Quantitation of the uncertainty in the prediction of flow fields induced by the spatial variation of the fracture aperture

Abstract

The problem of fluid flow in a single fracture with variable aperture is analyzed from a stochastic point of view. The spatially distributed fracture aperture is treated as a random variable and the random aperture process can be characterized as a zero-order intrinsic random process (nonstationary random process with homogeneous increments), resulting in nonstationary flow fields. To characterize the stochastic behavior of the flow process, the general solutions for the mean values of and variances of the pressure head and the integrated flux in the mean flow direction are derived. These derivations are performed using the Fourier-Stieltjes representation approach. The mean is an unbiased estimator and the variance can be used to characterize the uncertainty of the mean model. The theory is applied to the case where the process of random aperture can be described as fractal Brownian motion. The analysis shows that the larger the parameter H, the greater the variance of pressure head and integrated flux. The transfer function helps smooth out much of the small-scale variation caused by the random aperture field. The variance (uncertainty) of the integrated flux in x-direction around the mean flux (or the classical deterministic model solution) increases with distance. Estimation of total flux through a natural fractured system is often evaluated for the purpose of rational management of groundwater resources. The proposed stochastic theory for flow prediction under uncertainty under the natural field conditions as input to the flow model for a fractured system should provide useful information for regional groundwater resource management. To our knowledge, the subject of fluid flow in nonstationary random flow fields has not yet been addressed in fracture. Our findings will be useful for the interpretation of real field data and as a stimulus for further research in this area.