**1. Theoretical foundations**

#### **1.1. General considerations**

Flow through saturated or unsaturated soils is governed by Darcy's Law, which was original‐ ly proposed for saturated media. Research has demonstrated that this law is also applicable to the flow of water in unsaturated soils [1]. The main difference between these flows is that the hydraulic conductivity for saturated media is a constant value, but it varies as a function of the volumetric water content and also indirectly with changes in pore water pressure in unsatu‐ rated soils (**Figure 1**) [2, 3]. Darcy's Law [4] is often written as follows:

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$$w = k \, i \tag{1}$$

where *v* = Darcy's velocity, *k* = hydraulic conductivity, and *i* = total hydraulic head gradient.

**Figure 1.** Change in the permeability of a partially saturated medium [5].

The average velocity at which the water moves through a mass of soil is linear and is equal to the Darcy's velocity divided by the porosity of the soil. In an unsaturated soil, the average velocity is equal to the Darcy's velocity divided by the volumetric water content of the soil. The majority of analytical and numerical methods that are currently employed for solving water flow problems consider only the Darcy's velocity.

#### **1.2. Equation for steady-state flow (saturated porous media)**

The equation that describes steady-state flow in a porous medium is based on Darcy's Law [4] and on the principle of flow continuity (which states that the amount of water that enters the medium is equal to the amount that exits) and is known as the Laplace's equation (for a homogeneous and isotropic medium with *kx* = *ky* = *kz*):

$$\frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} + \frac{\partial^2 h}{\partial z^2} = 0 \tag{2}$$

where *h* = total hydraulic head, and *kx*, *ky*, and *kz* = hydraulic conductivities in the *x*-, *y*-, and *z*-directions, respectively.

The Laplace's equation is satisfied under the following conditions: (a) the flow is steady-state, (b) the soil is saturated, (c) the water and the solid particles are incompressible, (d) the flow does not modify the soil structure, and (e) there are no sources (via injection or extraction of water).

#### **1.3. Equation for transient flow (saturated or unsaturated porous media)**

In transient flow, the water levels vary as a function of time, and thus, water is either stored in or discharged from the medium. In these cases:

$$Flow\text{ that exists} = flow\text{that enters} - flow\text{discharging during a time interval }\Delta t \tag{3}$$

or:

*v ki* = (1)

where *v* = Darcy's velocity, *k* = hydraulic conductivity, and *i* = total hydraulic head gradient.

The average velocity at which the water moves through a mass of soil is linear and is equal to the Darcy's velocity divided by the porosity of the soil. In an unsaturated soil, the average velocity is equal to the Darcy's velocity divided by the volumetric water content of the soil. The majority of analytical and numerical methods that are currently employed for solving

The equation that describes steady-state flow in a porous medium is based on Darcy's Law [4] and on the principle of flow continuity (which states that the amount of water that enters the medium is equal to the amount that exits) and is known as the Laplace's equation (for a

where *h* = total hydraulic head, and *kx*, *ky*, and *kz* = hydraulic conductivities in the *x*-, *y*-, and

The Laplace's equation is satisfied under the following conditions: (a) the flow is steady-state, (b) the soil is saturated, (c) the water and the solid particles are incompressible, (d) the flow does not modify the soil structure, and (e) there are no sources (via injection or extraction of

¶¶¶ (2)

222 <sup>222</sup> <sup>0</sup> *hhh xyz* ¶¶¶ ++=

**Figure 1.** Change in the permeability of a partially saturated medium [5].

92 Groundwater - Contaminant and Resource Management

water flow problems consider only the Darcy's velocity.

homogeneous and isotropic medium with *kx* = *ky* = *kz*):

*z*-directions, respectively.

water).

**1.2. Equation for steady-state flow (saturated porous media)**

$$\text{Flow that exists} = \text{flow} \\ \text{that enters} + \text{flow} \\ \text{stored during a time interval } \Delta t \tag{4}$$

Equation (3) refers to the case of water drawdown, and Eq. (4) refers to the case of water filling. All the previous assumptions lead to the general mass balance equation:

$$\frac{\partial}{\partial \mathbf{x}} (k\_x \frac{\partial h}{\partial \mathbf{x}}) + \frac{\partial}{\partial y} (k\_y \frac{\partial h}{\partial y}) + Q = \frac{\partial \theta}{\partial t} \tag{5}$$

where *h* = total hydraulic head, *kx* = hydraulic conductivity in the *x*-direction, *ky* = hydraulic conductivity in the *y*-direction, *Q* = source term (applied boundary flux: injection or extrac‐ tion), *θ* = volumetric water content, and *t* = time.

Equation (5) is the so-called Richards's equation, and it describes transient flow in unsaturated soils. The term on the right (*∂θ*/*∂t*, the rate of change of the volumetric water content with respect to time) is related to the change in the water level with time. When there is no variation with time (*∂θ*/*∂t* = 0) and no source term (*Q* = 0), Eq. (5) becomes the Laplace's equation, which is used for steady-state flow in saturated soils. The Richards's equation, or any of its modified forms, has constituted the basis for the development of most numerical models to calculate infiltration through unsaturated porous media under transient-state flow conditions [6].
