Chaotic dynamics in flow through unsaturated fractured media

Abstract

Predictions of flow and transport within fractured rock in the vadose zone cannot be made without first characterizing the physics of unstable flow phenomena in unsaturated fractures. This paper introduces a new approach for studying complex flow processes in heterogeneous fractured media, using the methods of nonlinear dynamics and chaos––in particular reconstructing the system dynamics and calculating chaotic diagnostic parameters from time-series data. To demonstrate the application of chaotic analysis, this author analyzed the time-series pressure fluctuations from two water–air flow experiments conducted by Persoff and Pruess [Water Resour. Res. 31 (1995) 1175] in replicas of rough-walled rock fractures under controlled boundary conditions. This analysis showed that chaotic flow in fractures creates relaxational oscillations of liquid, gas, and capillary pressures. These pressure oscillations were used to calculate the diagnostic parameters of deterministic chaos, including correlation time, global embedding dimension, local embedding dimension, Lyapunov dimension, Lyapunov exponents, and correlation dimension. The results of the Persoff–Pruess experiments were then compared with the chaotic analysis of laboratory dripping-water experiments in fracture models and field-infiltration experiments in fractured basalt. This comparison allowed us to conjecture that intrinsic fracture flow and dripping, as well as extrinsic water dripping (from a fracture) subjected to a capillary-barrier effect, are deterministic-chaotic processes with a certain random component. The unsaturated fractured rock is a dynamic system that exhibits chaotic behavior because the flow processes are nonlinear, dissipative, and sensitive to initial conditions, with chaotic fluctuations generated by intrinsic properties of the system, not random external factors. Identifying a system as deterministically chaotic is important for developing appropriate short- and long-term prediction models, evaluating prediction uncertainty, assessing the spatial distribution of flow characteristics from time-series data, and improving chemical-transport simulations.