Non-ergodic solute transport in self-similar porous formations: the effect of conditioning
Introduction
Geologic media show hydraulic property variations on a multiplicity of spatial scales which identification and description is crucial for predicting the spreading of contaminants (e.g. [9], [15], [21]). Experimental data from several sources show variations of the hydraulic conductivities and transmissivities of increasing magnitude with the field length, and a similar effect is observed for the integral scales [24], [28]. In sedimentary formations this scaling-effect is engendered by the hierarchy of sedimentary processes operating in the depositional environment of the formation [22], [9], which at the regional scale leads to highly heterogeneous distributions of the hydraulic transmissivity, T. The ensuing variations of the hydraulic log transmissivity, Y=lnT, are often described by the following power law variogram (e.g. [28]):where is the two-point separation distance and β=2H, where H is the Hurst’s coefficient [18]. The model (1) describes a self-similar random space function (RSF), stationary in the increments, and with the variance growing unbounded with the observation scale [35]. It can be obtained as the weighted integral, over the correlation length, of an infinite hierarchy of stationary exponential or Gaussian models [28].
The model (1) has been criticized for the absence of an upper bound to the scales of variability [1]. In applications, an upper cutoff is introduced by geologic limits, boundary conditions, or scales of variability reproduced directly by the computational grid. To overcome this difficulty, which is particularly evident at the regional scale, we modify the model (1) as follows:where R is the upper limit cutoff of the spatial variability. The model (2) represents a stationary RSF with the covariance function given byAccording to (3), and consistently with the experimental findings discussed by Neuman [24], both the variance, σY2(R)=aRβ, and the integral scale,of the RSF increase with the cutoff R. As R→∞ (2) resumes the model (1) with unbounded variance and integral scale.
Transport of solutes in heterogeneous evolving-scale formations had been analyzed by Neuman [27] and Di Federico and Neuman [10], focusing on the dispersion of ensemble mean concentration. Plume-scale macrodispersion tensors had been obtained by Dagan [7], Rajaram and Gelhar [31], and Bellin et al. [2] by using the concept of relative solute dispersion introduced earlier in the turbulence context. All these studies concluded that solute spreading is greatly enhanced in presence of a multiplicity of heterogeneity scales, but none of them considered the effect of conditioning to available measurements of hydraulic transmissivity. Measurements of hydraulic transmissivity are an expensive source of information which should be used to render flow and transport models site-specific. In stochastic modeling this is accomplished by conditioning the hydraulic log transmissivity field on the available data, reducing the uncertainty due to the lack of information on the spatial variability at several scales. In this study we examine the influence of point measurements of the hydraulic transmissivity on the plume moments in a two-dimensional evolving-scale formation of the type described by (2), by using the Bayesian approach introduced by Dagan [3], [4] and further developed by Rubin [33]. Central to this method is the first-order approximation in σY2 of the velocity field, which ensures that the particle displacement pdf, conditional to the available data, is multivariate normal [23].
Section snippets
First-order analysis of conditional velocity covariances
The movement of water in saturated porous media under stationary flow conditions is described by the mass balance equation:and the Darcy’s law:where is the velocity, Hc is the hydraulic head, both conditional to the available data, and n is the porosity, which is assumed constant through the formation.
The hydraulic log-conductivity conditional to the available data, Yc is decomposed into the following three terms [19, p. 76]:where
Spatial moments
Let us consider an instantaneous release of solute with constant concentration C0 within the initial volume V0=A0b, where A0 is the horizontal projection of V0, and b is the aquifer’s thickness. Thus, the mass of solute released per unit of thickness is M0=C0A0n. Convenient quantitative measures of translation and spreading of the plume are provided by the trajectory of the plume’s centroid and the second-order spatial moments
Results and discussion
In evolving-scale formations of the type described by model (2), a single measurement can be very effective in reducing uncertainty because according to (4) the integral scale of the hydraulic log transmissivity is a significant fraction of the domain size. In this section we analyze how the position of the measurement with respect to the plume influences the plume moments. We explore the entire range of variability of H, which is bounded between 0 and 1, concentrating on the two cases of H
Conclusions
We have investigated the impact of conditioning to available data of hydraulic transmissivity on the concentration second-order moments. First-order solutions are obtained for effective and ensemble moments of a plume generated by an instantaneous release of solute in a two-dimensional evolving-scale formation, with the hydraulic transmissivity modelled as a self-similar RSF with the upper cutoff, R, to the scales of variability.
A considerable difference is observed between the effective and
Acknowledgements
The authors wish to thank Gedeon Dagan for the insightful review of the paper.
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