This unit introduces general concepts of spatial heterogeneities, principles of physical flow and transport processes in heterogeneous porous media, as well as how to set up simulations for flow and non-reactive transport in a 2D heterogeneous domain in CrunchFlow.
By the end of this lesson, you should be able to:
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If you have any questions, please post them to our Questions? discussion forum (not e-mail), located in Canvas. The TA and I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.
Flow and transport processes, including advection, dispersion and diffusion are described in the lesson Flow and Transport Processes in 1D System. In natural system we often need to consider these processes in multiple dimensions and in heterogeneous systems where the physical and geochemical properties of the subsurface are not evenly distributed. As an example, in Figure 1, we show that the distribution of permeability dictates the spatial distribution of injected tracer from injection wells during a biostimulation experiments at Old Rifle, Colorado (Li et al., 2010).
The general Advection-Dispersion Equation (ADE) in multiple dimensions is as follows:
Here C is the tracer concentration (mol/m3 pore volume), t is time (s), D is the combined dispersion–diffusion tensor (m2/s), v (m/s) is the flow velocity vector and can be decomposed into vL and vT in the directions longitudinal and transverse to the main flow in a 2D system. The dispersion-diffusion tensor D is defined as the sum of the mechanical dispersion coefficient and the effective diffusion coefficient in porous media De(m2/s). At any particular location (grid block), their corresponding diffusion / dispersion coefficients DL (m2/s) and DT (m2/s) are calculated as follows:
Here $\alpha_L$ and $\alpha_T$ are the longitudinal and transverse dispersivity (m). The dispersion coefficients vary spatially due to the non-uniform permeability distribution. Values of $\alpha_T$ are typically at least one order of magnitude smaller than $\alpha_L$ (Gelhar et al., 1992; Olsson and Grathwohl, 2007).
The terms homogeneity and heterogeneity are used to describe the uniformity and regularity in spatial distribution of geomaterial properties in natural subsurface systems. Homogeneity means spatially uniform-distributed properties. In natural systems, geological media are almost always heterogeneous. That is, their physical and (bio)geochemical properties vary spatially. Spatial heterogeneity can refer to both physical and geochemical properties. Physical properties include mineral grain size, porosity, and permeability / conductivity. Geochemical properties include, for example, mineral types, lithology, mineral surface area, and cation exchange capacity. Figure 2 shows a picture of an outcrop with layers of different types of geomaterials from the Macrodispersion site (MADE) in Columbia, Mississippi (Zheng and Gorelick, 2003).
Spatial heterogeneities can lead to significantly different chemical reactions and physical transport processes from their homogeneous counterparts. As a result, heterogeneities play an important role in determining processes and applications in subsurface systems. Examples include contaminant reactive transport (Li et al., 2011), oil and gas production (Chen et al., 2014; Hewett, 1986), and environmental bioremediation (Murphy et al., 1997; Song et al., 2014). For example, as shown in Figure 3, the different geomaterials in Figure 2 leads to orders of magnitude in the spatial variation of hydraulic conductivity (Figure 3A), and abnormal solute transport (Figure 3B) (Zheng et al., 2011).
Geostatistics has been well developed to quantitatively describe the characteristics of spatial physical heterogeneities, especially conductivity / permeability, in the natural subsurface. Here we briefly introduce a few geostatistical measures. Interested readers are referred to geostatistical books and software tools for more detailed information (Deutsch and Journel, 1992; Remy and Wu., 2009).
It measures the average water-conducting capability of porous media. There are different definitions widely used in subsurface hydrology:
Arithmetic mean $\kappa_T$ is defined as:
Here n is the total number of zones or subsystems; $\kappa_i$ is the local permeability of subsystem.
Harmonic mean $\kappa_h$ is defined as:
Geometric mean $\kappa_g$ is defined as:
Effective permeability$\kappa_e$ is derived from the Darcy’s Law:
where u is the average flow velocity (m/s) of the field; $\mu$ is the flow viscosity (Pa·s); $\Delta L$ is the length (m) along the main flow direction; $\Delta P$ is the pressure differential (Pa) along the main flow direction. The effective permeability describes the real water-conducting capability of the porous media. The upper bound of the effective permeability is often the arithmetic mean, and the lower bound is the harmonic mean. They correspond to flow through a perfectly layered systems, parallel and perpendicular to the layering, respectively (Heidari and Li, 2014; Zinn and Harvey, 2003).
The extent of deviation from the mean is calculated as follows:
where $\operatorname{Var}(\kappa)$ is the permeability variance (m4). Larger variance indicates larger extent of spatial heterogeneity. Permeability of geological media differs significantly in natural subsurface systems. Conductivity (K) is also commonly used as a measure of conducting capability of porous media. The Macrodispersion Experiment (MADE) site in Mississippi has both small silt and sand regions. It has a centimeter scale ln(K) variance as large as 20, while its variance at the meter-scale measured by borehole flowmeters is approximately 4.0, which is considerably smaller because small-scale variability is averaged out (Feehley et al., 2000; Harvey and Gorelick, 2000; Zinn and Harvey, 2003). The Wilcox aquifer in Texas also has high ln(K) variance of about 10, as does the Livermore site (ln(K) variance >20) (Fogg, 1986; LaBolle and Fogg, 2002). The Bordon site at Borden, Ontario, however, is relatively homogeneous with ln(K) variance around 0.20 (Mackay et al., 1986; Sudicky, 1986). Note that ln(K) mentioned here is the natural logarithm of hydraulic conductivity K (m/s).
Correlation length is a measure of correlation in spatial variation. Two points that are separated by a distance larger than the correlation length have fluctuations that are relatively independent, or uncorrelated. In contrast, properties of two points that are within correlation length are correlated. For detailed calculation and application of correlation length in subsurface fields, readers are referred to (Bruderer-Weng et al., 2004; Gelhar, 1986; Heidari and Li, 2014; Salamon et al., 2007). Correlation lengths differ significantly in natural subsurface systems, as shown in Table 1. Figure 4 shows the patterns of permeability fields (m2) with different correlation length generated from Gaussian Sequential Simulation Method.
Source | Geological Media | Correlation Length (m): Horizontal | Correlation Length (m): Vertical | Overall Length (m): Horizontal | Overall Length (m): Vertical |
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Socorro, New Mexico | fluvial sand | >3 | 0.1 | 14 | 5 |
Rio Grande Valley, New Mexico | fluvial sand | 7.6 | 760 | ||
Cape Cod, Massachusetts | glacial outwash sand | 5 | 0.26 | 20 | 5 |
East central Illinois | sandstone aquifer | 4.5×104 | -- | 5×105 | -- |
Las Vruces, New Mexico | alluvial silty-clay loam soil | 0.1 | -- | 6 | -- |
A series of hydraulic connectivity have been defined as integrative measures of spatial heterogeneity characteristics. Static connectivity can be calculated by the connective structure of the hydraulic conductivity or geological facies. Dynamic connectivity directly relates to physical flow or transport processes. For interested readers, we refer them to (Knudby and Carrera, 2005; Knudby and Carrera, 2006; Renard and Allard, 2013; Siirila-Woodburn and Maxwell, 2014; Willmann et al., 2008).
Setting up 2D heterogeneous domain for flow and transport calculation generally includes three primary steps. First, the 2D domain needs to be defined with the targeted size, dimension, and resolution. The relevant keyword blocks for setting up the domain include DISCRETIZATION, INITIAL_CONDITION, and BOUNDARY_CONDITION. The second step involves the set up for the calculation of flow, which involves the use of various keywords in the FLOW key block, including constant_flow, calculate_flow, read_permeabilityFile, and pressure. For 2D heterogeneous systems we always use “calculate_flow” to calculate flow by specifying a pressure gradient using “pressure” at the two main boundaries. The permeability distribution can be either defined in INITIAL_CONDITIONS with relevant permeability values in specific grid blocks defined under GEOCHEMICAL_CONDITIONS, or by reading in permeability distribution using the keyword read_permeabilityFile. Detailed format of the permeability file is provided in the manual. After that, the transport keywords can be specified in the TRANSPORT keyword block, similar to the ones discussed in the 1D homogeneous setup.
Click here [1] for CrunchFlow Files for Example 8.1 and other related files for Li et al. 2014
The following gives an example of setting up flow and transport calculations in 2D homogeneous and heterogeneous domains. Here we use the physical set up of 2D cross-sections of the Mixed and One-zone column in (Li et al., 2014). The authors packed 4 columns of 2.5 cm in diameter and 10.0 cm in length with relatively similar amount of magnesite and quartz distributed in different spatial patterns. Here we focus on two columns that represent two extreme cases: the Mixed column with uniform distributed magnesite and quartz, as well as the One-zone column with magnesite all gathered in one cylindrical zone of diameter of 1.0 centimeter. To keep it simple, here we will do the calculation for 2D cross-sections, instead of following the steps in section 3.2.4 in the paper to convert 2D to 3D. So our numbers might be different from the paper. We are also assuming that in the middle one zone magnesite is 100% of the solid phase. The 2D system has a size of 25 mm by 100 mm. A constant differential pressure is imposed at the boundaries in the z (vertical) direction, leading the primary water flow direction in the z direction from the bottom to the top. No flux boundaries are specified in the X direction (Li et al., 2014).
Columns | Mg zone |
Qtz zone | $^2a_L$ (cm) |
$^4a_T$ (cm) |
$\mathrm{\kappa}_e$ (×10-13m2) |
$\phi_{\mathrm{avg}}$ |
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Mixed | 0.05 | 0.005 | 8.26 | 0.44 | ||
One-zone | Width: 1.0 cm
$\Phi_{Mg:}\ 0.54$ |
Width: 1.5 cm $\Phi _{Qtz:}\ 0.38$ |
0.07 | 0.004 | 10.74 | 0.44 |
*The permeability of the pure sand columns of the same grain size was measured to be $8.7 \times 10^{-13} \mathrm{~m}^{2}$.
Please do the following:
Run the simulation at the grid block resolution of 1 mm by 1 mm.
Note: in plotting 2D figures, you can use softwares such as Tecplot or Matlab. Penn State provides free Matlab web access on WebLabs at Penn State [2]. You can access matlab from there. In addition, you can also get video tutorials from LinkedInLearning [3] at Penn State about using matlab.
In Heidari and Li (2014), 2D flow experiments and modeling were used to understand non-reactive solute transport in heterogeneous porous media with different spatial patterns. There are three two-dimensional (2D) sandboxes (21.9 cm × 20.6 cm) packed with the same 20% (v/v) fine and 80% (v/v) coarse sands in three patterns that differ in correlation length: Mixed, Four-zone, and One-zone. The Mixed cases contain uniformly distributed fine and coarse grains. The Four-zone and One-zone cases have four and one square fine zones, respectively (Figure 8).
Read the paper carefully and follow the example in this lesson to generate the 2D flow fields and tracer breakthrough curves for the three sandboxes of high permeability contrast (HC). All parameters and properties of the three flow cells are in Table 1 and Table 2 in the paper. You can compare your modeling output of the flow field and breakthrough curves with Figure 5 (Flow field and local breakthrough curves) and Figure 6 (overall breakthrough curves from 2D model) in the paper. Note that one pore volume is the same as one residence time.
Apologies for no original CrunchFlow files for this hw. you can follow Example 8.1 and similarly set it up for this homework.
In this lesson we studied general concepts and geostatistical measures of spatial heterogeneities, as well as how to set them up in 2D domains.
You have reached the end of Lesson 8! Please make sure you have completed all of the lesson activities before you begin Lesson 7.
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