Nanometer-scale characterization of microscopic pores in shale kerogen by image analysis and pore-scale modeling

Authors


Abstract

[1] Nanometer-scale scanning electron microscopy was applied in visualizing the microscopic pores within shale kerogen. Geometrical information of all individual pores was extracted by image analysis. Image segmentation and separation showed that most of the intrakerogen pores are discrete and isolated from each other, having relatively spherical morphology. These isolated intrakerogen pores result in huge challenges in gas production, because they are not effectively connected to natural and hydraulic fractures. Statistical results showed that nanopores, which have diameters smaller than 100 nm, make up 92.7% of the total pore number, while they make up only 4.5% of the total pore volume. Intrakerogen porosity and specific surface area are 29.9% and 14.0 m2/g, respectively. Accurate visualization and measurement of intrakerogen pores are critical for evaluation of gas storage and optimization of hydraulic fracturing. By lattice Boltzmann simulations, permeabilities and tortuosities were simulated in the three principal directions. Long tails were observed in breakthrough curves, resulting from diffusion of solute particles from low-flow-velocity pores to larger conduits at late times. The long-tailing phenomena at the pore scale are qualitatively consistent with those observed in real productions. Understanding the pore-scale transport processes between microscopic pores within kerogen and large fracture systems is of great importance in predicting hydrocarbon production. Upscaling methods are needed to investigate larger-scale processes and properties in shale reservoirs.

1. Introduction

[2] The consumption of hydrocarbon energy in the world has steadily increased during the past few decades. To meet the rising energy demand, production of unconventional hydrocarbons, including shale oil and gas, has attracted significant attention. However, recovery of hydrocarbons in tight reservoirs has proven extremely difficult due to the low porosity and permeability. Usually the permeability of organic-rich shale reservoirs ranges from nano to micro-Darcy, and porosity is below 10%. Horizontal wells with transverse hydraulic fractures are needed to produce hydrocarbons from shale reservoirs at a significant rate [Daneshy, 2009; Gaurav et al., 2012]. Therefore, it is necessary to well understand and consequently couple the multiscale transport processes between shale matrices and relatively large fractures [Balhoff et al., 2007, 2008].

[3] In the United States, important shale gas reservoirs include the Barnett Shale, Woodford Shale, Haynesville Shale, Fayetteville Shale, Marcellus Shale, and Eagle Ford Shale. A shale play is a geographic area containing an organic-rich fine-grained sedimentary rock. In gas-producing plays, kerogen, which is the organic matter for the source of hydrocarbons, can make up as much as 40% volume of the bulk rock [Passey et al., 2010]. Kerogen in shale gas reservoirs is converted into gas during a geological time scale, and the produced gas is retained in the source rock due to the extremely low permeability. In this case, the reservoir rock is the same as the source rock. Kerogen makes a significant contribution to porosity, permeability, and heterogeneity of the reservoir. It is crucial to obtain detailed information of the internal microstructure of shales, because existing research has shown that fine-scale heterogeneity greatly affects larger-scale transport properties [Farajzadeh et al., 2011]. Characterization of detailed internal microstructure of kerogen provides necessary information for predicting the transport properties of the shale, which is critical for evaluating hydrocarbon recovery.

[4] At the pore scale, the mechanisms for trapping gas molecules in kerogen are tortuosity trapping, adsorption trapping, and dissolution trapping. Tortuosity trapping refers to the process that free gas molecules are trapped in intrakerogen pores, where the transport is dominated by molecular diffusion. Adsorption trapping refers to the process by which gas molecules are adsorbed and fixed on the pore walls, while dissolution trapping refers to the process by which gas molecules dissolve in the kerogen bulk. While gas adsorption is a relatively slow process, it can account for up to 50% of the total gas production, as found in the Devonian shale [Lu et al., 1995]. This implies that adsorbed gas greatly affects the late-time production, and hence raises a necessity of accurate measurement of the internal porosity, specific surface area, and pore size distribution in shale kerogen.

[5] During the past few years, numerous researchers have employed scanning electron microscopy (SEM) to visualize the internal structure of shales at the nanometer scale [e.g., Charmers et al., 2009; Wang and Reed, 2009; Curtis et al., 2010]. However, there is limited research focusing on quantification of the geometrical information of individual pores in kerogen, as well as simulation of transport properties based on the high-resolution image data. In this study, we utilized a focused ion beam-scanning electron microscopy (FIB-SEM) facility to obtain the 3-D internal structure of kerogen in a shale sample extracted from a shale gas play. Image analysis methods were used to obtain detailed geometrical information of all individual pores. The lattice Boltzmann (LB) method was employed to estimate the absolute permeability and other transport properties of the pore space within the kerogen region.

[6] The workflow proposed in this study involves digital imaging, data analysis, and numerical modeling, which is of great importance in estimating hydrocarbon storage and enhancing gas production in shale reservoirs due to the following reasons. First, economic analysis is critical for the financial success of unconventional reservoir recovery. Accurate measurement of the geometrical information of intrakerogen pores has the great potential to achieve an accurate estimation of the total stored gas, which is important for the management and economic analysis of reservoir production. Second, a widely used enhanced production method in shale gas recovery is hydraulic fracturing, which is aimed at creating fractures that connect as many isolated intrakerogen pores as possible to achieve a larger stimulated reservoir volume (SRV). Understanding of the spatial distribution and geometrical information of intrakerogen pores will be significantly helpful in design and optimization of effective hydraulic fracturing. Third, decline curves predicted by conventional reservoir models are often unable to fit production curves in unconventional reservoirs. In order to match real production data, matrix permeability and SRV are usually increased artificially, which is inconsistent with the underlying physics. To correctly simulate hydrocarbon transport in unconventional reservoirs, it is important to take into account the multiscale transport properties in both the fractures and matrix, which depend primarily on accurate observation and understanding of nanometer-scale processes.

2. FIB-SEM Scanning

[7] The Helios NanoLab 650 fabricated by FEI Company was employed for FIB-SEM scanning. This instrument has a minimum resolution of 1 nm, but in order to obtain a balance between reasonable field of view and image quality, we chose a resolution of 12 nm. The single 2-D SEM image has a dimension of 1024 × 884 pixels in the x and y directions, respectively, as shown in Figure 1a. After a 2-D image was obtained, the ion beam removed one layer of the sample with thickness of 12 nm. Then the electron beam scanned the new sample surface to obtain the next 2-D image. A platinum layer with a thickness of 1 µm was deposited on the shale surface in order to prevent extra damage due to the ion beam. A fiducial mark was used as a registration point in order to align all 2-D images. There were 175 2-D slices imaged by the SEM, leading to a 3-D data set of 1024 × 884 × 175 voxels.

Figure 1.

(a) A 2-D raw image of shale obtained by SEM scanning, with a dimension of 1024 × 884 pixels in the x and y directions, respectively. The resolution is 12 nm per pixel length. Kerogen is shown in light gray, while the black holes are intrakerogen pores. (b) 3-D reconstruction of the whole data set. The subdomain contains kerogen (red) and intrakerogen pores (blue), and is of 280 × 120 × 100 voxels in the x, y, and z directions, respectively. The white material is pyrite, and all the other materials are set to transparent.

3. Image Analysis and Lattice Boltzmann Simulation

[8] A 3-D domain was cut within the kerogen region, which has a dimension of 280 × 120 × 100 voxels in the x, y, and z directions, respectively, as shown in Figure 1b. Gravitational force is in the z direction. The Blob3D package [Ketcham, 2005] was adopted for noise removal, segmentation, and extraction of the geometrical information of all individual pores inside the 3-D domain. Blob3D was developed in IDL (Interactive Data Language, Exelis Visual Information Solutions), based on the computational methods for quantitative analysis of 3-D features in geological specimens.

[9] In this study, because the FIB-SEM contrast was good, the median between solid and void peaks was selected as the threshold for phase segmentation. The processed image data were used as internal boundary conditions for solving pore fluid flow and solute transport by the LB method [Succi, 2001]. In this study, pressure is assumed to be below the dew point line, so gas exists in a single phase. Because of the small length scale and low-pressure gradient, gas can be treated as an incompressible fluid, and the LB method for simulating incompressible fluid flow was used (X. Yin, Colorado School of Mines, personal communication, 2013). The LB simulator used in this study was validated in simulating single-phase flow by direct comparison with analytical solutions and laboratory measurements as shown in the online supporting information of Chen et al. [2008], and then successfully extended to simulating multiphase [Chen and Zhang, 2009] and multiscale [Chen et al., 2010] flows. We employed the D3Q19 (19 velocity vectors in 3-D space) LB model since it has a good balance between computational stability and efficiency. The LB equation for fluid flow can be written as

display math(1)

where fi(x, t) is the particle distribution function specifying the probability that fluid particles at lattice location x and time t travel along the ith direction; ei is the lattice velocity vector corresponding to direction i, which is defined as (0, 0, 0)c for i = 0, (±1, 0, 0)c for i = 1, 2, (0, ±1, 0)c for i = 3, 4, (0, 0, ±1)c for i = 5, 6, (±1, ±1, 0)c for i = 7…10, (0, ±1, ±1)c for i = 11…14, and (±1, 0, ±1)c for i = 15…18. c is defined as math formula, where δx is the lattice spacing and δt is the time step. τ is the dimensionless relaxation time related to the kinematic viscosity by math formula. math formula is the equilibrium distribution function at location x and time t along the ith direction, which is chosen to recover the macroscopic Navier-Stokes equations and calculated by

display math(2)

where ωi is the weight coefficient. For the D3Q19 model, ωi = 1/3 for i = 0, ωi = 1/18 for i = 1.6, and ωi = 1/36 for i = 7.18. ρ and u are the macroscopic density and fluid velocity, respectively, and can be calculated by

display math(3)
display math(4)

[10] The LB simulator used in this study has been extended to simulate the transport of passive and conservative solute particles [Chen et al., 2009]. D3Q19 model was still adopted for solute transport simulation, in order to keep consistency with fluid simulation and prevent numerical instability at high Peclet numbers. In the transport simulation, a second distribution function gi is used:

display math(5)

where τs is the dimensionless relaxation time related to the solute diffusivity by math formula. math formula is the equilibrium distribution function at location x and time t along the ith direction, which is chosen to recover the macroscopic advection-diffusion equation and calculated by

display math(6)

where C is the macroscopic solute concentration and can be calculated by

display math(7)

[11] In simulating fluid flow which is governed by the Navier-Stokes equations, a pressure gradient was imposed in the longitudinal direction, and no-flow conditions were used at the four lateral sides and internal solid surface. In simulating solute transport which is governed by the advection-diffusion equation, zero-concentration-gradient conditions were used at the inlet and outlet sides. The unknown solute distribution functions streaming from the boundaries take the values in the previous time step. No-flux conditions were applied at the four lateral sides and internal solid surface. In this study, the solid-fluid boundaries are placed halfway between solid and fluid nodes, thus the bounce-back method can achieve second-order-accuracy Neumann-type boundary conditions for both fluid flow and solute transport [Mei et al., 1999, 2000]. Both the Reynolds and Peclet numbers were much smaller than one, implying that the flow was in Darcy regime and solute transport was diffusion dominated. The tortuosity, τ, defined as the ratio of the mean flow path length to the linear distance between inlet and outlet, was calculated by particle tracking based on the LB-simulated flow fields [Chen et al., 2009].

[12] The method of Zhang et al. [2000] was adopted to analyze the statistical representative elementary volume (sREV) of the kerogen structure. Specifically, in order to capture spatial heterogeneity we moved a cubic subdomain to every possible position within the kerogen region, and obtained the values of permeability for all subdomains by LB simulations. The mean and standard deviation of all LB-simulated permeabilities were calculated. Then we increased the subdomain size and repeated this process. Therefore, the mean and standard deviation of subdomain permeability can be plotted as function of the subdomain size. As the coefficient of variation (COV), defined as the ratio of standard deviation to mean, is smaller than 20%, the subdomain size is considered to be big enough to demonstrate homogeneous properties based on permeability. Similar sREV analysis was done as well based on porosity. Note that the sREV analysis was done in the kerogen region without including any matrix region, because intrakerogen porosity and permeability are what are important for estimation of hydrocarbon storage and optimization of hydraulic fracturing.

4. Results and Discussion

[13] A 2-D raw image of the shale sample obtained by SEM scanning is shown in Figure 1a, where kerogen, pyrite, and clay minerals are well resolved at the nanometer scale. Figure 1b shows the 3-D reconstruction of the whole data set, as well as the 3-D domain (280 × 120 × 100 voxels, i.e., 3360 nm × 1440 nm × 1200 nm) cut within the kerogen region for image analysis and LB simulations. The relatively spherical internal pores imply the kerogen is in the gas window, which implies that the source rock was heated in the temperature range of 150–200°C. There are 731 pores observed within the kerogen domain. In this study, an individual pore is defined as a collection of interconnected void voxels which are isolated from other pores. To provide valid permeability across the whole domain, there must exist at least one relatively large pore which intersects both the inlet and outlet sides. The morphology of two pores is shown in Figure 2. It can be seen that one has a maximum 1-D size of ∼80 nm, while the other one is ∼400 nm. The porosity of the 3-D kerogen domain was found to be 29.9%, which is close to the values measured in kerogen at a hermal maturity of 1.6%Ro, which implies a gas window [Loucks et al., 2009; Wang and Reed, 2009]. %Ro is the measured percentage of reflected light from a sample which is immersed in oil (%Ro = % reflectance in oil). It is an indicator of the vitrinite reflectance (VR) in the sediment which shows the maximum temperature history. Typically the VR value for oil generation is in the range of 0.5–1.1%Ro, and the value for gas generation is in the range of 1.0–3.0%Ro. The specific surface area, defined as the interface area between solid and void per unit volume, was calculated as 2.1 × 104 mm−1. With kerogen density being 1.5 g/cm3 [Ward, 2010], the specific surface area can be written as 14.0 m2/g, which is the interface area per unit mass. This value is close to the values measured in kaolinite and chlorite-rich rocks by the N2 Brunauer–Emmett–Teller (BET) method [Ji et al., 2012]. The high specific surface area plays an important role in adsorbing and trapping gas molecules.

Figure 2.

3-D morphology of two intrakerogen pores, with maximum 1-D size around (a) 80 nm and (b) 400 nm, respectively.

[14] The distribution of pore volume based on number fraction is shown in Figure 3a. To eliminate the resolution effect, only the pores consisting of at least 6 voxels were taken into account. It can be seen that relatively smaller pores make up the majority of the total pore number. Specifically, 74.6% of the pores have volume smaller than 105 nm3. These small pores are dispersed and isolated within kerogen, leading to huge difficulty for gas recovery because they are not effectively connected to natural and hydraulic fractures where hydrocarbon transport occurs. By Blob3D analysis, volume, surface area, and sphere normalized surface to volume ratio (SNSVR) are extracted for each individual pore. SNSVR is a number describing the 3-D morphology of an object, and defined as SNSVR = (SA/SAsphere)1.5, where SA is the surface area of the object and SAsphere is the surface area of a sphere that has the same volume of the object [Ketcham, 2005]. The larger the SNSVR, the more the shape of the object deviates from a sphere. The values of SNSVR for all pores were plotted as function of pore volume, as shown in Figure 3b. It is observed that only 0.5% of the pores have SNSVR values higher than 10, implying they are relatively large and interconnected void channels with extremely irregular shapes. 88.9% of the pores have the SNSVR value lower than 2, and 97.4% of them have it lower than 3, implying that most of the pores are relatively spherical. The ratios of surface area to volume for all pores were plotted as function of pore volume, as shown in Figure 3c. The solid line represents the surface area-to-volume ratio of a sphere as function of its volume. It can be seen that the relatively smaller pores have higher surface area-to-volume ratios. In fact, 88.8% of the pores have surface area-to-volume ratios higher than 0.1 nm−1, i.e., 105 mm−1. The high surface area-to-volume ratios play an important role in adsorbing gas molecules. Also, it was found that pores with volume smaller than 105 nm3 have surface area-to-volume ratios close to that of a sphere, implying that their morphology is relatively spherical. The relatively large pores with volume bigger than 107 nm3 have surface area-to-volume ratios much higher than that of a sphere, implying they have morphology that deviates from a sphere significantly.

Figure 3.

(a) Pore volume distribution based on number fraction, (b) sphere normalized surface to volume ratio (SNSVR), and (c) surface area-to-volume ratio, as function of pore volume.

[15] Blob3D is able to fit a relatively spherical object with an ellipsoid, and extract the maximum and minimum axis lengths. The aspect ratio is defined as the ratio between the maximum and minimum axis lengths, and its distribution based on the number fraction of all pores was plotted in Figure 4a. It can be found that 71.7% of the pores have aspect ratios lower than 2, implying that most of them are relatively spherical, which is consistent with the observations from Figures 3b and 3c. The size of minimum axis length is critical because it determines the capillary pressure. The distribution of minimum axis length based on number fraction is shown in Figure 4b. It was observed that 77.2% of the pores have minimum axis lengths smaller than 50 nm.

Figure 4.

Distributions based on (a) aspect ratio and (b) minimum axis length.

[16] Because a small number of extremely large pores cannot be fit by an ellipsoid, we define the equivalent pore diameter as the diameter of the sphere which has the same volume as the pore. It is a good approximation to use the equivalent pore diameter as the pore size indicator, since most of the pores are relatively spherical with low SNSVR values. Pore diameter distributions based on number and volume fractions are presented in Figures 5a and 5b, respectively. It is observed that the pores with diameters smaller than 100 nm make up 92.7% of the total pore number, while they make up only 4.5% of the total pore volume. Pores with diameters smaller than 100 nm are defined as nanopores, and are of great importance and interest for hydrocarbon recovery in shale reservoirs [Swami and Settari, 2012].

Figure 5.

Pore diameter distributions based on (a) number fraction and (b) volume fraction.

[17] By LB simulation of pore fluid flow, the permeabilities were found to be 3.92, 1.58, and 2.50 micro-Darcys (µD) in the z, y, and x directions, respectively. By sREV analysis, the mean and standard deviation of LB-simulated permeability in the z direction and image-analysis-based porosity as function of the subdomain size are shown in Figure 6. By using COV = 20% as the criterion, it was found that the 1-D size of sREV is 80 pixels based on permeability, because with this subdomain size the COV of permeability is 18.9%. In contrast, the 1-D size of sREV is 30 pixels based on porosity, since at this subdomain size, the COV of porosity is 11.1%. This implies that the domain size is large enough to demonstrate homogeneous properties relative to the characteristic length of pores. It should be noted that the sREV is resolution dependent. Should the larger-scale heterogeneity structure in the clay mineral region be included in the field of view, a different sREV size will be found. Thus, the resolution used should be determined from the characteristic length of the field of interest.

Figure 6.

Mean and standard deviation of (a) LB-simulated permeability and (b) image-analysis-based porosity, as function of the subdomain size.

[18] Ten thousand tracer particles were used for particle tracking in all the three principal directions based on the LB-simulated flow fields, and the distributions of tortuosity are shown in Figure 7a. The mean tortuosity values are 1.84, 2.54, and 2.65 in the z, y, and x directions, respectively. The standard deviations are 0.52, 0.60, and 0.51, respectively. It can be seen that the tortuosity in the z direction is obviously lower than those in the y and x directions. Considering that the permeability in this direction is the highest, we suspect that there exist relatively large and straight conduits in the z direction. Furthermore, it can be seen that the mean values of tortuosity are similar in the y and x directions, although the domain length in the x direction is more than twice as long as in the y direction. This implies that the 3-D kerogen domain is isotropic and scale invariant in both the y and x directions, since the tortuosity per unit sample length in these two directions are the same.

Figure 7.

(a) Cumulative distributions of tortuosity in the three principal directions and (b) breakthrough curves (BTCs) of passive and conservative solute in the z and y directions, where C* is the dimensionless outlet concentration normalized by the initial concentration C0.

[19] The mean pore diameter of all pores was calculated as 53 nm. The ratio of the mean pore diameter to spatial resolution is slightly higher than 4, which implies a considerable Knudsen number. However, most of the relatively small pores are isolated from each other, without providing any permeability for fluid flow. Only the large pores, which have diameters much bigger than 100 nm, contribute to the permeability of the whole domain. In these connected conduits, the Knudsen number is relatively small, thus the LB-simulated flow can be treated as a continuum flow [Kang et al., 2003]. If the permeability is controlled by channels with high Knudsen numbers, a slip boundary condition is needed at the fluid-solid interface [Succi, 2002; Yin et al., 2006]. It is also noted that a main issue for Blob3D is that it has relatively large error in calculating the geometrical information of extremely small 3-D objects that consists of only a limited number of voxels. However, this error is not critical for simulation of macroscopic fluid flow through the porous medium, since existing findings show that permeability in heterogeneous porous media is controlled by flow in large pores [O'Connor and Fredrich, 1999; Chen et al., 2008]. This is consistent with the Poiseuille law which states that the average flow velocity is proportional to the square of pore radius.

[20] Based on the LB-simulated pore flow fields in the z and y directions, instantaneous plane sources of passive and conservative solute were placed at the inlet sides of the domain at time zero, and the developments of solute plume in the pore space were simulated by the LB method [Chen et al., 2009]. The outlet concentration was averaged over the outlet plane perpendicular to the longitudinal direction, and reported against time, leading to breakthrough curves (BTCs) in the z and y directions, respectively, as shown in Figure 7b. It is found that the BTCs spanned a temporal range at the hundreds of microseconds scale, and had high initial peaks followed by long tails. This implies that the 3-D kerogen domain is highly open and porous, thus the solute particles migrated into the dead-end and low-flow-velocity regions from the relatively big conduits at early times and then were trapped there. After the solute concentration decreased in the big conduits, the trapped solute particles migrated back into the conduits, leading to long tails in the BTCs at late times.

[21] In the simulation of solute breakthrough, we did not take into account the trapping mechanisms of adsorption and dissolution because they are much slower compared to tortuosity trapping. We suspect that the tailing phenomena at late times will become more pronounced should we take into account these two mechanisms, considering the high specific surface area within kerogen [Ji et al., 2012; Zhang et al., 2012]. The simulated long tails in Figure 7b are qualitatively consistent with those observed in real gas productions [Baihly et al., 2010], where the actual production rates at late times are much higher than the predicted values. This implies that pore-scale desorption and diffusion of gas molecules from shale matrices to big fractures significantly affects the late-time production at the reservoir scale. Therefore, the combination of nanometer-scale visualization, image analysis, and numerical simulation is promising since it provides crucial information for predicting larger-scale hydrocarbon recoveries in shale reservoirs.

5. Conclusions

[22] In this study, we applied nanometer-scale SEM in visualizing the microscopic pores within shale kerogen. Geometrical information of all individual pores was extracted by image analysis. It was found that most of the intrakerogen pores are discrete and isolated, resulting in huge challenges in gas recovery. Statistical results showed that nanopores, which have diameters smaller than 100 nm, make up 92.7% of the total pore number, while they make up only 4.5% of the total pore volume. By lattice Boltzmann simulations and particle tracking, it was found that in the z direction, the permeability is slightly higher and tortuosity is lower, compared to the other two directions. This implies there exist relatively large and straight conduits in this direction. The permeabilities and tortuosities in the y and x directions are similar, implying that the kerogen structure is isotropic in these two directions at the scale of interest. Long tails were observed in the BTCs, resulting from diffusion of solute particles from dead-end and low-flow-velocity pores to larger conduits at late times. The long-tailing phenomena at the pore scale are qualitatively consistent with those observed in real gas productions at the reservoir scale.

[23] The simulated transport processes of conservative and passive solute particles reveal only the effect of tortuosity trapping resulting from the complexity of pore geometry. We suspect that the tailing phenomena at late times will be enhanced should the adsorption and dissolution trapping be taken into account. Since these trapping mechanisms are controlled by porosity, tortuosity, and specific surface area within kerogen, accurate visualization and measurement of intrakerogen pores are critical for estimation of the storage and recovery in shale reservoirs. Understanding the pore-scale transport processes between microscopic pores within kerogen and relatively larger fracture systems is of great importance in accurately predicting hydrocarbon productions in shale reservoirs.

[24] This study presents a promising workflow including both image analysis and numerical modeling for investigating the nanometer-scale geometrical and transport properties within shale kerogen. Accurate imaging of intrakerogen pores provides critical information for estimation of the total stored gas in an unconventional reservoir. Knowledge of the spatial and geometrical information of intrakerogen pores also benefits the design and optimization of hydraulic fracturing, which is aimed at connecting as many isolated intrakerogen pores as possible and consequently increases SRV. Pore-scale numerical modeling of transport processes provides necessary inputs for multiscale simulations of hydrocarbon recovery in fractured and heterogeneous reservoirs [Balhoff et al., 2007, 2008]. In this upscaling method, mortar coupling is employed to match the pressures and fluxes at the interfaces between pore and continuum-scale domains. More work is needed to better describe the adsorption and dissolution of gas molecules in intrakerogen pores and upscale them to the larger scales [Swami and Settari, 2012].

Acknowledgments

[25] We thank Halliburton for allowing the publication of this work. We thank Buddy McDaniel, Bob Sepulvado, and Mahmoud Eid for their help improving the manuscript. We thank the anonymous reviewers for the valuable comments which help improve the manuscript significantly.

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