# Unsaturated Hydraulic Conductivity in Composite Porous Media
## 1 Introduction  
Analyzing the water retention behavior coupled to the unsaturated hydraulic conductivity mechanism is essential to establish complete flow models and the derived effects in the deformability processes for unsaturated soils. The water retention curve (WRC) and the hydraulic conductivity, saturated, \(k_{s}\), and unsaturated, \(k_{u}\), are essential parametric functions used to generate practical numerical models of water's movement and the transport of pollutants in soils. In order to establish various theoretical frameworks and numerical adaptations of variably saturated and unsaturated water flow, it is necessary to adequately define these hydraulic functions of the soil [1]. Since the physical spectrum of the soil responds to the interaction mechanism between the volumetric water content (\(\theta\)) and the main hydraulic parameters [2], most mathematical models describe \(\theta\) in terms of effective saturation, \(\psi_{c}\). This parameter is traditionally defined as a scaled value between the water content in the saturation intervals and small fractions of the residual water content at the WRC dry limit [3, 4, 5]. Generally, it is incorrect to assume that unsaturated hydraulic conductivity, \(k_{u}\), is a soil property. In partially saturated soils, \(k_{u}\) is significantly affected by the degree of saturation and varies as a function of the matric suction value at any time \(t\). During the transition of soils to unsaturated states, the air partially replaces the macropores and micropores, where the flow paths reach high levels of tortuosity. Conservative increases in the matric suction of the soil produce decreases in the volume of pores occupied by thewater, increasing the shear resistance to the flow of the soil particles near the air-water interface and numerically reducing hydraulic conductivity [6]. Several empirical approximation methods can quantify porous media's flow and transport properties. However, although the approaches associated with the hydraulic properties of the soil are numerous, the choice of adjusted data for specific studies requires a considerable injection of time, labor, and resources. Therefore, in the last decades, many devices have been developed to measure the coefficient of permeability, saturated \(k_{\text{s}}\), and unsaturated \(k_{u}\) in the soil. The constant and variable head methods are widely used to measure \(k_{\text{s}}\) in different types of soils (Figure 1a,b). These models are, in particular, international standards and are universally approved [7; 8]. The flow rate flowing through soil beds is measured during both tests, either under constant or variable inlet heads. The variable system is often implemented on in situ tests adopted as outlet flow meters. However, the influence of the flow regime during the tests and the anisotropy of the soils produces qualitative results and little absolute \(k_{\text{s}}\). Additionally, numerous studies use variable hydraulic pressure mechanisms exclusively for materials with intrinsic permeabilities below \(10^{-10}\) m\({}^{2}\)[9; 10; 11]. Although permeameters are usually classified into two types, there is a broad typological scheme within these classifications. For the choice of the permeameter, it is vital to consider the anisotropy of the porous matrix and the division performance for the vertical (\(k_{\text{o}}\)) and horizontal (\(k_{h}\)) hydraulic conductivity. Currently, most of the predictive methods used to determine \(k_{\text{s}}\) require constitutive parameters of simple measurements, such as the pore size distribution curve (PaSD) and the particle size distribution curve (PSD). Although in theory, it is necessary to obtain detailed maps of how the continuous vacuum of soils is distributed and interconnected [3; 12; 13; 14; 15; 16; 17; 18]. On the other hand, experimental approaches to define \(k_{u}\) must be evaluated using the complete chain of measured or estimated water retention curves (WRC) [4; 19; 20; 21; 22; 23; 24; 25].