Shale distribution effects on the joint elastic–electrical properties in reservoir sandstone

We investigated the effect of shale distribution on the joint elastic wave and electrical properties of shaly reservoir sandstones using a dataset of laboratory measurements on 75 brine‐saturated (35 g/L salinity) rock samples (63 samples from the literature, 12 newly measured samples). All the data were collected using the ultrasonic (700 kHz) pulse‐echo measurement technique for P‐ and S‐wave velocities (Vp, Vs), attenuations (Qp−1, Qs−1), and a four‐electrode method for resistivity under elevated hydrostatic confining pressures between 10 and 50 MPa (pore fluid pressure 5 MPa). The distribution of volumetric shale content was classified by comparing the calculated dry P‐wave modulus to the modified Upper Hashin–Shtrikman bound for quartz and air mixtures, assuming pore‐filling shale. This scheme in particular allowed us to distinguish between pore‐filling and load‐bearing shale distributions according to idealized definitions, which provides new insight into the joint ultrasonic properties and resistivity behaviour for shaly sandstones. In resistivity–velocity space, the resistivity of load‐bearing shale increases with increasing velocity which form a more distinct trend with steeper gradient compared to those for partial pore‐filling shale and clean sandstones. Moreover, the pore‐filling shale trend straddles the clean sandstone trend and meets the load‐bearing shale trend between 100 and 150 apparent formation factors. In resistivity–attenuation space, the highest attenuations exist when the volumetric shale content is close to the frame porosity (for Qp−1 in particular), at the transition between pore‐filling and load‐bearing shales. The results will inform the development of improved rock physics models to aid reservoir characterization from geophysical remote sensing, particularly for joint seismic and controlled source electromagnetic surveys.


INTRODUCTION
Understanding the influence of shale distribution on the joint elastic and electrical properties of reservoir sandstones, and pressure dependence, is of great importance to hydrocarbon exploration, reservoir monitoring and characterization.It is well known that the presence of assemblages of clay mineral particles in a sandstone, often with their own porosity (i.e.shale), can influence both remotely sensed geophysical properties, such as seismic velocity and electrical resistivity, and reservoir properties, such as porosity and permeability (Han et al., 1986;Worthington, 1982).Clay distribution effects in sand-clay mixtures, and by analogy in shaly sandstones, are also well appreciated.Marion et al. (1992) showed how seismic velocity is controlled by shale content and distribution in the continuum from clean sand, through shaly sands (i.e.pore-filling shale) and sandy shales (i.e.load-bearing shale), to shale.Revil and Glover (1998) showed similar control of shale content and shale distribution in sands on permeability linked to electrical properties.However, there have been few observational studies of the joint properties of shaly sandstones and the effect of shale content and distribution.This knowledge is needed to validate theoretical models that could be used for joint inversion and interpretation of co-located seismic and electromagnetic survey data.Carrara et al. (1999) were among the first to measure laboratory joint elastic-electrical properties on sandstone samples.Their focus was to test an electro-seismic model to obtain fluid saturation and porosity; hence, they did not present any systematic relationships between seismic velocity and electrical resistivity.Several authors (Sheng & Callegari, 1984;Salem, 2001;Hacikoylu et al., 2006) used well-logging data to investigate the joint elastic and electrical properties of reservoir rocks.Others used electrical well-logging data for hydrocarbon exploration (Aladwani, 2021(Aladwani, , 2022a(Aladwani, , 2022b;;Aladwani et al.,2023).However, well-log results must be interpreted with caution because of spatial averaging effects for different rock properties and unknown parameters, compared with data collected on small rock samples in the laboratory where rock parameters can be better constrained.
To date, the most systematic and comprehensive laboratory study of joint elastic-electrical properties of shaly sandstones was reported by Han et al. (2011aHan et al. ( , 2011cHan et al. ( , 2015)).They measured the ultrasonic compressional (P-) and shear (S-) wave velocity and attenuation (denoted V p , V s , Q p −1 and Q s −1 respectively), and electrical resistivity (ρ), of 63 shaly sandstone samples under effective pressures (i.e. the difference between confining and pore fluid pressures) from 8 to 60 MPa.They observed a systematic influence of clay content on velocity-resistivity relations in particular (Han et al., 2011b), well described by an effective medium model (Han et al., 2011c), and on their pressure dependencies (Han et al., 2011a).Some systematic trends in resistivity-attenuation space were seen, but they were more uncertain.
Recent studies based on joint elastic-electrical properties have demonstrated a considerable reduction in rock resistivity caused by the clay minerals (Carcione et al., 2003;Lee, 2011;Carcione et al., 2012;Peng et al., 2018).Cilli and Chapman (2020) examined the effects of porosity on resistivity and elastic moduli using laboratory measurements on carbonate samples and a power-law relationship between porosity and pore aspect ratio.Pang et al. (2021) created an electrical model and suggested a dual-porosity clay parallel network, and to simulate the elastic properties, they also suggested differential effective medium equations and the Hashin-Shtrikman equations.Then, Pang et al. (2021) calculated the rock parameters based on saturation, clay content, total porosity and microcrack porosity using these two models.Additionally, a 3D elastic-electrical template was constructed using Poisson's ratio, resistivity and acoustic impedance.Zhang et al. (2022) studied the pore geometrical characteristics of tight sandstones and proposed a multiphase reformulated differential effective-medium model that uses the same pores or fractures with various aspect ratios and volume fractions as the unified pore geometry for both electrical and elastic modelling.The pore structure and mineral make-up of tight-oil rocks are complicated, with a significant percentage of clay (Lu et al., 2019;Gao et al., 2022).Pang et al. (2022) analysed the heterogeneity of tight-oil reservoir rocks from cores by using X-ray diffraction and casting thin sections, to determine the rock mineralogy and pore structure.The impacts of pores, microcracks and mineralogy on the elastic-electrical properties were then investigated using ultrasonic and resistivity studies under various confining pressures.Pang et al. (2021) created acoustic and electrical models based on effective-medium theories, the Cole-Cole and tripleporosity equations.Then, a 3D rock-physical template was constructed and calibrated using the well-log data and core samples (Pang et al., 2021).
Here, we seek insight into the role of shale distribution, in addition to shale content, on the joint properties by applying a quantitative shale classification scheme to the existing Han et al. data, as well as adding new laboratory measurements to validate and extend the previous study.The results both confirm original observations of Han et al. and also point to distinct effects of pore-filling and load-bearing shales on shaly sandstone joint properties.

Sample selection and preparation
A suite of 12 new shaly sandstone samples was chosen carefully to expand the Han et al. dataset.The samples were selected from a repository of several hundred rock samples that were measured previously in the laboratory ultrasonic pulse-echo system (Marks, 1994;Sharp, 1995).These samples came with petrophysical analysis results, including porosity, permeability and clay content, as well as ultrasonic velocity and attenuation that were used to select the best samples for this study.This was achieved by cross-plotting velocity, porosity and permeability for the Han et al. data, and the repository core data, to see where the repository core data could fill in or extend the clay content range of the Han et al. data.The chosen repository core samples are listed in Tables A1 and B1 in the appendices together with their previously measured petrophysical and ultrasound properties, with published sources indicated.
All samples were prepared according to standard procedures described in McCann and Sothcott (1992) and Han et al. (2011b).The samples were cored as 5 cm diameter cylindrical plugs, cut to a length of 2 cm with the two end faces ground flat and parallel to within ±0.01 mm.The samples were placed into an oven to be dried for 3 days at a temperature of 40˚C to conserve clay minerals as far as possible (temperatures above 60˚C will destroy clays).The cleaned and dried samples were weighed, their dimensions measured and then they were saturated in 35 g/L brine at an elevated pressure of 7 MPa for 3 days.The saturated samples were then held in a tank filled with the same brine before being transferred into the high-pressure rig for ultrasonic and electrical measurements.
The geophysical parameters P-and S-wave velocity and attenuation, and electrical resistivity, of each sample, were measured in quick succession at each effective pressure of 50, 40, 26, 15 and 8 MPa (the pore fluid pressure was maintained at 5 MPa); the pressure was allowed to equilibrate for 30 min before commencing measurements.The laboratory temperature was maintained at 19 ± 1˚C and relative humidity at 50% ± 1%.

Joint ultrasound and electrical resistivity tomography RIG
The ultrasonic pulse-echo method (Winkler & Plona, 1982;McCann & Sothcott, 1992) was used to measure P-and Swave velocities (V p , V s ) and attenuations (Q p −1 , Q s −1 ) at frequencies between 400 and 800 kHz.We used a dual P/S wave transducer giving velocity and attenuation coefficient measurement accuracies of ±0.3% and ±0.2 dB/cm, respectively (Best, 1992).The high-pressure rig and sample assembly were adapted for resistivity tomography measurements according to North et al. (2013).See Figure 1.
The rubber sleeve surrounding the sample contained 16 stainless steel electrodes in tetrapolar configuration, radially distributed in two rings around the sample.The electrodes allowed current injection and boundary voltage probing in various permutations, giving a resistivity measurement error under typical operating conditions of ≤0.1% at frequencies 1-500 Hz, for a sample electrical resistivity range of 1-100 Ω m.

SHALE CLASSIFICATION ANALYSIS
Here, we adapt the method described by Sørensen and Fabricius (2015) for the Han et al. data at the highest pressure of 60 MPa.The results of Han et al. (2011b) demonstrated that the pressure dependence of velocity, attenuation and resistivity are minimal at this high pressure and can be taken as representative of the intact lithology with negligible stress-release microcrack-related effects.Sørensen and Fabricius (2015) were able to delineate between four types of so-called clay distribution in a set of sandstones by comparing volumetric shale fraction to grain framework porosity obtained using the method of Gal et al. (1998).Here, "shale" is taken to mean solid clay mineral assemblages, including any associated microporosity and bound water; "clay" refers to the solid clay minerals only (without bound water).They defined the first group as clean sandstone with less than 2% shale measured by image analysis (thin section points counting), and the second group as sandstone with dispersed shale below the critical shale fraction, that is when the shale fraction is less than the sand grain frame porosity.The third group was for sandstone with shale at the critical shale fraction.The fourth group followed the Marion et al. (1992) idea of load-bearing shale, when shale volume exceeds the sand grain frame porosity.This classification depends on the modified upper Hashin-Shtrikman (MUHS) bound (Mavko & Mukerji, 1998;Gal et al., 1999) that connects the theoretical frame porosity to the elastic frame moduli of the sandstone.The majority of the shale was assumed to be authigenic and pore-filling (Sørensen & Fabricius, 2015).
Some factors led us to modify the approach adopted by Sørensen and Fabricius (2015).First, although shale fraction data from image analysis were available to Sørensen and Fabricius (2015), the Han et al. dataset had clay mineral content only from whole rock X-ray diffraction (XRD).Moreover, Sørensen and Fabricius (2015) made the assumption that the measured porosity (obtained by the helium porosimeter method) included the shale fraction porosity, including any bound water.This assumption is open to question because the helium porosimeter measures the effective porosity (i.e. the ratio of interconnected void spaces to the bulk volume), and the degree to which the fine shale pores could be penetrated by the helium gas is not known.Hence, Sørensen and Fabricius (2015) assumed that the difference between the theoretical framework porosity and the measured porosity gives the clay volume (without bound water), whereas, in our analysis, we assume that the difference between the framework porosity and the measured porosity gives the volumetric shale fraction (including any bound water).Sørensen and Fabricius (2015) assumed that the theoretical line (MUHS trend for air and quartz) represents clay, whereas, in nature, shale always comes with bound water and clay (minerals without bound water); so in our approach, the theoretical line in Figure 3 represents volumetric shale fraction, and the application scope is to widen the model to become applicable in natural reservoir shaly sandstone.
Hence, our "shale distribution" classification scheme shown in Figure 2 follows the Sørensen and Fabricius (2015) way of grouping the sandstones; only we use the difference between the theoretical frame porosity and the measured effective porosity from Figure 3 to calculate the shale fraction (see Equation 8).Then, we used the XRD-derived clay content (i.e.volume fraction of solid minerals only, excluding bound water and shale porosity), not shale content, to calculate the shale porosity (including bound water) (see Equation 9).Clean sandstones with less than 2% shale volume fraction are denoted as clean sandstones (Group 1).Group 2 comprises partially pore-filling shale with more than 2% shale volume fraction, and less than 90% of the frame porosity; this last percentage was chosen arbitrarily in an attempt to account for the shale porosity effect, valid for assumed shale porosity and bound water of 20%.Group 3 comprises homogeneous pore-filling shale (i.e. the shale fills the entire frame porosity); we chose an arbitrary interval between 90% and 110% of the frame porosity for this group.Group 4 is denoted loadbearing shale when the volumetric shale fraction is more than the upper limit of Group 3. Therefore, there is sufficient volumetric shale present for sand grains to be supported by a shale-grain framework (sandy shale).
The MUHS curve describes how the elastic moduli of clean sandstones evolve from deposition through compaction and cementation.It is not a rigorous bound on the elastic properties of clean sand, although sandstone moduli are almost always observed to lie on or below the MUHS curve (Mavko et al., 2009).Further, terms like "stiff pore shapes" and "soft pore shapes" are used usually to describe the data with respect to the MUHS curve.
The few data that were above the MUHS trend have significant amounts of minerals with higher elastic moduli than quartz such as carbonates or feldspars and belong to clean sandstones, so we assume the measured porosity is the frame porosity as there was a negligible shale fraction or none at all.
The frame porosity was calculated from the dry compressional wave modulus  dry for each sample at 60 MPa (Mavko et al., 2009).In Figure 3, the solid line represents the MUHS trend for air and quartz, which basically represents rocks with quartz minerals in framework and pores filled with air without shale or any other minerals, so the pores here are frame porosity.The samples with shale in pore space and in the framework lie below the solid line.The frame porosity can be computed for each shaly sandstone sample (see Figure 3 small The shale distribution classification scheme used in this study.a) represents clean sandstone with less than 2% shale, b) A rock with partial pore-filling shale in which the frame porosity is more than porosity, c) a rock with Homogenous pore-filling shale in which the frame porosity is filled with shale, d) a rock with Load-bearing shale is when the shale exceeding 110% of the frame porosity and some of the shales is part of the framework of the rock.Source: Adapted from Sorensen and Fabricius (2015).
graph, frame porosity fit) by fitting the samples to the solid line (MUHS).Then, frame porosity can be obtained.However, samples above the solid line contain shale less than 2%, so they belong to Group 1.
As the sandstones were measured under brine saturation conditions, the dry bulk modulus  dry was obtained using the following Gassmann fluid substitution (Gassmann, 1951): where ϕ is porosity,  o is the mineral bulk modulus,  f l is the pore fluid bulk modulus and  sat is the saturated bulk modus.
The dry compressional modulus is then calculated from where  is the dry frame shear modulus, equal to the measured saturated rock shear modulus according to Gassmann's theory.porosity and framework porosity are shale volume fraction and from here on we will use shale volume fraction in our analysis.The solid line is the MUHS bound for a critical porosity equal to 0.4 and a pure quartz compressional mineral modulus M 0 = 100 GPa (Koga et al., 1958).The MUHS bound represents the diagenetic trend for clean sandstone as the rock becomes progressively quartz cemented, with the quartz cement occupying the pore space and reducing porosity from the critical porosity.The MUHS curve is computed using the following equations:

Shale classification and group delineation
where  s is quartz grain bulk modulus (GPa),  s is quartz grain shear modulus (GPa),  b is quartz grain bulk modulus at the critical porosity (GPa),  b is quartz grain shear modulus at the critical porosity, ϕ b is the critical porosity,  sat is the saturated compressional modulus,  sat is the saturated bulk modulus and  is the shear modulus (Dvorkin et al., 1999).
Figure 3 illustrates that sandstone samples with low shale fraction are close to the MUHS trend line, and samples with higher shale fraction lie significantly below the MUSH trend.Thus, the measured rock porosity is lower than the idealized quartz frame porosity due to the presence of shale.Samples above the MUHS line might have minerals that have higher elastic moduli than quartz such as feldspars or carbonates.These samples have negligible shale fraction and fall within the clean sandstone group.We followed the Gal et al. (1999) assumption that the difference in porosity between the idealized frame porosity according to MUHS, ϕ frame , and the measured rock sample value ϕ e (or effective porosity) gives the shale volume fraction , in the pore space of the framework according to where  is shale fraction.The framework porosity ϕ frame for each sample in the dataset can be determined by comparing the measured porosity to the MUHS trend as illustrated in the small graph inset in Figure 3. Further, the volumetric shale fraction  in the pore space of the framework can be determined by The shale porosity ϕ sh (= Vϕ _sh /V sh , where V with subscripts ϕ _sh and V sh are the volumes of shale porosity and shale, respectively).For partial pore-filling shale and homogenous pore-filling shale, it can be determined from volumetric shale fraction Χ (= V sh /V rock , where V rock is the total rock volume) and solid clay fraction C (= V Cl /V rock , where V Cl is the solid clay mineral volume) from Then, the shale microporosity (ϕ sh ) was determined by Equation ( 9) where  is solid clay fraction without shale microporosity obtained from XRD, and  is shale fraction calculated from Equation ( 8).We will use only the shale fraction in the following analysis.
Figure 4 shows framework porosity against volumetric shale fraction.It illustrates the final shale distribution classification for 75 sandstone samples.Clean sandstones in Figure 4 (green triangle) are defined as those that contain less than 2% (the vertical blue line represents 2%) of shale by volume (Group 1).In order to account for the shale porosity effect, Group 2 consists of partially pore-filling shale with more than 2% shale volume fraction and less than 90% of the frame porosity; this last percentage was picked at random and is valid for an assumed shale porosity and bound water of 20% as shown in Figure 4 (pink square).We arbitrarily selected a range between 90% and 110% (the inclined red dashed line in Figure 4) of the frame porosity for Group 3 (red diamond in Figure 4), which consists of homogenous pore-filling shale, to represent this group.When the volumetric shale portion exceeds the top limit of Group 3, it is said to be load-bearing shale, which belongs to Group 4 (black circle in Figure 4).Because of this, therefore, there is enough volumetric shale present for a shale-grain structure to support sand grains (sandy shale).It can be seen that this particular dataset contains a reasonable spread of samples in all four shale classification groups.This allows us to draw some conclusions about the role of shale distribution on the joint elastic-electrical properties.
Figure 5 is a histogram showing the distribution of shale porosity that was obtained from Equation ( 9).The shale porosity distribution is in general negatively skewed if we exclude the data below 2% shale porosity; they have low shale content and were classified as clean sandstone.The mean value of shale porosity for all samples is 0.63.The mean value for partial pore-filling shale samples is 0.39, and for loadbearing shale, it is 0.78.The shale porosity distribution is consistent with the shale porosity distribution published by Hurst and Nadeau (1995) and Sørensen and Fabricius (2015).The mean value of 0.63 in this study is close to the mean values of 0.59 reported by Sørensen and Fabricius (2015) and 65% ± 14% reported by Vernik (1994).Figure 5 shows a maximum shale porosity of around 0.7 for sandstone with pore-filling shale, which is similar to that of Vernik (1994) and also agrees with the maximum shale porosity of about 0.7 of Sørensen and Fabricius (2015).Figure 6 shows an example of load-bearing and partial pore-filling shales in thin sections to visually validate the F I G U R E 5 Histogram of shale porosity obtained from Equations ( 9) and ( 10).assumption of load-bearing shale in the framework of the rock.In Figure 6a, the shale is insufficient to become part of framework, but shale is clearly part of the framework in Figure 6b.The Group 4 load-bearing shale might be dispersed within the pore space or laminated, as can be seen in Figure 6b.

RESULTS -THE EFFECT OF SHALE DISTRIBUTION ON JOINT ELASTIC-ELECTRICAL PROPERTIES
Velocity-resistivity (apparent formation factor F*) Han et al. (2010) were able to draw some conclusions on the joint velocity and resistivity relationship here called the velocity-resistivity relationship for short.Han et al. (2011c) expressed their resistivity results as apparent formation factor F*, defined as ρ 0 /ρ w , the resistivity of the sample saturated with brine (ρ 0 ) to the resistivity of the brine (ρ w ), and we will follow that convention here.Han et al. (2011b) were able to distinguish visually two limbs on the F*-velocity cross-plots corresponding to clayrich and clean sandstone trends with some scatter of data points.Here, we produce similar plots for F*-V p and F*-V s for this extended dataset and use clustering analysis to obtain a statistical estimate of the significance of different groupings of data points (Tan et al., 2005;Chen & Hoversten, 2012).
The K-mean clustering technique is a type of unsupervised training algorithm to treat the observations in data as objects having locations and distances from each other (Liu et al., 2004).The procedure involves choosing k initial centroids, where k is some clusters, usually the desired user-specified parameters; here, we tried k = 2-5 as shown in Figure 7. Kmean clustering then assigns each data point to its nearest centroid, and each group of data points assigned to a centroid becomes a cluster.The centroid of each cluster is then iteratively refined as new data points are assigned to the cluster.The algorithm finally converges when no further change occurs to the clusters (Chauhan et al., 2016;Lee et al., 2017).Figure 7 shows that the initial visual interpretation of Han et al. of two groups can be reasonably extended to three groups using K-mean clustering with k = 3.
In Figure 7, we compare the results of the K-mean clustering algorithm for k = 3 to the delineation of shale groups according to our shale classification scheme.Three clustering groups can be seen in Figure 7b: Cluster 3 (red colour) appears as a transition between Cluster 1 (blue) and Cluster 2 (black).
In general, F* increases with increasing V p (and V s in Figure 8) with shale-rich sandstones forming a steeper, separate trend than clean sandstones, as noted by Han et al. (2011b).Furthermore, we can now separate the shale-rich sandstone trend into two more trends, one dominated by homogeneous pore-filling shale and partial pore-filling shale (as indicated by Cluster 3) and the other by load-bearing shale (as indicated by Cluster 2).The partial pore-filling shale trend straddles the clean sandstone and load-bearing shale patterns.However, the load-bearing shale pattern is relatively tightly constrained, and the data points only overlap with a few of the pore-filling shale trend samples.We see similar trends for V p and V s in Figure 8 (Aladwani et al., 2016).

Porosity and joint properties
Porosity plays a critical role that affects both elastic and electrical properties of reservoir rocks (Archie, 1942;Han et al., 1986;Klimentos & McCann, 1990;Best et al., 1994;Han et al., 2011b).Porosity reduces the bulk and shear moduli of the solid framework of the reservoir rock which, in turn, affects the velocity of compressional and shear waves.In general, porosity increases attenuation due to the viscous interaction of pore fluids and solid framework (Biot, 1956;Murphy et al., 1986;Klimentos & McCann, 1990).Moreover, increasing porosity increases the electrical conductivity (decreases the electrical resistivity) when saturated with brine due to an increase in permeability allowing greater diffusion of free ions through the electrolyte (Klimentos & McCann, 1990).
Figure 9 shows cross-plots between apparent formation factor F* and P-and S-wave velocities, respectively, colourcoded by porosity.In general, apparent formation factor and velocity increase with decreasing porosity as noted by Han et al. (2011b).However, with our new analysis, we can see that there are three behaviours in the cross-plots between apparent formation factor F* and P-and S-wave velocities, respectively.The first behaviour is the trend in clean sandstone and sandstones with partial pore-filling shale that have apparent formation factors less than 70 and porosity more than 13%.The second behaviour is for sandstones with homogenous pore-filling and the partial pore-filling shales that have apparent formation factors above 70 and porosity around 12%.This trend is above the clean sandstone trend.The third behaviour is the load-bearing shale, showing a steeper trend for porosity less than 12%.
Figure 10 shows cross-plots of F* versus P-and Swave attenuations (F*-Q p −1 and F*-Q s −1 , respectively), colour-coded by porosity.In general, attenuation and porosity increase with F* up to a value of about F* = 100.The maximum attenuation occurs at porosities between 10% and 15% and then decreases.Low shale content sandstones appear to the left of the attenuation maximum and shale-rich sandstones to the right, as noted by Han et al. (2011b).We are now able to determine from our shale classification scheme that (i) the right-hand decreasing trend is dominated by loadbearing shale with porosity less than 10% with the lowest attenuation, and (ii) the left-hand trend is dominated by a mix of clean and partial pore-filling shale sandstones with porosity above 15%, and some overlap between trends.There are similar trends for Q p −1 and Q s −1 , although there is more scatter of low F* data points for Q s −1 as noted by Han et al. (2011b).In general, P-wave attenuation has the highest values when shale content is close to the frame porosity, at the transition between pore-filling and load-bearing shales.
Overall, the application of the shale classification scheme to the data in Figures 7 and 8 and Figures 9 and 10 shows that shale distribution within sandstones has a significant impact on the joint seismic-resistivity response, and this is related to the relative proportions of shale and sand grains.As can be seen in Figure 9, the trend in clean sandstone and sandstones with partial pore-filling shale, which has apparent formation factors less than 70 and porosity greater than 13%, is the first behaviour.The second behaviour is seen in sandstones with homogeneous pore filling and shale that partially fills some of the pores and has apparent formation factors above 70 and porosity around 12%.The clean sandstone trend lies underneath this one.The load-bearing shale exhibits the third behaviour, which displays a sharper tendency for porosity less than 12%.Thus, different proportions of shale and sand in pore space or in the framework of the rock exhibit different behaviours (Figure 9).However, the transition between pore-filling and load-bearing shales occurs where P-wave attenuation values are highest when shale content is close to the frame porosity in Figure 10.
In accordance with the present results, previous studies have demonstrated the effect of petrophysical properties such as porosity and clay content on the joint properties.having a strong influence on both elastic and electrical properties independently of each other.Because sandstone samples with similar porosity were found in both groups.However, I apply a statistical analysis (clustering method) and effective medium model (shale classification scheme) to the data and find out that the clustering technique divides the data into three categories (Figure 7b).The clustering techniques suggest further dividing the data into two groups in Han et al. (2011b) to further subdivide the dataset into more than two groups.The use of an effective medium model (shale classification scheme) to the data shows how the pore-filling and load-bearing shales affect joints' elastic-electrical properties.I concluded that the framework of the rock has significant influence on the joint properties whether the framework is sand-born (clean sandstone and partial pore-filling shale) or shale-born (load-bearing shale).In resistivity-velocity space (Figure 9), load-bearing shale forms a distinct trend to Group 1 (clean sandstone) and Group 2 (partial pore-filling shale) and gives rise to decreasing attenuation with increasing appar-ent formation factor.Moreover, in resistivity-velocity space, pore-filling shale touches the load-bearing shale trend and straddles the clean sandstone trend and gives rise to increasing attenuation with increasing resistivity up to a maximum when shale content is about equal to frame porosity, at the transition between pore-filling and load-bearing shales (Figure 10).

Pressure sensitivity and joint properties
Least-squares regression analysis was used to quantify the effect of shale distribution on the pressure dependence of elastic wave velocity and attenuation, and electrical resistivity, of shale-rich sandstones.Following Han et al (2011a), it is well established from various studies (Eberhart-Phillips et al., 1989;Jones, 1995;Khaksar et al., 1999;Kaselow et al., 2004;Han et al., 2011b) that an equation of the form can accurately describe the pressure dependence of all five geophysical parameters, that is P-and S-wave velocity (V p , V s ) and attenuation (Q p −1 , Q s −1 ) and electrical resistivity ρ.The variable  corresponds to the geophysical parameters of interest; ,  and  are the best fit regression coefficients, and  dif f is the effective pressure.Figure 11 shows the regression curves and data for the four newly measured reservoir rocks, colour-coded according to the shale classification scheme (V s and Q s −1 results have been omitted as they follow similar trends to those for V p and Q p −1 ).In general, these results agree with Han et al. (2011a), indicating that V p and ρ increase, and Q p −1 decreases, gradually until the rate of change converges to constant values at higher effective pressures.A possible explanation for pressure-dependent behaviour is the closure of small aspect ratio pores and micro pores (Meglis et al., 1996;Glover et al., 2000).Such low aspect ratio pores and micro pores could be due to cracks present either within a mineral grain or at grain contacts or might be related to shale minerals with their platy grains and associated microporosity.
A possible explanation for decreases in attenuation with pressure might be that microcrack squirt flow reduces as a result of cracks being closed according to mechanisms explained by Murphy et al. (1986) and Dvorkin et al. (1995).
The finite attenuation at higher effective pressure might be explained by background Biot-type losses (Biot, 1956) or might be due to clay-squirt flow (Best & McCann, 1995;Marketos & Best, 2010;Han et al., 2011b).By contrast with V p and Q p −1 in Figure 11a,b, electrical resistivity in Figure 11c shows systematic increases according to increasing shale content, from clean to partial pore-filling shale, to homogenous pore-filling shale, to load-bearing shale, with the highest resistivity in the load-bearing shale samples; F* also increases with pressure according to Equations ( 3)-( 11).A likely explanation for the increase in electrical resistivity with effective pressure is the closure of the narrow conduction pathways at grain contacts.Han et al. (2011b) noted that electrical resistivity is more pressure sensitive in shale-rich sandstones.Our results also show that the load-bearing samples are the most pressure sensitive, followed by those with partial pore-filling shale, then those with homogenous shale, then lastly clean sandstones in a clear succession.
Figure 12 shows the pressure dependence of the joint property F*-V p of the four newly measured sandstones, colour-coded according to the shale distribution scheme (F*-V s relationships are similar in Figure 12b).The gradients can be seen to increase systematically both as F* and V p magni- tudes increase, and with increasing shale content from clean sandstone through pore-filling and load-bearing shales.
Linear least-squares regression equations were derived for the joint elastic-electrical data according to The results are shown in Appendices A and B together with regression coefficients R 2 .Figure 11 reveals that there has been an increase in a pressure sensitivity of electrical resistivity in particular (relative to that of V p and V s ) from partial to homogenous pore-filling shale, to load-bearing shale (which has the highest sensitivity) (Aladwani et al., 2017); previously, Han et al. (2011b) proposed a similar effect for total clay content.Clean sandstone has the lowest resistivity due to the high ionic conduction pathway that move freely in pore space.We see a sharp increase in sensitivity between partial pore-filling and load-bearing shales, indicating different roles in quartz grain framework-dominated porosity versus shalegrain framework-dominated porosity.In the latter, the pores are more susceptible to pressure changes than in the former.
Figure 13 shows the pressure dependence of the joint property (a) F*-Q p −1 of the four newly measured sandstones, colour-coded according to the shale distribution scheme (F*-Q s −1 relationships are similar).The gradients can be seen to decrease systematically both as F*-Q p −1 magnitudes decrease, and with decreasing shale content from clean sandstone through pore-filling and load-bearing shale.However, the load-bearing shale shows steeper decreasing slope than the rest.Apparent formation factor increases with increasing differential pressure, whereas attenuation decreases with increasing differential pressure (Han et al., 2011a).The attenuation with differential pressure data conforms to the results of Jones (1995), Best and Sams (1997) and Han et al. (2011a).
Figure 13 shows the load-bearing shales have the highest apparent formation factor than the rest and have more steeper slope (note F*-Q s −1 shows similar effect).Figure 14 shows the pressure dependence of the joint property (a) V p -Q p −1 of the four newly measured sandstones, colour-coded according to the shale distribution scheme (V s -Q s −1 relationships are similar in Figure 14b).The clean sandstone and load-bearing shale for V p -Q p −1 , in particular, show a steeper decreasing gradient of velocity with increasing attenuation, whereas partial porefilling and homogenous pore-filling shales show a gentle decreasing gradient of velocity with increasing attenuation (note F*-Q s −1 shows similar effect).

Effect of shale distribution on seismic and electrical properties
In this section, we reproduce the parameter cross-plots of Han et al. (2011a), but with the new shale distribution classification imposed.Note that Equations ( 12)-( 14) express the pressure sensitivities in a normalized form, to aid comparisons of magnitudes of GN1, GN2 and GN3; Han (2010) used different, nonnormalized definitions for G1, G2 and G3 reported therein.Figure 15 shows GN1, the normalized pressure sensitivity of the relationship between resistivity ρ and P-wave velocity V p against porosity for the 75 shale-rich sandstones.In general, GN1 decreases with porosity, similar to observations of Han et al. (2011a) for G1; the scatter around this trend is approximately constant.Despite the transition from load-bearing shale samples to pore-filling shale with increasing porosity, the pressure sensitivity of the electrical resistivity −V p relation remains broadly constant on this log-linear scale.Hence, there is an exponential decrease in pressure sensitivity with porosity that is independent of shale distribution.
Figure 16 shows GN2, the normalized pressure sensitivity of the relationship between resistivity ρ and P-wave attenuation Q p −1 against porosity for the 75 shale-rich sandstones.Generally, GN2 decreases in magnitude (becomes less negative and approaches zero) with increasing porosity, from sandy shale (load-bearing shale) to shaly sandstones (i.e. with heterogeneous and critical pore-filling shale) and clean sandstones.However, there is a cluster of points for both sandy shales and shaly sandstones at about 10%-13% porosity that shows a much wider range of GN2 values than at lower and higher porosities.At this stage, it is unclear whether this observation is significant.
Figure 17 shows GN2 against attenuation for the 75 shaly sandstones.Han et al. (2011a) noted the same scatter of data points with no apparent correlation.However, when applying the shale classification scheme, we can see that pore-filling shale has generally lower pressure sensitivity (i.e. less negative and smaller magnitude) than load-bearing shale.This suggests that, in a load-bearing sandstone, the dominant pressure sensitivity is dominated by resistivity.Moreover, load-bearing shales seem to follow a curving trend with G2 increasing to a maximum around Q p −1 = 0.03 and then decreasing for higher Q p −1 values.It is difficult to discern trends for the shaly can clean sandstones as the scatter is high.
Figure 18 shows GN3, the normalized pressure sensitivity of the relation between P-wave velocity V p and attenuation Q p −1 against porosity.It can be seen that the GN3 magnitude decreases with increasing porosity (i.e.becomes more negative).In general, this trend is linear except for values between about 10%-13% porosity as noticed by Han et al. (2011a) for G3.
This indicates higher pressure sensitivity for elastic properties (GN3) around the critical porosity, whether load-bearing or pore-filling shale.This suggests some connection with the degree of disorganization (heterogeneity) of the mineral grains in the sandstone.The shale classification scheme adopted here is idealized, but around the critical porosity, we would expect possibly some load-bearing shales and some pore-filling shales distributed in patches throughout the sandstone.
The shale distribution classification scheme provides more insight in the relationship between quartz grain-born framework, shale-grain-born and the pressure sensitivity of the rock.Figure 15 shows GN1 against porosity, load-bearing shale is dominated at lower porosity, whereas at higher porosity, the clean and partial pore-filling shales are dominated.This suggests the significant impact of the shale-grain-born framework and of low aspect ratio pore in lower porosity, whereas quartz grain-born framework and high aspect ratio pore in high porosity.The relationship between the normalized velocity-attenuation (GN3) and porosity is complete because of the attenuation sensitivity to macropores in the clean sandstone (Group 1), partial pore filling (Group 2) and microporosity in homogenous pore-filling (Group 3) and load-bearing shales (Group 4) in Figure 18 because elastic velocity is not sensitive to various pore types.The GN3 covers less than two orders of magnitude, whereas GN1 and GN2 cover about three orders of magnitude.Therefore, the shrinkage of micro pores in homogenous pore-filling and load-bearing and macroporosity in clean sandstone and partial pore-filling due to changing of differential pressure is much larger in resistivity than attenuation.

CONCLUSIONS
This is the first time a shale classification scheme was applied on a comprehensive joint elastic-electrical dataset of 75 sandstones samples with a broad range of petrophysical properties and as a function of effective pressure from 8 to 60 MPa.We were able to investigate how load-bearing and pore-filling shales affect joint elastic-electrical properties.The following original conclusion can be drawn from the results: 1. Statistical cluster analysis in resistivity-velocity space on a semi-logarithmic scale provided evidence for at least three groups that coincide with the three substantial groups of a shale classification scheme (clean sandstones, shaly sandstones and sandy shales).Sandy shales (load-bearing shale, Cluster 3) have the highest values of F* and V p and show the tightest grouping and steepest trend of all three clusters.Cluster 2 is a clean sandstone and partial pore-filling shale.The pore-filling shale trend straddles the clean sandstone trend and touches the load-bearing shale trend in resistivity-velocity space (Cluster 1).All the three clusterings were approximately linearly correlated in resistivity-velocity space.2. Partial pore-filling shale seems to give rise to increasing attenuation with increasing electrical resistivity up to a maximum when shale content is about equal to frame porosity.However, load-bearing shale seems to give rise to decreasing attenuation with increasing resistivity.Overall, the highest attenuations occur when the volumetric shale content is close to the frame porosity (for Q p −1 in particular), at the transition between pore-filling and load-bearing shales.
3. Changes of parameters measured, such as compressional and shear wave velocity, attenuation and electrical resistivity with effective pressures, when applying shale classification scheme on the dataset, seems to agree with Equation ( 8) which is  =  − e − dif f , where N is petrophysical parameter of interest, ,  and  are the best fit coefficients and  dif f is the effective pressure.4. We demonstrated how load-bearing and partial porefilling shales affect the relationships between resistivity and velocity, resistivity and attenuation and velocity and attenuation with effective pressure.Resistivity has a high degree of sensitivity to pressure in sandstones with load-bearing shale compared to those with pore-filling shale. 5. We showed that elastic properties (V p and Q p −1 shown here, but the same result applies for S-waves), at around the critical porosity, seem to be more sensitive to effective pressure when there is a high degree of heterogeneity among the sandstone mineral grains.On the contrary, when dominated by load-bearing or by pore-filling shale, the grains might have a more homogeneously distributed shale, and the pressure sensitivity follows a more predictable trend when compared to, for example porosity.6. Pressure sensitivity is greatest for load-bearing shale when resistivity is involved in the cross-plot.7. The pressure sensitivity of ultrasonic velocity and attenuation cross-plot is dominated by the quartz grain framework and microcracks in shaly sandstones, and by load-bearing shale in sandy shales.
The results give further guidance for the development of joint property rock physics models, needed for better geophysical property inversions and improved reservoir characterization.These are needed to aid with interpreting geopressure changes in the reservoir and overburden shaly sandstones.

D A T A AVA I L A B I L I T Y S T A T E M E N T
The data that support the findings of this study are available in Appendix A.

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Schematic diagram of the experimental set-up, also showing the arrangement of electrodes around the rock sample.Scales are approximate (sample width is 5 cm).ERT, electrical resistivity tomography.Source: After Falcon-Suarez et al. (2017) and North et al. (2013).

Figure 3
Figure3shows dry compressional modulus and porosity, colour-coded by clay content for all the 75 sandstones.The sandstone samples were colour-coded by clay content just to show that samples with clay content lie significantly below the solid line.The differences between the measured

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Shale classification diagram according to shale fraction X, and frame porosity for the 63 shaly sandstone samples ofHan et al (2011b).Samples with star symbols represent the 12 new samples.The dashed red lines represent 90% and 110% of shale filling the framework porosity.Source: Method adapted fromSørensen and Fabricius (2015).

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I G U R E 6 Thin section images showing the shale distribution within (a) partial pore-filling shale (York 3) Group 2, and (b) load-bearing shale (W160.15H)Group 4.
Han et al. (2011b) divided the sandstone samples into two groups, clean sandstone with less than about 10% clay content and clay-rich sandstone with more than about 10% clay content.Han et al. (2011b) concluded that porosity does not control the joint velocity and resistivity properties, despite porosity F I G U R E 8 Scatter diagram showing S-wave velocity against apparent formation factors colour-coded by K-mean clustering methods: (a) K = 2, (b) K = 3, (c) K = 4 and (d) K = 5.

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Scatter diagram for the Han et al. (2011b) data showing (a) P-wave and (b) S-wave velocities against apparent formation factors colour-coded by porosity at differential pressure 50 MPa.The dark outlined points are new data collected in this study.

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Scatter diagram for the Han (2010) data showing (a) P-wave attenuation and (b) S-wave attenuation against apparent formation factors colour-coded by porosity at differential pressure 50 MPa.The dark outlined points are new data collected in this study.

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Experimental data and regression curves for four newly measured samples showing the dependence on effective pressure of (a) P-wave velocity, (b) P-wave attenuation and (c) electrical resistivity.S-wave velocity and S-wave attenuation show similar trends to those for P-waves.Samples are classified according to shale distribution.

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Examples of (a) P-wave velocity V p and (b) S-wave velocity V s against electrical formation factor F* for different shale distributions in sandstones, for effective pressures 8-60 MPa (for each sandstone, data points in sequence of 8, 20, 30, 40, 50 and 60 MPa from left to right).Arrows show the direction of increasing differential pressure.

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Examples of (a) P-wave attenuation 1/Q p and (b) S-wave attenuation 1/Q s against electrical formation factor F* for different shale distributions in sandstones, for effective pressures 8-60 MPa (for each sandstone, data points in sequence of 8, 20, 30, 40, 50 and 60 MPa from left to right).Arrows show the direction of increasing differential pressure.

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Examples of (a) P-wave velocity V p against P-wave attenuation 1/Q p and (b) S-wave velocity V s against S-wave attenuation 1/Q s for different shale distributions in sandstones, for effective pressures 8-60 MPa (for each sandstone, data points in sequence of 8, 20, 30, 40, 50 and 60 MPa from left to right).Arrows show the direction of increasing differential pressure.

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Scatter diagram showing the relationship between porosity and the GN1 parameter (pressure sensitivity of the resistivity-V p curve).GN1 for resistivity-V s against porosity shows a similar behaviour.The open symbols are(Han et al., 2011a) data, and solid symbols are new measurements from this study.

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Scatter diagram showing the relationship between porosity and the GN2 parameter (pressure sensitivity of the resistivity-Q p −1 curves).GN2 for resistivity-Q s −1 against porosity shows a similar behaviour.The open symbols (Han et al., 2011a) data, and solid symbols are new measurements from this study.

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Scatter diagram showing the relationship between GN2 and Q p −1 .A similar trend is seen for GN2 against Q s −1 .The open symbols are (Han et al., 2011a) data, and solid symbols are new measurements.F I G U R E 1 8 Scatter diagram showing the relationship between porosity and the GN3 parameter (pressure sensitivity of the V p -Q p −1 curves).GN3 for V p -Q s −1 against porosity shows a similar behaviour.The open symbols are(Han et al., 2011a) data, and solid symbols are new measurements.