Uncertainty propagation of arbitrary probability density functions applied to upscaling of transmissivities

Abstract

In many fields of study, and certainly in hydrogeology, uncertainty propagation is a recurring subject. Usually, parametrized probability density functions (PDFs) are used to represent data uncertainty, which limits their use to particular distributions. Often, this problem is solved by Monte Carlo simulation, with the disadvantage that one needs a large number of calculations to achieve reliable results. In this paper, a method is proposed based on a piecewise linear approximation of PDFs. The uncertainty propagation with these discretized PDFs is distribution independent. The method is applied to the upscaling of transmissivity data, and carried out in two steps: the vertical upscaling of conductivity values from borehole data to aquifer scale, and the spatial interpolation of the transmissivities. The results of this first step are complete PDFs of the transmissivities at borehole locations reflecting the uncertainties of the conductivities and the layer thicknesses. The second step results in a spatially distributed transmissivity field with a complete PDF at every grid cell. We argue that the proposed method is applicable to a wide range of uncertainty propagation problems.

Harmonic Analysis of Non-Phase-Locked Tides with Red Noise Using the red_tide Package

“…In this paper, plotted uncertainty bounds for derived quantities are estimated using a Monte Carlo approach, based on an ensemble population of x with the posterior Gaussian PDF. This approach is implemented due to its flexibility and simplicity compared with analytical solutions, which are not always in closed form, or with more complicated approximations such as piecewise linear discretization of PDFs (e.g., Lourens and van Geer 2016). The tidal amplitude is one such quantity, where Âm 5 â2 m 1 b2 m is an estimator of the true amplitude A m , which follows a noncentral x distribution when the standard deviations of a m and b m are equal (s am 5 s bm 5 s m ).…”

Stochastic upscaling of hydrodynamic dispersion and retardation factor in a physically and chemically heterogeneous tropical soil

“…The upscaling of hydraulic conductivity is well established in the literature and several approaches have been reported, showing the limitations and effectiveness of local and non-local upscaling methods for the reproduction of water flow patterns under different types of heterogeneity (Cadini et al, 2013;Cassiraga et al, 2005;Fernàndez-Garcia and Gómez-Hernández, 2007;Gómez-Hernández et al, 2006;Li et al, 2011a;Lourens and van Geer, 2016;Renard and de Marsily, 1997;Sánchez-Vila et al, 1996;Selvadurai and Selvadurai, 2014;Wen and Gómez-Hernández, 1996). However, upscaling hydraulic conductivity only is not enough to reproduce the fine-scale transport behavior at the coarse scale due to the loss of K heterogeneity present at the fine scale that influences solute transport behavior (Cassiraga et al, 2005;Journel et al, 1986;Scheibe and Yabusaki, 1998).…”

Improving hydrogeological models using the results of calibrated groundwater flow models

“…Performing elementary operations and kriging interpolation with piecewise linear PDFs is described in Lourens and van Geer [2016].…”

“…Subsequently, the PDF of the interpolation (D(u 0 )) and the PDF of the interpolation error are added to achieve a PDF containing all uncertainties. In previous work, this method is described in detail [Lourens and van Geer, 2016]. Generation of the PDF of the litho-layer thickness of the observations, and the choice of its attributes, like variance and shape, is described in Section 3.1.2 Ordinary kriging tends to generate negative weight factors, beside the positive ones, when the spatial distribution of the observations is somehow unbalanced around the estimation location, known as the screen effect.…”