Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity

TECTO-126250; No of Pages 14 Tectonophysics xxx (2014) xxx–xxx

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Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity Philippe Robion a,⁎, Christian David a, Jérémie Dautriat b, Jean-Christian Colombier a, Louis Zinsmeister c, Pierre-Yves Collin d a

Université de Cergy-Pontoise, Géosciences & Environnement Cergy, 5 mail Gay-Lussac, F-95031 Cergy-Pontoise, France CSIRO Earth Science and Resource Engineering, 26 Dick Perry Avenue, Kensington 6151 Western Australia, Australia IFP Energies nouvelles, Direction Ingénierie de Réservoir, 1 et 4 avenue de bois préau, 92852 Rueil-Malmaison Cedex, France d Université de Bourgogne, UMR CNRS 6282 Biogéosciences, Bâtiment Sciences Gabriel, 6 Bd Gabriel, 21000 Dijon, France b c

a r t i c l e

i n f o

Article history: Received 5 October 2013 Received in revised form 6 March 2014 Accepted 30 March 2014 Available online xxxx Keywords: Anisotropy Porosity Ferrofluid P-wave-velocity Carbonate Sandstone

a b s t r a c t The ferrofluid impregnation technique combined with anisotropy of magnetic susceptibility measurements (AMSff) is one of the ways to analyze the 3-D geometry of the pore space in a rock and indirectly to infer the anisotropy of permeability. We applied this method on different types of rocks (sandstones and carbonates) with a range of different porosity values (10–30%) and permeability (1 mD to 1 D). To get additional information on both the pore aspect ratio and the directional anisotropy we used another technique, measuring the anisotropy of P-waves velocity (APV) in dry and water saturated conditions. Comparing between both methods shows that despite the good agreement in directional data, inferring the true shape of the porosity is not straightforward. Modeling the presence of an elastic anisotropy in the solid matrix for sandstones allows one to get more consistent values for the pore aspect ratio obtained from both APV and AMSff. However for the carbonate rocks, due to an intricate distribution of microstructures, the aspect ratios obtained show significant discrepancies between the two methods. The ferrofluid method is very sensitive to the quality of the impregnation and suffers from a major drawback which is the threshold size of investigation, limited by the size of the magnetite nanoparticles (10 nm) and probably this method doesn't see all the porosity. On the other hand with acoustic methods, the range of porosity investigated is probably larger but several microstructural attributes can contribute to the elastic anisotropy which makes the pore shape effect more difficult to decipher. Therefore, we promote the combined use of both methods in order to get more reliable information on the pore shape in porous media. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Since a long time, it has been recognized that no unique and simple relationship exists between porosity and permeability although intuitively one would expect a correlation between these properties. Indeed, porosity being a scalar related to the volumetric fraction of voids does not carry any information on the pore space geometry and consequently any prediction of the permeability would require rock microstructure description (Carman, 1956; Guéguen and Palciauskas, 1994; Scheidegger, 1974). Furthermore, such prediction remains difficult due to the intrinsic rock heterogeneity observed at different scales, which may also induce anisotropic rock properties. Among the microstructural attributes responsible for fluid flow properties, such as grains preferred orientation or microcracks distribution along with multi-scale porosity distribution, e.g. dual porosity (Dautriat et al., 2011a), overall pore shape anisotropy may directly affect preferential flow directions. Based on this simple premise and given the complexity and variability of sedimentary rock microstructures, several petrophysical methods have been developed to ⁎ Corresponding author.

assess permeability anisotropy from a preferential shape of the porous network. Among them, a first approach is based on determination of dynamic elastic properties anisotropy inferred from acoustic velocity measurements. Different methods have been proposed based on the investigation of changes in ultrasonic velocity (P and-or S waves) induced by confining pressures (Rasolofosaon et al., 2000) and/or by different saturation states (Benson, 2004; Benson et al., 2003; Louis et al., 2003, 2004). The relationship between elastic property and pore shape anisotropy and permeability anisotropy is however not direct. A first drawback relates to a fourth order tensor description of the elasticity tensor (Rasolofosaon et al., 2000) whereas the permeability tensor is described by a second order symmetric tensor (Pfleiderer and Halls, 1994). Therefore, some simplifications on the shape of the ultrasonic velocity anisotropy are required; the latter is often reduced to a high order symmetry (elliptic, transverse isotropic or orthotropic symmetries). In addition, ultrasonic velocities are dependent on more microstructural parameters than permeability, including both compliant and non-compliant pores, connected and non-connected porosity and finally the solid matrix of the rock (Almqvist et al., 2011).

http://dx.doi.org/10.1016/j.tecto.2014.03.029 0040-1951/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

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P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

The second approach is based on the measurement of the anisotropy of magnetic susceptibility in rocks saturated with ferrofluid, following the pioneering work of Pfleiderer and Halls (1990). The ferrofluid is a colloidal liquid made of nanoparticles of magnetite suspended in a carrier fluid. The magnetic anisotropy of a ferrofluid-saturated rock will then mimic the anisotropy of the pore space. Indeed due to the very high intrinsic susceptibility of magnetite particles and their intrinsic isotropy, the anisotropy of magnetic susceptibility (AMS) of a pore filled with ferrofluid directly reflects the pore orientation (long axis vs. short axis). It has been shown that the magnetic susceptibility data requires a correction by a demagnetization factor because the shape of the AMS tensor is biased by demagnetization effects (Uyeda et al., 1963). By introducing the concept of equivalent pore to characterize the actual pore shape, Hrouda et al. (2000) proposed a correction, which have been extended by Jones et al. (2006) for different kinds of porosity shape. Consequently, the equivalent elliptical pore shape obtained from AMS measurements (Stacey, 1960) needs to be considered as an approximation of the actual shape of the porosity, which probably is much more complex. Provided that the mean magnetic susceptibility of a ferrofluid-impregnated sample is high enough to hide the signal of the non impregnated rock (which generally is several orders of magnitude less susceptible), the AMS of ferrofluid-saturated sample can be directly linked to the pore space. There is however a limitation of the method as the magnetite nanoparticles are restricted to invade only the pore fraction with throat size larger than the nominal diameter of the magnetic particles, which is 10 nm. In this paper we propose to investigate the efficiency of these two different approaches to estimate the pore shape anisotropy of sedimentary rocks. Ferrofluid impregnation is performed following the original protocol proposed by Pfleiderer and Halls (1990) that consists in a vacuum filling system. The measurements of ultrasonic anisotropy are based on the protocol proposed by Louis et al. (2003, 2004) that aims to measure the velocity difference under dry and water saturated conditions. Two lithology were investigated, carbonates and sandstones, with a large range of porosity (from 10% up to 30%) and permeability (from 1 mD up to 1 D). In the discussion section we compare the directional results obtained with each method and we infer from the P wave velocity anisotropy a mean pore shape to check the validity of the equivalent pore shape given by the ferrofluid impregnation method, using a model of effective elastic media developed by Tsukrov and Kuchanov (2000) including elliptical holes and anisotropic matrix.

2. Sampling and methodology For standardization's sake, we studied different rocks, both carbonates and sandstones, with various microstructures and covering a wide range of porosity and permeability. It must be emphasized here that we limited our study to rocks in which a satisfying ferrofluid impregnation was ensured. To investigate 3D spatial variations of rock properties, cylindrical samples drilled in three orthogonal directions from a single block. This approach was proposed by Louis et al. (2004) to overcome the difficulty of machining spheres from a block and was applied to prepare samples for investigating P-wave velocity anisotropy. 6 blocks were analyzed, including 3 carbonate rocks and 3 sandstones. The three blocks of carbonate rocks were sampled and oriented in the field in formations of Jurassic age in quarries located in the Western part of the Paris Basin. They belong to a carbonate platform environment formed in very shallow water conditions probably along with oolithic barriers. Blocks LVX6 and LVX4 come from Lavoux (France) and are of Lower Callovian age while block GRP1 comes from an oolitic formation of Bathonian age in Les Grippes (near the town of Chauvigny, France). The main constituents of these carbonates are ooids with very few dispersed bioclasts (Brachiopods and lamellibranches) cemented by sparite. Permeability and water porosity were measured by Zinsmeister

(2009). Permeability of LVX4 and LVX6 blocks are respectively 54 mD and 38 mD, significantly higher than that of GRP1 (1.4 mD). The three sandstone blocks were provided by Kocurek Industries (http://www.kocurekindustries.com). Two blocks came from quarries located in the vicinity of Boise (Southern Idaho, USA). They correspond to the Idaho Boise Gray and Idaho Boise Brown sandstones, referred to as IBG and IBB respectively, and belong to the Idaho formation of late Tertiary age. The Boise sandstone is considered as a standard rock for many rock mechanics and petroleum reservoir experiments (Wong et al., 1997; Zhang et al., 1990). Kovscek et al. (1995) reported a high porosity of 27% and a permeability of 910 mD in this formation. Prior to sample coring, porosity has been estimated on companion samples at IFPEN using both dry/wet mass measurements and High Pressure Mercury Injection techniques. IBG and IBB have a porosity of almost 27 and 20% respectively, and a permeability derived from Hg capillary pressure curves (Kamath et al., 1992) of 1.7 and 1.1 D respectively. The third selected block is the Castlegate sandstone (CAS), from the Mesaverde formation of Mesozoic age (Utah, USA). DiGiovanni et al. (2007) have shown that the Castlegate sandstone has an average grain size 0.2 mm, is poorly cemented by calcite and has a small clay content. The porosity and estimated permeability are respectively of 26% and 1.1 D. Ingraham et al. (2013) reported a moderate anisotropy of elastic wave velocities that are 10% faster in the direction perpendicular to the bedding than along the bedding plane. 2.1. Anisotropy of ferrofluid saturated rock samples The methodology used to impregnate our rock samples with ferrofluid is based on the protocol developed by Pfleiderer (1992). The first step consists in applying a vacuum down to 1 × 10−4 Pa to the sample. The vacuum equilibrium stage may require more or less time depending on the characteristics of the pore network of the samples. To ensure an efficient invasion of ferrofluid into the porous volume, the vacuum is applied for at least 48 h. Once the vacuum equilibrium is reached, the ferrofluid, a colloidal suspension of superparamagnetic magnetite particles of around 10 nm diameter, is introduced into the chamber and progressively invades the rock pore space. At this stage the vacuum is partly released up to almost 1 × 10−2 Pa. Then the sample is left immersed in these conditions for at least 12 h. The next stage is to measure the AMS of the impregnated sample from which the pore fabric can be estimated after correction of the demagnetization field effect. We call hereafter AMSff the anisotropy of magnetic susceptibility measured on a ferrofluid saturated sample. AMSff is measured with a KLY3S kappabridge designed by Agico Corporation. The specimen is slowly rotated in the bridge and the magnetic susceptibility is measured in 64 directions for three mutually perpendicular positions. The applied field was 80 Am−1. This procedure allows a very precise determination of magnetic susceptibility tensor K, computed with a program supplied with the device (Jelinek, 1978). In our experiments two different ferrofluids (provided by Ferrotec) have been used, EMG507 which is water-based ferrofluid with a density of 1120 kg/m3 and EMG909 which is an oil-based ferrofluid with a density of 1020 kg/m3. The vacuum impregnation method used here (Humbert et al., 2012; Louis et al., 2005; Pfleiderer and Halls, 1994; Pfleiderer and Kissel, 1994), despite its ability to reach theoretically the whole connected porosity, may induce a lack of ferrofluid invasion especially in the center of the sample. At the sample scale, this drawback generates an additional source of anisotropy that can significantly bias the estimation of pore space fabric derived from the method. We systematically checked the quality of the impregnation by reproducing the AMSff on three samples plugged in three orthogonal directions from a same block, assuming that the rock microstructures are homogenous at the scale of the three samples. Once projected in the geographic coordinates, the superimposition of the main anisotropy axes from one sample to the others illustrates the efficiency of the impregnation. Furthermore,

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

each sample has a cylindrical shape that respects a ratio close to 0.86 (14 mm diameter × 12 mm height) which is the standard aspect ratio used for cylinders to avoid demagnetization effects (Porath et al., 1966). This shape ratio involves a demagnetization factor similar to the one of an isotropic sphere. Due to the high intrinsic susceptibility of ferrofluid, about three orders of magnitude higher than the dry sample (magnetic susceptibility of our dry samples range from − 1 × 10− 5 (SI) for carbonates up to 80 × 10− 5 (SI) for IBG block), the AMSff signal largely exceeds that of the solid matrix, and therefore provides a direct measurement of the average effective anisotropy of the pore space. 2.2. Anisotropy of P-wave velocity Anisotropy of P-wave velocity (called APV hereafter) is measured with a fully automated device described by Robion et al. (2012), which has been adapted from the manual measurement scheme proposed by Louis et al. (2004). Following Thomsen's analysis (Thomsen, 1986) for transverse isotropic media which has been generalized for orthorhombic media (Tsvankin, 1997), Louis et al. (2004) concluded that the P-wave anisotropy in a weakly anisotropic rock can generally be approximated by a symmetric, second rank tensor. From this postulation, they proposed a methodology whereby the laboratory data of P-wave velocity in multiple directions are fitted to an ellipsoid, with the directions and magnitudes of the three principal axes corresponding to the eigenvectors and eigenvalues of a symmetric tensor. The samples are cored as for ferrofluid sample preparation from one single block in three mutually orthogonal directions (see also Fig. 4 in Louis et al., 2004) with standard dimensions 25 mm in diameter and 22.5 mm in length. The velocity is measured along the sample diameter for different positions after rotating the sample with an angular shift of 22.5° and 10° for sandstones and carbonates respectively. The complete revolution of 360° is then reduced to 180° (8 and 36 measurements per sample depending on the angular shift) by averaging each equivalent measured direction. Finally, with this scheme, the complete set of 24 (108) measurements on the three orthogonal samples provides properties along 21 (105) independent directions uniformly distributed in three orthogonal planes and 3 duplicate directions. Due to the natural rock heterogeneity of the sampled block, the properties of the three companion samples may slightly differ. To build the tensor as accurately as possible we need to make the assumption that velocities are the same in the duplicate directions. For this purpose, the differences in the velocity along the common directions are minimized by finding the best shifting of the data set thanks to a least square inversion scheme (Louis et al., 2004). 2.3. Water porosity Water porosity was measured by the triple-weight method giving the volume of connected pores (Norme-NF-EN-1936, 2007). Each sample was oven dried at 60 °C for 12 h and then saturated with distilled water. For this experiment, we measured the mass of the dry sample (mdry) after oven drying, the mass of the water saturated sample (msat) and the mass of the saturated sample immersed in water (mimm). Porosity ϕw is then obtained from these three measurements by the simple relationship ϕw = (msat − mdry)/(msat − mimm). 3. Results 3.1. Efficiency of ferrofluid saturation All petrophysical parameters related to ferrofluid saturation, magnetic susceptibility measurements and water porosity are shown in Table 1. In order to evaluate the efficiency of saturation

3

with ferrofluids, the measured ferrofluid porosity is compared to water porosity obtained by the standard triple weight method. The ferrofluid porosity ϕff is obtained by applying the following relationship (Pfleiderer, 1992):

ϕff ¼

msatff −mdry ρff V

ð1Þ

where mdry is the mass of the sample after drying for 12 h in an oven, msatff is the mass of the sample after saturation with ferrofluid, V is the volume of the sample and ρ ff is the density of the ferrofluid. For estimating the volume V we used the results of the triple weight method V = (msat − mimm)/ρw where ρw is the water density. This value is preferred to that obtained with a dial caliper which gives slightly overestimated values. If we compare the estimated volumes, the variations are of the order of 3%. This can be attributed first to the irregularities of the cylinder geometry, especially the sides that are not perfectly parallel, and second to the brittleness of the samples which can induce mass losses and influence the calculated volumes. The ferrofluid porosity tends to show lower values than the water porosity. The differences (Fig. 1, Table 1) are of the order of 3.6% but can reach 7.8%. Reversely, few samples (especially from LVX4 block) show higher ferrofluid porosity values than water porosity ones. It can be explained in different ways: 1) an overestimation of msatff − mdry; 2) an underestimation of V and 3) a decrease in ρff the density of the ferrofluid. However, it seems hardly possible that the density of the ferrofluid decreases during the experiments (because the ferrofluid that bathes the cell is reused) and the opposite effect is rather expected. This aspect has however not been controlled during our experiments. The most plausible explanation is the presence of small holes due to imperfections on the sample surface which apparently increases the mass of ferrofluid with the concomitant effect of reducing the volume measured by the triple weight method. Combining these two effects in equation (1) will result in an increase of ferrofluid porosity. In the more common situations where the ferrofluid porosity is lower, if water porosity is considered as a reference measurement, the lowest porosity values obtained with the ferrofluid could be a gauge of the efficiency of the vacuum impregnation method. Furthermore, the accuracy of the ferrofluid porosity estimation can be improved by applying the triple weight method with ferrofluid as saturating fluid. However, due to the high cost of ferrofluid, this method has not been tested to prevent any waste of the liquid. As two different fluids have been used during handling (EMG909 and EMG507) magnetic susceptibility should be influenced accordingly. Indeed, magnetic susceptibility of a 3.15 cm3 reference volume of EMG909 is about 1.6 times larger than ones of EMG507 (0.55 (SI) and 0.35 (SI) respectively). Fig. 2 shows the mean magnetic susceptibility Km of the saturated samples versus the mass of injected ferrofluid. For samples saturated with EMG909 (carbonates), the magnetic susceptibility increases fairly linearly with porosity. This trend is not so obvious with EMG 507 (sandstone and three samples of carbonate) for which the magnetic susceptibility values are more scattered. The two trends remain however in agreement with the contrast of magnetic susceptibility of the two ferrofluids. EMG909 samples exhibit indeed higher susceptibility than EMG507 ones. Despite the fact that the majority of the samples measured with EMG909 are carbonates, three of them were measured with EMG507 and show a clear decrease of susceptibility in relatively good agreement with the observed trends. The scatter in magnetic susceptibility measured with EMG507 is not related to the mass of ferrofluid injected in the sample which is better correlated to the water porosity. These differences are thus difficult to explain except by handling errors when measuring with the Kappameter. At this stage of the study we do not know the nature of these errors.

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

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P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

Table 1 Results of the ferrofluid impregnation, susceptibility measurements and water/ferrofluid porosity measurements. Type corresponds to the used ferrofluid: EMG 507 is water-based and EMG909 is oil-based (see density values in the text). V is the volume of the sample calculated with height and diameter. Vcalc is the volume of the sample calculated from the triple weight method. Minj is the mass of ferrofluid injected into the sample. ϕff is the ferrofluid porosity (see Eq. (1) in the text), K is the magnetic susceptibility; Pjel and T are eccentricity and shape parameters respectively (Borradaile and Jackson, 2004), ϕw is the water porosity and ρ is the density calculated from triple weight method with standard deviation σ . Ferrofluid impregnation

Magnetic susceptibility measurements

Water porosity measurements

Nom

Type

V (cm3)

Vcalc (cm3)

Minj (g)

Φff

K (SI)

Pjel

T

Φw

σ

ρ (g/cm3)

σ

CASX1 CASX3 CASY1 CASY2 CASZ1 CASZ3 IBBX3 IBBX4 IBBY1 IBBY2 IBBZ1 IBBZ4 IBGX1 IBGX5 IBGY1 IBGY5 IBGZ1 IBGZ5 GRP1_X GRP1_X2 GRP1_Y GRP1_Y2 GRP1_Z2 GRP1_Z22 LVX4_X1 LVX4_X2 LVX4_Y1 LVX4_Y2 LVX4_Z1 LVX4_Z2 LVX6_X1 LVX6_X2 LVX6_Y1 LVX6_Y2 LVX6_Z1 LVX6_Z2

EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG507 EMG909 EMG507 EMG909 EMG507 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909 EMG909

2.059 2.081 2.065 2.020 2.119 2.102 2.218 2.204 2.187 2.431 2.292 2.295 2.122 2.134 2.120 2.182 2.185 2.104 1.785 1.856 1.787 1.842 1.932 1.852 1.710 1.684 1.716 1.721 1.724 1.916 1.722 1.911 1.904 1.859 1.706 1.740

1.941 1.994 1.974 1.954 2.055 2.040 2.084 2.125 2.033 1.955 2.157 2.318 2.133 2.135 2.064 2.132 2.128 2.066 1.731 1.770 1.779 1.812 1.812 1.828 1.567 1.508 1.823 1.647 1.671 1.812 1.676 1.874 1.901 1.930 1.717 1.919

0.426 0.453 0.479 0.450 0.512 0.469 0.444 0.451 0.390 0.418 0.417 0.369 0.565 0.622 0.654 0.640 0.492 0.621 0.188 0.268 0.203 0.200 0.224 0.265 0.317 0.318 0.223 0.265 0.277 0.384 0.362 0.416 0.453 0.408 0.366 0.386

19.6 20.3 21.7 20.6 22.2 20.5 19.0 18.9 17.1 19.1 17.3 14.2 23.7 26.0 28.3 26.8 20.6 26.8 9.7 14.8 10.2 10.8 11.0 14.2 19.8 20.7 12.0 15.8 16.3 20.8 21.2 21.8 23.4 20.7 20.9 19.7

2.809E-02 3.105E-02 2.918E-02 2.978E-02 3.196E-02 3.717E-02 2.802E-02 4.376E-02 2.338E-02 5.335E-02 2.783E-02 3.709E-02 3.937E-02 3.826E-02 3.955E-02 3.659E-02 3.971E-02 4.146E-02 5.603E-03 3.256E-02 7.090E-03 2.313E-02 4.910E-03 3.174E-02 4.101E-02 3.510E-02 2.706E-02 2.953E-02 3.335E-02 3.515E-02 4.585E-02 5.346E-02 6.705E-02 5.136E-02 4.861E-02 4.458E-02

1.0310 1.0140 1.0340 1.0170 1.0100 1.0180 1.0210 1.0130 1.0280 1.0100 1.0150 1.0200 1.0300 1.0240 1.0140 1.0180 1.0320 1.0150 1.0061 1.0101 1.0211 1.0059 1.0027 1.0249 1.0105 1.0146 1.0115 1.0138 1.0180 1.0282 1.0140 1.0060 1.0110 1.0060 1.0100 1.0060

−0.6230 −0.8520 0.0560 −0.5100 −0.2210 −0.6870 0.1210 0.4190 −0.7920 0.8100 −0.0380 0.8820 0.5200 0.1970 0.0070 0.8750 0.8450 0.7940 0.0179 0.8249 −0.7089 0.8494 0.1118 −0.8188 −0.5741 −0.5483 −0.3726 −0.6858 −0.8373 −0.7472 −0.7780 −0.5620 −0.2420 0.5340 −0.8370 −0.3410

25.6

0.311

2.52

0.008

25.8

0.168

2.51

0.008

26.1

0.090

2.50

0.008

21.7

0.290

2.47

0.007

19.2

0.007

2.56

0.007

19.6

0.058

2.56

0.013

28.4

0.331

2.53

0.003

27.5

0.283

2.53

0.008

27.5

0.324

2.50

0.012

13.7

0.180

2.68

0.018

14.2

0.128

2.68

0.036

14.2

1.700

2.70

0.003

12.9

0.173

2.71

0.002

18.0

0.210

2.71

0.001

15.6

1.664

2.70

0.011

23.8

0.786

2.71

0.006

26.4

0.751

2.71

0.008

26.1

0.390

2.70

0.008

3.2. AMSff data Figs. 3 and 4 present the principal axes of magnetic susceptibility plotted in a lower hemisphere stereographic projection, corrected for the bedding-tilt (i.e. bedding pole corresponds to the center of the

stereograms) for sandstones and carbonates respectively. First of all, for each block taken individually, stereographic projection in a common reference indicates that after rotating the magnetic tensors AMSff axes distributions show either slight or no dependence with the coring direction. In the sandstones, the data reproducibility is verified. Among the

35

0.08 CAS IBB IBG GRP LVX4 LVX6

30

EMG507

0.07

EMG909

0.06

25

K([SI])

Φw(%)

0.05 20

0.04 0.03

15

0.02 10 0.01 5

0 5

10

15

20

25

30

35

Φff (%) Fig. 1. Water porosity (ϕw) from triple weight method versus ferrofluid porosity (ϕff) calculated with Equation (1) in the text.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mass (g) Fig. 2. Mass of injected ferrofluid in the samples versus magnetic susceptibility measured with KLY4. Samples in gray (white) correspond to carbonates (sandstones).

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

CAS

N Ferrofluid saturated

nsample= 6 Kfer= +1.13e−02[SI] K1 32.0 / 14.2 K2 123.1 / 4.4 K3 230.1 / 75.1 Pjmean= 1.0167 T mean= −0.3913

5

GRP

N

nsample= 6 Kfer= +9.48e−03[SI] K1 103.8 / 17.6 K2 355.9 / 44.2 K3 209.5 / 40.6 Pjmean= 1.0054 T mean= −0.5982

W

E

W

N

Ferrofluid saturated

nsample= 6 Kfer= +1.23e−02[SI] K1 257.6 / 5.3 K2 347.7 / 1.1 K3 89.3 / 84.6 Pjmean= 1.0128 T mean= 0.5777

LVX4

N

nsample= 6 Kfer= +1.92e−02[SI] K1 94.7 / 78.5 K2 266.7 / 11.3 K3 357.0 / 1.6 Pjmean= 1.0136 T mean= −0.9047

W

E

N

LVX6 Ferrofluid saturated

nsample= 6 Kfer= +2.88e−02[SI] K1 29.6 / 1.9 K2 299.2 / 11.0 K3 129.0 / 78.8 Pjmean= 1.0186 T mean= 0.8635

Ferrofluid-saturated

W

S W IBG

E

S W

S W IBB

Ferrofluid-saturated

nsample= 6 Kfer= +2.86e−02 [SI] K1 331.8 / 78.2 K2 206.0 / 6.9 K3 114.9 / 9.5 Pjmean= 1.0058 T mean= −0.8358

E

S W N Ferrofluid-saturated

W

W

S W Fig. 3. Stereograms of AMS for ferrofluid saturated samples for sandstones. Squares are for maximum magnetic susceptibility (K1), triangles for intermediate (K2) and circle for minimum magnetic susceptibility (K3). Black symbols are the mean directions calculated with Jelinek statistics [Jelinek, 1978 #77] and the corresponding confidence regions.

six independent tensors measured on each block, only one is moved away from the average site generally with an axis permutation. In the carbonates the same consistency is observed except for GRP1 samples. For this latter block, results are less clear-cut probably due to a lower magnetic anisotropy but the fabric remains interpretable. There is no relationship between erratic axis orientations of some samples

E

S W Fig. 4. Stereograms of AMS for ferrofluid saturated samples for carbonate samples. The symbols are similar to those reported in Fig. 3.

identified on the stereographic projections and a possible bias mentioned in the previous section. In our set of measurements, we do not find any particular answer to explain the behavior of nonreproducible samples. Based on these observations, we state that AMSff measurements are not affected by the drilling direction and we consider that the magnetic fabrics obtained with ferrofluid reflect a preferential orientation of the porosity.

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

6

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

For the sandstones, the magnetic fabrics appear to be approximately oblate and relate to the bedding plane. For the CAS block, the fabric is triaxial with the mean minimum direction (Kmin) close to the bedding pole. Both IBG and IBB blocks show a girdle distribution with maximum axes (Kmax) and intermediate axes (Kint) distributed along a plane close to the bedding plane. For the carbonate, the magnetic fabrics are less well defined as shown by wider confidence ellipses and more scattered data. For the three blocks, the fabrics are coaxial with the bedding plane measured in horizontal attitude in the field. The LVX4 and LVX6 blocks have a quite well defined distribution with a transverse isotropic symmetry about Kmax (prolate shape). This highlights a porosity vertically elongated (prolate distribution). As already mentioned, GRP1 block is less well defined with a distribution that approximately has a vertical position of Kmin axes and Kmax roughly within the bedding plane. Because we have used two different ferrofluids with distinct magnetic susceptibilities, in order to compare the porosity anisotropy between samples the values of eccentricity parameter Pjel and shape parameter T require a correction for the demagnetization factor. This correction was first proposed by Hrouda et al. (2000) by introducing the concept of equivalent pore, later deepened by Jones et al. (2006) by investigating the effect of different geometries. When the magnetic particle size becomes smaller than the single-domain grain size limit, the uniformly magnetized state represents a good first-order approximation of the true magnetization state. As long as individual particles are spherical, which is a reasonable assumption for ferrofluid, the demagnetization factor equals 1/3. When multiple particles are brought in close proximity (i.e. inter-particle space is of the order of the particle size), demagnetizing effects no longer concern individual particles but the whole aggregate. So, depending on the shape of the host porosity, for a direction of measurement, if intrinsic magnetic susceptibility of the ferrofluid is high enough, the magnetic susceptibility measured will be influenced by a demagnetization field opposed to the induced magnetic field. This demagnetization effect does not affect the principal directions of magnetic anisotropy but does impact the ratio of intensities along these principal directions, leading to an underestimation of the real anisotropy of the void space. The correction consists in the calculation of three demagnetization factors along the three principal axes

a)

assuming that the considered shape is ellipsoidal, which is obviously an over simplification. Osborn (1945) proposed analytical solutions for different kinds of ellipsoid which had been used by different authors (Benson, 2004; Benson et al., 2003; Hrouda et al., 2000; Jezek and Hrouda, 2007; Jones et al., 2006) to calculate an equivalent pore geometry from magnetic susceptibility measurements. In this work we developed our own program using SCILAB routines inspired from the work of Benson (2004). The results of these corrections are presented in Fig. 5 in a Borradaile and Jackson polar representation of Pjel vs T (Borradaile and Jackson, 2004). The corrections induce a significant increase of Pjel up to 1.31 (Fig. 5b). The shape parameter seems not to be significantly affected and the values of T on each block reflect fairly well the axes distribution observed in Figs. 3 and 4. The sandstone samples are generally included in the oblate domain (0 b T b 1) while the carbonate rocks are mostly in the prolate domain (−1 b T b 0). In details, the GRP1 samples having predominantly planar values are in good agreement with a fabric of revolution around the minimum axis Kmin and the CAS block that corresponds to a triaxial fabric is in the prolate domain. 3.3. APV data The anisotropy of P wave velocities is analyzed using the same convention used for AMSff data. As proposed by Louis et al. (2003, 2004), we present APV (Figs. 6 and 7) in dry conditions (stereogram with suffix _DRY), in water saturated conditions (suffix _SAT) and considering the difference between saturation and dry conditions (suffix _Dif). The investigation of APV between different saturation states helps to understand the contribution of the porosity on the shape and orientation of the anisotropy tensor as a complementary approach to the ferrofluid anisotropy. Each tensor on the stereograms corresponds to a triplet of individual samples processed using the method of Louis et al. (2003, 2004). Consequently, on each stereogram, the reported number of samples corresponds to the number of independent triplets that we can consider from a batch of samples drilled in the X, Y and Z directions of the studied blocks. For the sandstones (Fig. 6), the stereograms show a good reproducibility of measurements between the different states of saturation, the mean directions being slightly scattered, if at all. As for the AMSff data,

b) corrected for demagnetization factor

uncorrected for demagnetization factor

1

1.01

1.02

1.03

Pjel 1.04

1

1.05

+1 T=

+1

CAS IBB IBG GRP LVX4 LVX6

T=

CAS IBB IBG GRP LVX4 LVX6

1.1

1.15

1.2

1.25 Pjel 1.3

−1

T=

−1

T=

Fig. 5. Eccentricity parameter (Pjel) versus shape parameter (T) for AMSff plotted in a Borradaile and Jackson's diagram (Borradaile and Jackson, 2004). The T parameter ranges symmetrically from a prolate shape (−1 b T b 0) to an oblate shape. a) data uncorrected from demagnetization factor and b) corrected from demagnetization factor by using intrinsic susceptibility Ki for EMG909 = 0.55 (SI) and for EMG507 = 0.35 (SI).

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx N

CAS_DRY

CAS_SAT

Velocity

N

W

E

N

Velocity

N

Velocity

nsample= 18 V= +2.73e+03 km/s V1=83.9 / 36.8 V2=345.6 / 11.0 V3=241.7 / 51.0 Pjmean= 1.0187 T mean= −0.2102 E

N

S W IBB_Dif

N

E

IBG_SAT

Velocity

N

Velocity

nsample= 27 V= +3.34e+03 km/s V1=12.3 / 2.8 V2=102.3 / 1.6 V3=221.4 / 86.8 Pjmean= 1.0320 T mean= 0.7960 E

E

S W

IBG_Dif

N

Velocity

nsample= 27 V= +3.25e+02 km/s V1=209.4 / 81.2 V2=97.8 / 3.3 V3=7.3 / 8.1 Pjmean= 1.7385 T mean= −0.3299

W

S W

Velocity

W

S W

nsample= 27 V= +3.02e+03 km/s V1=8.2 / 6.2 V2=98.4 / 2.4 V3=209.7 / 83.3 Pjmean= 1.1000 T mean= 0.5931 W

E

nsample= 18 V= +2.36e+02 km/s V1=178.3 / 79.1 V2=66.3 / 4.1 V3=335.6 / 10.0 Pjmean= 1.4374 T mean= −0.6750

W

S W

Velocity

E

S W IBB_SAT

nsample= 18 V= +2.49e+03 km/s V1= 90.1 / 14.2 V2=356.6 / 13.7 V3=224.3 / 70.0 Pjmean= 1.0460 T mean= 0.7807 W

N

nsample= 27 V= +1.55e+02 km/s V1=202.9 / 70.5 V2=298.5 / 2.0 V3=29.2 / 19.4 Pjmean= 4.1846 T mean= 0.5710

W

S W

IBG_DRY

CAS_Dif

Velocity

nsample= 27 V= +2.43e+03 km/s V1=28.9 / 12.6 V2=191.0 / 76.7 V3=298.0 / 4.0 Pjmean= 1.0515 T mean= 0.1864

nsample= 27 V= +2.27e+03 km/s V1= 29.5 / 17.7 V2=176.0 / 69.1 V3=257.5 / 42.1 Pjmean= 1.1206 T mean= −0.8754

IBB_DRY

7

E

S W

W

E

S W

Fig. 6. Stereograms of Anisotropy of P-wave Velocity (APV) measured on sandstones in dry conditions (left), water saturated conditions (center), and differences between saturated and dry conditions (right). Same conventions as for AMS representation, squares are for maximum velocity (V1), triangles for intermediate (V2) and circle for minimum velocity (V3). Black symbols are mean directions calculated with Jelinek statistics (Jelinek, 1978) and corresponding 95% confidence cones around mean axes.

APV shows distinct behavior between CAS block on one hand and IBB and IBG blocks on the other hand. If we consider APV in dry conditions, CAS shows a prolate fabric with a maximum velocity close to the bedding plane and the minimum velocity (Vmin) and intermediate velocity (Vint) scattered in a plane perpendicular to the bedding. IBB and IBG blocks show well-defined planar fabric with Vmin close to bedding pole and consequently an acoustic foliation within the bedding. In saturated conditions, distributions are slightly modified but the APV mean directions keep approximately the same orientations. The saturated IBB block presents the weakest defined fabric dominated by axes orientation scattering. Carbonate rocks (Fig. 7) show similar axes distribution, although these are not so well defined with more erratic points appearing on the stereograms. In dry conditions, carbonate blocks exhibit distinct acoustic fabrics. The GRP1 block shows a prolate distribution with Vmax within the bedding plane while the LVX4 and LVX6 blocks are both characterized by a well-defined prolate fabric with Vmax normal to the bedding pole. The comparison of the principal axes' directions derived from AMSff with those of APV shows some similarities between the fabrics: – for IBB and IBG, fabrics are oblate with both methods; – for LVX4 and LVX6, both fabrics are prolate; – for the two other blocks CAS and GRP1, despite an apparent

difference, both fabrics present similar trends as it will be discussed in terms of shape parameters. Stereograms of the velocity differences inform indirectly on the preferential orientation of the porosity. However, these data should be considered carefully because interpretation assumes that the acoustic anisotropy is only supported by the connected porosity with a negligible contribution of both solid matrix and non-connected microcracks and/ or porosity (Louis et al., 2003). Under this assumption, as the velocity increases after saturation together with a decrease in anisotropy, the maximum velocity difference highlighted in Figs. 6 and 7 must be theoretically oriented along with the most compliant direction in the rock which is also the direction of minimum velocity (Kachanov et al., 1994; Louis et al., 2003). This is the case for the IBG, LVX6 and GRP1 blocks and to a lesser extent for the CAS and IBB blocks. Only LVX4 does not follow this rule. 3.4. Shape anisotropy The particular behavior of LVX4 can be explained taking into account the Vmean vs Pjel diagram presented in Fig. 8 in both saturated and dry conditions. Each point in the diagram represents an average tensor calculated from each stereogram of Figs. 6 and 7. Overall, the saturation increases the average velocity and reduces the velocity anisotropy,

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

8

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx GRP1_DRY

N

GRP1_SAT

Velocity

nsample= 36 V= +4.58e+00 km/s V1= 271.5 / 4.9 V2= 181.2 / 4.4 V3= 49.7 / 83.4 Pjmean= 1.0369 T mean= −0.3433 W

N

N

E

E

W

LVX4_SAT

Velocity

N

Velocity

nsample= 9 V= +5.17e+00 km/s V1= 3.8 / 85.1 V2= 232.0 / 3.2 V3= 141.8 / 3.6 Pjmean= 1.0665 T mean= −0.3348

W

W

N

LVX6_SAT

W

Velocity

W

S W

Velocity

E

S W LVX4_Dif

E

S W

nsample= 60 V= +3.81e+00 km/s V1= 289.5 / 86.8 V2= 29.7 / 0.6 V3= 119.7 / 3.1 Pjmean= 1.0685 T mean= −0.7590

Velocity

nsample= 9 V= +4.80e−01 km/s V1= 25.0 / 73.1 V2= 233.2 / 15.0 V3= 141.2 / 7.6 Pjmean= 1.5013 T mean= 0.2101

W

E

N

N

S W

nsample= 9 V= +4.69e+00 km/s V1= 201.5 / 86.7 V2= 106.9 / 0.3 V3= 16.9 / 3.3 Pjmean= 1.0354 T mean= −0.8989

LVX6_DRY

GRP1_Dif nsample= 36 V= +2.87e−01 km/s V1= 172.4 / 45.1 V2= 0.5 / 44.6 V3= 266.5 / 4.1 Pjmean= 1.3017 T mean= 0.6911

S W LVX4_DRY

Velocity

nsample= 36 V= +4.73e+00 km/s V1= 276.6 / 5.1 V2= 185.7 / 10.5 V3= 32.3 / 78.3 Pjmean= 1.0227 T mean= −0.0134

N

Velocity

S W LVX6_Dif

N

nsample= 60 V= +4.05e+00 km/s V1= 267.7 / 86.7 V2= 17.6 / 1.1 V3= 107.7 / 3.1 Pjmean= 1.0579 T mean= −0.7010

nsample= 60 V= +1.76e−01 km/s V1= 171.1 / 4.9 V2= 80.7 / 4.7 V3= 307.2 / 83.2 Pjmean= 1.3019 T mean= 0.4535

E

E

W

S W

S W

E

Velocity

W

E

S W

Fig. 7. Stereograms of Anisotropy of P-wave Velocity (APV) measured on carbonates. The symbols are similar to those reported in Fig. 6.

except for LVX4 samples (the anisotropy increases). The latter result is in good agreement with directional data which present usually more scattered distribution after water saturation. This scattering can be 1.14 CAS_sat IBB_sat IBG_sat GRP_sat LVX4_sat LVX6_sat

CAS_dry IBB_dry IBG_dry GRP_dry LVX4_dry LVX6_dry

1.12 1.1

Pjel

1.08 1.06 1.04 1.02 1 1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Vmean (km/s) Fig. 8. Mean velocity Vmean = (V1 + V2 + V3)/3 versus mean eccentricity parameter (Pjel) for both sandstones and carbonate rocks. The data are calculated for dry conditions (black and white symbols) and water saturated conditions (colored symbols).

interpreted as reflecting the composition of anisotropies superimposed to that induced by a preferential orientation of the porosity. Indeed, different anisotropy sources may also be expected: 1) an ‘anisotropic’ matrix, 2) the presence of microcracks which are not very sensitive to water saturation but has a significant influence on the P-waves propagation and/or 3) an occluded porosity which cannot be saturated by water. All of these sources can possibly impact the velocity measurements in saturated conditions. However, only a source modified by the presence of water (i.e. an increase of its contribution to the overall anisotropy) can be taken into account with the tensorial difference representation. This aspect will be discussed in the next section. For the other samples, the observed decrease of anisotropy is much more significant for the sandstones than for the carbonates. In the case of CAS block, the anisotropy reduction is quite pronounced and reaches 7%. For carbonates, anisotropies in dry conditions are lower and the decrease is relatively moderate. However, LVX6 exhibits a significant anisotropy of 7% before saturation with a decrease of only 1% (Pjel = 1.07 and P jel = 1.06 in dry and saturated conditions respectively) after saturation which can be interpreted as a lack of drop in anisotropy. These results indicate that the anisotropy measured with acoustic methods may be mainly related to the porosity in the case of sandstones and to lesser extent for the carbonate rocks too. LVX blocks show clearly a more complicated behavior that could be explained by the composite sources of sample anisotropy listed above.

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

It is worth noting that the shapes of ellipsoids obtained with P waves velocity (especially in dry conditions) are roughly similar to those obtained with AMSff (T vs Pjel in Fig. 5). The sandstones are generally oblate while carbonates are prolate. As for AMSff, CAS is rather prolate and GRP1 has the most oblate values for carbonates. 4. Discussion The initial petrophysical properties of the studied samples were chosen in order to cover a wide range of porosity from 10% up to 30% and different kinds of microstructures and mineralogy, i.e. sandstones and carbonates. The different approaches used to characterize the preferred orientation of the porosity show similar results both on orientation and shape, which is an indicator of the efficiency of our ferrofluid impregnation method. Furthermore, the results obtained on the three perpendicular samples show a good reproducibility with only few samples to discard. Thus, this test remains the prerequisite to further investigate preferred orientation of porosity with the ferrofluid impregnation method. All in all our experimental data are qualitatively in good agreement, but some of them need to be discussed in more details. Particularly, both experiments, i.e. ferrofluid and acoustic, provide opportunity to discuss the relevance of the pore anisotropy measured by two separate methods. We will organize the discussion below following three main aspects: a) the pore volume impregnation, b) the anisotropy directions and c) the pore shape anisotropy. a) Efficiency of pore volume impregnation. The comparison between the porosity inferred from both water and ferrofluid saturation must inform us about the efficiency of the ferrofluid impregnation. Fig. 1 shows that the water porosity is almost always higher than the ferrofluid porosity with a maximum difference of about 7.5%. Regarding the range of water porosity, this high difference value is not satisfying, particularly when the sample porosity is particularly low, i.e. less than 15% in our case. As highlighted in Fig. 1, this meaningful difference seems to be not controlled by the nature of the rock (sandstone vs carbonates), or by the range of porosity (or permeability), nor by the viscosity of the fluid (water-based or oil-based ferrofluids). Therefore, the colloidal nature of the ferrofluid coupled with the efficiency of the vacuum impregnation may explain such discrepancies. The minimal size of 10 nm for the magnetic particles evidently represents the lowest pore throat potentially invaded by the ferrofluid and thus only the pore space connected by larger throats could be investigated. Depending on the pore throat geometry this threshold is probably much higher. On several occasions (results not presented here), cutting the samples in half after ferrofluid impregnation has revealed an ineffective impregnation, with ferrofluid only present near the outer surface whereas the inner core has not been invaded at all. Fortunately these observations are not systematic but clearly indicate that the impregnation must be carefully performed prior to AMSff measurements. This protocol could be improved by modifying some parameters in the sample preparation, such as extending the duration of vacuuming. Another suggested improvement consists in a systematic measurement of the porosity by the triple weight method, replacing water by ferrofluid that would make measure of ferrofluid porosity more reliable. Furthermore, comparing the efficiency of ferrofluid invasion between the vacuum method and saturation under pressure, which is sometimes preferred to ensure optimal saturation in low permeability samples. This could probably also help to better understand the origin of the lack of saturation close to the sample core. b) Comparison of anisotropy directions. Even if the comparison between water and ferrofluid porosities suggests that the pore space may not always be completely invaded by the ferrofluid, the orientations of anisotropy obtained with AMSff appear to match properly with those obtained with APV. This

9

observation highlights the fact that both methods are roughly sensitive to the distribution of pore space. At least four blocks (IBG, IBB, LVX4 and LVX6) exhibit AMSff axes' distributions consistent with those measured with APV in dry conditions. As a rule of thumb, the long axis of the porosity preferential orientation (Kmax for AMSff) is close to the orientation of the maximum velocity and short axes (Kmin for AMSff) coincide with the minimum velocity direction. However, this result is not systematic; for instance, in the case of CAS block a permutation between the intermediate and minimum axes is observed. By analyzing the tensor difference between dry and saturated conditions (Louis, 2003), i.e. the direction of the maximum difference marking the direction of the maximum pore compliance, in most of the cases, the direction of the maximum (minimum) difference coincides with the orientation of the short (long) axis of the pores. If we compare the tensor differences with ferrofluid directions, only LVX4 does not follow this rule, with the largest increase of P-wave velocity being parallel to the long-axis of porosity inferred from ferrofluid measurements. This behavior is also highlighted by plotting the anisotropy of P-wave velocity versus mean velocity (Fig. 8) where both anisotropy and velocity increase after water saturation of LVX4. In the latter case, it is impossible to determine clearly the origin of the observed anisotropy with P-wave velocity, and a significant source, which is enhanced after saturation, comes probably from the complex arrangement of the microstructures in oolitic carbonates (see for exemple Casteleyn et al. (2011)). Therefore this data set indicates that the anisotropy of P-wave velocity may result from composition of superimposed microstructural sources, which makes the interpretation of directional data more difficult. A direct interpretation of acoustic anisotropy data in terms of preferential pore orientations must be considered carefully and we recommend first to ensure that the mean velocity increases with fluid saturation and is simultaneously associated with an anisotropy decrease, as shown in a diagram P vs. mean velocity (Fig. 8). c) Pore shape analysis. The direct use of the acoustic measurements to characterize the shape of the void space remains challenging, mainly because the elastic properties are dependent on both solid matrix and pore inclusions within the former. By comparison, the magnetic susceptibility of a ferrofluid-saturated pore will depend only on one parameter that is the demagnetization factor in the direction of measurement. Nevertheless, in order to use quantitative information on the void space anisotropy, which is relatively easy to get from magnetic measurements, we are going to test if AMSff and APV can converge towards a common description of the pore space geometry. Among the numerous models proposed to link the changes in elastic properties of a solid to the presence of more or less anisotropic inclusions within the solid matrix (Guéguen and Kachanov, 2011; Kachanov and Sevostianov, 2005), we have chosen the 2D model of Tsukrov and Kuchanov (2000) which predicts the variation of elastic properties of an anisotropic solid with more or less elliptical holes having arbitrary distributed main axes orientation. The Tuskrov–Kachanov's model (2D-TK model hereafter) assumes that the cavities do not interact with others, which seems to be valid for relatively low porosity (b25%) in the case of holes with high aspect ratio. The 2D-TK model links Young's modulus (E) and Poisson's ratios (ν) to the porosity ϕ and its orientation with respect to the anisotropy of the matrix. For one family of parallel elliptical holes of index k inclined at an angle φ with respect to a reference axis of the anisotropic matrix (the bedding plane), one can express a symmetric second rank hole density tensor: β

ðkÞ πX  2 2 a nn þ b tt : k A

ð2Þ

A is the fractional pore in the area (ratio of the area of the elliptical pores on the area of the solid matrix which can be taken equal to

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

10

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

0.24

unity), a and b, the long-axis and short-axis of the ellipse along directions of unit vectors t and n respectively. The components of β, βnn ¼ ðkÞ2

ð1=AÞπ∑ aðkÞ2 and βtt ¼ ð1=AÞπ∑ b

can be recast by introducing

0.2

the porosity ϕ and aspect ratio α = b/a, as proposed by Louis et al. (2003). One obtains βnn = π/α and βtt = πα. From Eq. (31) of Tsukrov and Kuchanov (2000): E and ν are then expressed as following:

0.18

k

k

0.16

0

∥

⊥

∥

0.893 (1-3) b/a=0.950 0.895 (1-3) 0.938 (1-2) 0.932 (1-2)

0.04

ð3Þ

⊥

0.977 (2-3)

0.975 (2-3) b/a=1.000

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Porosity 0.3

K0=21.7 n0=0.135 G0=20.9 E0=47.5 E 0/E 0=1.0 n 0/n 0=1.0 a=b/a

0.28 0.26 0.24

b/a=0.800

0.22

E∥ E⊥

0.792 (1-3)

0.2

IBB IBG

0.18

γ

0.16

b/a=0.850 0.833 (2-3)

0.14 0.12

0.850 (1-3)

b/a=0.900

0.1 0.879 (2-3)

0.08 0.06

0.946 (1-2)

0.04

b/a=0.950

0.955 (1-2)

0.02 b/a=1.000

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Porosity 0.45

K0=19.5 n0=0.128 G0=19.3 E0=43.55 E 0/E 0=1.0 n 0/n 0=1.0 a=b/a

0.4

0.633 (1-3)

0.35 0.3

b/a=0.700

b/a=0.750

Cas

0.711 (1-2)

0.25

b/a=0.800

γ

allel to the bedding. The case where E0⊥ = E0∥ and ν0⊥∥ = ν0∥⊥ corresponds to the equation of Kachanov et al. (1994) derived for an isotropic matrix which had been used by Louis et al. (2003).To express the anisotropy of P-wave velocity in terms of elastic moduli we borrow the notations of Louis et al. (2003). We used the P-wave modulus M = ρV2P = E(1 − ν)/(1 + ν)/(1 − 2ν) where ρ is the density of the porous material and the anisotropy of P-wave modulus γ = 2(M∥ − M⊥)/ (M∥ + M⊥). Consequently, a negative value of γ will indicate a velocity normal to the bedding higher than parallel to it. To compare our measurements obtained in 3D (APV and AMSff) with the prediction of the 2D-TK model we need to identify on each block the plane containing both maximum and minimum velocities. Furthermore, this plane must be perpendicular to the bedding orientation in order to better estimate the role of anisotropic matrix. The passage from 3D to 2D treatment is acceptable if one considers transverse isotropy symmetry (TI medium). Most of the samples generally exhibit such isotropy (Figs. 6 and 7) and thus satisfy this condition in dry conditions. Indeed, sandstones IBB and IBG (oblate shape) and carbonates LVX4 and LVX6 (prolate shape) show the plane of isotropy around the maximum velocity and within the bedding plane. For GRP1 (prolate shape), the plane of isotropy appears normal to the bedding symmetry axis, corresponding to the maximum velocity, but the minimum velocity is conveniently oriented along the bedding pole. For CAS (prolate shape), both maximum and minimum velocities are very close to the bedding plane. For this specific case, we decided to use the plane defined by the intermediate and maximum velocities to calculate gamma. This remains a moderate limiting factor because the intermediate and minimum velocity values are quite similar. As a starting point for our discussion, we consider the case of an isotropic matrix where we take E0⊥ = E0∥ and ν0⊥∥ = ν0∥⊥ (Fig. 9). We then selected E0 = 95GPa and v = 0.28 for the carbonates by considering a pure calcite composition (Mavko et al., 2009), E0 = 47.5GPa and v = 0.135 for Boise sandstones and E0 = 43.5GPa and v = 0.135 for Castlegate sandstone. The moduli of the sandstone matrix have been calculated using a Voigt–Reuss–Hill average based on the modal analyses published by Zhang et al. (1990) (Quartz = 67%, feldspar = 16%; mica = 3%; clays = 13%) for Boise and by Digiovanni et al. (2007) for Castlegate. In Fig. 9, different curves are plotted for various values of αim (for isotropic matrix) ranging from 1 up to 0.65, αim being the pore aspect ratio for an isotropic matrix. In the latter case, we used the absolute value of γ that is completely independent of any directions in the matrix. Our calculations show logically that the higher

0.961 (2-3)

0.02

the long-axis of the elliptical holes and the bedding direction; L ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ν 0 1 2 ffi − E0∥⊥ þ pffiffiffiffiffiffiffi with G0⊥ ∥ the shear modulus in the direction par0 0 G0 ∥

0.912 (1-2)

0.06

where superscript 0 indicates the properties of the matrix, the subscript ∥ and ⊥ are the properties of the matrix related to the direction parallel and normal to bedding respectively; φ is the angle between

∥⊥

0.884 (1-3)

0.12

0.08

0

b/a=0.850

b/a=0.900

0.1

 qffiffiffiffiffiffi −1 E⊥ −1 2 2 0 ¼ 1 þ ϕ þ ϕL E cos φ þ αsin φ α ⊥ E0⊥ ν ∥⊥ ν ∥⊥ ϕ ν ν ϕ ¼ 0 þ qffiffiffiffiffiffiffiffiffiffiffi ; ⊥∥ ¼ ⊥∥ þ qffiffiffiffiffiffiffiffiffiffiffi 0 E∥ E∥ E 0 0 E⊥ ⊥ E E E0 E 0

GRP1 LVX4 LVX6

0.14

γ

 qffiffiffiffiffi −1 E∥ −1 2 2 ¼ 1 þ ϕ þ ϕL E0∥ α sin φ þ αcos φ 0 E∥

K0=72.0 n0=0.280 G0=37.1 E0=95.0 E 0/E 0=1.0 n 0/n 0=1.0 a=b/a

0.22

0.2 b/a=0.850

0.15 b/a=0.900

0.1

0.893 (2-3) b/a=0.950

0.05

b/a=1.000

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Porosity Fig. 9. Variations of the anisotropy of P-wave modulus γ versus porosity ϕ for different aspect ratios (α = b/a) of elliptical pore inclusions in the case of a porosity embedded in an isotropic matrix. The solid curves are calculated with Tsukrov and Kuchanov (2000) Eq. (31). Three calculations are performed depending on the mechanical properties as input parameter, a) carbonates, b) Boise sandstones (IBB and IBG), c) Castlegate sandstone (CAS). Symbols (1–3), (1–2), (2–3) indicate the plane of anisotropy that bears V1–V3, V1– V2, V2–V3 respectively (see also Figs. 6 and 7 for location).

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx Table 2 Comparison of aspect ratios obtained with different methods, i.e. acoustic and ferrofluid. α −1 im is the inverse of aspect ratio α = (b/a) calculated from Tsukrov and Kuchanov (2000) for isotropic matrix; α −1 am inverse of aspect ratio calculated for anisotropic SAT −1 −1 matrix with anisotropy equal to ratio of P-wave modulus MSAT am ⊥ α ∥ /M⊥ α im . M SAT ∥ measured in saturated conditions (∥ and corresponding to the direction of SAT M⊥

bedding and the direction normal to bedding respectively); Pff is the equivalent pore ratio (long-axis/short-axis) measured with AMSff and its standard deviation.

Cas IBB IBG GRP LVX4 LVX6

α −1 im

SAT MSAT ∥ /M⊥

α −1 am

Pff

1.406 1.176 1.263 1.119 1.117 1.131

1.107 1.023 1.064 1.060 1.132 1.118

1.218 1.134 1.160 1.020 1.264 1.069

1.159 1.134 1.175 1.051 1.058 1.033

± ± ± ± ± ±

0.079 0.049 0.052 0.045 0.021 0.012

values of αim correspond to the higher values of γ. These values are also reported in Table 2 for a direct comparison with the equivalent pore aspect ratio (Pff). Pff is simply derived from the values of Pjel corrected for the demagnetization factor correction and averaged for each block (Fig. 5). The comparison is also shown in Fig. 11. The differences between Pff and α−1 im (for isotropic matrix) are significant and the largest differences are observed for sandstones (CAS and IBG). The consideration of an anisotropic inclusion embedded within an isotropic matrix does not allow a correct match with the anisotropy obtained from AMSff. Indeed, the 2D-TK model overestimates systematically the measured pore aspect ratio. To remedy this discrepancy, a certain amount of anisotropy is introduced into the matrix. From the expression (3), the degree of anisotropy of the matrix may be input into the model thanks to the ratio of Young's modulus E0∥ /E0⊥. As we do not have a proper way to estimate this ratio, we used the elastic modulus ratio measured on sat0

SAT

urated samples under the assumption that EE0∥ ¼ M∥SAT . In Fig. 10, two cases ⊥

M⊥

are considered for the orientation of the matrix anisotropy: 1) the maximum velocity is within bedding plane with E0∥ N E0⊥ (as for CAS, IBB, IBG and GRP1) and φ = 0; 2) the maximum velocity is perpendicular to bedding E0∥ b E0⊥ (as for LVX4 and LVX6) and wherein φ = 90°. With the former, φ = 0° involves that the pore elongation is parallel to the bedding while with the latter, φ = 90°involves that the pore elongation is normal to the bedding. The corrected values for α−1 am (for anisotropic matrix) are also reported in Table 2, as well as the associated MSAT ∥ / MSAT ⊥ input in the model. The computed curves (Fig. 10) show that the introduction of an anisotropic matrix with φ = 0 requires reducing the value of the pore anisotropy to explain the measured anisotropy of γ. However the interpretation of the results for the sandstones seems to be more straightforward than for carbonates. For IBB and IBG, the corrections for anisotropic matrix match well α−1 am and Pff as illustrated in Fig. 11, where the numerical values computed in Fig. 10 are presented. For CAS, the value considered for the matrix anisotropy in the model appears not sufficient to fit the anisotropy of the porosity inferred from ferrofluid measurements. To reach the values of Pff, the corrected ratio (E0∥ /E0⊥) should be 1.2, i.e. almost twice higher than the one measured in saturated conditions (20% compared to 10%). This discrepancy can be explained by a lower sensitivity of the ferrofluid measurements to cracks compared to the elastic methods. We think that the ferrofluid method highlights preferentially the larger pores, which represent the main fraction of the porosity, rather than the cracks, which account for a small fraction of the mean porosity, despite a small aspect ratio. In this case, the orientation of crack distributions has to be added to the anisotropy of the porosity, and we can imagine a distribution of cracks whose long-axis is parallel to the pore long-axis preferential orientation. For the carbonates, αam is systematically higher than 1 which means that the long-axis of the porosity is tilted in the direction normal to the maximum compliance. So, in the case of carbonates, the observed APV should be interpreted as indicating a long-axis of pore shape within

11

the bedding, in total disagreement with AMSff results. To overcome this inconsistency, we can decrease the value of the anisotropy of the matrix (E0∥ /E0⊥ in the model) so as to reduce the value of γ calculated and to lower α below unity. In this case the anisotropy of the matrix and the anisotropy of the porosity would be oriented in the same direction in accordance with our directional data obtained with both APV and AMSff. The unexpected results obtained for carbonates could also be interpreted as reflecting the presence of voids which are not detected by the ferrofluid impregnation method. In oolitic carbonates, dual (and non connected) porosity can be expected (Casteleyn et al., 2010, 2011; Dautriat et al., 2011b) and can clearly impact the flow properties, and therefore the ferrofluid impregnation processes. Moreover, even if observations on thin sections do not indicate the presence of cracks, the preferential orientation of the contacts between grains may produce such additional anisotropy as proposed for sandstones by Louis et al. (2003). However, this interpretation must be confirmed according to finer microstructural investigations. In addition, the fact that anisotropy measured in saturated condition is not sufficient to compensate the whole anisotropy must be elucidated. This observation probably relates to the limitation to use the anisotropy of P-wave modulus(γ) to simulate the anisotropy of Young modulus.

5. Conclusions The results obtained in this study indicate that the characterization of the anisotropy of the porosity with ferrofluid impregnation is a reliable method but requires some methodological cautions. Even if the directions of anisotropy are reproducible from one method to another, comparison of water porosity and ferrofluid porosity shows that impregnation with ferrofluid is generally less efficient. To overcome the problem of poor impregnation, a preliminary test consists in measuring the anisotropy at least on three orthogonal samples. This test already performed by some authors (Benson et al., 2003; Humbert et al., 2012; Louis et al., 2005) should be generalized. An additional precaution (also proposed by Pfleiderer and Kissel (1994)) is to cut the sample in half to visually check the quality of the impregnation by the ferrofluid. Furthermore, a good reproducibility of the measurements does not necessarily prove that all the porosity was invaded. Indeed the throat size distribution that usually controls the connectivity of the pore space in rocks may be very broad and if one remembers that the size of the oxide grains in the ferrofluid is 10 nm, the threshold limit of investigation may be higher than the lowest pore-throat accessible in the rock. The ability to effectively leave the colloidal fluid flow through rock is theoretically controlled by the pore throat diameter and how these pore throats are organized, and probably the threshold limit is higher than the size of ferrofluid particles. We did not find in our data any correlation between the mineralogical nature of the rock (sandstone versus carbonates), the fluid viscosity (water based versus oil-based ferrofluid) and the permeability of the rock to explain the more or less good impregnation. Thus, the origin of this discrepancy should mainly relate to the impregnation protocol. Some attention has to be paid in the duration of the vacuum stage as well as the soaking period in the cell. The observed differences between ferrofluid porosity and water porosity could be decreased by using impregnation under pressure. A comparative study remains to be done between these two methods. Our directional data set highlights the good agreement between AMSff and APV methods. This is a strong argument to say that the anisotropy of the porosity revealed by the two approaches is mainly driven by the coarse fraction of porosity. One can also say that whereas a successful ferrofluid impregnation is certainly driven by methodological considerations, the P-wave velocity method is sensitive to multiple sources of anisotropy. We have seen in the selected model that the presence of an elastic anisotropy in the solid matrix would enhance the whole anisotropy. However, as few laboratory data are available that allow us to characterize the value of the elastic anisotropy, we have

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

12

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

LVX4_1−3

b/a=1.300

0.1

b/a=1.250 b/a=1.200

0

b/a=1.150 1.264 b/a=1.100

−0.1

b/a=1.050 b/a=1.000

−0.2

γ

b/a=0.950

−0.3

b/a=0.900 b/a=0.850

−0.4

b/a=0.800

−0.5

b/a=0.750

−0.6

E0

b/a=0.700

E0

−0.7 0

0.05

0.10

0.15

0.2

0.25

0.3

0.35

0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.3 −0.35 −0.4 −0.45 −0.5 −0.55 −0.6

K0=72.0; ν0=0.280; G0=37.1; E0=95.0 E 0/E 0=0.89 ; ν 0/ν 0=.89 φ=90

b/a=1.300 b/a=1.250 b/a=1.200 b/a=1.150 b/a=1.100 b/a=1.050

1.069

b/a=1.000 b/a=0.950 b/a=0.900 b/a=0.850 b/a=0.800

E0 b/a=0.700

0

0.4

b/a=0.750

E0

0.05

0.10

0.15

Porosity

0.6

LVX6_1−3

γ

0.2

K0=72.0; ν0=0.280; G0=37.1; E0=95.0 E 0/E 0=0.88 ; ν 0/ν 0=.88 φ=90

0.2

0.25

0.3

0.35

0.4

Porosity

K0=72.0; ν0=0.280; G0=37.1; E0=95.0 E 0/E 0=1.06 ; ν 0/ν 0=1.06 φ=0

GRP1_1−3

0.7

K0=20.0; ν0=0.128; G0=19.8; E0=44.6 E 0/E 0=1.11 ; ν 0/ν 0=1.11 φ=0

Cas_1−2

b/a=0.650

0.5

b/a=0.600

0.6

b/a=0.700

b/a=0.650

b/a=0.750

0.4

0.5

b/a=0.700

b/a=0.800

0.3

b/a=0.750

0.4

b/a=0.850

b/a=0.800 b/a=0.900

0.2

0.3

2=0.822

γ

γ

b/a=0.950 b/a=1.000

1.020

0.1

b/a=0.950 b/a=1.000

b/a=1.100

0

b/a=0.900

0.2

b/a=1.050

0.1

b/a=1.150

b/a=1.050 b/a=1.100

b/a=1.200

−0.1

0

b/a=1.250

b/a=1.150

b/a=1.300

−0.2

b/a=1.200 b/a=1.250

−0.1

E0

E0

E0

b/a=1.300

E0

−0.3

−0.2 0

0.05

0.10

0.15

0.2

0.25

0.3

0.35

0.4

0

0.05

0.1

Porosity

IBB_1−3 b/a=0.600

0.5

b/a=0.650 b/a=0.700

0.4

b/a=0.750

0.3

b/a=0.800 b/a=0.850

0.2

γ

b/a=0.900 0.882 b/a=0.950 b/a=1.000

0

b/a=1.050 b/a=1.100

−0.1

b/a=1.150 b/a=1.200 b/a=1.250

−0.2

0

E

b/a=1.300

E0

−0.3 0

0.05

0.10

0.15

0.2

0.25

Porosity

0.2

0.25

0.3

0.35

0.4

Porosity

K0=22.0; ν0=0.135; G0=21.2; E0=48.2 E 0/E 0=1.02 ; ν 0/ν 0=1.02 φ=0

0.1

0.15

0.3

0.35

0.4

0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2

K0=22.0; ν0=0.135; G0=21.2; E0=48.2 E 0/E 0=1.06 ; ν 0/ν 0=1.06 φ=0

IBG_1−3

b/a=0.650 b/a=0.700 b/a=0.750 b/a=0.800 b/a=0.850 0.862

γ

0.6

b/a=0.850

b/a=0.900 b/a=0.950 b/a=1.000 b/a=1.050 b/a=1.100 b/a=1.150 b/a=1.200 0

E

b/a=1.250

E0

0

0.05

b/a=1.300

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Porosity

Fig. 10. Variations of the anisotropy of P-wave modulus (γ) versus porosity for different aspect ratios (α = b/a) of elliptical pore inclusions in the case of a porosity embedded in an E0 anisotropic matrix. Each plot corresponds to different input parameters in the model: the elastic moduli of the solid matrix, Bulk modulus (K0) and Poisson ratio (v0); the ratio E0∥ for matrix ⊥ anisotropy is taken as the ratio of the P-wave moduli corresponding to V1 and V3 in saturated conditions; the angle φ between the long-axis of the porosity and the orientation of E0∥ ; aspect ratios α = (b/a). The values indicated on each plot correspond to the calculated aspect ratio.

Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029

P. Robion et al. / Tectonophysics xxx (2014) xxx–xxx

13

1.5 Isotropic matrix

CAS IBB IBG GRP LVX4 LVX6

1.4

Anisotropic matrix

CAS IBB IBG GRP LVX4 LVX6

Pff

1.3

1.2

1.1

1.0 1

1.1

1.2

1.3

1.4

1.5

α−1 Fig. 11. Comparison between equivalent pore ratio obtained with ferrofluid (Pff) and the inverse pore aspect ratios α−1obtained in case of anisotropic matrix (blue data) and isotropic matrix (black and white symbols).

arbitrarily chosen the anisotropy of P wave velocity moduli after saturation which is not always satisfactory. Previous studies related to ferrofluids have so far paid relatively little attention to the real meaning of the aspect ratio, or they did not even take into account the correction for demagnetization factor. An accurate estimation of the pore aspect ratio may be however of great interest in seismic inversion and hydrological simulations and the ferrofluid impregnation remains a robust method to provide a direct assessment of the 3D distribution of porosity for rocks presenting a wide range of pore-throat sizes, from a ten of nanometers up to few hundreds of micrometers. For our sedimentary rocks, the pore aspect ratio is greater than 0.85, with less anisotropic porosity for carbonates than for sandstones. We have seen that our results converge in a number of instances, but not for all. The consideration of elastic anisotropy of the solid matrix may be a limiting factor to fully control the aspect ratios inferred from acoustic method. Ultimately, it seems easier to get a reliable aspect ratio with ferrofluid impregnation under a strict impregnation protocol. Acoustic methods allow rather the investigation of a larger range of porosity, but the interpretation remains more complicated. Therefore, the joint implementation of both APV and AMSff is of particular interest to get reliable information on the shape of the porosity. Acknowledgments The authors would like to thank both the anonymous reviewers as well as Associate Editor Bjarne Almqvist for their comments which helped to improve the manuscript. References Almqvist, B.S.G., Mainprice, D., Madonna, C., Burlini, L., Hirt, A.M., 2011. Application of differential effective medium, magnetic pore fabric analysis, and X-ray microtomography to calculate elastic properties of porous and anisotropic rock aggregates. J. Geophys. Res. Solid Earth 116, B01204. Benson, P., 2004. Experimental Study of Void Space, Permeability and Elastic Anisotropy in Crustal Rock Under Ambient and Hydrostatic Pressure. PhD University of London (273 pages). Benson, P.M., Meredith, P.G., Platzman, E.S., 2003. Relating pore fabric geometry to acoustic and permeability anisotropy in Crab Orchard Sandstone: a laboratory study using magnetic ferrofluid. Geophys. Res. Lett. 30, 19. Borradaile, G.J., Jackson, M., 2004. Anisotropy of magnetic susceptibility (AMS): magnetic petrofabrics of deformed rocks. Geological Society, London, Special Publications 238, 299–360.

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Please cite this article as: Robion, P., et al., Pore fabric geometry inferred from magnetic and acoustic anisotropies in rocks with various mineralogy, permeability and porosity, Tectonophysics (2014), http://dx.doi.org/10.1016/j.tecto.2014.03.029