A peridynamic model of flow in porous media
Introduction
Naturally occurring soils, especially fine-textured ones, exhibit shrinking and swelling behavior [1], [2], [3]. These soils tend to swell when their moisture content increases, and shrink when it decreases. At the field scale, this behavior leads to tensile stresses that may exceed the soil’s failure limit and trigger the formation and evolution of cracks during drying phases. Cracks may in turn close during infiltration phases when the soil becomes wetter and swells [4], [5], [6] giving them a dynamic nature leading to highly nonlinear responses. These desiccation cracks have a length scale of ten to a hundred centimeters and their effect on the hydraulic properties of the soil is not captured by standard laboratory tests using a Representative Elementary Volume (REV) with a length scale of a few centimeters.
Desiccation cracking has a wide spectrum of environmental, agricultural, and hydrological impacts. The movement of moisture and solutes into and within the soil increases due to the presence of these cracks that act as preferential pathways for rapid water movement to deeper layers [7], [8], [9], [10], [11]. This rapid movement may lower the effectiveness of irrigation [12] and causes fast seepage of nutrients and pesticides away from the plants into deeper layers reducing the contaminants’ residence time in the unsaturated zone where they are usually absorbed by the plants and degraded by bacteria, and increasing the probability of ground water and/or surface water contamination, depending on the relief. In addition, desiccation cracks can have a dramatic effect on processes of surface water movement and flood dynamics by altering the partitioning of rainfall between infiltration and runoff, which is an important issue to consider when modeling and forecasting flood events.
Desiccation cracks also have engineering and geotechnical impacts with potentially very serious environmental and public safety repercussions. For example, desiccation cracks developing at the surface of a slope may trigger the onset of a landslide. If they develop in the core of an earth dam, cracks act as preferential moisture flow paths, increasing the moisture content of the dam and, with it, the pore water pressure which eventually leads to its failure [13]. Clay barriers used in landfills and nuclear waste disposal sites are also subject to desiccation cracking which reduces the barrier’s containment effectiveness [14], [15].
In this paper, we present a peridynamic model for transient moisture flow in unsaturated, heterogeneous, and anisotropic soils. The model is an alternative to the classic Richard’s equation and is based on Silling’s reformulation of the theory of elasticity for solid mechanics [16], [17]. In the proposed model, we replace the classic, local, continuum mechanics formulation by a nonlocal integral functional. The model is free of spacial derivatives, and the flow is driven by the hydraulic potential field instead of the gradient of the hydraulic potential field. Katiyar et al. [18] have derived similar peridynamic formulations for saturated steady state flow.
Due to the lack of spacial derivatives, this model is capable of handling the spurious formation of cracks, which translate into points of singularities in the parameter and hydraulic potential fields, within the simulation domain without failing. This would allow us to couple the derived model with a peridynamic mechanical model and simulate the formation of desiccation cracks and their dynamics and assess the potential of such an approach on evaluating their impact on flow and solute transport. This coupling is however the subject of a subsequent paper.
We would like to point out that the nonlocal aspect of the proposed model is related to the mechanism of state change in the domain. In classic nonlocal formulations [19], [20], [21], [22], [23], [24], [25], the new value of a state is the one with the maximum likelihood and the change is driven by some statistical measure of the gradient of the driving field within the surrounding region. On the other hand, in peridynamic models such as this one, the change of state at a point is driven by the influence of the value of some field at points that are at some finite distance away.
We will start by presenting the peridynamic model concept and derive the peridynamic expression for the rate of change of moisture content. We will then derive the peridynamic equations of flow power dissipation and moisture flux, which we will use in deriving the relationship between the peridynamic hydraulic conductivity density and the classic hydraulic conductivity for unsaturated, homogeneous, heterogeneous, isotropic, and anisotropic soils. We will also show that the peridynamic model equations of moisture flow and flux converge to the classic Richard’s and Darcy’s equations at the limit of vanishing horizon. This will be followed by a presentation of the numerical implementation and validation of the model in one and two dimensions.
Section snippets
Peridynamic flow model
Consider the homogeneous and isotropic body of soil in Fig. 1, where each point in represents a differential volume [], and is at some total hydraulic potential [L]. Suppose the change in moisture content at every point in is driven by pairwise interactions with all other points in despite the finite distance separating each pair points.
Suppose also that these pairwise interactions are equivalent to a one dimensional resistive pipe that acts as a conduit and does not
Flow power and moisture flux
We will now define the peridynamic expressions for moisture flux, and for the power dissipated by moisture flow. These are important quantities in analyzing flow problems in general, and are particularly useful in deriving the relationship between the peridynamic hydraulic conductivity and the classic conductivity K. We will use the peridynamic equation for moisture flux during our derivation of the relationship between and the classic conductivity K for the case of one-dimensional flow,
Peridynamic hydraulic conductivity
In this section we would like to relate the non-measurable peridynamic hydraulic conductivity to the classic hydraulic conductivity of the soil. This entails finding an expression for as a function of the classic hydraulic conductivity that results in equal values for variables such as moisture flux or flow power in the peridynamic and classic model.
So far, the only parameter influencing the degree of locality of the model is , the radius of the horizon. The smaller is, the more local the
Numerical implementation
Numerical implementation of the peridynamic model was based on a medium discretization into nodes using a regular grid. Fig. 4 shows a section of a discretized soil column in 1D and 2D respectively with a grid spacing of and a horizon radius with, where m is the horizon radius in multiples of grid lengths. Each node in the grid represents a volume of for the 1D case, or for the 2D, and has a moisture content , an associated hydraulic potential and a hydraulic
Model validation
In order to validate the proposed model, and analyze the effects that the horizon radius (), the density of points per horizon radius (m), and the type of influence function (uniform or linear) have on the performance of the model, several 1D and 2D scenarios are simulated. Due to the lack of an analytical solution of the flow problem, the same scenarios are also simulated using the finite element models HYDRUS-1D and HYDRUS 2D/3D [32], [33] that solve the classic Richard’s equation in one
Conclusion
In this paper we derived a nonlocal derivative free alternative of the Richards equation. We replaced the deferential flow equation by an integral functional, where the absence of spacial derivative allows for the simulation of domains with internal evolving singularities such as desiccation cracks, a feature of soils with a high shrink/swell potential. Expressions relating the peridynamic hydraulic conductivity to the measurable classic hydraulic conductivity were derived for problems in one-
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