Influence of lithology and compaction on the pore size distribution and modelled permeability of some mudstones from the Norwegian margin

Marine and ?Wroleum Geology ELSEVIER

Marine and Petroleum Geology 15 (1998) 163-I 75

Influence of lithology and compaction on the pore size distribution and modelled permeability of some mudstones from the Norwegian margin Yunlai Yang, Andrew C. Aplin* Fossil Fuels and Environmental Geochemistry Postgraduate Institute: NRG, University of Newcastle, Newcastle upon Tyne. NE1 7RU, U.K

Received 12 December 1996; revised 31 December 1997; accepted 5 January 1998)

Abstract We have derived a permeability model which uses pore shape, pore throat size distribution and pore alignment as key inputs. The pore shape is two frustra of cones connected at their base. Both the pore shape and alignment change with increasing compaction, developing a higher aspect ratio and becoming increasingly perpendicular to the direction of maximum stress. The uncalibrated model predicts the vertical permeability of some experimentally compacted muds to a factor of three. The model is used to estimate the vertical and horizontal permeability of eleven mudstones from the Norwegian Margin for which we have also determined porosity, pore size distribution, grain size distribution and specific surface area. Samples were chosen in order to investigate the influence of both compaction and lithology on pore size distribution and permeability. Porosity is lost mainly by the collapse of those larger pores which also contribute most of the permeability. Modelled vertical permeabilities of samples buried to between 855 and 3605 m vary from 3.3 x lo-l9 to 1.2x IO-“m2 and are not simply related either to porosity or effective stress. Permeability is strongly influenced both by porosity and detailed lithology, as described for example by the % < 2 pm particles. The modelled permeability of lithologically similar mudstones decreases logarithmically with decreasing porosity but at a single level of effective stress permeabilities of lithologically different mudstones vary by more than two orders of magnitude. At lower levels of effective stress (< 10 MPa) coarser grained (siltier) mudstones appear to have a greater range of pore radii, a much larger mean pore radius and much higher permeabilities than finer grained mudstones. Vertical permeabilities calculated using the equation derived in this paper are between three and seventy times lower than permeabilities calculated from the Kozeny-Carman equation, assuming a value of one hundred for the product of the tortuosity and shape factors in the equation. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords:

Mudstone ; Permeability ; Pore size distribution ; Compaction

1. Introduction

Permeability measures the ease with which fluid flows through a porous medium and is the coefficient k in Darcy’s law :

q=kAh L

(1)

where q = flow rate (units of L3T-‘); k = coefficient of permeability or hydraulic conductivity (LT- ‘) ; A = cross sectional area (L’) ; L = length along flow path (L) and Ah = fluid head (pressure) loss along the flow path (15). Ah/L is the hydraulic gradient i and is dimen-

*Corresponding author. Tel. : +44 191 222 6426; fax: +44 191 222 5431. 902&I-8172/98$19.00 0 1998Elsevier Science Ltd. AI1rights reserved PII: SO264-8 172(98)00008-7

sionless. In Darcy’s law the permeability is a constant for a given medium and fluid. To separate the influence of fluid from that of the porous medium, the specific or absolute permeability K is defined which only describes the permeability of the porous medium (see Leonards, 1962 for a review) :

K,kl P

(2)

where K = specific permeability (L2), q = coefficient of dynamic viscosity of the fluid (TFLe2) and p = unit weight of the fluid (ICC3). This paper concerns the permeability of mudstones, the accurate assessment of which is critical to the evaluation of any process involving fluid flow and fluid retention (e.g. overpressure) in sedimentary basins. However, mudstone permeability and its relationship to detailed

164

Y. Yang, A.C. AplinlMarine and Pmoleum

lithology and consolidation is an area of considerable ignorance. Published data are few but indicate that permeabilities vary by 10 orders of magnitude, with a range of three orders of magnitude at a single porosity value (Neuzil, 1994). Such great variability, and our poor understanding of why the variation is so large, are major obstacles to the accurate description of fluid flow in sedimentary basins. The extent of the problem is well illustrated by the wide range of proposed porosity : permeability relationships for ‘standard shales’ within the fluid flow modules of basin models. The permeability for the models’ ‘standard shales’ vary by up to 5-6 orders of magnitude at a single porosity. One form of porosity :permeability relationships within basin models is : k = k&/rJ

(3)

where k, and e, are the reference permeability and void ratio of the sediment and e is the current void ratio. Like other permeability :void ratio functions, eqn (3) requires values for its constants k, and c, which are very poorly known and which, given the highly variable lithological nature of mudstones, are likely to be very variable. One purpose of this paper is to investigate the importance of mudstone lithology, as defined for example by a grain size parameter, on permeability. Perhaps the main reason for the lack of permeability data is the difficulty of measuring the sub microDarcy values which characterise mudstones with porosities below thirty percent (Neuzil, 1994). Permeability is thus commonly estimated from more easily determined physical properties of mudstones such as porosity, pore size distribution or specific surface area (see Olsen, 1962 ; Leonards, 1962 ; Scheidegger, 1974 ; Garcia-Bengochea et al., 1979 and Lapierre et al., 1990 for reviews). The Kozeny-Carman equation, which describes permeability (I,‘) in terms of porosity (4) and specific surface area, is one such equation which was developed initially for sands : K=--

1

4’ k&, (I- 4’)s’

where k, and k, are poorly defined shape and tortuosity factors, and S = specific surface area (L-l). The KozenyCarman equation assumes that all pores are capillary tubes of identical cross section and its applicability to mudstones has not been properly tested. More sophisticated models describe permeability as functions of pore size distributions (Leonard% 1962 ; Scheidegger, 1974 ; Garcia-Bengochea et al., 1979 ; Pittman, 1992) or as network models, systems of interconnected capillaries with randomly assigned radii (e.g. Fatt, 1956 ; van Brakel, 1975 ; Dullien, 1992 ; O’Carroll and Sorbie, 1993). Pore size data are especially useful in petroleum systems, not only because they can be used to infer permeability but

Geology 15 (1998) 163-l 75

because they are required for the assessment of capillary entry pressures and thus seal capacity and failure. However, there are few, if any, systematic studies of the way in which the pore size distributions of mudstones are influenced by lithology and compaction. Indeed, there are few published pore size distributions for deeply buried mudstones (Heling, 1970; Borst, 1982; Katsube et al., 1991), most pore size data having been measured on relatively shallow-buried muds (e.g. Griffiths and Joshi, 1989 ; Lapierre et al., 1990). This paper has two purposes. Firstly, we derive a new permeability model which uses pore throat size distribution, pore shape and pore alignment as key inputs. The model attempts to account for the platy nature of clay minerals, which leads to permeability anisotropy arising from particle realignment during burial (Al-Tabbaa and Wood, 1987 ; Dewhurst et al., 1996). Secondly, we present pore size, specific surface area and grain size data for eleven mudstone samples from the Norwegian Margin and use these data to calculate permeabilities. The samples were carefully chosen so that we could investigate the influence of both consolidation and lithology on the pore size distribution and permeability of mudstones.

2. Permeability model 2.1. Existing permeability models

Two general approaches have been used to assess the permeability of fine-grained sediments in terms of their porosity and pore size distribution. Firstly, laboratory measurements can be used to build empirical relationships between permeability and porosity (e.g. Tavenas et al., 1983a, b ; Nagaraj et al., 1993 ; Tokunaga et al., 1994). The relationship is normally described by a lithology dependent parameter but the approach is fundamentally limited by the difficulty of measuring the very low permeabilities typical of mudstones. Another approach is to build a mathematical permeability model based on the Hagen-Poiseuille equation, which describes Newtonian flow in a straight tube of circular cross section (Leonards, 1962) :

where p = the unit weight of the fluid, r = the radius of the tube, I = the hydraulic gradient, ye= coefficient of dynamic viscosity of the fluid. The Hagen-Poiseuille equation has been used extensively as the basis for deriving permeability from pore size distributions. Many models (see Garcia-Bengochea et al., 1979 and Lapierre et al., 1990 for reviews) are based on the Hagen-Poiseuille equation and are similar to that described by Scheidegger (1974) :

Y. Yang, A.C. AplinlMarine and Petroleum Geology I5 (1998) 163-175

D = diameter of pores I$I= porosity, = pore size density function by volume.

where f(D)

165

and

This model assumes that all pores are capillary tubes with circular cross sections and that one third of pores are aligned along the direction of flow. More sophisticated models include a description of the connections between pores (e.g. Juang and Holtz, 1986). There are limitations in both empirical and mathematical approaches. The existing empirical models are based on data from mechanically compacted soils and may not be applicable to more deeply buried mudstones. The mathematical descriptions assume that pores are continuous tubes whereas in practice pore size distributions measured with mercury intrusion determine the size distribution of pore throats. The pore throat is smaller than the pore and therefore in a real pore system more than one modelled pore tube is required to account for the volume of the real pore. As a result the modelled permeability will be overestimated (Lapierre et al., 1990). Furthermore, the models do not consider permeability anisotropy, although this is likely to be important for deeply buried mudstones (Al-Tabbaa and Wood, 1987 ; Dewhurst et al., 1996). 2.2. New permeability model In this section we develop a permeability model based on a distribution of pores with defined shapes and measured distributions of throat sizes. The basis of the model is as follows. The total flow rate through an individual pore is calculated and then summed to give the flow rate of a mudstone layer with the thickness of one pore. The nature of pore connections are not considered (Juang and Holtz, 1986) because the mercury injection technique measures a connected network of pore throats. Using Icm diameter mudstone chips containing typical pore sizes of 50 nm, the mercury injection technique samples around lo5 pores across the sample. We assume here that those lo5 pores are representative of the pores in a layer of mudstone with the thickness of one pore. In this case each layer has the same permeability as all others and pore connections are not required to describe the permeability of the bulk sample. Finally, in order to explore the possible evolution of permeability anisotropy during burial, we allow the pore angle to change as an assumed function of effective stress. The alignment of pores is represented by their average angle with respect to the bedding of the sediments. 2.2.1. Pore shape The pore shape comprises two frustra of cones connected at their base (Fig. 1). It is characterised by its half

Fig. 1. Pore shape used in the permeability model developed in the text. R is the radius of the widest part of the pore and r is the pore throat radius. L is the half length of the pore.

length L, its largest radius R and its pore throat radius Y, which together allow us to define the pore shape by two ratios : J,

,fi

(74

r

and J2

~5

(7b)

r

J, and J2 are assumed to be identical for all the pores in

a given specimen so that the volume and surface areas of a pore can be defined (see Appendix 1). 2.2.2. Pore throat size distribution Based on many measured pore throat size distributions of fine grained sediments, we have established that the following equation accurately describes the pore throat size distribution of many mudstones by volume : F(x) =

1

1+ exp(a-bx-cx3)

(8)

where F is the cumulative distribution function by volume ; a, b and c are positive coefficients and x = ln(r-0.5)

(9)

where r is the pore radius in nanometres. We have chosen 0.5 nm as the lower limit of pore throat size radius because it corresponds to half the interlayer spacing of glycolated smectite. Nitrogen, which is used to measure the specific surface area of our samples, does not penetrate the interlayer spacing of clay minerals (van Olphen, 1977), which is therefore not included in the surface area measurement. The nature of eqn (8) is such that when r --f 0.5 nm (minimum pore radius), x -+ - CO,F+ 0, and when x -+ maximum pore throat size, F approaches one. The coefficients a, b and c in eqn (8) can be evaluated by

Y. Yang, A.C. AplinlMarine

166

and Petroleum

regression of measured pore throat size distributions. In some cases the regression results in a slightly negative value of c, in which case the function F(x) will not approach one when SCapproaches the radius of largest pores. This problem can be avoided if one allows c to equal zero in these cases. Equation (8) gives an excellent fit to the pore throat size distributions of mudstones that we have measured (typically r = 0.990; see Fig. 2 as an example). The distribution density function of pore radii is derived by differentiating eqn (8) : f’(x) = s

dx

=

(b + 3cx2) exp (a - bx - cx’)

(10)

[l+ exp(a-bx-cx3)]2

and the mean pore radius, r(p) is : +)

=

‘ml f(x)[ s .~x,

exp (x) + OS] dx

(11)

where xmaxcorresponds to the largest pore radius and exp(x) + 0.5 is the pore radius. 2.2.3. Flow rate in a pore. In deriving the permeability model for pores with the shape defined above, we assume that flow obeys the Hagen-Poiseuille equation at any pore cross section. Applying the Hagen-Poiseuille equation to the defined pore shape we obtain a flow rate through the pore :

(12) where dh = the hydraulic head drop across dx, dhldx = I = hydraulic gradient, z = the radius at a distance x along the pore (Fig. 1). The pore radius z can be expressed as :

100

&

$j 80 8 g 60 .S 46 tii -Z E 20 3 0

10

100

1000

IO

Pore throat radius nm

Geology

15

i 1998)

163-175

R-r z=R----.X L

(13)

Combining eqns (12) and (13) we obtain : (14) the solution of which gives the flow rate in a pore with throat radius r : npIr4

‘=

8~

3Ji’ I+J,+J~

where I = the overall hydraulic gradient across the pore = Ah/L, where Ah is the hydraulic head drop over the half length (L) of the pore. Note that if the pore is a tube with a circular section, then R = r, .I, = 1 and eqn (15) is transformed into the Hagen-Poiseuille eqn. The validity of equation (15) requires laminar flow and also that there is no stagnation zone along the flow path through the pore. Both assumptions are reasonable for the pore sizes, shapes and flow rates relevant to this study (Azzam and Dullien, 1977). 2.2.4. Permeability model of a porous medium Equation (15) gives the flow rate in a single pore of defined shape and defined pore throat radius. The flow rate through a porous medium is the integrated flow through all the pores, which for pores with throat radii between r and (r + dr) is : f’M-4 dx vi q1

(16)

where V is the unit volume of sediment, qi is the flow rate in one pore with radius r and Vi is the volume of the pore with throat radius r. Thus, V4f(x)dx/V, is the number of pores with throat radii between r and (r + dr). In order to ascribe direction to the flow, something of particular interest in highly compacted and probably anisotropic mudstones, we need to define the average angle, IX,at which pores lie with respect to the bedding plane of the sediment. The relation between the hydraulic gradient within individual pores, Zand the overall hydraulic gradient of the porous medium, i is : I = ices(a) for flow parallel to the bedding direction and I = kin(a) for flow perpendicular to the bedding direction. The contribution to total flow rate in a pore inclined at an angle tl to the bedding direction is: q_T= qcos(cr) and to the flow rate perpendicular to the bedding direction is : q, = qsin(@). The flow rate perpendicular to the bedding direction is then : .‘max V$f(x)q, sin (a) dx

Fig. 2. Example pore throat size distribution of a mudstone from the Norwegian Margin (continuous line) with modelled fit using eqn (8).

(15)

4? =

j - JJ

v,

(17)

167

Y. Yang, A.C. AplinjMarine and Petroleum Geology 15 (1998) 163-175

Substituting eqns (15) and (A. 1) into eqn (17) and noting that I = i* sin(a), we obtain : 9Y =

9 V+i( sin (a))*

Jf

161

J2( 1+ J, + Jf>’

(18)

s hai

f(x)[exp(x)+0.5]dx -co

and since :

Kc411

e=

elOO-filn

& (

(23) )

where e is the void ratio of the sediment at effective stress CJ’,elOOis the void ratio at effective stress 100 kPa and D is the compaction coefficient of the sediment. Beta and elOOvary with lithology, increasing in more clay-rich, finer-grained mudstones (Skempton, 1970 ; Burland, 1990; Aplin et al., 1995). Our unpublished data suggest that elOO is related to clay particle ( < 2 pm) content as follows :

Aip

K, = 9*10-‘“&sin(a))* I 16

elOO= 0.71 +0.563*clay+2.308*clay2

J: J2U

f(x)[ exp (x) + 0.51dx -‘x

+J,

(20) +w

(m’)

where V and A are, respectively, the unit volume and cross sectional area of the sediment and K, is the absolute permeability of the sediment along the direction perpendicular to the bedding. The pore throat radius is in nanometres. Similarly the absolute permeability along the bedding direction, K, is : K = 9*10-‘“#(cos(a))’

x

16

J? J*(l +J, +J:)’

(21)

%ax

Jp

j(x)[ exp (x) + 0.51dx --5

With increasing compaction and loss of porosity, we anticipate that J, will decrease and J2 will increase, since in clay-rich sediments it is likely that pores will flatten but not decrease in length (increased aspect ratio) and that the widest part of the pore will compact more than the pore throat. We assume here (see Appendix B for derivation) that : for 0’ > 100 kPa (25) J2 = J,

foro’ < 100 kPa

Since /?ln(a’/lOO) is equal to eloo- e [eqn (23)], eqn (25) can be rewritten as : J2-J,

(m’)

= 10.24(e,,,-e)

for g’ > 100 kPa

J2 = J,

In eqns (20) and (21) permeability is estimated in terms of porosity, the average alignment of pores, the pore shape and the pore throat size. The modelled permeability is also predicted to be directly related to the mean pore throat size [see eqn (1 l)]. Using a numerical simulation of flow in a random, three dimensional capillary network, O’Carroll and Sorbie (1993) also concluded that permeability is a function of mean tube size (in this case mean was defined by number). The permeability anisotropy, yk is the ratio of K-KY :

(22) 2.2.5. Relationship between pore shape, pore angle and effective stress Use of the permeability model requires a description of the way that pore shape (i.e., J, and JJ and pore alignment (i.e. a) change with increasing consolidation. Our approach is to assume that the rate at which J,, J2 and CLchange with effective stress is described by similar equations to those which describe the loss of porosity with increasing effective stress for normal consolidation (Skempton, 1970 ; Craig, 1995) :

(24)

for 6’ < 100 kPa

(26)

Equation (26) is preferred to eqn (25) since it does not involve the evaluation of effective stress. The equations show that the pore shape flattens as the void ratio of the mud decreases and also, since fl and elM)are strong functions of lithology (Burland, 1990), that the change in shape is related to lithology. We also assume that the permeability anisotropy, as measured by the angle a between the long axis of the pore and the bedding plane, increases with increasing effective stress in a manner similar to that described for porosity. The way in which a decreases as a function of effective stress and lithology (/I) is described as (see Appendix B) : a =

45”-10.24.Bln

&j (

= 45”-10.24”~(e,,,-e) 1 for cr’> 100 kPa

a = 45”

for (r’d 100 kPa (27)

Using the model described above, the increase in calculated permeability anisotropy with increasing effective

Y. Yung, A.C. AplinlMarine and Petroleum Geology 15 (1998) 163-175

168

161

lE-17 r

I

/A

%lE-18 y" $lE-19 5 z $lE-20

0

10

20

30

40

Effective stress MPa Fig. 3 Permeability anisotropy (yk) calculated with eqn (22) and assuming the average inclination of pores (u) to decrease with effective stress according to the expression in eqn (27). Each curve defines the anisotropy of mudstones with different compressibility coefficients (/I). Finer-grained muds have higher fi values.

stress is shown in Fig. 3. The probable importance of lithology on the anisotropy of permeability is clearly seen, with only minor increases in coarser, siltier muds but very striking increases in finer grained muds. Currently there are no directional permeability data with which to validate the proposed model. However, ratios of horizontal to vertical permeability between one and three have been measured in clays subjected to relatively low levels of stress (Lumb and Holt 1968 ; Larsson 1981 ; Tavenas et al., 1983a; Al-Tabbaa and Wood 1987; Chandler et al., 1990; Leroueil et al., 1990). The formats of eqns (26) and (27) need to be verified by the acquisition of a large dataset of mudstone permeability. Here, however, the assumptions we have made concerning the evolution of J,, J2 and CIwith increasing effective stress now allow us to calculate, using eqns (A.3) (20) (21) and (25) respectively, the values of Jr, J2, K,., and K, for mudstones for which there are specific surface area and pore throat size distribution data. 2.3. Comparison qf modelled and measured permeabilities Thorough validation of the permeability model requires the acquisition of permeability data for well characterised mudstones with known pore throat size distributions, specific surface area and clay particle content. One of the few available datasets is that of Dewhurst et al. (1998) for the London Clay. Dewhurst et al. measured the vertical permeability and corresponding pore throat size distribution of two samples of London Clay compacted in the laboratory to a range of effective stresses between 1.5 and 33 MPa. We have used seven of Dewhurst et al.‘s published permeabilities and pore throat size distributions as an initial test of the model presented here. Dewhurst et al.‘s permeabilities include some directly measured values and some which

lE-21v lE-21

'

' lE-20

I

I

I

IE-19

IE-18

lE-17

Modelled Ky m2 Fig. 4. Comparison between measured and modelled vertical permeability of experimentally compacted samples of London Clay (measured permeabihties from Dewhurst et al. (1998).

were estimated from consolidation tests. The latter are systematically greater than the directly measured permeabilities and we have corrected these permeability estimates back to the values which can be inferred from the trend of the directly measured permeabilities. Figure 4 shows that the modelled and measured permeabilities agree to within a factor of three with no systematic error as a function of lithology, porosity or effective stress. These results are very encouraging and give us the confidence in the next section to use the model to estimate the permeability of some natural mudstones. Nevertheless, more validation data are required to confirm the model’s general applicability. 3. Application to Norwegian margin mudstones 3.1. Samples and method&

Eleven samples of picked cuttings and sidewall cores were taken from Tertiary, Cretaceous and Jurassic formations from two wells on the Norwegian Margin (Table 1). The samples are a subset of those studied by Aplin et al. (1995). The samples were chosen in order to investigate the influence of both effective stress and lithology on pore size distribution and permeability. Two selections were made: (1) a lithologically homogeneous series of mudstones with 50-55% clay particles (~2 pm), buried to different levels of stress and (2) a series of lithologically distinct mudstones buried to a similar level of stress. Most of the analytical methods have been described elsewhere (Aplin et al., 1995; Yang and Aplin, 1997). Briefly, samples were gently disaggregated using a freezethaw technique. Grain densities were measured by the small pycnometer method at 20°C and grain size distributions between 0.1 and 80 pm were determined by a combination of the sedimentation pipette method (British Standards, 1377, part 6, 1990) and a laser scattering par-

Y. Yang, A.C. AplinjMarine and Petroleum Geo1og.v IS (1998) 163-175

Table 1 Samples Well

Depth (m)

B/3-2

867

813-2

885

8/3-Z 1607 813-2 1744 8/3-2 1853 S/3-2 2055 813-2 2284 30/2-t 855 30/2- 1 2060 30/2-l 2645 30/2-l 3605

Series

Group

Lower-middle Eocene Lower-middle Eocene Aptian Hauterverian Hauterverian Ryazanian Oxfordian Oligocene Palaeocene

Hordaland

Oxfordian

Formation

Hordaland Cromer Knoll Cromer Knoll Cromer Knoll Bokntjord Bokntjord Hordaland Lista Shetland Brent

Sola Valhall Valhall Sauda Egersund Lista Ness

title sizer (Malvern Mastersizer). Specific surface area was measured on intact, freeze-dried samples using a Micromeritics Flowsorb 2300 surface area analyser (e.g. Ross and Olivier, 1964 ; Gregg and Sing, 1967). Absolute surface areas are recorded from both the adsorption and desorption of nitrogen, although the surface area data reported here are taken from the desorption readings. Repeat analyses were made and the level of repeatability was observed to be within ca 2%. The mineralogy of the samples was determined semiquantititively by X-ray diffraction analysis of whole sediments, spiked with 10% aluminium. The size distribution of pore throats was measured with a mercury porosimeter, assuming the surface tension of mercury to be 0.48 N/m and the contact angle between mercury and the particle surface to be 141’ (Heling, 1970 ; Lapierre et al., 1990). In this case pore radius r is given by : 746,000 r=----P

(28)

where r is the pore throat radius (nm) andp is the pressure in kPa. The distributions were corrected for artefacts which may arise during sample preparation, potentially resulting in the formation of large cracks. Such cracks are observed on pore size frequency distributions as a small, second peak at large pore radii. The second mode is rejected and the true distribution is assumed to end at the turning point between the two peaks. The in situ effective stress 0’ was calculated using the density log and the most realistic estimate of pore fluid pressure (e.g. measured in adjacent sands or from drilling mud weight). 3.2. Results and discussion The TOC contents of the samples range from 0.694.69%. The samples are mineralogically similar, pri-

169

marily comprising mixed-layer illite-smectite, illite-mica, kaolinite and quartz with minor pyrite and feldspar (Table 2). Two samples contain minor (< 10%) calcite. 3.2.1. Pore throat size distributions Between 16 and 28 MPa in situ effective stress, the porosity of the mudstones with 50-55% clay decreases from 27%-12% (Fig. 5). The mean pore throat radius also declines, from around 3&8nm (Fig. 6). The pore throat size distributions of muds with 50-55% clay are shown in Fig. 7. The cumulative pore throat size distributions indicate that there are very few pores larger than 100 nm and that most of the porosity loss occurs through the collapse of pores larger than 15 nm. Porosity loss by the collapse of larger pores has been frequently observed in muds which have been experimentally compacted at lower stress levels (Delage and Lefebvre, 1984 ; Delage, 1987 ; Lapierre et al., 1990 ; Griffiths and Joshi, 1990; Vasseur et al., 1995; Dewhurst et al., 1998). The same process of porosity loss appears to continue at higher levels of effective stress. The pore throat size distributions of two siltier samples (3640% clay particles) are shown in Fig. 8. The distribution of the shallower sample is much less well sorted than the deeper samples, and its mean pore radius is an order of magnitude larger (Fig. 6). The deeper sample has a surprisingly low porosity and has a mean pore radius which is similar to those of more clay-rich samples (Fig. 6). The influence of lithology, represented by the clay particle (% particles < 2 pm diameter) content, on porosity and pore throat size distributions is explored in Fig. 9, which displays pore throat size data for three lithologically different samples which have been buried to similar levels of in situ effective stress (6.47.4 MPa). The distributions are strikingly different. The siltier sample has a much higher porosity, a less well sorted pore throat size distribution and a mean pore radius which is almost two orders of magnitude greater than that of the finest sample (Fig. 6; Table 2). Eighty percent of the pores in the most clay-rich sample are smaller than 20 nm compared to only 8% for the most clay-poor sample. The range of mean pore throat radius displayed by these three samples alone is similar to that reported by Borst (1982) for a much larger dataset, suggesting the importance of detailed mudstone lithology as an important control on pore throat size distributions. 3.2.2. Perrneabilitl Vertical permeabilities calculated using equation (20) are plotted against porosity in Fig. 10 and against in situ effective stress in Fig. 11. Calculated vertical permeabilities range between 3.3 x lo-I9 and 1.2 x lo-*’ m*, decreasing logarithmically with porosity. This range is within that reported by Neuzil (1994) for permeabilities determined both experimentally and inferred for field

Table Sample

2 data

Well Depth (m) Effective stress (MPa) % Clay’ Grain density Specific surface area (m’g-‘) Void ratio at 100 kPa effective stress (elm) % TOC Mineralogy

Porosity Mean pore throat sire (nm) Pore throat size standard deviation (nm) K-C permeability (m*)’ Vertical permeability (m’)’ Horizontal permeability (m’)’ Porosity contributing 90% permeability (X)

_ 813-2 861 1.2 60.1 2.53 24.0 1.788

813-2 885 7.4 72.2 2.69 48.7 2.245

4.69 I/S : l/M ; Q:K

I/S ; I/M

0.294 53 108 1.24x IO-‘” I .69 x IO- ‘I’ 4.7 x lo-‘” 41

8/3-2 1607 15.9 51.8 2.81 25.6 1.527

1.55

;

Q;K 0.286 8 11 2.65 x ION-‘” 1.8 x IO ?’ 7.6x 10 ” 58

’Percentage of particles with diameter smaller than two microns. 2Kozeny-Carman permeability. ‘Modelled permeability using equations in this paper. Mineralogy in decreasing order of abundance: I/S = Illite-Smectite;

1.04 I/S ; I/M Q:C:P; KF:PF 0.267 27 46

:

7.4 x 10 1o 1.38 x IO-‘” 3.26 x IO “I 57

I:M = Ilhte/Mica.

Q =

813-2 1744 17.5 54.2 2.75 19.4 1.599

813-2 1853 18.7 52.4 2.84 22.0 1.545

813-2 2055 21.0 55.4 2.69 20.7 1.636

1.01 I/S; I/M; Q;K;Ch: KF:C 0.201 19 21

0.8 I/M : US ; Q;K:Ch P; KF 0.201 15 30

2.59 I/M ; l/S ; Q:K:Ch; P;KF 0.171 33 100

4.6 x IO-”

3.6x IO-‘"

5.6x IO-"

4.8x 10 ” 1.27 x IO-‘” 53

1.52 x IO-“’ 59

Quartz;

K = Kaohnite,

8/3-2 2284 23.4 39.8 2.60 6.9 I .206

2.3 x 10~“’ 7.8 x IO-” 2.27 x lo-:” 37

Ch = Chlorite;

3.45

0.054 15 31 5.0x lomz’ 1.6x10 ” 3.7 x 10 I’ 57

KF = K feldspar:

30/2-l 855 6.4 36.8 2.55 7.0 1.136

30/2- 1 2060 16.4 50.7 2.15 16.3 I.495

0.85 Q; I/S: I/M. K: KF:P 0.478 452 1425

1.1 I/S; Q; I/M;K;C PF;KF 0.262 35 58

1.12x10 ” 3.3 x IO_‘” 3.8 x 10 ” 31

PF = Plagioclase

30/2-l

30/2-l

2645

3605

20.5 51.8 2.74 24.3 1.527 0.69 I/S; Q; l/M;K. PF;Ch:KF 0.193 15 12

1.7x lo-” 1.41 x IO_?” 3.25 x IO “’ 51

Feldspar:

P = Pyrite;

2 6x IO-“’ 4.7 x 10 ?’ 1.23x lo-‘” 67

C = CaIclte

27.5 52.9 2 82 17.0 1.560 2.52

0.114 8 9 9.0x IO 71 1.2 x IO_” 3.6x 10 ” 66

Y. Yang, A.C. Aplinlktarine and Petroleum Geology 15 (1998) 163-175

171

0.5

0.30

E 'Z 0.4 ,o

%

~ 0.25 .% $ 0.20 s a 015

0

??

g 0.3 F

0

5

0

0.10

I

I

I

15

20

25

,

0.2

E s 0.1 0

0

0

,

,

,

,

,

*

,

,

/

/

t

/

,

,

C-------

,#

I

10

OIL

30

10

In situ effective stress MPa

1000

100

10000

100000

Pore throat radius nm

Fig. 5. Porosity vs effective stress for samples containing SO-55% < 2 pm particles.

Fig. 8. Cumulative porosity distributions of mudstones with 36 40% < 2 pm particles. Numbers refer to in situ (maximum) effective stress of sampies (MPa).

c -g

0.4

, 8’

b a 0.3 F

‘)

,

,

/#

L--

,__---I

---

_j 'S 0.2 E 3 0.1 0

fi

r”

1 01

I

I

I

I

5

10

15

20

I

25

0’

30

1

I

I

10

100

In situ effective stress MPa

I

1000

I

10000

lot000

Pore throat radius nm

Fig. 6. Mean pore throat radius of mudstones as a function of effective stress. Percentages are the < 2 pm contents of the samples.

Fig. 9. Cumulative porosity distributions of three hthologically distinct mudstones, buried to an effective stress of about 7 MPa. Numbers refer to % < 2 pm particles.

lE-18

0.30 r

r-l

r37-40%

A

??50-55%

@PE-19 L-l 0 60% * 73%

G

1 E-20

1 Ol

I 10

I

100

I 1000

I

10000

Pore throat radius nm Fig. 7. Cumulative porosity distributions of mudstones with 56 55% < 2 pm particles. Numbers refer to in situ effective stress of samples (MPa).

lE-21;

o1

0.2

0.3

0.4

0.5

Porosity Fig. 10. Vertical permeability, calculated using eqn (20), as a function of porosity. Samples with differing percentages of < 2 pm particles shown by different symbols.

112

Y. Yang, A.C. AplinlMarine and Petroleum Geology 15 (1998)

lE-18

size contribute to both permeability and surface area. Figures 12-l 4 show cumulative distributions of porosity, permeability and surface area for three lithologically different mudstones, as a function of pore throat size.

A

C,IlE-19 E 2

163-175

lE-20

4.OE-19

0.5

-12

.-,-‘-‘-‘I “E

3.0E-1g

:,J-

I

.’

.’

- 0.4

??

lE-21;

5

,.

15

20

25

30 I

x” al 2.0619

/

.z JS 2

- 0.3

i l.OE-19

- 0.2

i

settings by inverse analysis of pressure or flow data. The permeability range at a single porosity is about a factor of five for the 5&55% clay samples, compared to a published range of three orders of magnitude for mudstones in general (Neuzil, 1994). This implies that the porosity : permeability relationships of specific mudstone lithologies, defined for example by grain size, may be more predictable and less variable than for mudstones in general. There is some evidence that the two siltier samples have a slightly higher permeability than the finer samples at a given porosity. The permeabilities of the 5(r55% clay samples decrease fogarithmicalfy with increasing effective stress (Fig. 11). The siltier sample at 23 MPa falls on the same trend as the finer samples but the more shallow buried, siltier sample lies above the trend defined by the 5&55% clay samples. This very limited dataset is consistent with that of Dewhurst et al. (1998) who showed experimentally that the rate of loss of permeability with effective stress was greater in more coarse grained muds than in finer grained muds. Figure 11 also illustrates the important influence of detailed lithology on the permeability of mudstones at a specific level of effective stress. The modelled permeability of three mudstones, buried to about 7 MPa effective stress but with a range of clay particle contents between 36 and 73%, varies by more than two orders of magnitude. The differences are due to variations in both porosity and pore throat size distributions and reinforce the key point that the rate at which both porosity and permeability are lost during compaction is strongly influenced by detailed mudstone lithology (Aplin et al., 1995). These data cast serious doubt over the use in basin models of simple equations which use a single coefficient to describe the relationships between porosity, effective stress and permeability for all mudstones. The pore shape and permeability models described earlier can be used to determine how pores of different

0.1 ;

O.OE+OO

0.1

$ -8

g ‘s =

- 4

1

10 Pore

100 throat

1000 radius

10000

_ 0

l&

.z 9 3m

i

;

i

E rJY

a

In situ effective stress MPa Fig. I I, Vertical permeability, calculated using eqn (20) as a function of effective stress. Samples with differing percentages of < 2 nrn particles shown by different symbols.

I

,g

E ;

nm

Fig. 12. Cumulative specific surface area (dot&dash line), porosity (thin line) and permeability (thick line) of a mudstone with 36.8% clay particles buried to effective stress = 6.4 MPd (30/2-l. 855 m). Most of the permeability is contained within the largest pores whereas most of the surface area resides within the smallest pores.

Pore throat

radius nm

Fig. 13. Cumulative specific surface area (dot-dash line), porosity (thin line) and permeability (thick line) of a mudstone with 72.2% clay particles buried to effective stress = 7.4 MPa (g/3-2, 885 m). Most of the permeability is contained within the largest pores whereas most of the surface area resides within the smallest pores.

LOE-21

I @k 4.OE-21 $‘3.OE-21

0.20 - 0.15 -

$ p 3 5

- 0.10 2.OE-21

-

l.OE-21

-

.’ /

- 0.05

0 I

O.OE+OO 0.1

I 1 Pore

10 throat

100 radius

100:

nm

Fig. 14. Cumulative specific surface area (dot-dash line), porosity (thin line) and permeability (thick line) of a mudstone with 51.8% clay particles buried to effective stress = 20.5 MPa (30/2-l, 2645 m). Most of the permeability is contained within the largest pores whereas most of the surface area resides within the smallest pores.

Y. Yang, A.C. AplinlMarine and Petroleum Geology I5 (1998) 163-175

Independent of either effective stress or lithology, the vast majority of surface area is contributed by pores less than 10 nm in radius ; much of the surface area resides in pores smaller than those measured by our mercury injection technique (4 nm radius). In contrast, most of the permeability is contributed by the relatively small number of large pores which have a low specific surface area ; this follows inevitably from the Hagen-Poiseuille equation which shows that flow rate is proportional to the fourth power of pore radius [eqn (5) ; see also Olsen, 1962 and David, 19931. The relative importance to permeability of the largest pores depends on both lithology and effective stress. For the sample with 37% clay particles at 7 MPa effective stress, 50% of the permeability resides in pores which only contribute 1% of the surface area, and 90% of the permeability resides in pores which make up 2% of the surface area (Fig. 12). The equivalent numbers for the sample with 72% clay particles at the same effective stress are 2 and 20% (Fig. 13) and the values for the 52% clay sample buried to 20.5 MPa are 6 and 23%, respectively (Fig. 14). The data in Table 2 suggest that for most mudstones (i.e. < 60% clay size particles) buried to a kilometre or less, 90% of the permeability will be contributed by 3&40% of pores, except in exceptionally fine-grained muds. At greater depths (> 15 MPa effective stress ; I .5-2 km burial depth), 90% of the permeability is contributed by 50-70% of the pores, for both the 5& 55% clay samples and the single, siltier sample. The specific permeabilities perpendicular to and parallel to the bedding direction of the Norwegian Margin samples (k, and k,, respectively), calculated from the model proposed in this paper are given in Table 2 along with the specific permeabilities calculated from the modified Kozeny-Carman equation. In the absence of known shape and tortuosity factors, we have assumed that the product of k. and k, to be 100, the maximum value reported by Cornell and Katz (1953) and Dolch (1959) for limestones. Despite the high values assumed for tortuosity and shape, the vertical permeabilities predicted by our model are three to seventy times smaller than those predicted from the Kozeny-Carman equation. Horizontal permeabilities calculated from the mode1 are between one and thirty times smaller than the KozenyCarman permeabilities. The discrepancies between the calculated permeabilities highlight the uncertainties of assigning accurate permeability values to mudstones using more easily measured parameters such as porosity and specific surface area. Certainly, the disparity between those pores contributing permeability and those which contribute surface area must force an evaluation of the applicability of the KozenyCarman equation as a way of estimating mudstone permeability, since it includes specific surface area as a key variable. Furthermore, the Kozeny-Carman equation assumes that all pores are capillary tubes with identical cross sectional areas, whereas pore throat radii

173

in mudstones typically vary by two orders of magnitude (Figs 7-9 ; see also Olsen, 1962). 4. Concluding remarks We have developed a mode1 which estimates the permeability of mudstones based on measured pore throat size distributions plus assumed pore alignments and pore shapes. Although the model must be checked and calibrated against permeabilities measured on well characterised mudstones, it does predict the permeability of a series of experimentally compacted muds to within a factor of three. The mode1 outputs imply that changes in mudstone permeability can be understood in terms of the way that pore throat size distributions change with increasing consolidation. The pore throat size distributions of the deeply buried (> 800 m) mudstones presented here are influenced both by consolidation and detailed lithology. With increasing effective stress, porosity is lost by the collapse of relatively large pores, resulting in a decrease in the mean pore throat size. The pore throat size distribution of mudstones at a specific effective stress appears to vary with lithology ; of three samples buried to - 7 MPa, the siltiest mudstone has the highest porosity and much the largest pores. Vertical permeabilities calculated using the model developed here range from 3.3 x lo-l9 and 1.2 x 10m2’m2, three to seventy times lower than those calculated using the KozenyCarman equation. Published data (e.g. Neuzil, 1994) show that the permeability of mudstones is only loosely related to porosity. Our model data indicate that the observed variability can be at least partly explained by variations in detailed lithology, as described for example by the % < 2 pm particles in the mud. Permeability is thus a function of both porosity and detailed lithology. The restricted data presented here suggest that the permeability of lithologically similar mudstones decreases logarithmically with decreasing porosity. Similarly, there is no unique relationship between mudstone permeability and effective stress ; three lithologically distinct mudstones at a similar level of effective stress have permeabilities which vary by over two orders of magnitude, In a previous paper (Aplin et al., 1995) we showed that lithology exerts a major influence on the rate at which mudstones compact with effective stress; that is, the relationship between porosity and effective stress. It appears that lithology is also a highly influential control on the relationship between mudstone permeability and porosity. The use in basin modelling of single (‘default’) constants describing relationships between (1) porosity and effective stress, and (2) porosity and permeability in mudstones is an oversimplification and is likely to yield major errors in modelled calculations relating to fluid flow and pressure development in sedimentary basins.

174

Y. Yang, A.C. AplinlMarine and Petroleum Geology 15 (1998) 163-175

Acknowledgements

Much of this work was supported by the IBS project (Interdisciplinary Basin Studies), part of the Joule research programme funded by the Commission of the European Communities (contract no. JOU2-CT-920110). YY has recently been supported by NERC ROPA award number GR3/R95 16. Statoil generously donated samples and ancillary data. We thank journal reviewer Ben Clennell for his detailed and thoughtful comments of the manuscript, Dave Dewhurst for his comments on an earlier version of the manuscript, and also Ken Sorbie and Steve MacDougall for their suggestions on the derivation of flow rate in a pore.

Appendix A : Volume and surface area of pore

The volume V, and surface area Si of a pore with the shape defined in Fig. 1 are : = +cJ2(l +J, +J:)r3

V, = +cL(R2+rR+r2)

(A.1)

which J, and J2 change can be described by analagous equations to those relating the decrease in porosity with effective stress :

for c’ > 100 kPa J, = Jz = C,,

foro’ < 100 kPa (B.1)

where C,, A, and B are positive coefficients. In this way J, and J, are related to effective stress and lithology, which is represented in eqn (23) by the compaction coefficient b (Aplin et al., 1995). J, and J, are related as :

J2-J,

= (B+AMln(&j)

WI

By assuming arbitrarily that J2-J, = 30 when 0’ = 35,000 kPa and 0 = 0.5, we get: A+B

= 10.24

(B.3)

and and thus : S, = ~~c(Y+R)JL~+(R-~)~ = 2n(l+ J,)dm’r’

(A.2)

If V is the volume of a specimen and 4 its porosity, the total volume of all pores with radius between r and r + dr is VqZf(x)dx and the number of pores in this volume is V@f(x)dx/ Vi.The mass of the solid phase of the specimen is V(1 - 4)G,, where f(x) is the pore throat size distribution density function as specified in eqn (10) and G,, the grain density (g/cm’). The specific surface area is related to the pore throat size distribution as : ~),~~,~ Vb?f(x)S, d y s,=

s

vi

.

-x W - 4)G.t (1 + J,)(,/m

30004 = (1 -$)G\

= 10.24pln

& (

for rr’> 100 kPa (B.4)

1

J2 = J,

for (T’d 100 kPa

Since ,!IIn (o’/lOO) is equal to elO,,- e [eqn (23)], eqn (B.4) can be rewritten as : J2-J,

= 10.24(e*-e)

fora’>

Jz = J,

100kPa

for r? < 100 kPa

(B.5)

We also assume that the permeability anisotropy, as measured by the angle CIbetween the long axis of the pore and the bedding plane, increases with increasing effective stress in a manner similar to that described for porosity. Assuming that at < 100 kPa the pores have no preferred alignment (i.e. 51= 45”) a decreases as a function of effective stress and lithology (p) as :

Jz(l +J, +J:)

s ‘m,.,

X

J2-J,

J’W

_~[exp(x)+0.5]dx(m*‘s)

cr=45”-C,/3ln

Appendix B : Evolution of pore shape and pore angle during burial

We anticipate that during burial, mudstone pores will flatten but not decrease in length with increasing effective stress. In modelling this process, we have chosen to assume firstly that when the effective stress is less than 100 kPa, R = L or J, = J2 and secondly that the way in

CI= 45”

for 0’ > 100 kPa

&, (

(A.3)

(B.6)

1

for c’ < 100 kPa

Assuming arbitrarily that CI.= 15” when 0’ = 35,000 kPa and p = 0.5, we have: = 45”- 10.24”*(e,,,-e) for 0’ > 100 kPa for c’ d 100 kPa (B.7)

Y. Yang, A.C. AplinlMarine and Petroleum Geology 15 (1998) 163-175

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for the disaggregation

of