decay measurements (Egermann et al.,
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2003; Lenormand and Fonta, 2007), correlations with the liquid limit (Casey et al., 2013), and
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numerical modeling based on microstructural measurements (e.g., Haghshenas et al., 2016). One
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such method that is widely used is that of Swanson (1981), in which the shape of the mercury
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intrusion capillary pressure (MICP) curve is correlated with permeability. This method is well
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validated in consolidated reservoir rocks, but its applicability has not been demonstrated in
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marine muds. In this note, we show that Swanson’s method works well in marine muds over
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nearly 5 orders of magnitude of permeability. The simple method can therefore be used to
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determine permeability of old, unpreserved samples, or even cuttings.
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2. Swanson’s method Swanson’s method correlates the shape of the MICP curve with permeability. During an
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MICP measurement, a dried, evacuated sample is immersed in mercury, and as the mercury
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pressure increases incrementally, the pore system of the sample is filled with mercury. As
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mercury is nonwetting on most geologic media, the intrusion is a capillary drainage process, and
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is affected both by the size distribution of pores and their connectivity (Purcell, 1949; Blunt,
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2017). The MICP curve, which plots volume of mercury intruded versus mercury pressure,
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therefore contains important pore structure information. Thomeer (1960) introduced an empirical
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parameterization of the MICP curve based on the observation that when the total volume fraction
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of the sample occupied by mercury is plotted against mercury pressure on log-log scales, the
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curve approximates a hyperbola in many instances. The Thomeer hyperbola is expressed as
66 67
= exp −
,
(1)
68
3
69
where Vb(Pc) is the bulk volume fraction of sample occupied by mercury at pressure Pc (ratio of
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intruded mercury volume to total sample volume), Vb∞ is the bulk volume fraction of mercury at
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infinite pressure (equal to the total interconnected porosity), Pd is the displacement pressure, and
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G is a pore geometrical factor (Fig. 1a). Thomeer (1960) showed correlations between
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permeability and various parameters in Eq. 1, thus their significance in describing pore structure. Swanson (1981) drew on contemporary work (e.g., Schowalter, 1979; Swanson, 1979)
74 75
demonstrating the relationship between the shape of the MICP curve and percolation processes,
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in particular the pressure at which a connected network of mercury-filled pores forms and first
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allows transport across the sample. Identifying this point directly on the MICP curve can be
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challenging and different methods exist in the literature (e.g., Schowalter, 1979; Thompson et al.,
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1987; see also Daigle et al., 2019, §4.1). Swanson (1981) used the point corresponding to the
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vertex of the Thomeer hyperbola (Fig. 1b) and showed that the ratio Vb/Pc at this point was
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strongly correlated with permeability in 56 samples of sandstone and carbonate reservoirs. The vertex of the Thomeer hyperbola as used by Swanson (1981) may be determined by
82 83
recasting Eq. 1:
84 85
log =
.
.
(2)
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Defining ξ = log10Vb∞/Vb(Pc) and υ = log10Pc/Pd, Eq. 2 describes a hyperbola that is symmetric
88
about the line υ = ξ:
89 90
= . !.
(3)
91
4
The vertices of this hyperbola are located at ±#
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parameterization, ξ ≥ 0 and υ ≥ 0, so the vertex of interest is located at % = #
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#
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.0 10,⁄ . , giving
.
, ±#
92
. In the Thomeer
.
.
and
=
. The corresponding values of Vb and Pc are thus &' = &'( 10+,⁄ . and ./ =
.
96
97
= 10
1 2.33
+ #
(4)
98 99
at the vertex. Swanson (1981) found that the permeability of his 56 samples could be modeled as
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a power law function of the right-hand side of Eq. 4 with a standard deviation of ±0.67 orders of
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magnitude.
102 103 104
3. Samples We compared predicted permeability using Swanson’s method with measured
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permeability for a suite of 35 natural and resedimented marine mud samples. The samples
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included 2 from the deepwater Gulf of Mexico, 14 from the Nankai Trough, 6 from offshore
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southern Alaska, 4 resedimented mixtures of silt and Boston Blue Clay, and 9 resedimented
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mixtures of Nankai Trough mud and silt. All samples had MICP data to at least 55,000 psi (380
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MPa) intrusion pressure. Permeabilities in all cases were determined from uniaxial consolidation
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tests performed on fully saturated samples, and porosities were determined from the difference
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between wet and dry masses with grain densities constrained by helium pycnometry or from dry
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mass and dry volume following International Ocean Discovery Program Method C (Blum, 1997) 5
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and therefore represent total porosities. Relevant sample properties are given in Table 1, and
114
detailed values are given in the supporting information.
115 116
4. Results and discussion
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4.1 Swanson’s original method
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We fit Thomeer hyperbolas to the MICP curves for all samples and modeled permeability
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as a function of Vb/Pc at the vertex of the hyperbola as given by Eq. 4. The resulting power law is
120 121
.:
4 = 243 8 9
,
(5)
122 123
where k is permeability in m2, Vb is in decimal, and Pc is in Pa (Fig. 2). When k is in mD, Vb in
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percent, and Pc in psi as in Swanson (1981), the leading coefficient is 391 and the exponent
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remains 2.53. These values are similar to those reported by Swanson (1981) for his suite of
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sandstones and carbonates (355 and 2.005). However, the fit to our dataset is poor (R2 = 0.598)
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and the predicted permeability differs from the measured value by up to 2 orders of magnitude
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for some samples.
129 130 131
4.2 Modified Swanson method Swanson (1981) presented his method as a way of correcting for a sample-size effect in
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MICP data. Intact samples such as cm-scale core plugs often display a distinct plateau region in
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their MICP curves, but smaller samples like drill cuttings or crushed samples tend to lose this
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plateau (e.g., Schowalter, 1979; Mishra and Sharma, 1988) (Fig. 3). Purcell (1949) introduced a
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method of determining permeability from the MICP curve by considering flow through a bundle 6
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of parallel capillary tubes whose size distribution was given by the MICP curve. However, as
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Swanson (1981) noted, the difference in the shape of the MICP curve for smaller samples
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yielded errors when using this method. The position of the vertex of the Thomeer hyperbola, on
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the other hand, is assumed not to vary much with sample size, making it a more useful parameter
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to correlate with permeability. This approach is similar to that of Daigle (2016), who showed that for marine muds the
141 142
permeability may be predicted by
143 144
4=
;2 =+>? 8 9 , < +>?
(6)
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where φ is total porosity, xt is the bulk volume occupied by mercury at percolation, and rc is the
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critical pore size determined from the capillary pressure at xt. Here, percolation is identified from
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the plateau region in the MICP curve as the point of minimum slope, and Daigle (2016)
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presented a method of correcting MICP data for sample-size effects to allow determination of
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this parameter when a plateau was indistinct. This method requires having an independent
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measurement of the true pore size distribution, for example from microtomography images or
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gas sorption isotherms, and is therefore not practical when only MICP data are available.
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Let us postulate a function similar to Eq. 6 in which Vb and Pc at the vertex of the
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Thomeer hyperbola take the places of xt and 1/rc. Permeability should thus be modeled using
155 156
=+ B
4 = @ 8A
+
9 ,
(7)
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7
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where A and B are constants. Since the quantity in parentheses on the right-hand side of Eq. 7 has
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units of [LT2/M] and k has units of [L2], A must have units of [L2MB/LBT2B]. In SI units A would
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therefore be in m2·PaB. Using Eq. 7, we obtain a much better match to the measured
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permeability, with A = 1300 m2·Pa2.73 and B = 2.73 (R2 = 0.837) (Fig. 4). The standard deviation
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of this fit is ±0.93 orders of magnitude, which compares favorably to the standard deviation
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Swanson (1981) obtained using his method and dataset.
164 165 166
4.3 Reconciling the two methods There are a few possible reasons why the modification to Swanson’s method was
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necessary and why Eq. 7 offers a better fit to the data than Eq. 5. First, the marine muds in our
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dataset have high porosities (minimum 0.25, maximum 0.72, mean 0.50) and the values of Vb at
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the vertex are on average 38% of the porosity value. While Swanson (1981) did not report
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specific values of these properties for his samples, the few examples he provided indicate that the
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porosities and Vb at the vertex were much lower (< 0.22 and < 0.13) and Vb at the vertex was a
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much larger fraction of the porosity value, approaching 50%. In a situation where Vb is small and
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approximately half of the porosity, the quantity C − &' ⁄1 − &' in Eq. 7 can be approximated
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simply as Vb, in which case Eq. 7 reduces to the original Swanson method. Samples that deviate
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from this situation, however, will display a poor fit when using Eq. 5 to predict permeability.
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Another possibility lies in differences in pore structure. Both the original Swanson
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method and the modification we propose for marine muds use the ratio of a pore volume to a
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capillary pressure, or equivalently the product of a pore volume and a pore size, to predict
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permeability. The main difference between these methods is the pore volume that is considered.
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The original Swanson method follows reasoning from Purcell (1949) and Thomeer (1960) that
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permeability should be a function of total interconnected pore volume. In pore systems that are
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well connected, the pore volume or Vb in the Swanson case can be used in such a relationship.
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Indeed, similar conclusions have been drawn in modeling electrical conductivity of networks and
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rocks using effective medium theory (e.g., Kirkpatrick, 1973; Sen et al., 1981; Mendelson and
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Cohen, 1982; Ghanbarian et al., 2014). However, when the connectivity is poor and/or the pore
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size distribution is broader, consideration of the percolation threshold is necessary (Stauffer and
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Aharony, 1992). Pore networks in marine muds are generally more complex and poorly
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connected than conventional reservoir rocks (e.g., Daigle and Dugan, 2011), so this could be an
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additional contributing factor to the poor fit shown in Fig. 2.
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Finally, it should be noted that Swanson (1981) based his method on a correlation to
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horizontal permeability, that is, permeability in the direction orthogonal to the borehole axis,
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which the permeability values in our dataset, and those typically measured in the scientific ocean
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drilling community, are vertical permeabilities. MICP measurements are usually omnidirectional,
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and correlating them to directional quantities will always introduce some degree of scatter.
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Vertical permeabilities of marine sediments that have not been significantly sheared or rotated
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are typically smaller than horizontal permeabilities by a factor of less than 2 (Daigle and
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Screaton, 2015), but this difference combined with the inherent anisotropy of marine muds
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caused by alignment of platy clay grains (Milliken and Reed, 2010) likely contribute to the
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differences between ours and Swanson’s methods.
200 201 202 203
5. Conclusions Using MICP curves and measured permeability from a suite of natural and resedimented marine muds, we presented a modification of the method of Swanson (1981) to determine
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permeability directly from MICP data. The modified Swanson method takes the form of Eq. 7
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with A = 1300 and B = 2.73. Some possible reasons for the modification were presented. The
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method allows prediction of permeability for samples that are irregularly shaped and not fully
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saturated, which includes drill cuttings and legacy scientific ocean drilling cores. This will vastly
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improve our understanding of fluid flow in the marine subsurface.
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Acknowledgments Support for HD was provided by the University of Texas at Austin. This research used
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samples and/or data provided by the Integrated Ocean Drilling Program (IODP). The University
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of Texas GeoFluids Consortium (supported by 12 energy companies) and the Schlanger Ocean
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Drilling Fellowship from the Consortium for Ocean Leadership provided funding. Comments
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from 2 anonymous reviewers helped strengthen this work.
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322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337
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Table 1. Marine mud properties Depth range (mbsf)† 21.81 – 522.14
Porosity range 0.33 – 0.72
Permeability range (m2) -18 2.2 x 10 – 2.9 x 10-16
Permeability reference Daigle and Dugan (2014)
Deepwater Gulf of Mexico silty clays
105.5 – 259.7
0.42 – 0.49
1.9 x 10-18 – 1.2 x 10-17
Southern Alaska silty clays Resedimented Nankai Trough silty clay + silica powder (Min-U-Sil 40) Resedimented glaciomarine clay (Boston Blue Clay) + silica powder (MinU-Sil 40)
492.63 – 936.32
0.34 – 0.41
1.2 x 10-17 – 9.4 x 10-17
N/A
0.25 – 0.62
1.6 x 10-20 – 2.2 x 10-16
Long et al. (2008); Daigle and Dugan (2009) Daigle and Piña (2016) Reece et al. (2013)
N/A
0.44 – 0.52
8.2 x 10-17 – 1.0 x 10-16
Description Nankai Trough silty clays
339
Schneider (2011); Schneider et al. (2011)
MICP reference Daigle and Dugan (2014) Daigle et al. (2019)
Daigle et al. (2019) Daigle et al. (2019)
Daigle et al. (2019)
†
mbsf = m below seafloor
340 341 342 343 344 345 346 347 348 349 350 351 352
16
353
Figure captions
354
Figure 1. (a) MICP data with Thomeer fit (Eq. 1) for a mud sample from the Nankai Trough.
355
Thomeer parameters are given. (b) Thomeer hyperbola from (a) transformed to illustrate location
356
of the vertex, indicated by the black circle. The symmetry line of the hyperbola is indicated by
357
the dashed line.
358 359
Figure 2. Measured permeability versus the ratio of Vp/Pc at the vertex of the Thomeer
360
hyperbola. Dashed line is the power law regression given in Eq. 5, which represents the original
361
Swanson method.
362 363
Figure 3. Effect of sample size on the character of the MICP curve. Note the loss of the distinct
364
plateau as sample size decreases. Figure based on data presented by Schowalter (1979).
365 366
Figure 4. Permeability prediction using Eq. 7. Measured data are black dots, and regression line
367
using Eq. 7 is dashed line.
17
a.
b.
1000
1 Vb∞ = 0.53 Pd = 5.2 MPa G = 0.18
Pc (MPa)
log10(Pc/Pd)
100
10
1 0.01
0.1 Vb Measured
1 Thomeer fit
0.5
0 0
0.5 log10(Vb∞/Vb)
1
10-15 R2 = 0.598
k (m2)
10
-16
10-17 10-18 10-19 10-20 10-9
10-8 10-7 (Vb/Pc)Vertex (1/Pa)
10-6
Capilary pressure →
Decreasing sample size
Plateau location 1
0
Mercury saturation (fraction of pore volume)
10-15 R2 = 0.837
k (m2)
10
-16
10-17 10-18 10-19 10-20 10-9
10-8
(
1 φ - Vb Pc 1 - Vb
10-7
(
(1/Pa)
Vertex
10-6
•
MICP measurements performed on unpreserved samples can help determine permeability
•
We modified Swanson’s method to yield accurate predictions in marine muds
•
The modifications account for the particular pore structure inherent in muds
Hugh Daigle: Conceptualization, methodology, data collection and analysis, writing Julia Reece: Data collection and analysis Peter Flemings: Supervision, data curation
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0264-8172(19)30607-5
DOI:
https://doi.org/10.1016/j.marpetgeo.2019.104155
Reference:
JMPG 104155
To appear in:
Marine and Petroleum Geology
Received Date: 2 October 2019 Revised Date:
22 November 2019
Accepted Date: 28 November 2019
Please cite this article as: Daigle, H., Reece, J.S., Flemings, P.B., A modified Swanson method to determine permeability from mercury intrusion data in marine muds, Marine and Petroleum Geology (2019), doi: https://doi.org/10.1016/j.marpetgeo.2019.104155. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
1
A modified Swanson method to determine permeability from mercury intrusion data in
2
marine muds
3
Hugh Daigle1*, Julia S. Reece2, Peter B. Flemings3
4
1
5
Austin, Austin, Texas, USA
6
2
Department of Geology and Geophysics, Texas A&M University, College Station, Texas, USA
7
3
Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas, USA
8
*Corresponding author. Email: [email protected] Tel: +1-512-471-3775
Hildebrand Department of Petroleum and Geosystems Engineering, The University of Texas at
9 10
Abstract
11
The permeability of shallow marine sediments is an extremely important parameter to
12
constrain, as it affects fluid and nutrient transport near the sediment-water interface, mediates
13
mass exchange between igneous basement and oceans, and plays a role in seismicity along
14
convergent margins. Determining the permeability of these sediments in the laboratory is
15
difficult because existing methods typically require fully saturated, intact samples of large
16
volume (tens of cm3), which are usually not collected with high spatial resolution in scientific
17
ocean drilling operations. We demonstrate how mercury injection capillary pressure (MICP) data
18
may be used to predict the permeability of marine muds using a modification of the widely used
19
Swanson method. Our results show that MICP measurements performed on small, irregular, and,
20
most importantly, unpreserved samples can yield important permeability information. This will
21
improve the spatial resolution of permeability data in the shallow marine subsurface and allow
22
analyses to be performed on the significant quantities of existing legacy core.
23
Key words: permeability, marine sediments, ocean drilling
1
24 25
1. Introduction The prevalence of muds in sedimentary sequences – comprising 70% of sediments in
26
most basins (Dewhurst et al., 1998) – mean that their permeability controls fluid and chemical
27
fluxes at the basin scale (Neuzil, 1994; Bethke, 1989; Person et al., 1996). This in turn affects
28
processes on both active and passive margins, including fault localization, sediment accretion,
29
and seismic slip on active margins (Davis et al., 1983; Moore and Saffer, 2001), and
30
overpressure generation, seafloor fluid expulsion, and submarine slope failure on passive
31
margins (Dugan and Sheahan, 2012). The importance of marine mud permeability has led to
32
extensive characterization efforts based on samples from scientific ocean drilling (e.g., Daigle
33
and Screaton, 2015, and references therein).
34
Permeability of marine sediments is usually determined in the laboratory. Measurement
35
techniques include steady-state flow-through tests, uniaxial consolidation tests, and transient
36
pulse decay measurements (Daigle and Screaton, 2015). All three laboratory methods typically
37
require several days to perform (e.g., Daigle, 2011; Reuschle, 2011). An additional complication
38
in the laboratory is the need for fully saturated, intact, regularly shaped samples several tens of
39
cm3 in volume. In the scientific ocean drilling community, these requirements have limited the
40
number of permeability measurements conducted, as significant amounts of core must be
41
dedicated to such efforts, and analysis of older cores, particularly archived, unpreserved material,
42
is not possible.
43
Various methods have been developed to determine permeability on samples that are
44
small, irregularly shaped, and, in the case of samples from oil and gas wells, initially saturated
45
with two or more fluids. These include pressure-pulse decay measurements (Egermann et al.,
46
2003; Lenormand and Fonta, 2007), correlations with the liquid limit (Casey et al., 2013), and
2
47
numerical modeling based on microstructural measurements (e.g., Haghshenas et al., 2016). One
48
such method that is widely used is that of Swanson (1981), in which the shape of the mercury
49
intrusion capillary pressure (MICP) curve is correlated with permeability. This method is well
50
validated in consolidated reservoir rocks, but its applicability has not been demonstrated in
51
marine muds. In this note, we show that Swanson’s method works well in marine muds over
52
nearly 5 orders of magnitude of permeability. The simple method can therefore be used to
53
determine permeability of old, unpreserved samples, or even cuttings.
54 55
2. Swanson’s method Swanson’s method correlates the shape of the MICP curve with permeability. During an
56 57
MICP measurement, a dried, evacuated sample is immersed in mercury, and as the mercury
58
pressure increases incrementally, the pore system of the sample is filled with mercury. As
59
mercury is nonwetting on most geologic media, the intrusion is a capillary drainage process, and
60
is affected both by the size distribution of pores and their connectivity (Purcell, 1949; Blunt,
61
2017). The MICP curve, which plots volume of mercury intruded versus mercury pressure,
62
therefore contains important pore structure information. Thomeer (1960) introduced an empirical
63
parameterization of the MICP curve based on the observation that when the total volume fraction
64
of the sample occupied by mercury is plotted against mercury pressure on log-log scales, the
65
curve approximates a hyperbola in many instances. The Thomeer hyperbola is expressed as
66 67
= exp −
,
(1)
68
3
69
where Vb(Pc) is the bulk volume fraction of sample occupied by mercury at pressure Pc (ratio of
70
intruded mercury volume to total sample volume), Vb∞ is the bulk volume fraction of mercury at
71
infinite pressure (equal to the total interconnected porosity), Pd is the displacement pressure, and
72
G is a pore geometrical factor (Fig. 1a). Thomeer (1960) showed correlations between
73
permeability and various parameters in Eq. 1, thus their significance in describing pore structure. Swanson (1981) drew on contemporary work (e.g., Schowalter, 1979; Swanson, 1979)
74 75
demonstrating the relationship between the shape of the MICP curve and percolation processes,
76
in particular the pressure at which a connected network of mercury-filled pores forms and first
77
allows transport across the sample. Identifying this point directly on the MICP curve can be
78
challenging and different methods exist in the literature (e.g., Schowalter, 1979; Thompson et al.,
79
1987; see also Daigle et al., 2019, §4.1). Swanson (1981) used the point corresponding to the
80
vertex of the Thomeer hyperbola (Fig. 1b) and showed that the ratio Vb/Pc at this point was
81
strongly correlated with permeability in 56 samples of sandstone and carbonate reservoirs. The vertex of the Thomeer hyperbola as used by Swanson (1981) may be determined by
82 83
recasting Eq. 1:
84 85
log =
.
.
(2)
86 87
Defining ξ = log10Vb∞/Vb(Pc) and υ = log10Pc/Pd, Eq. 2 describes a hyperbola that is symmetric
88
about the line υ = ξ:
89 90
= . !.
(3)
91
4
The vertices of this hyperbola are located at ±#
93
parameterization, ξ ≥ 0 and υ ≥ 0, so the vertex of interest is located at % = #
94
#
95
.0 10,⁄ . , giving
.
, ±#
92
. In the Thomeer
.
.
and
=
. The corresponding values of Vb and Pc are thus &' = &'( 10+,⁄ . and ./ =
.
96
97
= 10
1 2.33
+ #
(4)
98 99
at the vertex. Swanson (1981) found that the permeability of his 56 samples could be modeled as
100
a power law function of the right-hand side of Eq. 4 with a standard deviation of ±0.67 orders of
101
magnitude.
102 103 104
3. Samples We compared predicted permeability using Swanson’s method with measured
105
permeability for a suite of 35 natural and resedimented marine mud samples. The samples
106
included 2 from the deepwater Gulf of Mexico, 14 from the Nankai Trough, 6 from offshore
107
southern Alaska, 4 resedimented mixtures of silt and Boston Blue Clay, and 9 resedimented
108
mixtures of Nankai Trough mud and silt. All samples had MICP data to at least 55,000 psi (380
109
MPa) intrusion pressure. Permeabilities in all cases were determined from uniaxial consolidation
110
tests performed on fully saturated samples, and porosities were determined from the difference
111
between wet and dry masses with grain densities constrained by helium pycnometry or from dry
112
mass and dry volume following International Ocean Discovery Program Method C (Blum, 1997) 5
113
and therefore represent total porosities. Relevant sample properties are given in Table 1, and
114
detailed values are given in the supporting information.
115 116
4. Results and discussion
117
4.1 Swanson’s original method
118
We fit Thomeer hyperbolas to the MICP curves for all samples and modeled permeability
119
as a function of Vb/Pc at the vertex of the hyperbola as given by Eq. 4. The resulting power law is
120 121
.:
4 = 243 8 9
,
(5)
122 123
where k is permeability in m2, Vb is in decimal, and Pc is in Pa (Fig. 2). When k is in mD, Vb in
124
percent, and Pc in psi as in Swanson (1981), the leading coefficient is 391 and the exponent
125
remains 2.53. These values are similar to those reported by Swanson (1981) for his suite of
126
sandstones and carbonates (355 and 2.005). However, the fit to our dataset is poor (R2 = 0.598)
127
and the predicted permeability differs from the measured value by up to 2 orders of magnitude
128
for some samples.
129 130 131
4.2 Modified Swanson method Swanson (1981) presented his method as a way of correcting for a sample-size effect in
132
MICP data. Intact samples such as cm-scale core plugs often display a distinct plateau region in
133
their MICP curves, but smaller samples like drill cuttings or crushed samples tend to lose this
134
plateau (e.g., Schowalter, 1979; Mishra and Sharma, 1988) (Fig. 3). Purcell (1949) introduced a
135
method of determining permeability from the MICP curve by considering flow through a bundle 6
136
of parallel capillary tubes whose size distribution was given by the MICP curve. However, as
137
Swanson (1981) noted, the difference in the shape of the MICP curve for smaller samples
138
yielded errors when using this method. The position of the vertex of the Thomeer hyperbola, on
139
the other hand, is assumed not to vary much with sample size, making it a more useful parameter
140
to correlate with permeability. This approach is similar to that of Daigle (2016), who showed that for marine muds the
141 142
permeability may be predicted by
143 144
4=
;2 =+>? 8 9 , < +>?
(6)
145 146
where φ is total porosity, xt is the bulk volume occupied by mercury at percolation, and rc is the
147
critical pore size determined from the capillary pressure at xt. Here, percolation is identified from
148
the plateau region in the MICP curve as the point of minimum slope, and Daigle (2016)
149
presented a method of correcting MICP data for sample-size effects to allow determination of
150
this parameter when a plateau was indistinct. This method requires having an independent
151
measurement of the true pore size distribution, for example from microtomography images or
152
gas sorption isotherms, and is therefore not practical when only MICP data are available.
153
Let us postulate a function similar to Eq. 6 in which Vb and Pc at the vertex of the
154
Thomeer hyperbola take the places of xt and 1/rc. Permeability should thus be modeled using
155 156
=+ B
4 = @ 8A
+
9 ,
(7)
157
7
158
where A and B are constants. Since the quantity in parentheses on the right-hand side of Eq. 7 has
159
units of [LT2/M] and k has units of [L2], A must have units of [L2MB/LBT2B]. In SI units A would
160
therefore be in m2·PaB. Using Eq. 7, we obtain a much better match to the measured
161
permeability, with A = 1300 m2·Pa2.73 and B = 2.73 (R2 = 0.837) (Fig. 4). The standard deviation
162
of this fit is ±0.93 orders of magnitude, which compares favorably to the standard deviation
163
Swanson (1981) obtained using his method and dataset.
164 165 166
4.3 Reconciling the two methods There are a few possible reasons why the modification to Swanson’s method was
167
necessary and why Eq. 7 offers a better fit to the data than Eq. 5. First, the marine muds in our
168
dataset have high porosities (minimum 0.25, maximum 0.72, mean 0.50) and the values of Vb at
169
the vertex are on average 38% of the porosity value. While Swanson (1981) did not report
170
specific values of these properties for his samples, the few examples he provided indicate that the
171
porosities and Vb at the vertex were much lower (< 0.22 and < 0.13) and Vb at the vertex was a
172
much larger fraction of the porosity value, approaching 50%. In a situation where Vb is small and
173
approximately half of the porosity, the quantity C − &' ⁄1 − &' in Eq. 7 can be approximated
174
simply as Vb, in which case Eq. 7 reduces to the original Swanson method. Samples that deviate
175
from this situation, however, will display a poor fit when using Eq. 5 to predict permeability.
176
Another possibility lies in differences in pore structure. Both the original Swanson
177
method and the modification we propose for marine muds use the ratio of a pore volume to a
178
capillary pressure, or equivalently the product of a pore volume and a pore size, to predict
179
permeability. The main difference between these methods is the pore volume that is considered.
180
The original Swanson method follows reasoning from Purcell (1949) and Thomeer (1960) that
8
181
permeability should be a function of total interconnected pore volume. In pore systems that are
182
well connected, the pore volume or Vb in the Swanson case can be used in such a relationship.
183
Indeed, similar conclusions have been drawn in modeling electrical conductivity of networks and
184
rocks using effective medium theory (e.g., Kirkpatrick, 1973; Sen et al., 1981; Mendelson and
185
Cohen, 1982; Ghanbarian et al., 2014). However, when the connectivity is poor and/or the pore
186
size distribution is broader, consideration of the percolation threshold is necessary (Stauffer and
187
Aharony, 1992). Pore networks in marine muds are generally more complex and poorly
188
connected than conventional reservoir rocks (e.g., Daigle and Dugan, 2011), so this could be an
189
additional contributing factor to the poor fit shown in Fig. 2.
190
Finally, it should be noted that Swanson (1981) based his method on a correlation to
191
horizontal permeability, that is, permeability in the direction orthogonal to the borehole axis,
192
which the permeability values in our dataset, and those typically measured in the scientific ocean
193
drilling community, are vertical permeabilities. MICP measurements are usually omnidirectional,
194
and correlating them to directional quantities will always introduce some degree of scatter.
195
Vertical permeabilities of marine sediments that have not been significantly sheared or rotated
196
are typically smaller than horizontal permeabilities by a factor of less than 2 (Daigle and
197
Screaton, 2015), but this difference combined with the inherent anisotropy of marine muds
198
caused by alignment of platy clay grains (Milliken and Reed, 2010) likely contribute to the
199
differences between ours and Swanson’s methods.
200 201 202 203
5. Conclusions Using MICP curves and measured permeability from a suite of natural and resedimented marine muds, we presented a modification of the method of Swanson (1981) to determine
9
204
permeability directly from MICP data. The modified Swanson method takes the form of Eq. 7
205
with A = 1300 and B = 2.73. Some possible reasons for the modification were presented. The
206
method allows prediction of permeability for samples that are irregularly shaped and not fully
207
saturated, which includes drill cuttings and legacy scientific ocean drilling cores. This will vastly
208
improve our understanding of fluid flow in the marine subsurface.
209 210 211
Acknowledgments Support for HD was provided by the University of Texas at Austin. This research used
212
samples and/or data provided by the Integrated Ocean Drilling Program (IODP). The University
213
of Texas GeoFluids Consortium (supported by 12 energy companies) and the Schlanger Ocean
214
Drilling Fellowship from the Consortium for Ocean Leadership provided funding. Comments
215
from 2 anonymous reviewers helped strengthen this work.
216 217
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Integr. Ocean Drill. Program, 322, 1-26. https://doi.org/10.2204/iodp.proc.322.205.2013/
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Reuschle, T., 2011. Data report: permeability measurements under confining pressure,
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Expeditions 315 and 316, Nankai Trough. Proc. Integr. Ocean Drill. Program, 314/315/316,
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1-17. https://doi.org/10.2204/iodp.proc.314315316.205.2011.
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Schneider, J., 2011. Compression and permeability behavior of natural mudstones. Ph.D. thesis, University of Texas at Austin, Austin, TX. Schneider, J., Flemings, P.B., Day-Stirrat, R.J., Germaine, J.T., 2011. Insights into pore-scale
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controls on mudstone permeability through resedimentation experiments. Geology, 39(11),
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1011-1014. https://doi.org/10.1130/G32475.1.
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Schowalter, T.T., 1979. Mechanics of secondary hydrocarbon migration and entrapment. AAPG
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Bull., 63(5), 723-760. https://doi.org/10.1306/2f9182ca-16ce-11d7-8645000102c1865d.
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Sen, P.N., Scala, C., Cohen, M.H., 1981. A self-similar model for sedimentary rocks with
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application to the dielectric constant of fused glass beads. Geophysics, 46(5), 781-795.
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Stauffer, D., Aharony, A., 1992. Introduction to Percolation Theory, 2nd ed. Taylor and Francis, London. Swanson, B.F., 1979. Visualizing pores and nonwetting phase in porous rock. J. Petrol. Technol., 31(1), 10-18. https://doi.org/10.2118/6857-PA.
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pressures. J. Petrol. Technol., 33(12), 2498-2504. https://doi.org/10.2118/8234-PA.
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Thomeer, J.H.M., 1960. Introduction of a pore geometrical factor defined by the capillary
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Thompson, A.H., Katz, A.J., Raschke, R.A., 1987. Mercury injection in porous media: a
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https://doi.org/10.1103/PhysRevLett.58.29.
322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337
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Table 1. Marine mud properties Depth range (mbsf)† 21.81 – 522.14
Porosity range 0.33 – 0.72
Permeability range (m2) -18 2.2 x 10 – 2.9 x 10-16
Permeability reference Daigle and Dugan (2014)
Deepwater Gulf of Mexico silty clays
105.5 – 259.7
0.42 – 0.49
1.9 x 10-18 – 1.2 x 10-17
Southern Alaska silty clays Resedimented Nankai Trough silty clay + silica powder (Min-U-Sil 40) Resedimented glaciomarine clay (Boston Blue Clay) + silica powder (MinU-Sil 40)
492.63 – 936.32
0.34 – 0.41
1.2 x 10-17 – 9.4 x 10-17
N/A
0.25 – 0.62
1.6 x 10-20 – 2.2 x 10-16
Long et al. (2008); Daigle and Dugan (2009) Daigle and Piña (2016) Reece et al. (2013)
N/A
0.44 – 0.52
8.2 x 10-17 – 1.0 x 10-16
Description Nankai Trough silty clays
339
Schneider (2011); Schneider et al. (2011)
MICP reference Daigle and Dugan (2014) Daigle et al. (2019)
Daigle et al. (2019) Daigle et al. (2019)
Daigle et al. (2019)
†
mbsf = m below seafloor
340 341 342 343 344 345 346 347 348 349 350 351 352
16
353
Figure captions
354
Figure 1. (a) MICP data with Thomeer fit (Eq. 1) for a mud sample from the Nankai Trough.
355
Thomeer parameters are given. (b) Thomeer hyperbola from (a) transformed to illustrate location
356
of the vertex, indicated by the black circle. The symmetry line of the hyperbola is indicated by
357
the dashed line.
358 359
Figure 2. Measured permeability versus the ratio of Vp/Pc at the vertex of the Thomeer
360
hyperbola. Dashed line is the power law regression given in Eq. 5, which represents the original
361
Swanson method.
362 363
Figure 3. Effect of sample size on the character of the MICP curve. Note the loss of the distinct
364
plateau as sample size decreases. Figure based on data presented by Schowalter (1979).
365 366
Figure 4. Permeability prediction using Eq. 7. Measured data are black dots, and regression line
367
using Eq. 7 is dashed line.
17
a.
b.
1000
1 Vb∞ = 0.53 Pd = 5.2 MPa G = 0.18
Pc (MPa)
log10(Pc/Pd)
100
10
1 0.01
0.1 Vb Measured
1 Thomeer fit
0.5
0 0
0.5 log10(Vb∞/Vb)
1
10-15 R2 = 0.598
k (m2)
10
-16
10-17 10-18 10-19 10-20 10-9
10-8 10-7 (Vb/Pc)Vertex (1/Pa)
10-6
Capilary pressure →
Decreasing sample size
Plateau location 1
0
Mercury saturation (fraction of pore volume)
10-15 R2 = 0.837
k (m2)
10
-16
10-17 10-18 10-19 10-20 10-9
10-8
(
1 φ - Vb Pc 1 - Vb
10-7
(
(1/Pa)
Vertex
10-6
•
MICP measurements performed on unpreserved samples can help determine permeability
•
We modified Swanson’s method to yield accurate predictions in marine muds
•
The modifications account for the particular pore structure inherent in muds
Hugh Daigle: Conceptualization, methodology, data collection and analysis, writing Julia Reece: Data collection and analysis Peter Flemings: Supervision, data curation
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: