Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 2. The solute transport

Abstract

The problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two-and three-dimensional flows. Approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity. The solute concentration, regarded as a random variable, is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty. The uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale. In any case the concentration expectation does not obey a diffusion type equation in the case of two-dimensional flows, unless the solute body has traveled a distance larger than a few tens transmissivity integral scales. This distance may be exceedingly large in many conceivable applications. streamlines become tortuous even if the flow is uniform in the average. According to our basic approach of Part 1, heterogeneity is modeled as random because of uncertainty. The main 835 aim of the study is to determine the various possible paths of the solute body in probabilistic terms and subsequently the distribution of the expectation and variance of concentration in space and time. SOLUTE TRANSPORT IN HETEROGENEOUS POROUS FORMATIONS GeneralThe last topic of the present study is the problem of transport of a solute carried by water which flows in a large, heterogeneous, porous formation. Only an inert solute, which does not change significantly the density and viscosity of water to affect its flow and which preserves its total mass (no decay or adsorption), is considered here.The problem has been studied intensively in the past for flows through homogeneous porous media at laboratory scale under the designation of hydrodynamic dispersion. It has been found that the macroscopic concentration satisfies a convection-diffusion type of equation in which the molecular diffusion coefficient is supplemented by the pore scale dispersion coefficients (for a review, see, for example, Fried [1975]). The investigation can be traced back to the chemical engineering literature, where such problems are encountered in industrial applications. Hydrologists were attracted to the area later, in the early sixties, in relation to problems of salt water intrusion and waste disposal in aquifers. Unfortunately, the approach which is appropriate to laboratory columns or sand boxes has been applied to problems of solute transport in large natural formations, while not realizing that the change in scale might alter the nature of the phenomenon (see, as an example of hydrological applications, Fried [1975]). Field tests, costly and complicated, have shown, however, that the apparent dispersion coefficients in natural formations are larger by a few orders of magnitude than those determined in laboratory. It has been suggested, consequently, that the same dispersion-convection equations can be used in aquifers but w...

Comment on “A note on the recent natural gradient tracer test at the Borden site” by R. L. Naff, T.‐C. Jim Yeh, and M.W. Kemblowski

“…In particular, this is true for a two-dimensional transport model [Dagan, 1982[Dagan, , 1984[Dagan, , 1987Freyberg, 1986] or a three-dimensional one for a anisotropic log conductivity covariance [Gelhar and Axness, 1983]. This writer completely agrees with the authors reasoning and herein are some additional arguments, based on recent publications.…”

“…Indeed, in a periodic stratified formation, Iv can be viewed as the distance between layers of same conductivity. With these assumptions, it has been found that theoretical results for X22 based on a twodimensional model [Dagan, 1982[Dagan, , 1984[Dagan, , 1987, after averaging over Iv lead to values consistent , w,t., .c.d measurements [Freyberg, 1986]. This discussion shows that additional field experiments or theoretical developments are necessary in order to draw more definite conclusions about the accuracy of transport models in representing the three spatial moments of solute plumes.…”

Nonlocal nonreactive transport in heterogeneous porous media with interregional mass diffusion

“…In the last 2 decades, stochastic theory has been extensively applied to study flow and solute transport in heterogeneous porous media [Dagan, 1989;Gelhar, 1993;Cushman, 1997]. The theory is mainly used to study chemical dispersion processes caused by the spatial velocity variation, which is, in turn, caused by the heterogeneity of hydraulic conductivity [e.g., Dagan, 1982Dagan, , 1984 ]. In these stochastic studies the distribution of hydraulic conductivity or sorption coefficient is assumed to be normal or lognormal with a single mean and variance.…”

On stochastic modeling of groundwater flow and solute transport in multi-scale heterogeneous formations

“…If the flow field is stationary, then g will be zero, and (23) and (24) return to the Dagan's results [Dagan, 1982;1984]. From (24), we can obtain the covariance of τ (L; a) as…”