Elsevier

Computers and Geotechnics

Volume 40, March 2012, Pages 45-52
Computers and Geotechnics

Evaluation of hydraulic conductivity in 3D random and heterogeneous particulate materials using network model

https://doi.org/10.1016/j.compgeo.2011.09.007Get rights and content

Abstract

In this study we present a three-dimensional network model for the evaluation of hydraulic conductivity using new pore structure characterization algorithms. We define a porous medium from the sphere packing generated by discrete element method (DEM). The network model is based on connectivity of pore chambers linked along discretized pathway using a harmonic mean radius value computed from a series of radius-varying tubes that represent the irregularly shaped pore channel. Particle packing cases under investigation include regular packings, monodispersed, and polydispersed cases with varying porosity. Finite element analysis is implemented to obtain hydraulic conductivity values for regular packing cases for comparison. Laboratory experimentation results corroborate numerically obtained values with good agreement. Observations show the pore space should be tessellated once pore chambers are established in order to preserve the geometrical uniqueness of randomly shaped pore channel, avoid redundancy, and minimize pore channel overlap. Also, it is observed that limits must be defined for the total number of grains needed for representativeness and discretization density for the quantification of random heterogeneous pore structure.

Introduction

Hydraulic conductivity is a key macro-scale parameter used to characterize a fluid flow in geomaterials. It is often estimated from the volumetric fraction of pore space and the finer fraction of soils and is routinely measured by laboratory experimentation and in situ testing (e.g., constant or falling head tests, in situ pumping test). Yet, the fluid flow in porous media is mainly controlled by pore structure and network, with pore connectivity playing a crucial role. The accurate estimation of hydraulic conductivity is fundamental to many geo-engineering applications; groundwater seepage, hydrocarbon recovery, geological disposal of nuclear waste, deep geothermal energy exploitation and geological sequestration of carbon dioxide [1], [2], [3], [4].

Recent advances in numerical computation and image processing techniques enable the investigation of particle-scale morphology and detailed description of fluid migration in porous media. In particular, the pore characteristics of granular materials are readily obtained through various methods: series of 2D thin sections and their 3D stacking [5], [6], computer aided X-ray tomography [7], [8], and stochastic reconstruction of 3D pore images [9], [10]. Estimates of hydraulic conductivity at the pore-scale can be made using the Lattice Boltzmann method based on particle motion and collision on a lattice typically computed using a scanning grid [11], [12]. However, it is computationally expensive and hampers development of applications using high resolution 3D image data [13].

Alternatively, semi-empirical correlations such as the Kozeny-Carman equation may be used as a first order approximation. An improved expression of the empirical Hazen formula that indirectly considers the particle-scale information described in Ref. [14] is as follows:k=ρwgηw(1/CK-C)(1/Sa2)e31+ewhere ρw is the density of water, g is the acceleration of gravity, ηw is the viscosity of water, e is the void ratio, CKC is an empirical coefficient, and Sa is the surface area per unit volume of particles which is sensitive to the choice of particle diameter. Finite element analysis may also be employed to determine the hydraulic conductivity of a 3D granular system by solving the Navier–Stokes equation at the pore-scale and averaging the computed local velocities based on the discretized 3D images obtained by micro-CT or discrete element method (DEM) [15], [16].

The network model considers the idealization of irregularly shaped pore geometry into the pore chamber enclosed by adjacent particles and the pore channel that connects associated pore chambers. The pore space is often tessellated into tetrahedra for a Finney packing (i.e., close random packing of spheres with equal radii). This tessellation may be used to help define the pore bodies [17], [18] whereby the flow path between pore bodies is divided into cylindrical tubes inscribed to fit the shape of the pore channel or defined by the effective radius equal to the arithmetic mean of the largest inscribed radii [19]. As the packing heterogeneity increases, the free boundary is not fully captured by the channel radius and the method tends to diverge from correct estimation of hydraulic conductivity. Also, the regularly spaced lattice structure of pore network using harmonic mean of pore channel tube for clean sand shows the overestimation of conductivity values, likely due to the difference in the representation of the pore channel [20].

Therefore, the complexity of pore geometry makes the accurate quantification of pore structure and its conductivity challenging [19], [20], [21], [22]. Hydraulic conductivity may be computed for a network of pore channels where each channel is divided into a series of tubes acting as sub-channels. Cumulative fluid flow through such a series of tubes may be described by the harmonic mean that represents the conductance of a constitutive tube acting along the entire pore channel [23]. The validity and accuracy of this approach depend on the representative pore structure and the map of pore connectivity for a given 3D image.

This study presents the construction of solid particles and the idealization of corresponding pore structures based on a 3D image analysis followed by the implementation of the network model to evaluate the hydraulic conductivity of particulate materials. The proposed model approach adopts the finely tuned characterization of irregularly shaped pore space using the harmonic mean of a series of radius-varying tubes that connect distinct pore bodies. We consider a wide range of packing configuration types and porosity cases. Then, the comparative studies with experimentation are presented.

Section snippets

Overview of network model of hydraulic conductivity

An overview of the network model employed for the estimation of hydraulic conductivity and the importance of properly characterizing the pore structure are presented next.

Idealization of pore structure

The construction of 3D pore structures of particulate materials through the idealization of radius-varying tubes that interconnect larger pore chambers is presented next.

Finite element analysis of regular packing systems

The discretized images of the regular packing cases are used in finite element analyses to simulate the fluid flow through the pore space. Darcy’s law can be derived from more fundamental particle-scale governing equations based on Stokes’ law [29]. The incompressible Navier–Stokes equation describes the viscous motion of an incompressible fluid within the saturated pores of a particulate system.vt+(v·)v+gp-ηρw2v=-gz

The empirical Darcy’s law can be found by averaging the incompressible

Results and discussion

The hydraulic conductivity of the sand used in this study (i.e., Ottawa 20-30 sand) is measured by the constant head test (ASTM D 2434) for a range of porosity values from ∼0.35 to ∼0.40. The sand sample is fully saturated with water in a cylindrical container (diameter D = 14 cm, length L = 8 cm). Fig. 7 shows the experimentally measured and estimated hydraulic conductivity values obtained by all the methods presented in this study. Comparison of results above for common mean grain diameter 0.72 mm

Conclusions

The feasibility of the network model described herein is thoroughly evaluated for a wide range of packing configurations over a typical porosity range in soils. Results were compared using the well-known Kozeny-Carman relation, finite element solutions for regular packing cases, physical experimentation performed on Ottawa 20-30 sand, and the proposed network model approach applied to random (i.e., mono and polydispersed packing) and regular packing cases. Success of the network model depends

Acknowledgements

This research was conducted with the support provided to T.S. Yun by the basic science research program through National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0005593).

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