Calculation of Permeability of Clay Mineral in Natural Slope by Using Numerical Analysis

Abstract

A natural landslide is mainly occurred by rainfall, snowmelt, earthquakes and construction works. Especially, the role of rainfall or snowmelt in slope stability is very important because it causes a decrease in shear strength by reducing the soil cohesion. If clay exists in the weathered soil, the physical characteristics such as viscosity and permeability are generally different from the condition without the clay. In this case, changes of permeability or viscosity due to the rainfall or snowmelt are dependent on the content of clay in soil. In order to calculate the variation in permeability according to the content of clay in soil, many researchers have conducted laboratory experiments or in-situ tests in the field. However, it is difficult to determine the property of the clay such as a viscosity because of its poor crystalline property. In order to solve this problem and to calculate permeability of clay under various dry densities, we used molecular dynamic (MD) simulation to examine the viscosity of micro scale and homogenization analysis (HA) method to expand micro material property to macro scale. In this research, we determined the permeability of clay with various dry densities due to the rainfall or snowmelt conditions by using MD/HA method.

Molecular Structure of Kaolinite

Radioactive waste disposal facilities have been planned in formations containing kaolinite, for example in the Opalinus Clay of Switzerland, because of their low permeability and resultant diffusion-dominant characteristics. For safely isolating radioactive substances for a long time, it is essential to fully understand the physical and chemical properties of the host rock. For this purpose, authors here present a unified procedure of molecular dynamics (MD) simulation and homogenization analysis (HA) for water-saturated kaolinite clay. This MD/HA procedure was originally developed for analyzing seepage, diffusion and consolidation phenomena of bentonite clay. In the current research, a series of MD calculations were performed for kaolinite and kaolinite-water systems, appropriate to a saturated deep geological setting. Then, by using HA, the seepage behavior is determined for conditions of the spatial distribution of the water viscosity associated with some configuration of clay minerals. The seepage behavior is calculated for different void ratios and dry densities.

Kaolinite is a 1:1 clay mineral; composed of alternating silica tetrahedral and aluminum octahedral sheets. To achieve charge balance, the apical oxygens of the silica tetrahedra are incorporated into the octahedral sheet. In the plane of atoms common to both sheets, two-thirds of the atoms are oxygens and are shared by both silicon and the octahedral aluminum cations. The remaining atoms in this plane comprise hydroxyl molecules (OH), located in such a way that each sheet is directly below the base of a silica tetrahedron. A diagrammatic sketch of the kaolinite structure is shown in Fig. 1. The structural formula is Si4Al4O10(OH)₈, and the charge distribution is indicated in Fig. 2. Mineral particles of the kaolinite subgroup consist of these basic units, stacked in the c-direction. The bonding between successive layers is by both van der Waals forces and hydrogen bonds. The extensive hydrogen bonding, in particular, is sufficiently strong that there is no interlayer swelling. Because of a slight difference of oxygen-to-oxygen distance in the tetrahedral and octahedral layers there is some distortion of the ideal tetrahedral network. As a result, kaolinite is triclinic instead of monoclinic.

Fig. 1

Diagrammatic sketch of the structure of kaolinite

Fig. 2

Charge distribution in kaolinite

Homogenization Analysis for the Seepage

Seepage Problem by HA

A two-scale HA is introduced for a macro-domain Ω0 using the coordinate system x ⁰ and a micro-domain Ω1 using the coordinate system x ¹. Both coordinates are related as x = x ⁰/ε by the scale factor ε.

To represent incompressible flow, the following Stokes equation is introduced:

$$ \rho \left(\frac{\partial {V}_i^{\varepsilon }}{\partial t}+{V}_j^{\varepsilon}\frac{\partial {V}_i^{\varepsilon }}{\partial {x}_j}\right)=-\frac{\partial {P}^{\varepsilon }}{\partial {x}_i}+\frac{\partial }{\partial {x}_j}\left(\eta \frac{\partial {V}_i^{\varepsilon }}{\partial {x}_j}\right)+{f}_i\kern1em \mathrm{in}\kern1em {\Omega}_f $$

(1)

$$ \frac{\partial {V}_i^{\varepsilon }}{\partial {x}_i}=0\kern1em \mathrm{in}\kern1em {\Omega}_f $$

(2)

where ρ is the density of water, V ^ε is the water velocity in the fluid domain Ωf, P ^ε is the pressure, fi is the body force, and η is the water viscosity. Note that we consider a steady state, and hence the convective term Vj∂Vi/∂xj of the left-hand side (LHS) of (1) vanishes in the perturbation procedure, and we can ignore the LHS terms from the beginning. The superscript ε implies a variable which varies rapidly in the microscale domain. The viscosity distribution calculated by MD is shown in Fig. 3 for an isolated kaolinite layer. Naturally the velocity vanishes on the fluid-solid interface Γ:

Fig. 3

Diffusivity and viscosity of water in the neighbourhood of a silicate surface (a) band gibbsite surface (b)

$$ {V}_i^{\varepsilon }=0\kern1em \mathrm{on}\kern1em \Gamma $$

(3)

Now we introduce ‘two-scale domains‘for homogenization analysis (Fig. 4); that is, the macro-domain Ω0 and micro-domain Ω1. Coordinate systems that were set x ⁰ in Ω0 and x ¹ in Ω1 which are related by

Fig. 4

Two-scale domains for homogenization analysis

$$ {x}^1=\frac{x^0}{\varepsilon } $$

(4)

Since the two-scale coordinates are employed, the differentiation is changed to

$$ \frac{\partial }{\partial {x}_i}=\frac{\partial }{\partial {x}_i^0}+\frac{1}{\varepsilon}\frac{\partial }{\partial {x}_i^1} $$

(5)

By using the parameter ε we introduce the following perturbations:

$$ \begin{array}{l}{V}_i^{\varepsilon }(x)={\varepsilon}^2{V}_i^0\left({x}^0,{x}^1\right)+{\varepsilon}^3{V}_i^1\left({x}^0,{x}^1\right)+\dots \\ {}{P}^{\varepsilon }(x)={P}^0\left({x}^0,{x}^1\right)+\varepsilon {P}^1\left({x}^0,{x}^1\right)+\dots \end{array} $$

(6)

The perturbed terms of the right-hand sides (RHS) of (6) are assumed to be periodic:

$$ \begin{array}{l}{V}_i^{\alpha}\left({x}^0,{x}^1\right)={V}_i^{\alpha}\left({x}^0,{x}^1+{X}^1\right)\\ {}{P}^{\alpha}\left({x}^0,{x}^1\right)={P}^{\alpha}\left({x}^0,{x}^1+{X}^1\right)\end{array} $$

(7)

where X ¹ is the size of a microscale unit cell.

Equations (5) and (6) were substituted into (1) and (2) to get a set of perturbation equations. Due to the ε⁻¹-term resulting from (1) we understand that p ⁰ is a function of only the macro-coordinates x ⁰. The ε⁰-term resulted from (1) is given by

$$ -\frac{\partial {P}^1}{\partial {x}_{{}_i}^1}+\frac{\partial }{\partial {x}_{{}_j}^1}\left(\eta \frac{\partial {V}_i^0}{\partial {x}_{{}_j}^1}\right)=\frac{\partial {P}^0}{\partial {x}_{{}_i}^0}-{f}_i $$

(8)

The body force f usually works in the macro-domain Ω0(f = f(x ⁰)), and the RHS terms of (8) are functions only of x ⁰. Thus we can introduce the separation of variables into p ¹(x ⁰, x ¹) and V ⁰(x, x ¹) as

$$ \begin{array}{l}{V}_i^0\left({x}^0,{x}^1\right)=-\left|\frac{\partial {P}^0\left({x}^0\right)}{\partial {x}_j^0}-{f}_j\left({x}^0\right)\right|\;{v}_i^j\left({x}^1\right)\\ {}{P}^1\left({x}^0,{x}^1\right)=-\left|\frac{\partial {P}^0\left({x}^0\right)}{\partial {x}_j^0}-{f}_j\left({x}^0\right)\right|\;{p}^j\left({x}^1\right)\end{array} $$

(9)

These are substituted into (8), and we get the following microscale incompressible Stokes’ equations:

$$ \begin{array}{l}-\frac{\partial {P}^k}{\partial {x}_{{}_i}^1}+\frac{\partial }{\partial {x}_{{}_j}^1}\left(\eta \frac{\partial {v}_i^k}{\partial {x}_{{}_j}^1}\right)+{\delta}_{ik}=0\kern1em \mathrm{in}\kern1em {\Omega}_{1f}\\ {}\frac{\partial {v}_i^k}{\partial {x}_{{}_j}^1}=0\kern1em \mathrm{in}\kern1em {\Omega}_{1f}\end{array} $$

(10)

where v(x ¹) and p ^k(x ¹) are characteristic functions for velocity and pressure, respectively (δ _ik is Kronecker’s delta). By solving (10) under periodic conditions the characteristic functions, which reflect a complex geometry of the microscale domain, were obtained. Let us operate an integral average of (9) in the micro-domain, producing Darcy’s law as,

$$ \begin{array}{l}{\tilde{V}}_i^0\left({x}^0\right)=\left\langle {V}_i^0\left({x}^0,{x}^1\right)\right\rangle =-{K}_{ji}\left|\frac{\partial {P}^0\left({x}^0\right)}{\partial {x}_j^0}-{f}_j\left({x}^0\right)\right|\\ {}{K}_{ij}\equiv \left\langle {V}_j^i\left({x}^1\right)\right\rangle =\frac{1}{\left|{\Omega}_1\right|}\;{\displaystyle {\int}_{{\Omega_1}_f}{V}_j^i\left({x}^1\right)}\;d{x}^1\end{array} $$

(11)

where |Ω₁| is the volume of the micro-domain Ω1 and represents an averaging operation in the micro-domain. We call K _{i j} the HA-permeability.

Equation (8) gives a mass conservation relationship between the micro-domain and the macro-domain. When averaging this in the micro-domain, the second term of LHS vanishes due to the periodicity; then substituting Darcy’s law (9) yields the macroscale incompressible permeability equation:

$$ \frac{\partial {\tilde{V}}_i^0}{\partial {x}_i^0}=0\Rightarrow -\frac{\partial }{\partial {x}_i^0}\left[-{K}_{ji}\left\{\frac{\partial {P}^0\left({x}^0\right)}{\partial {x}_j^0}-{f}_j\left({x}^0\right)\right\}\right]=0 $$

(12)

The water velocity and pressure are approximated as

$$ {V}_i^{\varepsilon}\left({x}^0\right)\simeq {\varepsilon}^2{V}_i^0\left({x}^o,{x}^1\right),\kern1.25em {P}^{\varepsilon}\left({x}^o\right)\simeq {P}^0\left({x}^o\right) $$

(13)

It should be remembered that, in HA, the distribution of velocity and pressure are calculated in the micro-domain. The procedure to solve the total HA-seepage problem is summarized as follows: first, we solve micro-scale equation (10) and get V ^k _i and P ^k, then determine Darcy’s coefficient K _ij from (11). Next by solving macro-scale equation (12), we get the macro-pressure and velocity fields. In classical geomechanics, Darcy’s law is written as

$$ {\widehat{V}}_i^{*}=-{K}_{ij}^{*}\frac{\partial \phi }{\partial {x}_j^0},\kern1em \phi =\frac{p}{\rho g}+\xi $$

(14)

Where ϕ is the total head, p/ρg is the pressure head, ξ is the elevation head, and g is the gravity constant. K ^* _ij is called C-permeability and is defined as

$$ {K}_{ij}^{*}={\varepsilon}^2\rho g{K}_{ij} $$

(15)

Conclusions

Macroscale and microscale models of the kaolinite-water permeability system analyzed here are shown in Fig. 7. The number of mineral layers in one stack is assumed to be eight. It is known that one layer is connected to others by hydrogen bond, and the separation distance is calculated as 0.8 nm by the MD simulation. The density of solid part of the mineral is also calculated by MD as 2.56 g/cm³. A distribution of viscosity calculated by MD is shown in Fig. 3. The distance between two stacks is determined by the overall dry density of clay. We assume that the kaolinite stacks are randomly distributed; then the averaged permeability K* is estimated as K* = K ₁₁*/3. The relationships between the C-permeability (K*), the void ratio (e) and the dry density (ρ _d ) are plotted in Figs. 5 and 6. These relations can curve fit as

Fig. 5

Relationship between C-permeability (K*) and void ratio (e)

Fig. 6

Relationship between C-permeability (K*) and dry density (ρ _d )

$$ \begin{array}{l} \log {K}^{*}=1\times {10}^{-9}{\rho}_d^{-6.709}\\ {} \log {K}^{*}=3\times {10}^{-10}{e}^{2.775}\end{array} $$

(16)

where the unit of K* is in m/s.

In this paper, a unified MD/HA method for analyzing the seepage problem in kaolinite was presented. The results obtained by this method are similar to the experimental data, which supports the validity of the method (Fig. 7).

Fig. 7

Permeabilities given by HA and by Pusch