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Time-lapsed visualization and characterization of shale diffusion properties using 4D X-ray microcomputed tomography Yulai Zhang, Peyman Mostaghimi, Andrew Fogden, Adrian Sheppard, Alessio Arena, Jill Middleton, and Ryan Troy Armstrong Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b03191 • Publication Date (Web): 02 Jan 2018 Downloaded from http://pubs.acs.org on January 2, 2018
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Energy & Fuels
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Time-lapsed visualization and characterization of shale diffusion properties using 4D X-ray
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microcomputed tomography
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Yulai Zhanga, Peyman Mostaghimia, Andrew Fogdenb, Adrian Sheppardc, Alessio Arenab, Jill Middletonc,
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and Ryan T. Armstronga*
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a
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b
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c
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*Corresponding author:
[email protected] School of Petroleum Engineering, The University of New South Wales, NSW 2052, Australia; FEI Oil & Gas, Suite 102, Level 1, 73 Northbourne Avenue Canberra, ACT 2600 Australia;
Department of Applied Mathematics, The Australian National University, Acton ACT 2601, Australia
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Abstract
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Diffusion is an important mass transport mechanism in shale matrix which usually has pore sizes ranging
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from molecular dimensions to micrometers. Better characterization of the diffusion properties is helpful
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in understanding the multi-physical mass transport process in shale. We present a method for
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measuring local effective diffusivity of shale core plugs using 4D X-ray micro-computed tomography
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(micro-CT). Liquid-liquid diffusion of X-ray opaque diiodomethane CH2I2 from a Permian Basin shale core
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plug into the surrounding X-ray transparent toluene is monitored by 4D micro-CT imaging. The time-
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sequenced diffusion tomograms enable 4D visualization of the dynamic process. Local directional
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effective diffusivities are measured numerically from the micro-CT data using a mathematical method.
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The measured data are analysed with relation to compositional variations of the sample. Dykstra
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Parsons coefficient is used to quantify the degree of heterogeneity of the measured data at the sub-core
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scale. We find that the diffusion in the Permian Basin sub-plug is uneven and influenced by matrix
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heterogeneities. Dense materials, e.g. pyrite, have low porosity and low horizontal effective diffusivity of
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around 10-15 m2/s or below; light materials, e.g. fossil, have high porosity and high horizontal effective
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diffusivity of around 10-14 m2/s or above. Compositional variation of the sample leads to porosity and
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mass transport property changes. 4D imaging and local diffusivity measurements identify the true
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heterogeneity of the shale sample, which is advantageous over static imaging. The measured local
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effective diffusivity enables us to infer smaller scale characteristics and thus provides a means to relate
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microscale shale rock structure to macroscale transport properties.
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1. Introduction
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Shale gas has significantly changed the United States energy sector and also has great potentials in many
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other countries around the world1-4. Although advanced drilling techniques and large scale hydraulic
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fracturing have significantly increased gas production rates from low permeable shale rocks, our
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fundamental knowledge of shale rocks remains far from complete1, 5, 6. Compared with conventional
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sandstones, shale rocks are more heterogeneous in composition with complex pore space, e.g. shale
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pores may range from molecular dimensions to micrometres, varying over several orders of magnitude7-
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13
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transport properties of shale is needed. As diffusion is a major mass transfer mechanism in the matrix
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where the pore sizes are usually at nanometer length scale8, 14, diffusion studies in shale matrix are of
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practical importance. The main purpose of this paper is to study the diffusion properties of shale rocks
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within the shale matrix by using time-lapsed X-ray microcomputed tomography (micro-CT). The method
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and validation for measuring effective diffusion coefficients from micro-CT experiments is published
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elsewhere15. In this paper, we focus on the characterization of the shale sample by applying the time-
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lapsed methodology to relate effective diffusivity values to shale properties and heterogeneity.
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Molecular diffusion is the movement of molecules without bulk motion, and it is a result of the random
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movement of molecules and molecule-molecule collisions according to Brownian motion. In a system
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with different types of species, molecules tend to diffuse from higher concentration to lower
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concentration, resulting in mixing of different substances, until an equilibrium state is reached.
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Molecular diffusion is described by Fick’s law at the continuum scale, which states that diffusive flux is
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proportional to the concentration gradient under the steady state condition and is defined as
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= − ∙
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where is diffusive flux, is the concentration of the diffusing substance and is the diffusion
. Better understandings on how the tremendous variation in length scales will affect the mass
(1)
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coefficient. The theories and equations for the determination of for gases16, 17 and liquids are different
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and since we are studying liquid-liquid diffusion in shale, only liquid diffusion fundamentals are
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introduced here. For liquid diffusion, which is much slower than gas-gas diffusion, can be estimated
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from the Stokes-Einstein equation, defined as
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=
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where is the Boltzmann constant, is solvent viscosity and is radius of the diffusing particle. In eq 2,
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(2)
does not depend on pressure, this is because liquids are only slightly compressible and pressure 2 ACS Paragon Plus Environment
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changes do not have a significant impact on the rate of collisions between molecules. In a binary system
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with two types of species, may not be a constant and changes with the concentration of the diffusing
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substance18, 19. The Stokes-Einstein equation (eq 2) is only applied to the diffusion of pure fluids. For
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diffusion in porous media, due to reduced void space and torturous pore paths, the measured
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becomes smaller and thus an effective diffusion coefficient is commonly used to incorporate the
confining effect of a porous media. The ratio between and , which is known as diffusive
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conductance R, is related to accessible porosity , tortuosity , and constrictivity 20-23. A common
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relationship for diffusive conductance is
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=
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which attempts to relate the macroscopic properties of a porous media to the confining effect that is
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microscopic in nature. From the literature, R ranges from 0.05 to 0.28 for liquid diffusion in
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unconsolidated porous media with porosity ranging from 0.1 to 0.3920. For gas diffusion in shale, R
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values ranging from 0.013 to 0.038 were calculated using the lattice Boltzmann method (LBM) for
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volumes constructed from scanning electron microscopy (SEM) images14. However, these volumes were
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tiny, e.g. 500nm×500nm×1250nm, so the porosity (from 0.18 to 0.27) may not be representative. More
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direct approaches use the method of volume averaging to consider the microstructure for determining
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in porous medium24-26. Overall, eq 3 is a general relationship that can be used to provide insight into
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the microscale structure from macroscale measurements.
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For diffusion in shale rocks, the previous researches can be roughly divided into two categories:
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=
(3)
(1) Building mathematical models that incorporate multiple physics, e.g. Darcy flow, slip flow,
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Knudsen diffusion, adsorption and desorption, and then running simulations to determine the
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relative importance of each mechanism in terms of their contribution to total gas flow.
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(2) Measuring the diffusion coefficient of the porous media for a particular pair of fluids, typically
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using adsorption/desorption isotherm studies for gas diffusion, and contact-mixing experiment
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of two solutions for liquid diffusion.
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In Category 1, Mehmani et al.27 considered both Knudsen diffusion and slip flow for each pore throat in a
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pore network model and found that the apparent permeability of the pore network was significantly
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increased at lower pressures. Huang et al.28 also took the pore network modelling approach and treated
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gas conductivity enhancement by adding the Klinkenberg correlation. Chen et al.14 employed the LBM to 3 ACS Paragon Plus Environment
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run nanoscale simulations on reconstructed pore structures of shale, from which effective Knudsen
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diffusion coefficients were predicted. In Category 2, adsorption/desorption isotherm experiments are
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frequently used to measure the diffusion coefficient of gases, typically CH4 or CO2, in crushed shale
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samples29, 30. For this method, either a unipore model31 or bidisperse model32, 33 is used to fit the
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experimental data, with the latter being capable of measuring two diffusion coefficients, namely macro-
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pore and micro-pore diffusion coefficients. These methods are also routinely used for coals34, 35. Another
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method is X-ray micro-computed tomography (micro-CT) imaging, which has been used extensively to
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characterize porous rocks36-40 , study morphology and topology41, 42, model reactive transport43, 44, and
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visualize multiphase flow45-49. Reviews on micro-CT techniques and their applications on porous
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medium can be found in the works of Wildenschild and Sheppard50, Mostaghimi et al.51, Bultreys et al.52
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and Blunt et al.53.
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For studies of diffusion in porous medium using micro-CT, only a few papers can be found in the
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literature. Guerrero-Aconcha and Kantzas54 measured the diffusion coefficient of propane in heavy oil
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using computed assisted tomography. Liu et al. measured the local diffusion coefficients along the 1D
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diffusion path of CO2 in n-decane saturated porous media55. Also, Tidwell et al. investigated the effects
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of heterogeneous porosity on liquid diffusion in dolomite matrix using X-ray absorption imaging56. Polak
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et al. studied the vertical diffusion of a liquid tracer from a fracture into the surrounding matrix of a
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fractured chalk using a CT scanner and estimated the diffusion coefficient57. Vega et al. monitored the
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diffusion of krypton gas in a coal sample using micro-CT58. However, in these studies the problems were
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often simplified to 1D profiles for selected time steps. Although Agbogun et al. measured the diffusion-
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accessible porosity of a dolostone sample and calculated temporal tracer concentrations in 3D during
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diffusion, no diffusivity was measured59. Zhang et al.15, 60 measured the local directional effective
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diffusivity of CH2I2 in toluene in shale plugs using 4D imaging using a new mathematical method. The
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method treats each voxel as a continuum and is able to calculate local effective diffusivity by calculating
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the local diffusive flux and local concentration gradient. However, the measurement was carried out
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only on 2D domains, which would have a degree of uncertainty due to diffusion in the 3rd dimension.
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Moreover, as the studied 2D domains were limited in size, the measured diffusivity data may not be
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abundant enough to provide a statistical representation of the rock the sample. Therefore, full 4D
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analysis of diffusion in shale is needed, which allows for studying the time evolution of the process and
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directional anisotropy and thus, provides more reliable and abundant data. In addition, the link between
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effective diffusion coefficient and rock heterogeneity also needs to be addressed.
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Herein, we present 4D experimental data that provides an unprecedented wealth of information on
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diffusion in shale rock. Similar to the experiment in Zhang et al.15, we use 4D micro-CT imaging to study
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the liquid-liquid diffusion of diiodomethane CH2I2 from a Permian Basin shale core plug surrounded by
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toluene. The spatial and temporal progress of shale diffusion is visualized in sequenced 3D tomograms –
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4D imaging. The novelty of this work is that we are able to quantify diffusion coefficients at the
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micrometer length scale while visualizing larger more representative 3D samples rather than previous
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works that are only 1D or 2D. Imaging techniques commonly suffer from the trade-off between field of
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view and image resolution7. The technique proposed herein bridges this gap and thus provides a means
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to relate shale rock structure to macroscale transport properties. We quantify the variability of diffusion
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coefficients within a shale core plug, and then we link the observed macroscopic phenomenon to
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microscopic pore structures and heterogeneities. Moreover, by measuring the local effective diffusivity
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of the whole plug in 3D, we are able to make more reliable statistical characterization and quantification
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of sample anisotropy. The presented technique is not limited to shale rock and could be applied to any
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porous system and in particular highly heterogeneous materials. Our work provides a mean to develop
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large scale models that directly link to the underlying structure and sub-scale heterogeneity of porous
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systems.
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2. Materials and Methods
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A shale sub-plug from the Permian Basin formation was used. The sub-plug was drilled perpendicular to
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the bedding plane out of a centimeter-sized core plug. The sub-plug was 4.3mm in diameter and 6.2mm
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in height. Bulk analysis of sister sample material gave a porosity of 7.8% from Helium porosimetry and a
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total organic carbon (TOC) content of 3.0 wt% from the LECO method. The thermal maturity of this
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siliceous shale lies in the early-mid oil window, and its mineral composition from XRD is listed in Table 1.
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The sub-plug was scanned over its full height, at a voxel size of 1.9 µm, using a double-helically
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trajectory on a HeliScan micro-CT scanner61 in a sequence of 2 states, namely 1) after cleaning and
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drying and 2) after saturation with CH2I2. Cleaning was performed using toluene and methanol to
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remove hydrocarbons, soluble bitumen, water and salts. Saturation of the cleaned and dried sample
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with CH2I2 was performed in an off-line process using vacuum infiltration followed by isostatic
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pressurization to 8000 psi for 10 days. In particular, CH2I2 is used as a contrast agent due to its high
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opacity; however, this also tends to produce imaging artefacts such as beam hardening, which must be
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dealt with during image processing.
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A series of time-sequenced tomograms during diffusion were acquired by continuous circular scanning
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(at lower resolution, 5 μm/voxel) of a vertical sub-section of the initially CH2I2-saturated sub-plug after
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its immersion in a toluene-filled holder under ambient conditions. This series of micro-CT images
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monitors the diffusion of CH2I2 from the sub-plug into the surrounding toluene. Table 2 lists the duration
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of each circular scan, number of circular scans, image size, resolution of tomograms, and the field of
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view for the experiment. All tomograms were corrected for beam hardening, masked, spatially aligned,
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and linearly rescaled such that the grayscale values (attenuation) of solid grains matched across all
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imaged states. Beam hardening correction (BHC) was performed on the dry-state and saturated-state
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tomograms and the first tomogram in the diffusion series, to flatten their radial profiles, and the same
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correction to this first diffusion tomogram was applied to all subsequent tomograms in the series.
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Accurate 3D-3D image registration was accomplished using MANGO software package62, which provides
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voxel-by-voxel precision. Further details on sample preparation and image analysis are provided in a
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previous conference paper by Zhang et al. (2017)57.
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Table 1. Sample characterization data with minerals presented as wt% of total mineral Porosity
TOC
Calcite
(%)
(wt%)
(wt%)
7.8
3.0
0
Illite/
Illite/
Quartz
Smectite
Muscovite
Plagioclase
(wt%)
(wt%)
(wt%)
5
25
65
Pyrite (wt%) 5
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Table 2. Micro-CT acquisition parameters Duration of each circular scan (min) 36
Number of circular scans 60
Height of field of view (mm) 2.6
Tomogram size
Resolution
(voxels)
(µm/voxel)
871×865×520
5.0
165 166 167
The mathematical method for measuring local effective diffusivity values is presented and validated in
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our previous work.15. Herein, we extend our previous work by implementation of the conjugate gradient
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methods to solve the system of equations for a large 3D domain. This allows for quantification of vertical
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and horizontal effective diffusivities, which were not possible in our previous validation work.13
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Therefore, only a brief introduction of the method is provided below.
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Relative concentration and relative density fields of CH2I2 are calculated from the grayscale values of the
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tomograms. We calculate relative mass concentration of CH2I2 ( ) as
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(!,#,$,%) = &
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where -./011(2,3,4,5) is the grayscale at location (6, 7, 8) of the diffusion tomogram at time t, and
& '())(!,#,$,%) *& '#(!,#,$)
(4)
&+ , (!,#,$) *& '#(!,#,$)
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-.9:;) is obtained from
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?(!,#,$,%) =
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where the constant -.AB;C; is the grayscale of a voxel in a resolved pore in the state of full saturation
& '())(!,#,$,%) *& '#(!,#,$)
(5)
& &+ , *& @(
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with CH2I2 and -.D0= is the corresponding value in the dry (air-filled) state. Relative density values ranges
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between 0 and the local porosity of the voxel.
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Fick’s second law, eq 6, is used to model diffusion. To measure the local effective diffusion coefficients,
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the spatial variation of is considered as
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E?(!,#,$,%) E%
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where is the effective diffusion coefficient at each location and each has three components, i.e. H, I
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= F ∙ ( F(!, #, $, %))
(6)
and J directions. Here we assume the spatial variation of is only a result of pore geometry changes in
the space, i.e. does not change with local compositional changes. So is assumed to be constant for
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each location and for each direction. To solve for , a scalar potential field, P, is introduced as
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FK = = − F(!, #, $, %)
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where J is diffusive flux. Then eq 6 becomes
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−
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where the left-hand side can be estimated by the temporal difference of two density fields using the
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finite difference method, i.e. >5LM5 and >5 . eq 8, which is formally known as Poisson’s equation, leads to
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a large system of linear equations after discretising using finite difference method (FDM) for the 3D
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images (871×865×520 voxels) obtained. The conjugate gradient (CG) method is used to solve for the
E?(!,#,$,%) E%
(7)
= F K(!, #, $, %)
(8)
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solutions, which takes around 3 hours on 8 CPU cores and 100GB of memory. After solving for the
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potential field, P, the effective diffusion coefficient at each voxel in each direction can be calculated by
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( )( = − (F)( , ( = N, O, P
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where i signifies all 3 dimensions. However, the local diffusion coefficient results are more suitably
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presented in terms of horizontal local effective diffusion coefficient Q and vertical local effective
(FK)
(9)
(
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diffusion coefficient R . R is the J direction local diffusion coefficient. Q is in the HI plane, parallel to
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bedding, and calculated by
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S(!,#,$) =
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where N(!,#,$) and O(!,#,$) are the diffusive flux at location (6, 7, 8) in H and I direction, respectively.
TN(!,#,$) LO(!,#,$)
(10)
TFN(!,#,$) LFO(!,#,$)
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Also, FN(!,#,$) and FO(!,#,$) are the concentration gradient at location (6, 7, 8) in H and I direction,
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respectively.
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Although the pore spaces in shale are sub-resolution at 5 μm/voxel, the local porosities can be
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estimated from tomograms by dividing density (>), eq 5, by concentration of CH2I2 (φ), eq 4, which
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provides
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(!,#,$)
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where (2,3,4) is the local porosity of each voxel and does not change with time. Dykstra-Parsons (DP)
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coefficients63 are calculated to quantify the variation of local parameters in 3D space by using
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K =
215
where [̅ is the mean value of a given variable, which could be mean grey-scale, porosity or effective
=
& &+ , (!,#,$) *& '#(!,#,$)
(11)
& &+ , *& @(
V *'WX.Z ' V '
(12)
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diffusion coefficient, i.e. a variable value with 50% probability. Also, []^._ is the value of the same
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parameter with 84.1% probability, which is mean plus one standard deviation.
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3. Results and Discussion
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The images in Figure 1 show the same central longitudinal slice of the helically-scanned tomogram of the
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Permian Basin sub-plug in its (a) dry and clean state, and (b) CH2I2 saturated state after registration. In 8 ACS Paragon Plus Environment
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Figure 1 (a), the prevalent bright features comprise dense mineral, which from Table 1 is pyrite. Aside
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from a couple of visible fractures, which taper in from the sub-plug periphery and appear to be coring-
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induced, the darkest features are chambers with slightly brighter interiors – presumably comprising low
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density authigenically-formed mineral – which more commonly occur in the upper half. In some
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instances, mainly in the lower half, this interior mineral phase has grown and densified to leave only a
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thin dark ring around it. These dark regions, and those in thin, short seams, may comprise pore or
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organic matter or a mixture of both of these low attenuating phases. The remainder of the sub-plug is a
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fairly featureless, uniform grayscale in Figure 1(a), within which the voxels presumably contain a fine
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mixture of mineral grains, clays, porosity and organic matter11.
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After saturation with CH2I2, the brightening of voxels from Figure 1(a) to Figure 1(b) directly reflects the
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local connected porosity at that location. All dark features in Figure 1(a) strongly brighten and thus
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possess substantial porosity. The low density mineral interiors of chambers are also highly porous, while
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the denser interiors naturally brighten to a lesser degree. The uniform grey background regions in Figure
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1(a) also brighten due to matrix porosity, to reveal features and textures that were not apparent in the
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dry state. In particular, the saturation exposes two distinct rock types in the upper and lower halves of
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the sub-plug in Figure 1(b). The upper half has somewhat lower porosity, which is also more
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homogeneously distributed. Occasional dark spots in the matrix correspond to solid grains of
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quartz/plagioclase (from Table 1). Although these grains are much larger than the voxel size of 1.9 µm,
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they cannot be distinguished in the dry state due to the similar attenuation of the surrounding matrix of
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clays of illite and smectite (from Table 1), but become apparent in Figure 1(b) due to the brightening of
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the latter. The porosity in the lower half is somewhat higher and is more heterogeneously distributed,
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exhibiting darker sub-laminations of low porosity that generally coincide with pyrite-lean bands. Note
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that the above-mentioned fractures from the sub-plug extremities are mainly not brightened since CH2I2
244
drained from these regions during acquisition of the saturated-state tomogram, driven by the large
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density difference between CH2I2 and air. However, a number of even finer fractures within the sub-plug
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interior become apparent due to the saturation treatment, as discussed further below.
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1mm 247 248
(a)
(b)
249 250
Figure 1. Central longitudinal slice (4.3mm x 6.2mm) of the helically-scanned tomogram of the Permian Basin sub-plug (a) in its
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circular scans during the subsequent diffusion experiment.
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The yellow rectangle in Figure 1, straddling the two rock types, shows the reduced vertical field of view
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of the circular scans of this sub-plug during the diffusion experiment. Figure 2(a) and 2(f) display this
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same slice of these two tomograms after cropping to this reduced height. Figure 2 (b-e) show the
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corresponding registered tomogram slice at four times during 4D imaging of the diffusion experiment,
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namely at 2h 42m, 5h 42m, 17h 6m, and 33h 18m, respectively. These time-sequenced tomograms
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display the progress of diffusion by the darkening of the images from Figure 2(a) to (e), due to the local
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loss of attenuation arising from the surrounding toluene (dark) entering the sub-plug to mix with and
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dilute the CH2I2 (bright).
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At the very start of the diffusion series (e.g. Figure 2(b)), more fine fractures are brightened than in the
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saturated state of Figure 2(a), since the 4D imaging (starting from the CH2I2 re-saturated state) involves
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less fracture drainage owing to the reduced fluid density difference and the faster imaging. The diffuse
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darkening of the fractures near the sub-plug periphery at these early stages is driven by fast diffusion of
clean & dry state and (b) after saturation with CH2I2 and registration. The yellow rectangle indicates the field of view of the
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toluene along these pathways, while radial ingress and mixing of toluene through the unfractured matrix
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regions lags behind. Over time, the diffusive darkening advances and proceeds most rapidly where the
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toluene meets up with fine interior fracture segments and/or locations where matrix porosity is higher.
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The high porosity chambers only admit toluene once their surrounds do, and thus are not preferred
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pathways. Diffusion is faster in the lower rock type; midway through the imaged experiment the
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darkening has spanned this lower half while the upper half remains bright across its centre. Near the
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end of the imaging series, the grayscales in the lower half in Figure 2(e) approach those in the dry state
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of Figure 2(f), since toluene attenuates only slightly greater than air, while diffusion remains incomplete
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in middle of the upper half, especially apparent from the chambers there in which their CH2I2 remains
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substantially undiluted.
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Figure 2. Central longitudinal slice (4.3mm x 2.6mm) of the registered tomogram series of the Permian Basin sub-plug, in its (a)
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CH2I2 saturated state and (f) clean & dry state, reduced from Figure 1 to the field of view of the diffusion experiment, and (b)-(e)
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Figure 3 displays the same field of view of these same six states for the corresponding difference
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tomograms, i.e. CH2I2-saturated state minus diffusing-state. This difference serves to subtract the
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contributions from all solid components (minerals and organic matter) to isolate the changes in CH2I2
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concentration in pores due to the dilution of CH2I2 by toluene. In particular, these images locally
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brighten in proportion to the loss in volume of CH2I2 at each voxel. Near the start of the diffusion
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imaging (e.g. Figure 3(b)), the difference is only bright where diffusion of toluene into the sub-plug has
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commenced, primarily along and out from fractures. The remainder is dark, indicating no CH2I2 dilution
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at this stage. The darkest fractures correspond, as mentioned above, to those that drained in the
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saturated-state image and thus contain more CH2I2 at the start of the diffusion series. The brightening of
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the difference tomograms continues from Figure 3(b) to 3(e), proceeding fastest along the more
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preferred pathways, i.e. fractures. This last image is very similar in its lower half to the difference CH2I2-
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saturated state minus dry state in Figure 3(f), while the dark cloud over the center of the upper half
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demonstrates the incompleteness of diffusion there. The contribution of locally high porosity and/or
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fractures to diffusion, based on the above-mentioned calculation of concentration and density from
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such tomogram differences, is quantified below.
showing diffusion tomogram at time of 2h42m, 5h42m, 17h6m, and 33h18m, respectively.
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(a)
(b)
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Figure 3. Central longitudinal slice (4.3mm x 2.6mm, with field of view corresponding to Figure 2) of the registered tomograms
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saturated minus diffusing-state difference tomograms at time of 2h42m, 5h42m, 17h6m, and 33h18m, respectively.
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We selected a late time diffusion tomogram, 30 hours after the start of the diffusion experiment, for full
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3D simulation using eq 4 – eq 9. This particular time was selected to make sure that diffusion has
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occurred in most of the voxels such that we have a concentration gradient across the sample that allows
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for measurement of local diffusion coefficients. In Figure 4 (a) and (b) we display horizontal ( Q ) and
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vertical ( R ) effective diffusion coefficients in 3D for the entire field of view. These results provide a
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direct means to evaluate the spatial variation of effective diffusivities with relatively high resolution.
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With red colors indicating higher values while blue colors indicating lower values, heterogeneities are
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clear in both of the two volumes, which we will discuss in detail.
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Special attention is required when interpret the measurement results because they may be affected by
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the mathematical method and/or tomogram noise. Three types of data are thought to be unreliable
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with our current methods and may lead to erroneous diffusivity measurements. The 1st type of error is
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from data in the middle of the core plug, i.e. around the central axis. Within this region Q values are
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of the Permian Basin sub-plug in (a) its dry state, (f) its CH2I2-saturated minus dry-state difference; (b)-(e) 4 slices of the CH2I2-
smaller compared with those in the outer regions. These smaller Q do not necessarily mean low 13 ACS Paragon Plus Environment
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diffusivity, they are a result of the method. For our diffusion experiment, in which CH2I2 diffuses radially
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from the sub-plug into the surrounding toluene, the central axis has a symmetrical no-flux boundary
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condition. As a result, the diffusive flux and concentration gradient near the central axis are too small to
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be accurately captured. The 2nd type of error is in the outer region along the perimeter of the R volume.
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The obtained R values are obviously smaller than compared with those in the inner region, which is not
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a true reflection of the diffusion properties of the sample. The reason for this is thought to be that
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vertical diffusion in this region is too small to be accurately captured. For our experiment set-up, the
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principal diffusion is in the radial direction and vertical diffusion is only a result of heterogeneous radial
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diffusion. In other words, unless there is vertical concentration gradient caused by uneven horizontal
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diffusion, there will not be vertical diffusion. Because the outer region is close to the boundary, diffusion
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is more complete in all directions and like a steady state which leads to relatively homogenous
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concentration field. As a result, vertical diffusion is small. By contrast, in the inner region which is far
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from the boundary, diffusion is more controlled by fractures. Fractures create fast radial diffusion from
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deep inside the sub-plug, which results in uneven horizontal diffusion and heterogeneous concentration
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field. So, vertical concentration gradient and vertical diffusion in the inner region are larger in the inner
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region. The 3rd type of error is evident as white regions in the Q and R volumes, which are too high
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values, e.g. infinity. They are prone to occur in regions where almost no concentration gradient of CH2I2
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exists and no diffusion occurs. These regions correspond to materials that have too low porosity, like
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pyrite and quartz. Because their porosity is very low, the attenuation change due to CH2I2 diffusion is too
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small and easily blurred by image noise. As a result, erroneous concentration gradient of CH2I2 is
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calculated which leads to faulty diffusivity measurement. For the remainder of our analyses, we
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disregard regions of the sample that have erroneous local diffusivity measurement.
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In Figure 4 (c) and (d) we display the histograms of the effective diffusivity data in Figure 4 (a) and Figure
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4 (b), respectively after removing data along the boundaries. That means only Q data that have a
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horizontal distance to the central axis larger than 100 voxels are used for Figure 4 (c) and only the R
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data that have a horizontal distance to the central axis smaller than 300 voxels are used for Figure 4 (d).
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The histograms show that about 98% of Q values are between 10-15 and 10-13 m2/s while R values are
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more distributed with about 80% in the range between 10-16 and 10-14 m2/s. Overall, the R is generally
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lower than the Q , indicating anisotropic diffusion property, which corresponds to the sample being cut
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parallel to the bedding plane.
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Figure 4. Local effective diffusion coefficient calculation results in 3D of the full field of view of the diffusion experiment with (a) showing the S and (b) showing the ` . Each data volume has a size of 430×520 (radius × layer) voxels; (c) and (d) are the histograms of the S and ` volume, respectively.
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Figure 5 and Figure 6 show (a) two cross-sections of the Q volume of the Permian Basin sub-plug, and
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their corresponding (b) porosity maps, (c) dry tomograms, and (d) CH2I2 saturated tomograms. These
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images are taken from a plane parallel to the bedding. In Figure 5 (a) the Q data appears relatively
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homogeneous without too many abrupt variations. The dry and CH2I2 saturated tomograms, Figure 5 (c)
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and (d), also show a relatively homogeneous cross-section in terms of attenuation, which correspond 15 ACS Paragon Plus Environment
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well to the porosity map of Figure 5 (b). The vast gray background in the dry tomogram, Figure 5 (c),
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corresponds to the shallow green areas in Figure 5 (b) that have porosities between 5% and 10% and the
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continuous areas that are mostly red in Figure 5 (a), which have local effective diffusion coefficients of
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around 10-14 m2/s. As discussed earlier, they presumably contain a fine mixture of mineral grains, clays,
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porosity and organic matter. The bright spots in the dry tomogram, which are most likely dense pyrite
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crystals, have lower porosities below 5% and also lower local effective diffusion coefficients of around
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10-15 m2/s. They correspond to the small blue clusters in Figure 5 (a). The more porous (with porosities
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above 20%) and less dense chambers, presumably comprising low density authigenically-formed mineral,
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are very bright in the CH2I2 saturated tomogram with more CH2I2 admitted. They are dark red in the Q
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map with high local effective diffusion coefficients above 10-14 m2/s. One common feature of these
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heterogeneities is that they all have a relatively small correlation length. No clear evidence of fractures
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is observed for this particular cross-section, which suggests a relatively homogeneous diffusion with only
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small-scale heterogeneities that are a result of spatial compositional changes in the sub-plug. The DP
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coefficient of the Q data in Figure 5 (a) is 0.57.
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(d)
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Figure 5. A cross-section of the Permian Basin sub-plug with (a) showing the measured local effective S in unit of m /s, (b) 2
showing the porosity map, (c) showing the dry & clean tomogram and (d) showing the CH2I2 saturated tomogram.
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The images provided in Figure 6 demonstrate a region of the shale sample that is quite different from
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the one in Figure 5. One most obvious example is the large blue regions that have lower local effective
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diffusion coefficients in the upper part of Figure 6 (a). They correspond to the lower-porosity less-
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permeable area in the porosity map, Figure 6 (b). These regions are also evident in the CH2I2 saturated
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tomogram, Figure 6 (d). For shale rock, the pore space of these regions may have pore throats that are
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only several nanometers in diameter and diffusive pathways may be highly torturous and not well
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connected. Although the materials for these regions have similar density as other regions, judging from
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the moderate and homogeneous attenuation values in the dry and clean tomogram, they have lower
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porosity, which leads to slow mass transport. This change of composition is clearly captured by the Q
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map and CH2I2 saturated tomograms. This once again suggests that the attenuation properties of the
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clean and dry tomograms fail to identify heterogeneity. The importance of identifying these low porosity
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materials is that these regions will have a negative effect both on gas storage and gas flow in a shale gas
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reservoir, quantification of the weight or volumetric percentage of these materials will give us a rough
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idea of the quality of the sample in terms of reservoir rock. It would also be helpful for the economic
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evaluation of a shale gas play if a large enough number of samples can be analysed. It is interesting that
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even in the low porosity region we can find some high porosity high permeable regions, i.e. the dark red
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spots in the upper region of Figure 6 (a) which are yellow with porosities above 20% in Figure 6 (b). This
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reflects the complex composition and structure of the sample. The DP coefficient of Q of the cross-
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section in Figure 6 is 0.59, which is higher than the one presented in Figure 5, although the two cross-
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section planes are only 0.34 mm apart.
402 403
(a)
(b)
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1mm 404 405 406 407 408
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Figure 6. A cross-section of the Permian Basin sub-plug, 0.34mm below the one shown in Figure 5, with (a) showing the measured local effective S in unit of m /s, (b) showing the porosity map, (c) showing the dry & clean tomogram and (d) 2
showing the CH2I2 saturated tomogram.
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To study the variation of the degree of heterogeneity between cross-sections, DP coefficient of effective
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horizontal diffusivity Q is calculated for each cross-section from top to bottom of the field of view, with
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results shown in Figure 7 (a). The results show fairly evident changes in the DP coefficient for effective
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diffusivity within the scanned section of the sub-plug, which is only 2.6 mm in length. The upper section
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has clearly lower DP coefficients than the lower section, indicating that the lower section is more
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heterogeneous. This corresponds well to the observations we have in the high-resolution saturated
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tomogram, Figure 1 (b), in which the rock type in the lower half of the sub-plug is more heterogeneous.
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These findings demonstrate the heterogeneity in different layers and may also be helpful in studying the
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laminations of shale on a micrometer scale. For comparison studies, the DP coefficients of grayscale
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values of the CH2I2 saturated and dry and cleaned tomograms are also calculated and shown in Figure 7
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(b). However, quite different to Figure 7 (a), Figure 7 (b) demonstrates much more homogeneity since
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the DP coefficients are generally much smaller and do not change much between cross-sections. So 4D
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imaging and local diffusivity measurements identify the true heterogeneity of the shale sample.
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Dykstra Parsons coefficient 0.55
0.6
0.65
Dykstra Parsons coefficient
0.7
0
422 423 424 425 426
0.05
0.1
0.15
1 Cross-section number
1
Cross-section number
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101 201 301
101 201 301
401
401
501
501 Effective diffusivity
Grayscale - wet tomo
(a)
Grayscale - dry tomo
(b)
Figure 7. DP coefficient of each cross-section of (a) the effective S volume, (b) the CH2I2 saturated and dry & clean tomogram of the Permian Basin sub-plug, from top to bottom.
In Figure 8, R maps of vertical central longitudinal slices of the Permian Basin sub-plug are cut from the
427
3D volume. The R maps in Figure 8 are more heterogeneous than the Q maps in Figure 5 and Figure 6,
428
with large blue regions having low effective diffusion coefficients. However, as discussed earlier, the
429
blue areas of the sub-plug may not necessarily have low diffusivities. The blue areas near the left and
430
right boundaries in Figure 8 (a) and (b) are due to the fact that the diffusion flux is too small to be
431
accurately captured within these regions, which is a result of the experimental setup where diffusion is
432
mostly in the horizontal direction pointing to the zero CH2I2 concentration perimeter boundary. Apart
433
from this, two near horizontal blue bands are observed in the R maps that are presumably influenced
434
by fractures. The measured R values of a horizontal fracture may be low due to the reason that in such
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fractures, where the principal diffusion is in the horizontal direction, vertical diffusion is too small to be
436
accurately captured similar to the effect along the perimeter boundary. However, low R values may
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also be the result of zero flux symmetry boundary conditions along the center plane between two
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fractures. The domain that locates between two parallel fractures is a source for CH2I2 for the two
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fractures, so the center plane between the fractures would have zero flux in the vertical direction due to
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symmetry. This is a similar phenomenon to the lower Q values around the central axis of the Q
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volume in Figure4 (a). Apart from the influence of fractures, local heterogeneities are also responsible
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for the pattern showing in the R maps, although a more specific explanation may be difficult given
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tomogram resolution of 5 micrometers/voxel. 3D focused ion beam SEM (FIB-SEM) imaging may be
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added in the future to identify the local heterogeneities that have low effective diffusion coefficient.
1mm 445 446 447 448 449
(a)
(b)
Figure 8. (a) Local effective ` results of a central longitudinal slice of the Permian Basin sub-plug, same as the one in Figure 2
and Figure 3; (b) local effective ` results of another central longitudinal slice of the Permian Basin sub-plug, perpendicular to the one in (a).
450
4. Conclusions
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We study the liquid-liquid diffusion in shale using 4D micro-CT imaging. The time-sequenced diffusion
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tomograms enable us to visualize the dynamic process in 4D, which not only provides detailed
453
information of the diffusion but also makes it possible to identify local heterogeneities of the shale plug.
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The 4D results revealed that diffusion occurring in the Permian Basin sub-plug is uneven and influenced
455
by matrix heterogeneities that are mainly due to compositional changes within the matrix. Local
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diffusivity is generally positively related to porosity, i.e. materials with higher porosity have higher
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effective diffusivity. By measuring the local effective diffusion coefficients, the low porosity materials
458
that cannot be distinguished from density/attenuation in the dry and clean tomograms are clearly
459
identified. Identification of those heterogeneities not only helps us to better characterize the diffusion
460
properties of shale rock but also helps us to evaluate the quality of a shale sample in terms of reservoir
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rock. Heterogeneity is evident in horizontal layers that are parallel to the bedding, and the degree of
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heterogeneity varies between layers in the vertical direction, which may be a reflection of the laminated
463
nature of shale rocks. Diffusion anisotropy of the shale sample is evidenced by 3D diffusivity
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measurement, which shows lower vertical effective diffusion coefficients than horizontal ones in general.
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Although shale pore structures are unable to be seen at current tomogram resolution, the measured
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local effective diffusion coefficient enables us to infer smaller scale characteristics over a voxel volume 20 ACS Paragon Plus Environment
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(around one hundred cubic micrometers). For instance, high diffusivity indicates larger porosity, better
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pore connectivity, and smaller tortuosity.
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For future work, SEM images and FIB-SEM volumes on selected locations will be obtained to show the
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nanometer scale pore structures, which will then be linked to their effective diffusivity measured here.
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Then we are going to run pore scale simulations of diffusion on shale FIB-SEM images with different
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porosity and calculate their effective diffusivity. The aim is to get the correlation between effective
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diffusivity and porosity, which in combination with local porosity distribution from micro-CT can be used
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to map local effective diffusivity for a whole sample plug. Then plug scale simulations of diffusion can be
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carried out and effective diffusivity for the whole plug can be calculated. In this way, we can accomplish
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a complete workflow of characterization and diffusion property upscaling for shale matrix from
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nanometer to plug scale.
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Acknowledgement
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This research/project was undertaken with the assistance of resources and services from the National
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Computational Infrastructure (NCI), which is supported by the Australian Government. We acknowledge
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funding from the member companies of the ANU/UNSW Digicore research consortium.
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References
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