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A simple effective medium approach for the bulk electrical and elastic properties of organic-rich shales
Journal of Applied Geophysics 169 (2019) 98–108
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A simple effective medium approach for the bulk electrical and elastic properties of organic-rich shales Kelvin Amalokwu a,⁎,1, Kyle Spikes a, Kevin Wolf b a b
Department of Geological Sciences, John A. and Katherine G. Jackson School of Geosciences, The University of Texas at Austin, Austin, TX 78712, United States of America BP America, 501 Westlake Park Blvd, Houston, TX 77079, United States of America
a r t i c l e
i n f o
Article history: Received 6 December 2018 Received in revised form 16 March 2019 Accepted 10 June 2019 Available online 21 June 2019
a b s t r a c t Shales are widespread in the Earth's crust, and organic-rich shales have become an important energy resource. Geophysical data are becoming increasingly important for characterizing shale reservoirs to find zones of good quality, typically characterized by high kerogen and low clay content. Away from well control, most geophysical characterization of shale reservoirs are based on the effects of kerogen on the elastic properties of the shale rock. However, both clay and kerogen act to reduce the stiffness of shale rocks, and their effects might be inseparable using seismic properties alone. A combination of the elastic and electrical properties of organic-rich shales might improve our ability to locate zones of good quality by decoupling the effects of kerogen and clay. Kerogen is considerably more resistive than clay and should, in theory, increase the bulk resistivity of the shale when present. The effects of shale composition on the joint elastic-electrical properties of shales can be explored using a combination of suitable well data and rock physics modeling. A comprehensive suite of well data could be used to observe trends, with a joint model aiding trend analysis. Effective medium models lend themselves well to estimating joint effective properties of a heterogeneous composite. This is because using the same approach for both the elastic and electrical properties would ensure an equivalent microstructure for both properties, as would be expected in practice. In this study, a joint effective medium model was calibrated and constrained using well data, achieving good correlation with elastic and electrical well log data from two wells. The results show that both kerogen and clay tend to reduce the stiffness of shales as expected, however; kerogen plays a significant role in increasing the bulk electrical resistivity of shales, whereas clay has the opposite effect. This joint elastic-electrical approach could potentially help discriminate kerogen-rich from clay-rich shales, in addition to constraining each individual property. © 2019 Published by Elsevier B.V.
1. Introduction Organic rich shales have become an attractive source of unconventional hydrocarbons, which has led to a drive to understand shale properties better. Through a combination of horizontal drilling and hydraulic fracturing, oil companies are able to access shale plays that were considered uneconomic a few years ago (Glaser et al., 2013). Shale reservoirs typically have low porosity and extremely low permeability, leaving the hydraulic fractures to provide the primary pathways for fluid flow into the well. Therefore, drilling and hydraulic fracturing jobs have to be designed to coincide with zones of good reservoir quality. Geophysical data is becoming increasingly important for characterizing shale reservoirs to find reservoir zones of good quality (so-called “sweetspots”).
⁎ Corresponding author. E-mail address: [email protected] (K. Amalokwu). 1 Work was done while the corresponding author was at UT Austin.
https://doi.org/10.1016/j.jappgeo.2019.06.005 0926-9851/© 2019 Published by Elsevier B.V.
High kerogen content and low clay content are important characteristics of sweet-spots in shale reservoirs (Josh et al., 2012). Most geophysical characterization methods for shale reservoirs are based on the effects of kerogen on the elastic properties of the shale rock. However, both clay and kerogen are more compliant than the minerals that are predominant in reservoir rocks (e.g., quartz, calcite and dolomite), and as such they generally act to reduce the stiffness of these rocks. Although individual clay platelets are stiffer than kerogen (e.g., Mavko et al., 2009), their effects on the overall bulk property of the rock might be indistinguishable from the characterization of elastic properties alone, which is usually in the form of elastic wave velocities from seismic data. This study looks at the possibility of de-coupling the effects of clay and kerogen by looking at their contrasting electrical properties. Because kerogen is significantly more resistive than clay (e.g., Yang et al., 2016), in theory, they should have opposite effects on the bulk resistivity of the shale. Many studies have looked at the effect of kerogen on the bulk elastic properties of shales, but only a few studies have looked at the joint elastic-electrical response of shales (e.g., Bachrach, 2011;
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Woodruff et al., 2015; Sayar and Torres-Verdín, 2016). However, these joint elastic-electrical studies of shales focused mainly on constraining the elastic properties using electrical properties, rather than the effect of kerogen on the bulk electrical response, as far as the authors have been able to determine. This study analyses what effect kerogen has on the bulk electrical response of shales, in addition to constraining elastic properties using electrical properties (and vice versa). Effective medium models have been used widely and successfully to model the effective properties of heterogeneous sedimentary rocks as they can incorporate different geometries and volume fractions of constituents. The self-consistent approximation (SCA) and differential effective medium (DEM) methods are the two more commonly used methods for estimating the effective properties of microheterogeneous rocks (Jakobsen et al., 2000). The DEM method preserves the connectivity of the initial background phase (Sheng, 1990; Berryman, 1992) while other constituents are added as isolated inclusions; as a result, the DEM method depends on the order in which the constituents are added. The SCA method on the other hand has no preferred background or inclusion and treats the constituents symmetrically, e.g., clay+water is the same as adding water+clay. However, within the SCA scheme, the constituents are only connected within a volume concentration (from 0.4– 0.6). Therefore, in a case where there are biconnected phases, neither the SCA nor the DEM is entirely satisfactory on its own, but their combination has been shown to have the capability of modeling the joint electrical and elastic properties of isotropic porous rocks (e.g., Han et al., 2011) and the elastic properties of anisotropic porous rocks (e.g., Hornby et al., 1994; Jakobsen et al., 2000). We do not account for cracks/fractures or other stress-dependent microstructural details in other to keep the modeling approach fairly simple. The effective medium models mentioned above along with orientation distribution functions have been used to account for cracks in shale models (e.g., Sayar and Torres-Verdín, 2016; Ren and Spikes, 2016). The noninteracting approximation has also been show to be well suited to account for cracks/fractures in rocks (e.g., Kachanov, 1993; Sayers and Kachanov, 1995; Sarout and Guéguen, 2008). We use a rich suite of well log data to calibrate our shale model, which is a combination of the SCA and DEM methods. The joint modeling approach ensures the elastic and electrical properties of the composite have the same microstructure (e.g., Han et al., 2016; Sarout et al., 2017). We do not focus on the modeling approach; rather, we focus on analyzing the effect of kerogen on the bulk elastic and electrical properties of the shales by using a joint elastic-electrical model constrained by petrophysical data, and by the joint (elastic-electrical) modeling approach. More importantly, we use the data to observe the influence of kerogen on these bulk properties and the calibrated model to analyze the controlling factors. The results suggest electrical properties could improve remote discrimination of kerogen in organic-rich shales where good quality seismic and electrical data are available. 2. Models The details of the combined SCA + DEM approach and its motivations are well described in the literature (e.g., Sheng, 1990; Hornby et al., 1994; Jakobsen et al., 2000; Han et al., 2011), so we give a brief summary here. The combined SCA + DEM approach can be used to obtain a bi-connected two-phase composite at any volume concentration. The approach begins by combining the two phases (phase 1 and phase 2) using the SCA method at a volume concentration where the phases are connected in the SCA method (0.4–0.6), which we will refer to as the “critical concentration φc”. The mixture at the critical concentration is then taken to be the background for the next step. If the volume fraction of phase 2 in the target effective medium is less than the critical concentration, the appropriate amount of phase 1 is added, else the appropriate amount of phase 2 is added.
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Fig. 1 shows a schematic illustrating this approach, which can be extended to include more than two phases, but only two phases can be combined at each step. The same process applies to both the elastic and the electrical case. Although not show in this manuscript, different approaches to combine the individual mineral constituents using effective medium theory were attempted. Using DEM alone with either clay, kerogen, quartz or carbonates as the background material proved sufficient for the elastic properties but not for the electrical properties. Using orientation distribution functions in conjunction with the DEM model still proved insufficient. The model with clay background under-predicted the electrical resistivity whereas those with a quartz, keorgen or carbonate background over-predicted the electrical resistivity. Introducing a single step of combined SCA + DEM approach can improve the results depending on how the individual constituents are combined. However, the electrical resistivity predictions were still poor. The approach we found to work well after trying many configurations is shown in Fig. 2. The first step of the approach has two separate parts done in parallel. One part combines kerogen and clay using DEM with kerogen as the background. The aspect ratio of clay used here was 0.2. The other part of the first step starts with creating a bi-connected sandstone by combining quartz and the effective fluid (brine+gas) using the SCA + DEM approach, to which the remaining minerals are added using the DEM method. The aspect ratios for all the components in this step were set to 1.0. The resulting composite is then combined with the kerogen-clay mixture using another step of the SCA + DEM approach. The aspect ratio for the kerogen-clay mix was set to 0.2, while that for the sanstone mixture was set to 1.0. The effective fluid (bulk modulus and electrical conductivity) was obtained using Bries average (Brie et al., 1995) with exponent e = 2, as opposed to Reuss or Voigt averages, which did not work as well for the electrical resistivity. The effective electrical conductivity of a gas-liquid mix appears to follow an exponential-type relationship. Bries law was chosen since it has been used widely for computing effective fluid bulk modulus of liquid-gas mixture, with the moduli replaced with the conductivities for computing the effective fluid electrical conductivity of the brine+gas mix.
2.1. Elastic models For the elastic case, the anisotropic DEM and SCA formulations by Hornby et al. (1994) will be used. The DEM equation is given by dC DEM ¼
dvi C i −C DEM Q i ; 1−vi
ð1Þ
where h i−1 ^ C i −C DEM ; Qi ¼ I þ G
ð2Þ
where Ci is the stiffness tensor of the ith component with volume fraction vi; I is the identity matrix; CDEM is the stiffness tensor of the mate^ is a fourth-rank Eshelby tensor associated with the aspect rial; and G ratios of inclusions. The SCA equation is given by (Hornby et al., 1994; Jakobsen et al., 2000)
C
SCA
¼
( N X i¼1
)( vi C i Q i
N
∑ v jQ j j¼1
)−1 ð3Þ
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Fig. 1. Schematic showing the combined SCA + DEM approach (both elastic and electrical). The SCA model is used to calculate the effective properties at a critical concentration φc. This then becomes the background for the DEM step to calculate the final effective properties at any volume concentration. Image adapted from Han et al. (2011).
Fig. 2. Schematic showing the modeling approach used in this study. It contains two stages of the SCA + DEM approach. The first step of the approach has two separate parts done in parallel, one combining kerogen and clay using DEM, the other creating a fluid-saturated sandstone using the SCA + DEM approach and adding other minerals using DEM. The final step combines the two resulting composites from the first step by using SCA + DEM.
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2.2. Electrical models
where
The anisotropic DEM equation for electrical conductivity is given by e.g., (Mendelson and Cohen, 1982; Dürr et al., 2002)
L¼
dσDEM
h i−1 dvi −1=2 −1=2 ¼− ðσi −σDEM Þ 1 þ σDEM i σDEM ðσi −σDEM Þ ; 1−vi
ð4Þ
where σi is the conductivity tensor of the ith component with volume fraction v; σDEM is the conductivity tensor of the material; and the matrix i is related to the ith phase through the depolarization vectors. For oblate spheroids with semiaxes b1 b b2 = b3, i simplifies to a diagonal matrix of depolarization vectors L1, L2 = L3. The depolarization vectors are given by (Landau and Lifshitz, 1960; Mendelson and Cohen, 1982)
L1 ¼ L; L2 ¼ L3 ¼
ð1−LÞ ; 2
ð5Þ
1 þ e3 e− tan−1 e ; e2
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ð6Þ
where e = [(b2/b1)2 − 1]1/2. For oblate spheroids, the SCA equation for electrical conductivity for a two-component material is given by (e.g., Hornby et al., 1993; Berryman and Hoversten, 2013) 2 X v j σ j −σ SCA i ¼0 1 þ Γi σ j −σ SCA i j¼1
ð7Þ
where this is evaluated for i = 1,2,3, corresponding to the directions of the principal axes of the grain (oblate spheroid), which is the same as the principal axes of the effective conductivity of the composite medium (Mendelson and Cohen, 1982). Hence, i = 2 and i = 3 are equivalent for the case of spheroidal inclusions. Γi = Li/σSCA i , where Li is the same as in the DEM case.
Fig. 3. Mineralogy data from ECS logs from a) Well 1 and b) Well 2. Well 1 contains about eleven minerals, with the reservoir portion (bxx40m) dominated mainly by clay, quartz and kerogen. Kerogen is only present within the interval ∼xx21 − xx40m. The ECS logs from Well 2 shows a total of five mineral components, with the dominant minerals within the interval being kerogen, clay, QFM (quartz, feldspar, mica).
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Fig. 4. Well log data from Well 1 showing (from left to right) gamma ray, formation electrical resistivity, total porosity, water saturation, formation bulk density, P-wave velocity and Swave velocity. This interval represents the reservoir unit and is the interval that has been most characterized in the well. Note that there are missing data points at depths of ∼ xx40-xx41 m and Nxx42m for S-wave velocity (Vs), making it look like the data has been chopped at those depths.
3. Data Well log data from two different shale reservoirs were used for this study. The reservoirs are well characterized in terms of mineralogy and fluid content, which provided important constraints for the modeling inputs. Fig. 3 shows mineralogy data for both wells obtained from elemental capture spectroscopy (ECS) logs. Fig. 4 and Fig. 5 show log measurements of other properties in the reservoirs for Well 1 and Well 2, respectively. Note that the kerogen content has been calibrated to log data, and a separate log for kerogen content was provided for both wells. Mineralogy data from ECS shows Well 1 contains about eleven minerals within the selected interval (xx15 − xx45m), with kerogen only present within the interval ∼xx21 − xx40m. Above xx40m, kerogen, clay, quartz, and K-feldspar are the more dominant minerals, with dolomite, plagioclase and pyrite present in smaller amounts, and calcite, anhydrite, siderite and apatite occurring in trace amounts. Below xx40m, calcite and dolomite dominate, with other minerals occurring in smaller amounts (e.g., quartz, clay and plagioclase). In Well 2, kerogen is present everywhere along the selected interval of xx26 − xx58m, and the ECS data shows a total of six minerals in the well. The amount of siderite in the well is insignificant (b1%), so there are effectively five mineral components from the ECS. The dominant
minerals in Well 2 are kerogen, clay, QFM (quartz, feldspar, mica), with small amounts of carbonate and pyrite. Fig. 4 and Fig. 5 show gamma ray (GR), formation electrical resistivity (res.), total porosity (ϕtotal), water saturation (Sw), formation bulk density (ρb), P-wave velocity (Vp) and S-wave velocity (Vs) for both wells. The intervals shown here are the reservoir units, which are the intervals with comprehensive well log data. For model calibration, all the logs shown here are used except the GR logs. The ϕtotal and Sw logs are used as inputs for the model, and the remaining (resistivity, Vp, Vs, ρb) are used for comparison and calibration. 4. Results We present the results of applying the same joint electrical-elastic model to two different wells. Mineral composition and volume fractions were obtained from ECS logs, and pore fluid (gas saturation) information from saturation logs. Kerogen content and fluid saturation from logs were not just estimated from well logs, but also calibrated to core data, thereby improving the accuracy of the estimates. The properties of individual minerals were taken from the literature (Telford et al., 1990; Mitchell, 2004; Bandyopadhyay, 2009; Mavko et al., 2009; Hu et al., 2014; Yang et al., 2016). Well 1 brine resistivity, bulk modulus and density are 0.125 Ωm, 3.127 GPa and 1.086 kgm−3, respectively.
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Fig. 5. Well log data from the reservoir interval in Well 2 showing (from left to right) gamma ray, formation electrical resistivity, total porosity, water saturation, formation bulk density, Pwave velocity and S-wave velocity.
For Well 2, brine resistivity, bulk modulus and density are 0.028 Ωm, 3.127 GPa and 1.086 kgm−3, respectively. For both wells, gas resistivity, bulk modulus and density are 108 Ωm, 0.0184 GPa and 0.0838 kgm−3, respectively. These values were obtained using salinity, pressure and temperature information from the wells as input into an in-house software package. Table 1 shows the input electrical conductivities, aspect ratio, and bulk and shear moduli for the different minerals. Using these inputs in the workflow shown in Fig. 2, the elastic (bulk and shear modulus) and electrical (electrical resistivity) model responses can be computed. Brine salinity measured from cores along with pressure and temperature information were used to estimate brine resistivity, bulk modulus and bulk density. The main parameters left to fit the model with become the exponent in Bries formula and aspect ratio
of clay and the kerogen+clay mix (note that kerogen was the background in the first step of the DEM), set to be the same. We quickly found that a value of 0.2 (within the range used in the literature) gave good results for Well 1, without having to use any optimization techniques. The same aspect ratios were then used for Well 2. Strictly speaking, the properties of some individual minerals (including aspect ratios) could well vary with depth, but we have assumed they are constant across the entire length of the wells for simplicity. The idea was to come up with a fairly simple modeling approach capable of generalizing to different wells. Figs. 6 and 7 show the model fit (red curve) to the well log data (black curve). The parameters shown are P-wave modulus in GPa, which represents the C33 component of the elastic stiffness tensor; Swave modulus in GPa, which represents the C44 componenet of the
Table 1 Table showing the bulk modulus, shear modulus, electrical conductivity and aspect ratio of the different minerals used in the effective medium models (see Telford et al., 1990; Mitchell, 2004; Bandyopadhyay, 2009; Mavko et al., 2009; Hu et al., 2014; Yang et al., 2016).
K (GPa) μ(GPa) σ(S/m) Aspect ratio
Clay
Kerogen
Quartz
Calcite
Dolomite
Siderite
K-feldspar
Plagioclase
Anhydrite
Apatite
Pyrite
27 7 0.05 0.2
6.6 3.2 10−7 0.2
37 44 10−11 1
76.8 32 10−11 1
94.9 45 10−11 1
123.7 51 0.014 1
37.5 15 10−4 1
75.6 25.6 10−4 1
56.1 29.1 10−8 1
89 44.5 10−6 1
138.3 109.8 3.33 1
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Fig. 6. Model comparison to well log data for Well 1. The figure shows comparisons for (from left to right) P-wave modulus, S-wave modulus and electrical resistivity, with the fourth pane showing the mineral composition as a function of depth. The model fits the data pretty well for both elastic moduli and for the electrical resistivity.
elastic stiffness tensor; and electrical resistivity in Ωm, which represents horizontal resistivity ρ11 of the electrical tensor. In Well 1, the overall fit between data and model (Fig. 6) is very good for the P- and S- moduli and for the electrical resistivity. There are missing data points at depths of ∼ xx40-xx4 m and N xx42 m for the Smodulus; however, the other well logs have measurements for these depth. The P- modulus ranges from ∼ 39–140 GPa, while the S- modulus ranges from ∼ 13–29 GPa, and the electrical resistivity ranges from 1 to 1709 Ωm. Both elastic moduli stay fairly constant until depths of ∼ xx40 m where there is an increase in moduli that corresponds to higher contents of carbonates (calcite and dolomite) and is representative of the underlying rock layer. The formation electrical resistivity, however, shows more variation with depth (compared to the elastic moduli), suggesting that the “soft components” (kerogen, clay and pore fluids) have different effects on the bulk electrical resistivity. In Well 2, the same modeling approach used in Well 1 gives a good fit for the P-wave and S-wave moduli and for the electrical resistivity. Apart from the over-prediction in electrical resistivity at ∼ xx27 m and ∼ xx50 m, the model matches the data both in magnitude and trend quite well. The values for P- wave modulus, S-wave modulus and electrical resistivity are lower in this well relative to Well 1, ranging from ∼ 20–67 GPa, 6–23 GPa, and ∼ 16–653 Ωm, respectively. These lower values (compared to Well 1) are most likely due to the higher clay content in Well 2 (see Fig. 3). There is not a great deal of variation in both elastic moduli, as the volume fractions of the “harder components”
(carbonates) are not significant and do not vary much along the interval. The formation electrical resistivity shows a little bit more variation with depth for similar reasons as in Well 1. However, the variation is not very significant because the volume fractions of the “soft components” (kerogen, clay and pore fluids) do not vary quite as much with depth as in Well 1. 5. Discussion The model gives a good fit to the well data for both the elastic and electrical properties in the two wells, both in magnitude and trend. Because we want to evaluate the possibility of decoupling the “soft” elastic response of kerogen and clay by their different electrical responses, we focus more on the effect of kerogen on the bulk electrical resistivity of shales. Fig. 8 shows a crossplot of electrical resistivity and P-wave modulus for Well 1, color coded by kerogen content. The log data (Fig. 8a) is generally well reproduced by the model (Fig. 8b). It can be seen that shales with higher kerogen content tend to have higher electrical resistivity values compared to the lower electrical resistivity values of clay-rich portions. We attribute this to be mainly due to the contrast in intrinsic resistivity of clays versus kerogen as we will aim to show using thought experiments in subsequent paragraphs. Both the kerogen-rich and clay-rich portions have lower values of P-wave modulus (compared to where their concentrations are low) but are distinguishable by the contrast in electrical resistivity values.
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Fig. 7. Comparison between model and well log data for Well 2. The results show a good fit between model and data for (from left to right) P-wave modulus, S-wave modulus and electrical resistivity. The fourth pane shows the mineral composition as a function of depth.
Fig. 8. Kerogen effects in Well 1 evaluated using a cross-plot of electrical resistivity vs. P-wave modulus. a) Well log data b) Model output. The model reproduces the data fairly well. Higher kerogen content tends to give rise to higher electrical resistivity values. Also, softer zones (high kerogen and clay content) have lower values of P-wave modulus as expected.
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Fig. 9. Well 2 cross-plot of electrical resistivity vs. P-wave modulus for a) well log data b) model output. Again, higher kerogen content tends to produce higher electrical resistivity values as seen in Fig. 8, which is fairly well reproduced by the model.
Fig. 10. Plots showing results of replacing kerogen with an equivalent volume fraction of clay in the model and comparing the results to the well log data for Well 1. From left to right, the figure shows P-wave modulus, S-wave modulus and electrical resistivity. The model grossly under-predicts the electrical resistivity of areas where kerogen has been replaced but only slightly over-predicts both elastic moduli.
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A similar crossplot for Well 2 is shown in Fig. 9. Again, the well log data (Fig. 9a) and the model (Fig. 9b) are in good agreement. Kerogen is present along the entire length of the well interval used for this study (see Fig. 3b); however, the general trend is still higher kerogen content tends to plot toward higher values of electrical resistivity. The variation in P-wave modulus is small because the amount of stiffer minerals (e.g., carbonates, quartz) do not vary very significantly, as pointed out earlier. The calibration of the model (as described in the Section 4), with the help of pertinent petrophysical properties (i.e., mineralogy and fluid properties) from a comprehensive suite of well data from two wells, allows further exploration of kerogen effects on the bulk resistivity of shales beyond observing trends seen in the data. Using the calibrated model, we can explore “what if” scenarios, such as replacing the kerogen with clay. This exercise was done to show that kerogen influences the bulk electrical resistivity significantly, when present. In this thought experiment, the elastic moduli, the electrical conductivity and density of kerogen is replaced by those of clay, while every other aspect of the model remains the same. Figs. 10 and 11 show results of replacing the volume fraction occupied by kerogen with clay in the model, compared with the actual well log data. In both wells, it can be seen that kerogen has a significant influence on the bulk resistivity of the shales, as the kerogen-free resistivity predicted is very low and nowhere close to the measured bulk
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resistivity. The elastic properties, however, are slightly over-predicted, but the effect on the elastic properties is not as high as the effect on the electrical properties. In Well 1, resistivity of zones without kerogen (bxx21m and Nxx40m) are in good agreement with the model (Fig. 10), which gives us more confidence in the predictions. A similar comparison cannot be made in Well 2 because kerogen is present along the entire interval. It should be pointed out that although kerogen has a significant effect on the bulk resistivity of the shales, it is not the only controlling factor, as the pore fluids also play an important role as expected. However, it is clear that kerogen has a significant impact on the bulk resistivity of the shales in these wells. The simplicity of the modeling approach is attractive, but there are some limitations as a result. There is an interplay between the aspect ratio (geometrical detail) and the orientation of the clay platelets, and changing either can have similar effects. However, we did not use an orientation distribution function to avoid introducing another free parameter, which was unnecessary, looking at the model-data fit. The implication of this is that both effects (geometrical and orientation) are lumped into one parameter (aspect ratio), affecting our estimates of the actual aspect ratios and anisotropy. This is a reasonable tradeoff as we do not have data on either aspect ratio or anisotropy, and it keeps the model fairly simple. The modeling approach does not account for fractures/cracks, but this does not seem to have affected the
Fig. 11. Well 2 results when kerogen is replaced with an equal volume fraction of clay in the model and compared to the well log data. The results show that again, the electrical resistivity of areas where kerogen has been replaced is grossly under-predicted, but the over-prediction of both moduli is not as significant.
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goodness of fit significantly. This could suggest the effect of fractures on the measured properties is not significant in the measurement direction due to the orientation of the fractures (expected to be vertical) relative to the direction of measurement. Note that the measured elastic property is that in the vertical direction, while the electrical property measured is that in the horizontal direction. However, this situation might not always be the case, and this modeling approach would need to be tested on different shale types to determine its generality. Accounting for fractures/cracks could improve the model predictions but in the process introduce additional poorly constrained parameters. Also, it is not clear how to calculate the effective fluid electrical conductivity of a mutiphase fluid. Although the success of Bries formula suggests an exponential-type relationship between gas saturation and electrical conductivity, more work is needed to establish the true nature of this relationship. The step in the model where a bi-connected sandstone is created is meant to illustrate that if we have quartz in the system, the quartz aggregates when distributed in the final shale sample behave like sandstones which typically exhibit interconnected behavior. However, it is hard to track the actual connectivities in the final shale composite and it is important avoid over-interpreting the model because although models are helpful, they are still idealized representations. Although the reservoirs used in this study are relatively quartz-rich, it can be seen from Fig. 6 that the model still works fairly well for the underburden in Well 1 which is carbonate-rich. Further model validation in reservoirs with different lithology would be helpful in determining generality of modeling approach. The ultimate goal is to be able to invert for these rock and fluid properties through a joint-inversion of surface seismic and electrical data, but obtaining data (especially electrical) with such resolution to do this might not be possible in practice presently. However, identification of zones that could potentially be richer in kerogen from surface data (electrical and seismic) might be possible given the right conditions, and rock physics modeling could help constrain the interpretation. For example, absolute values can be ambiguous, but a normalization with background/ kerogen-free response (obtained from modeling and/or data) could be attempted, separating the different responses into different zones, possibly improving interpretability. 6. Conclusion The effect of kerogen on the joint elastic-electrical properties of kerogen-rich shales was analyzed in two wells by using well data and rock physics modeling. This joint approach provides a means of not only constraining the individual properties, but more importantly, could help decouple the effects of kerogen and clay, characterized by their similar ‘soft’ elastic responses. Using the same effective medium modeling approach for both properties ensured an equivalent microstructure of the shale composite for both the elastic and electrical properties. The results show that the higher electrical resistivity of kerogen does tend to increase the bulk electrical resistivity of shales whereas clay tends to reduce it. This characteristic could potentially be used to improve sweet-spot identification in organic-rich shales or aid in the identification of zones with higher hydrocarbon potential during exploration/appraisal, provided good quality electrical resistivity data can be obtained. Our analyses suggest that if other constituents remain relatively constant, shales that are clay-rich (negligible kerogen content) have significantly lower electrical resistivity than when kerogen is present, but elastic properties might not be as significantly affected.
Acknowledgments The authors wish to thank BP for funding this project and for supporting the publication of this work. We would also like to thank the reviewers for helpful comments and suggestions. References Bachrach, R., 2011. Elastic and resistivity anisotropy of shale during compaction and diagenesis: joint effective medium modeling and field observations. Geophysics 76 (6), E175–E186. Bandyopadhyay, K., 2009. Seismic Anisotropy: Geological Causes and its Implications to Reservoir Geophysics (Phd thesis). Berryman, J.G., 1992. Single-scattering approximations for coefficients in biot’s equations of poroelasticity. J. Acoust. Soc. Am. 91 (2), 551–571. Berryman, J.G., Hoversten, G.M., 2013. Modelling electrical conductivity for earth media with macroscopic fluid-filled fractures. Geophys. Prospect. 61 (2), 471–493. Brie, A., Pampuri, F., Marsala, A., Meazza, O., 1995. Shear sonic interpretation in gasbearing sands. SPE Annual Technical Conference and Exhibition. Dürr, O., Dieterich, W., Nitzan, A., 2002. Effective medium theory of conduction in stretched polymer electrolytes. J. Phys. Chem. B 106 (24), 6149–6155. Glaser, K.S., Miller, C.K., Johnson, G.M., Toelle, B., Kleinberg, R.L., Miller, P., Pennington, W.D., 2013. Seeking the sweet spot: reservoir and completion quality in organic shales. Oilfield Review 25 (4), 16–29. Han, T., Best, A.I., MacGregor, L.M., Sothcott, J., Minshull, T.A., 2011. Joint elastic-electrical effective medium models of reservoir sandstones. Geophys. Prospect. 59 (4), 777–786. Han, T., Clennell, M.B., Cheng, A.C., Pervukhina, M., 2016. Are self-consistent models capable of jointly modeling elastic velocity and electrical conductivity of reservoir sandstones? Geophysics 81 (4), D377–D382. Hornby, B.E., Schwartz, L.M., Hudson, J.A., 1993. Effective medium modeling of the electrical and elastic properties of anisotropic porous media. SEG Tech. Prog. Expanded Abstracts 1993, 786–791. Hornby, B.E., Schwartz, L.M., Hudson, J.A., 1994. Anisotropic effective-medium modeling of the elastic properties of shales. Geophysics 59 (10), 1570–1583. Hu, H., Dai, L., Li, H., Jiang, J., Hui, K., 2014. Electrical conductivity of k-feldspar at high temperature and high pressure. Mineral. Petrol. 108 (5), 609–618. Jakobsen, M., Hudson, J.A., Minshull, T.A., Singh, S.C., 2000. Elastic properties of hydratebearing sediments using effective medium theory. J. Geophys. Res. 105 (B1), 561–577. Josh, M., Esteban, L., Delle Piane, C., Sarout, J., Dewhurst, D.N., Clennell, M.B., 2012. Laboratory characterisation of shale properties. J. Pet. Sci. Eng. 8889, 107–124. Kachanov, M., 1993. Elastic solids with many cracks and related problems. Advances in Applied Mechanics. 30. Elsevier, pp. 259–445. Landau, L.D., Lifshitz, E.M., 1960. Electrodynamics of Continuous Media. Pergamon Press, Oxford. Mavko, G., Mukerji, T., Dvorkin, J., 2009. The Rock Physics Handbook. Cambridge University Press. Mendelson, K.S., Cohen, M.H., 1982. The effect of grain anisotropy on the electrical properties of sedimentary rocks. Geophysics 47 (2), 257–263. Mitchell, B.S., 2004. An Introduction to Materials Engineering and Science for Chemical and Materials Engineers. John Wiley and Sons. Ren, Q., Spikes, K.T., 2016. Modeling the effects of microscale fabric complexity on the anisotropy of the eagle ford shale. Interpretation 4 (2), SE17–SE29. Sarout, J., Guéguen, Y., 2008. Anisotropy of elastic wave velocities in deformed shales: part 2modeling results. Geophysics 73 (5), D91–D103. Sarout, J., Cazes, E., Delle Piane, C., Arena, A., Esteban, L., 2017. Stress-dependent permeability and wave dispersion in tight cracked rocks: Experimental validation of simple effective medium models. J. Geophys. Res. Solid Earth 122 (8), 6180–6201. Sayar, P., Torres-Verdín, C., 2016. Using anisotropic effective medium theories to quantify elastic properties of sandstone-shale laminated rocks. Geophysics 81 (4), D315–D333. Sayers, C., Kachanov, M., 1995. Microcrack-induced elastic wave anisotropy of brittle rocks. J. Geophys. Res. Solid Earth 100 (B3), 4149–4156. Sheng, P., 1990. Effective-medium theory of sedimentary rocks. Phys. Rev. B 41, 4507–4512. Telford, W.M., Geldart, L.P., Sheriff, R.E., 1990. Applied Geophysics. vol. 1. Cambridge University Press. Woodruff, W.F., Revil, A., Prasad, M., Torres-Verdín, C., 2015. Measurements of elastic and electrical properties of an unconventional organic shale under differential loading. Geophysics 80 (4), D363–D383. Yang, A., Firdaus, G., Heidari, Z., 2016. Electrical resistivity and chemical properties of kerogen isolated from organic-rich mudrocks. Geophysics 81 (6), D643–D655.
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Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
A simple effective medium approach for the bulk electrical and elastic properties of organic-rich shales Kelvin Amalokwu a,⁎,1, Kyle Spikes a, Kevin Wolf b a b
Department of Geological Sciences, John A. and Katherine G. Jackson School of Geosciences, The University of Texas at Austin, Austin, TX 78712, United States of America BP America, 501 Westlake Park Blvd, Houston, TX 77079, United States of America
a r t i c l e
i n f o
Article history: Received 6 December 2018 Received in revised form 16 March 2019 Accepted 10 June 2019 Available online 21 June 2019
a b s t r a c t Shales are widespread in the Earth's crust, and organic-rich shales have become an important energy resource. Geophysical data are becoming increasingly important for characterizing shale reservoirs to find zones of good quality, typically characterized by high kerogen and low clay content. Away from well control, most geophysical characterization of shale reservoirs are based on the effects of kerogen on the elastic properties of the shale rock. However, both clay and kerogen act to reduce the stiffness of shale rocks, and their effects might be inseparable using seismic properties alone. A combination of the elastic and electrical properties of organic-rich shales might improve our ability to locate zones of good quality by decoupling the effects of kerogen and clay. Kerogen is considerably more resistive than clay and should, in theory, increase the bulk resistivity of the shale when present. The effects of shale composition on the joint elastic-electrical properties of shales can be explored using a combination of suitable well data and rock physics modeling. A comprehensive suite of well data could be used to observe trends, with a joint model aiding trend analysis. Effective medium models lend themselves well to estimating joint effective properties of a heterogeneous composite. This is because using the same approach for both the elastic and electrical properties would ensure an equivalent microstructure for both properties, as would be expected in practice. In this study, a joint effective medium model was calibrated and constrained using well data, achieving good correlation with elastic and electrical well log data from two wells. The results show that both kerogen and clay tend to reduce the stiffness of shales as expected, however; kerogen plays a significant role in increasing the bulk electrical resistivity of shales, whereas clay has the opposite effect. This joint elastic-electrical approach could potentially help discriminate kerogen-rich from clay-rich shales, in addition to constraining each individual property. © 2019 Published by Elsevier B.V.
1. Introduction Organic rich shales have become an attractive source of unconventional hydrocarbons, which has led to a drive to understand shale properties better. Through a combination of horizontal drilling and hydraulic fracturing, oil companies are able to access shale plays that were considered uneconomic a few years ago (Glaser et al., 2013). Shale reservoirs typically have low porosity and extremely low permeability, leaving the hydraulic fractures to provide the primary pathways for fluid flow into the well. Therefore, drilling and hydraulic fracturing jobs have to be designed to coincide with zones of good reservoir quality. Geophysical data is becoming increasingly important for characterizing shale reservoirs to find reservoir zones of good quality (so-called “sweetspots”).
⁎ Corresponding author. E-mail address: [email protected] (K. Amalokwu). 1 Work was done while the corresponding author was at UT Austin.
https://doi.org/10.1016/j.jappgeo.2019.06.005 0926-9851/© 2019 Published by Elsevier B.V.
High kerogen content and low clay content are important characteristics of sweet-spots in shale reservoirs (Josh et al., 2012). Most geophysical characterization methods for shale reservoirs are based on the effects of kerogen on the elastic properties of the shale rock. However, both clay and kerogen are more compliant than the minerals that are predominant in reservoir rocks (e.g., quartz, calcite and dolomite), and as such they generally act to reduce the stiffness of these rocks. Although individual clay platelets are stiffer than kerogen (e.g., Mavko et al., 2009), their effects on the overall bulk property of the rock might be indistinguishable from the characterization of elastic properties alone, which is usually in the form of elastic wave velocities from seismic data. This study looks at the possibility of de-coupling the effects of clay and kerogen by looking at their contrasting electrical properties. Because kerogen is significantly more resistive than clay (e.g., Yang et al., 2016), in theory, they should have opposite effects on the bulk resistivity of the shale. Many studies have looked at the effect of kerogen on the bulk elastic properties of shales, but only a few studies have looked at the joint elastic-electrical response of shales (e.g., Bachrach, 2011;
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Woodruff et al., 2015; Sayar and Torres-Verdín, 2016). However, these joint elastic-electrical studies of shales focused mainly on constraining the elastic properties using electrical properties, rather than the effect of kerogen on the bulk electrical response, as far as the authors have been able to determine. This study analyses what effect kerogen has on the bulk electrical response of shales, in addition to constraining elastic properties using electrical properties (and vice versa). Effective medium models have been used widely and successfully to model the effective properties of heterogeneous sedimentary rocks as they can incorporate different geometries and volume fractions of constituents. The self-consistent approximation (SCA) and differential effective medium (DEM) methods are the two more commonly used methods for estimating the effective properties of microheterogeneous rocks (Jakobsen et al., 2000). The DEM method preserves the connectivity of the initial background phase (Sheng, 1990; Berryman, 1992) while other constituents are added as isolated inclusions; as a result, the DEM method depends on the order in which the constituents are added. The SCA method on the other hand has no preferred background or inclusion and treats the constituents symmetrically, e.g., clay+water is the same as adding water+clay. However, within the SCA scheme, the constituents are only connected within a volume concentration (from 0.4– 0.6). Therefore, in a case where there are biconnected phases, neither the SCA nor the DEM is entirely satisfactory on its own, but their combination has been shown to have the capability of modeling the joint electrical and elastic properties of isotropic porous rocks (e.g., Han et al., 2011) and the elastic properties of anisotropic porous rocks (e.g., Hornby et al., 1994; Jakobsen et al., 2000). We do not account for cracks/fractures or other stress-dependent microstructural details in other to keep the modeling approach fairly simple. The effective medium models mentioned above along with orientation distribution functions have been used to account for cracks in shale models (e.g., Sayar and Torres-Verdín, 2016; Ren and Spikes, 2016). The noninteracting approximation has also been show to be well suited to account for cracks/fractures in rocks (e.g., Kachanov, 1993; Sayers and Kachanov, 1995; Sarout and Guéguen, 2008). We use a rich suite of well log data to calibrate our shale model, which is a combination of the SCA and DEM methods. The joint modeling approach ensures the elastic and electrical properties of the composite have the same microstructure (e.g., Han et al., 2016; Sarout et al., 2017). We do not focus on the modeling approach; rather, we focus on analyzing the effect of kerogen on the bulk elastic and electrical properties of the shales by using a joint elastic-electrical model constrained by petrophysical data, and by the joint (elastic-electrical) modeling approach. More importantly, we use the data to observe the influence of kerogen on these bulk properties and the calibrated model to analyze the controlling factors. The results suggest electrical properties could improve remote discrimination of kerogen in organic-rich shales where good quality seismic and electrical data are available. 2. Models The details of the combined SCA + DEM approach and its motivations are well described in the literature (e.g., Sheng, 1990; Hornby et al., 1994; Jakobsen et al., 2000; Han et al., 2011), so we give a brief summary here. The combined SCA + DEM approach can be used to obtain a bi-connected two-phase composite at any volume concentration. The approach begins by combining the two phases (phase 1 and phase 2) using the SCA method at a volume concentration where the phases are connected in the SCA method (0.4–0.6), which we will refer to as the “critical concentration φc”. The mixture at the critical concentration is then taken to be the background for the next step. If the volume fraction of phase 2 in the target effective medium is less than the critical concentration, the appropriate amount of phase 1 is added, else the appropriate amount of phase 2 is added.
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Fig. 1 shows a schematic illustrating this approach, which can be extended to include more than two phases, but only two phases can be combined at each step. The same process applies to both the elastic and the electrical case. Although not show in this manuscript, different approaches to combine the individual mineral constituents using effective medium theory were attempted. Using DEM alone with either clay, kerogen, quartz or carbonates as the background material proved sufficient for the elastic properties but not for the electrical properties. Using orientation distribution functions in conjunction with the DEM model still proved insufficient. The model with clay background under-predicted the electrical resistivity whereas those with a quartz, keorgen or carbonate background over-predicted the electrical resistivity. Introducing a single step of combined SCA + DEM approach can improve the results depending on how the individual constituents are combined. However, the electrical resistivity predictions were still poor. The approach we found to work well after trying many configurations is shown in Fig. 2. The first step of the approach has two separate parts done in parallel. One part combines kerogen and clay using DEM with kerogen as the background. The aspect ratio of clay used here was 0.2. The other part of the first step starts with creating a bi-connected sandstone by combining quartz and the effective fluid (brine+gas) using the SCA + DEM approach, to which the remaining minerals are added using the DEM method. The aspect ratios for all the components in this step were set to 1.0. The resulting composite is then combined with the kerogen-clay mixture using another step of the SCA + DEM approach. The aspect ratio for the kerogen-clay mix was set to 0.2, while that for the sanstone mixture was set to 1.0. The effective fluid (bulk modulus and electrical conductivity) was obtained using Bries average (Brie et al., 1995) with exponent e = 2, as opposed to Reuss or Voigt averages, which did not work as well for the electrical resistivity. The effective electrical conductivity of a gas-liquid mix appears to follow an exponential-type relationship. Bries law was chosen since it has been used widely for computing effective fluid bulk modulus of liquid-gas mixture, with the moduli replaced with the conductivities for computing the effective fluid electrical conductivity of the brine+gas mix.
2.1. Elastic models For the elastic case, the anisotropic DEM and SCA formulations by Hornby et al. (1994) will be used. The DEM equation is given by dC DEM ¼
dvi C i −C DEM Q i ; 1−vi
ð1Þ
where h i−1 ^ C i −C DEM ; Qi ¼ I þ G
ð2Þ
where Ci is the stiffness tensor of the ith component with volume fraction vi; I is the identity matrix; CDEM is the stiffness tensor of the mate^ is a fourth-rank Eshelby tensor associated with the aspect rial; and G ratios of inclusions. The SCA equation is given by (Hornby et al., 1994; Jakobsen et al., 2000)
C
SCA
¼
( N X i¼1
)( vi C i Q i
N
∑ v jQ j j¼1
)−1 ð3Þ
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Fig. 1. Schematic showing the combined SCA + DEM approach (both elastic and electrical). The SCA model is used to calculate the effective properties at a critical concentration φc. This then becomes the background for the DEM step to calculate the final effective properties at any volume concentration. Image adapted from Han et al. (2011).
Fig. 2. Schematic showing the modeling approach used in this study. It contains two stages of the SCA + DEM approach. The first step of the approach has two separate parts done in parallel, one combining kerogen and clay using DEM, the other creating a fluid-saturated sandstone using the SCA + DEM approach and adding other minerals using DEM. The final step combines the two resulting composites from the first step by using SCA + DEM.
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2.2. Electrical models
where
The anisotropic DEM equation for electrical conductivity is given by e.g., (Mendelson and Cohen, 1982; Dürr et al., 2002)
L¼
dσDEM
h i−1 dvi −1=2 −1=2 ¼− ðσi −σDEM Þ 1 þ σDEM i σDEM ðσi −σDEM Þ ; 1−vi
ð4Þ
where σi is the conductivity tensor of the ith component with volume fraction v; σDEM is the conductivity tensor of the material; and the matrix i is related to the ith phase through the depolarization vectors. For oblate spheroids with semiaxes b1 b b2 = b3, i simplifies to a diagonal matrix of depolarization vectors L1, L2 = L3. The depolarization vectors are given by (Landau and Lifshitz, 1960; Mendelson and Cohen, 1982)
L1 ¼ L; L2 ¼ L3 ¼
ð1−LÞ ; 2
ð5Þ
1 þ e3 e− tan−1 e ; e2
101
ð6Þ
where e = [(b2/b1)2 − 1]1/2. For oblate spheroids, the SCA equation for electrical conductivity for a two-component material is given by (e.g., Hornby et al., 1993; Berryman and Hoversten, 2013) 2 X v j σ j −σ SCA i ¼0 1 þ Γi σ j −σ SCA i j¼1
ð7Þ
where this is evaluated for i = 1,2,3, corresponding to the directions of the principal axes of the grain (oblate spheroid), which is the same as the principal axes of the effective conductivity of the composite medium (Mendelson and Cohen, 1982). Hence, i = 2 and i = 3 are equivalent for the case of spheroidal inclusions. Γi = Li/σSCA i , where Li is the same as in the DEM case.
Fig. 3. Mineralogy data from ECS logs from a) Well 1 and b) Well 2. Well 1 contains about eleven minerals, with the reservoir portion (bxx40m) dominated mainly by clay, quartz and kerogen. Kerogen is only present within the interval ∼xx21 − xx40m. The ECS logs from Well 2 shows a total of five mineral components, with the dominant minerals within the interval being kerogen, clay, QFM (quartz, feldspar, mica).
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Fig. 4. Well log data from Well 1 showing (from left to right) gamma ray, formation electrical resistivity, total porosity, water saturation, formation bulk density, P-wave velocity and Swave velocity. This interval represents the reservoir unit and is the interval that has been most characterized in the well. Note that there are missing data points at depths of ∼ xx40-xx41 m and Nxx42m for S-wave velocity (Vs), making it look like the data has been chopped at those depths.
3. Data Well log data from two different shale reservoirs were used for this study. The reservoirs are well characterized in terms of mineralogy and fluid content, which provided important constraints for the modeling inputs. Fig. 3 shows mineralogy data for both wells obtained from elemental capture spectroscopy (ECS) logs. Fig. 4 and Fig. 5 show log measurements of other properties in the reservoirs for Well 1 and Well 2, respectively. Note that the kerogen content has been calibrated to log data, and a separate log for kerogen content was provided for both wells. Mineralogy data from ECS shows Well 1 contains about eleven minerals within the selected interval (xx15 − xx45m), with kerogen only present within the interval ∼xx21 − xx40m. Above xx40m, kerogen, clay, quartz, and K-feldspar are the more dominant minerals, with dolomite, plagioclase and pyrite present in smaller amounts, and calcite, anhydrite, siderite and apatite occurring in trace amounts. Below xx40m, calcite and dolomite dominate, with other minerals occurring in smaller amounts (e.g., quartz, clay and plagioclase). In Well 2, kerogen is present everywhere along the selected interval of xx26 − xx58m, and the ECS data shows a total of six minerals in the well. The amount of siderite in the well is insignificant (b1%), so there are effectively five mineral components from the ECS. The dominant
minerals in Well 2 are kerogen, clay, QFM (quartz, feldspar, mica), with small amounts of carbonate and pyrite. Fig. 4 and Fig. 5 show gamma ray (GR), formation electrical resistivity (res.), total porosity (ϕtotal), water saturation (Sw), formation bulk density (ρb), P-wave velocity (Vp) and S-wave velocity (Vs) for both wells. The intervals shown here are the reservoir units, which are the intervals with comprehensive well log data. For model calibration, all the logs shown here are used except the GR logs. The ϕtotal and Sw logs are used as inputs for the model, and the remaining (resistivity, Vp, Vs, ρb) are used for comparison and calibration. 4. Results We present the results of applying the same joint electrical-elastic model to two different wells. Mineral composition and volume fractions were obtained from ECS logs, and pore fluid (gas saturation) information from saturation logs. Kerogen content and fluid saturation from logs were not just estimated from well logs, but also calibrated to core data, thereby improving the accuracy of the estimates. The properties of individual minerals were taken from the literature (Telford et al., 1990; Mitchell, 2004; Bandyopadhyay, 2009; Mavko et al., 2009; Hu et al., 2014; Yang et al., 2016). Well 1 brine resistivity, bulk modulus and density are 0.125 Ωm, 3.127 GPa and 1.086 kgm−3, respectively.
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Fig. 5. Well log data from the reservoir interval in Well 2 showing (from left to right) gamma ray, formation electrical resistivity, total porosity, water saturation, formation bulk density, Pwave velocity and S-wave velocity.
For Well 2, brine resistivity, bulk modulus and density are 0.028 Ωm, 3.127 GPa and 1.086 kgm−3, respectively. For both wells, gas resistivity, bulk modulus and density are 108 Ωm, 0.0184 GPa and 0.0838 kgm−3, respectively. These values were obtained using salinity, pressure and temperature information from the wells as input into an in-house software package. Table 1 shows the input electrical conductivities, aspect ratio, and bulk and shear moduli for the different minerals. Using these inputs in the workflow shown in Fig. 2, the elastic (bulk and shear modulus) and electrical (electrical resistivity) model responses can be computed. Brine salinity measured from cores along with pressure and temperature information were used to estimate brine resistivity, bulk modulus and bulk density. The main parameters left to fit the model with become the exponent in Bries formula and aspect ratio
of clay and the kerogen+clay mix (note that kerogen was the background in the first step of the DEM), set to be the same. We quickly found that a value of 0.2 (within the range used in the literature) gave good results for Well 1, without having to use any optimization techniques. The same aspect ratios were then used for Well 2. Strictly speaking, the properties of some individual minerals (including aspect ratios) could well vary with depth, but we have assumed they are constant across the entire length of the wells for simplicity. The idea was to come up with a fairly simple modeling approach capable of generalizing to different wells. Figs. 6 and 7 show the model fit (red curve) to the well log data (black curve). The parameters shown are P-wave modulus in GPa, which represents the C33 component of the elastic stiffness tensor; Swave modulus in GPa, which represents the C44 componenet of the
Table 1 Table showing the bulk modulus, shear modulus, electrical conductivity and aspect ratio of the different minerals used in the effective medium models (see Telford et al., 1990; Mitchell, 2004; Bandyopadhyay, 2009; Mavko et al., 2009; Hu et al., 2014; Yang et al., 2016).
K (GPa) μ(GPa) σ(S/m) Aspect ratio
Clay
Kerogen
Quartz
Calcite
Dolomite
Siderite
K-feldspar
Plagioclase
Anhydrite
Apatite
Pyrite
27 7 0.05 0.2
6.6 3.2 10−7 0.2
37 44 10−11 1
76.8 32 10−11 1
94.9 45 10−11 1
123.7 51 0.014 1
37.5 15 10−4 1
75.6 25.6 10−4 1
56.1 29.1 10−8 1
89 44.5 10−6 1
138.3 109.8 3.33 1
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Fig. 6. Model comparison to well log data for Well 1. The figure shows comparisons for (from left to right) P-wave modulus, S-wave modulus and electrical resistivity, with the fourth pane showing the mineral composition as a function of depth. The model fits the data pretty well for both elastic moduli and for the electrical resistivity.
elastic stiffness tensor; and electrical resistivity in Ωm, which represents horizontal resistivity ρ11 of the electrical tensor. In Well 1, the overall fit between data and model (Fig. 6) is very good for the P- and S- moduli and for the electrical resistivity. There are missing data points at depths of ∼ xx40-xx4 m and N xx42 m for the Smodulus; however, the other well logs have measurements for these depth. The P- modulus ranges from ∼ 39–140 GPa, while the S- modulus ranges from ∼ 13–29 GPa, and the electrical resistivity ranges from 1 to 1709 Ωm. Both elastic moduli stay fairly constant until depths of ∼ xx40 m where there is an increase in moduli that corresponds to higher contents of carbonates (calcite and dolomite) and is representative of the underlying rock layer. The formation electrical resistivity, however, shows more variation with depth (compared to the elastic moduli), suggesting that the “soft components” (kerogen, clay and pore fluids) have different effects on the bulk electrical resistivity. In Well 2, the same modeling approach used in Well 1 gives a good fit for the P-wave and S-wave moduli and for the electrical resistivity. Apart from the over-prediction in electrical resistivity at ∼ xx27 m and ∼ xx50 m, the model matches the data both in magnitude and trend quite well. The values for P- wave modulus, S-wave modulus and electrical resistivity are lower in this well relative to Well 1, ranging from ∼ 20–67 GPa, 6–23 GPa, and ∼ 16–653 Ωm, respectively. These lower values (compared to Well 1) are most likely due to the higher clay content in Well 2 (see Fig. 3). There is not a great deal of variation in both elastic moduli, as the volume fractions of the “harder components”
(carbonates) are not significant and do not vary much along the interval. The formation electrical resistivity shows a little bit more variation with depth for similar reasons as in Well 1. However, the variation is not very significant because the volume fractions of the “soft components” (kerogen, clay and pore fluids) do not vary quite as much with depth as in Well 1. 5. Discussion The model gives a good fit to the well data for both the elastic and electrical properties in the two wells, both in magnitude and trend. Because we want to evaluate the possibility of decoupling the “soft” elastic response of kerogen and clay by their different electrical responses, we focus more on the effect of kerogen on the bulk electrical resistivity of shales. Fig. 8 shows a crossplot of electrical resistivity and P-wave modulus for Well 1, color coded by kerogen content. The log data (Fig. 8a) is generally well reproduced by the model (Fig. 8b). It can be seen that shales with higher kerogen content tend to have higher electrical resistivity values compared to the lower electrical resistivity values of clay-rich portions. We attribute this to be mainly due to the contrast in intrinsic resistivity of clays versus kerogen as we will aim to show using thought experiments in subsequent paragraphs. Both the kerogen-rich and clay-rich portions have lower values of P-wave modulus (compared to where their concentrations are low) but are distinguishable by the contrast in electrical resistivity values.
K. Amalokwu et al. / Journal of Applied Geophysics 169 (2019) 98–108
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Fig. 7. Comparison between model and well log data for Well 2. The results show a good fit between model and data for (from left to right) P-wave modulus, S-wave modulus and electrical resistivity. The fourth pane shows the mineral composition as a function of depth.
Fig. 8. Kerogen effects in Well 1 evaluated using a cross-plot of electrical resistivity vs. P-wave modulus. a) Well log data b) Model output. The model reproduces the data fairly well. Higher kerogen content tends to give rise to higher electrical resistivity values. Also, softer zones (high kerogen and clay content) have lower values of P-wave modulus as expected.
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Fig. 9. Well 2 cross-plot of electrical resistivity vs. P-wave modulus for a) well log data b) model output. Again, higher kerogen content tends to produce higher electrical resistivity values as seen in Fig. 8, which is fairly well reproduced by the model.
Fig. 10. Plots showing results of replacing kerogen with an equivalent volume fraction of clay in the model and comparing the results to the well log data for Well 1. From left to right, the figure shows P-wave modulus, S-wave modulus and electrical resistivity. The model grossly under-predicts the electrical resistivity of areas where kerogen has been replaced but only slightly over-predicts both elastic moduli.
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A similar crossplot for Well 2 is shown in Fig. 9. Again, the well log data (Fig. 9a) and the model (Fig. 9b) are in good agreement. Kerogen is present along the entire length of the well interval used for this study (see Fig. 3b); however, the general trend is still higher kerogen content tends to plot toward higher values of electrical resistivity. The variation in P-wave modulus is small because the amount of stiffer minerals (e.g., carbonates, quartz) do not vary very significantly, as pointed out earlier. The calibration of the model (as described in the Section 4), with the help of pertinent petrophysical properties (i.e., mineralogy and fluid properties) from a comprehensive suite of well data from two wells, allows further exploration of kerogen effects on the bulk resistivity of shales beyond observing trends seen in the data. Using the calibrated model, we can explore “what if” scenarios, such as replacing the kerogen with clay. This exercise was done to show that kerogen influences the bulk electrical resistivity significantly, when present. In this thought experiment, the elastic moduli, the electrical conductivity and density of kerogen is replaced by those of clay, while every other aspect of the model remains the same. Figs. 10 and 11 show results of replacing the volume fraction occupied by kerogen with clay in the model, compared with the actual well log data. In both wells, it can be seen that kerogen has a significant influence on the bulk resistivity of the shales, as the kerogen-free resistivity predicted is very low and nowhere close to the measured bulk
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resistivity. The elastic properties, however, are slightly over-predicted, but the effect on the elastic properties is not as high as the effect on the electrical properties. In Well 1, resistivity of zones without kerogen (bxx21m and Nxx40m) are in good agreement with the model (Fig. 10), which gives us more confidence in the predictions. A similar comparison cannot be made in Well 2 because kerogen is present along the entire interval. It should be pointed out that although kerogen has a significant effect on the bulk resistivity of the shales, it is not the only controlling factor, as the pore fluids also play an important role as expected. However, it is clear that kerogen has a significant impact on the bulk resistivity of the shales in these wells. The simplicity of the modeling approach is attractive, but there are some limitations as a result. There is an interplay between the aspect ratio (geometrical detail) and the orientation of the clay platelets, and changing either can have similar effects. However, we did not use an orientation distribution function to avoid introducing another free parameter, which was unnecessary, looking at the model-data fit. The implication of this is that both effects (geometrical and orientation) are lumped into one parameter (aspect ratio), affecting our estimates of the actual aspect ratios and anisotropy. This is a reasonable tradeoff as we do not have data on either aspect ratio or anisotropy, and it keeps the model fairly simple. The modeling approach does not account for fractures/cracks, but this does not seem to have affected the
Fig. 11. Well 2 results when kerogen is replaced with an equal volume fraction of clay in the model and compared to the well log data. The results show that again, the electrical resistivity of areas where kerogen has been replaced is grossly under-predicted, but the over-prediction of both moduli is not as significant.
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goodness of fit significantly. This could suggest the effect of fractures on the measured properties is not significant in the measurement direction due to the orientation of the fractures (expected to be vertical) relative to the direction of measurement. Note that the measured elastic property is that in the vertical direction, while the electrical property measured is that in the horizontal direction. However, this situation might not always be the case, and this modeling approach would need to be tested on different shale types to determine its generality. Accounting for fractures/cracks could improve the model predictions but in the process introduce additional poorly constrained parameters. Also, it is not clear how to calculate the effective fluid electrical conductivity of a mutiphase fluid. Although the success of Bries formula suggests an exponential-type relationship between gas saturation and electrical conductivity, more work is needed to establish the true nature of this relationship. The step in the model where a bi-connected sandstone is created is meant to illustrate that if we have quartz in the system, the quartz aggregates when distributed in the final shale sample behave like sandstones which typically exhibit interconnected behavior. However, it is hard to track the actual connectivities in the final shale composite and it is important avoid over-interpreting the model because although models are helpful, they are still idealized representations. Although the reservoirs used in this study are relatively quartz-rich, it can be seen from Fig. 6 that the model still works fairly well for the underburden in Well 1 which is carbonate-rich. Further model validation in reservoirs with different lithology would be helpful in determining generality of modeling approach. The ultimate goal is to be able to invert for these rock and fluid properties through a joint-inversion of surface seismic and electrical data, but obtaining data (especially electrical) with such resolution to do this might not be possible in practice presently. However, identification of zones that could potentially be richer in kerogen from surface data (electrical and seismic) might be possible given the right conditions, and rock physics modeling could help constrain the interpretation. For example, absolute values can be ambiguous, but a normalization with background/ kerogen-free response (obtained from modeling and/or data) could be attempted, separating the different responses into different zones, possibly improving interpretability. 6. Conclusion The effect of kerogen on the joint elastic-electrical properties of kerogen-rich shales was analyzed in two wells by using well data and rock physics modeling. This joint approach provides a means of not only constraining the individual properties, but more importantly, could help decouple the effects of kerogen and clay, characterized by their similar ‘soft’ elastic responses. Using the same effective medium modeling approach for both properties ensured an equivalent microstructure of the shale composite for both the elastic and electrical properties. The results show that the higher electrical resistivity of kerogen does tend to increase the bulk electrical resistivity of shales whereas clay tends to reduce it. This characteristic could potentially be used to improve sweet-spot identification in organic-rich shales or aid in the identification of zones with higher hydrocarbon potential during exploration/appraisal, provided good quality electrical resistivity data can be obtained. Our analyses suggest that if other constituents remain relatively constant, shales that are clay-rich (negligible kerogen content) have significantly lower electrical resistivity than when kerogen is present, but elastic properties might not be as significantly affected.
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