Non-ergodic solute transport in self-similar porous formations: the effect of conditioning

Abstract

We analyze the influence of conditioning to hydraulic transmissivity measurements on transport of a conservative tracer in a two-dimensional evolving-scale formation with a large-scale cutoff of the hydraulic transmissivity. First-order analytical solutions of ensemble and effective plume moments, conditional to a single measurement of hydraulic transmissivity, are provided for an instantaneous release of solute within a linear source normal to the mean flow direction. The proposed solutions show that both ensemble and effective moments are significantly affected by the additional information brought into flow and transport models through conditioning to available data. We conclude that the effect of conditioning to available measurements is more pronounced on the ensemble moments than on the effective moments, and uncertainty is not reduced in the same proportion for longitudinal and transverse effective second moments, since the former shows the largest reduction of uncertainty for a given measurement location. This result confirms the experimental findings showing larger uncertainty in the estimation of the transverse moments. Furthermore, for all moments the impact of the measurement reduces with the distance of the measurement point from the mean plume trajectory.

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]):

$$\text{γ}{}_{\mathrm{Y}}\text{(r)=}\text{1}\text{2}\text{〈[Y(}\text{x}\text{)−Y(}\text{x}{}^{\mathrm{\prime }}\text{)]}{}^2\text{〉=ar}{}^{\mathrm{\beta }}\text{,}$$

where $\text{r=|}\text{x}\text{−}\text{x}{}^{\mathrm{\prime }}\text{|}$ 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:

$$\text{γ}{}_{\mathrm{Y}}\text{(r)=}\text{ar}{}^{\mathrm{\beta }}\text{for}\text{r⩽R,}\text{aR}{}^{\mathrm{\beta }}\text{for}\text{r>R,}$$

where R is the upper limit cutoff of the spatial variability. The model (2) represents a stationary RSF with the covariance function given by

$$\text{C}{}_{\mathrm{Y}}\text{(r)=}\text{σ}{}_{\mathrm{Y}}{}^2\text{(R)}\text{1−}\text{r}\text{R}{}^{\mathrm{\beta }}\text{for}\text{r⩽R,}\text{0}\text{for}\text{r>R.}$$

According to (3), and consistently with the experimental findings discussed by Neuman [24], both the variance, σ_Y²(R)=aR^β, and the integral scale,

$$\text{I}{}_{\mathrm{Y}}\text{=}\text{β}\text{1+β}\text{R,}$$

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 σ_Y² of the velocity field, which ensures that the particle displacement pdf, conditional to the available data, is multivariate normal [23].