## 1. INTRODUCTION

[2] The intriguing word “chaos” has attracted the attention of many scientists and nonscientists for centuries. Hesiod was probably the first to introduce the word chaos in his *Theogony* (written in ∼700 B.C.) as nothing but void, formless matter, infinite space. He appears to associate chaos with the great chasm, as some sort of gap between earth and sky; but chaos also represented the underworld or the earth. The first modern scientific application of the word chaos belongs to *Li and Yorke* [1975], who described the mathematical problem of a time evolution with sensitive dependence on initial conditions. Since the pioneering work of *Li and Yorke* [1975], chaos has been used in the scientific literature not only to define randomness but also to define the characteristics of chaotic dynamics generated by predominately deterministic processes. Chaotic dynamics, known popularly as chaos, is among the most fascinating new fields in modern science, reforming our perception of order and pattern in nature [*Gleick*, 1987]. Chaos theory has become a widely applied scientific concept, used in such larger constructs as “complexity theory” [*Nicolis and Prigogine*, 1989], “complex systems theory,” “synergetics” [*Haken*, 1983], and “nonlinear dynamics” [*Abarbanel*, 1996]. Nonlinear dynamics is a fast developing field of physical sciences, with a wide variety of applications in such fields as biology [e.g., *May*, 1981; *Olsen et al.*, 1994], physics, chemistry, medicine, economics, earth sciences, and geology [e.g., *Hide*, 1994; *Turcotte*, 1997]. Chaotic dynamics is one of the fields within nonlinear dynamics. The term chaos theory is also used as a popular pseudonym for dynamic systems theory [*Abraham et al.*, 1997]. Note that chaos and turbulence (which were often used as synonyms in the scientific literature) represent two different types of processes [*Haken*, 1983].

[3] Some examples of dynamic systems that display nonlinear deterministic chaotic behavior with aperiodic and apparently random variability include atmospheric [*Lorenz*, 1963; *Nicolis*, 1987], geologic [*Turcotte*, 1997], geochemical [*Ortoleva*, 1994], and geophysical [*Dubois*, 1998; *Read*, 2001] processes, avalanches resulting from the perturbation of sandpiles of various sizes [*Rosendahl et al.*, 1993], falling of water droplets [*Cheng et al.*, 1989], river discharge and precipitation [*Pasternack*, 1999], oxygen isotope concentrations [*Nicolis and Prigogine*, 1989], viscous fingering in porous media [*Sililo and Tellam*, 2000], oscillatory fluid release during hydrofracturing in geopressured zones buried several kilometers in actively subsiding basins [*Dewers and Ortoleva*, 1994], thermal convection in porous media at large Rayleigh numbers [*Himasekhar and Bau*, 1986], and instabilities at fluid interfaces [*Moore et al.*, 2002]. It is noteworthy that experimental and theoretical investigations have shown that different physical, biological, mechanical, and chemical systems exhibit very similar (even universal) patterns, typical for deterministic chaotic systems. Nonetheless, up to now the practical application of chaos theory remains as much an art as a science.

[4] For many years the general approach to flow investigations in a fractured environment has been based on using stochastic methods to describe random-looking data sets [e.g., *Gelhar*, 1993], without considering that deterministic chaotic processes could cause apparent randomness of experimentally observed data.

[5] In his theoretical analysis of steady groundwater flow in a fully saturated, heterogeneous aquifer (with no discontinuities and a particular model for the spatial variability of hydraulic conductivity and based on Darcy's law), *Sposito* [1994] demonstrated the absence of chaos for groundwater flow as related to the impossibility of closed flow paths. The possibility of chaos is expected to occur in partially saturated, heterogeneous structured soils and fractured rock with discontinuity effects and drastic differences in permeability and flow mechanisms (such as those between high-conductivity flow channels and low-conductivity matrix).

[6] The goals of this paper are to review the theoretical concepts, present the results, and provide perspectives on investigations of flow and transport in unsaturated heterogeneous soils and fractured rock, using the methods of nonlinear dynamics and deterministic chaos. The paper is structured as follows: Section 2 presents a review of basic theoretical concepts and models of nonlinear dynamics. Section 3 provides an analysis of two key elements generating nonlinear dynamic and chaotic processes, geometry and physics of flow, and shows that unsaturated flow processes satisfy the criteria of a chaotic system. Section 4 provides some examples of chaos using pertinent results from laboratory and field infiltration experiments. Finally, section 5 gives concluding remarks and perspectives on using nonlinear dynamics in investigating unsaturated flow processes.