Aldo Fiori

Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications

Flow and transport take place in a formation of spatially variable conductivity K(x). The latter is modeled as a lognormal stationary random space function. With Y=lnK, the structure is characterized by the mean 〈Y〉, the variance σY2, the horizontal and vertical integral scales Ih and Iv. The fluid velocity field V(x), driven by a constant mean head gradient, has a constant mean U and a stationary two-point covariance. Transport of a conservative solute takes place by advection and by pore-scale dispersion (PSD), that is assumed to be characterized by the constant longitudinal and transverse dispersivities αdL and αdT. The local solute concentration C(x, t), a random function of space and time, is characterized by its statistical moments. While the mean concentration 〈C〉 was investigated extensively in the past, the aim here is to determine the variance σC2, a measure of concentration fluctuations. This is achieved in a Lagrangean framework, continuous limit of the particle-tracking procedure, by adopting a few approximations. The present study is a continuation of a previous one (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595–1606) and extends it as follows: (i) it is shown that the indepence of the advective component of a solute particle trajectory from the trajectory component associated with PSD, is a rigorous first-order approximation in σY2. This independence, that was conjectured in the work of Dagan and Fiori (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595–1606), simplifies considerably the solution; (ii) the covariance of two-particle trajectories, needed in order to evaluate σC2, is rederived, correcting for an error in the previous work. The general results are applied to determining CVC=σC/〈C〉 at the center of a small solute body, of initial size much smaller than Ih=Iv, as function of σY2, t′=tU/I and Pe=UI/DdT=I/αdT. Though PSD reduces considerably CVC as compared with advective transport (Pe=∞), its value is still quite large for time intervals of interest in applications. This finding is in agreement with the analysis of field data by Fitts (Fitts, C.R., 1996. Uncertainty in deterministic groundwater transport models due to the assumption of macrodispersive mixing: evidence from the Cape Cod (Massachussets, USA) and Borden (Ontario, Canada) tracer tests. J. Contam. Hydrol., 23, 69–84).

Finite Peclet Extensions of Dagan's Solutions to Transport in Anisotropic Heterogeneous Formations

Dagans [1984] landmark solution for transport in heterogeneous porous formations is extended to the case of finite Peclet numbers. It is suggested that pore-scale dispersion matters only with reference to transverse spreading, and that Dagans solution, valid for Pe=∞ is an adequate approximation in a wide range of finite Peclet numbers.

The influence of pore‐scale dispersion on concentration statistical moments in transport through heterogeneous aquifers

Transport of an inert solute in a heterogeneous aquifer is governed by two mechanisms: advection by the random velocity field V(x) and pore-scale dispersion of coefficients Ddij. The velocity field is assumed to be stationary and of constant mean U and of correlation scale I much larger than the pore-scale d. It is assumed that Ddij=αdijU are constant. The relative effect of the two mechanisms is quantified by the Peclet numbers Peij=U/Ddij=I/αdij, which as a rule are much larger than unity. The main aim of the study is to determine the impact of finite, though high, Pe on 〈C〉 and σC2, the concentration mean and variance, respectively. The solution, derived in the past, for Pe=∞ is reconsidered first. By assuming a normal X probability density function (p.d.f.), closed form solutions are obtained for 〈C〉 and σC2. Recasting the problem in an Eulerian framework leads to the same results if certain closure conditions are adopted. The concentration moments for a finite Pe are derived subsequently in a Lagrangean framework. The pore-scale dispersion is viewed as a Brownian motion type of displacement Xd of solute subparticles, of scale smaller than d, added to the advective displacements X. By adopting again a normal p.d.f. for the latter, explicit expressions for 〈C〉 and σC2 are obtained in terms of quadratures over the joint p.d.f. of advective two particles trajectories. While the influence of high Pe on 〈C〉 is generally small, it has a significant impact on σC2. Simple results are obtained for a small V0, for which trajectories are fully correlated. In particular, the concentration coefficient of variation at the center tends to a constant value for large time. Comparison of the present solution, obtained in terms of a quadrature, with the Monte Carlo simulations of Graham and McLaughlin [1989] shows a very good agreement.

Steady Flow Toward Wells in Heterogeneous Formations: Mean Head and Equivalent Conductivity

We consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw (Figure 1). The hydraulic conductivity K is modeled as a three-dimensional stationary random space function. The two-point covariance of Y = In (K/KG) is of axisymmetric anisotropy, with I and Iυ, the horizontal and vertical integral scales, respectively, and KG, the geometric mean of K. Unlike previous studies which assumed constant flux, the well boundary condition is of given constant head (Figure 1). The aim of the study is to derive the mean head 〈H〉 and the mean specific discharge 〈q〉 as functions of the radial coordinate r and of the parameters σy2, e = I/Iυ and rw/I. An approximate solution is obtained at first-order in σy2, by replacing the well by a line source of strength proportional to K and by assuming ergodicity, i.e., equivalence between , , space averages over the vertical, and 〈H〉 〈q〉, ensemble means. An equivalent conductivity Keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge Q as the actual one, for the given head Hw in the well and a given head in a piezometer at distance r from the well. This definition corresponds to the transmissivity determined in a pumping test by an observer that measures Hw, , andQ. The main result of the study is the relationship (19) Keq = KA(1 − λ) + Kefuλ, where KA is the conductivity arithmetic mean and Kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer. The weight coefficient λ 10, λ has the simple approximate expression λ* = ln (r/I)/ In )r/rw). Near the well, λ ≅ 0 and Keq ≅ KA, which is easily understood, since for rw/I ≪ 1 the formation behaves locally like a stratified one. In contrast, far from the well λ ≅ 1 and Keq ≅ Kefu the flow being slowly varying there. Since KA > Kefu, our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well. However, the difference between KA and Kefu is small for highly anisotropic formations for which e ≪ 1 . A nonlocal effective conductivity, which depends only on the heterogeneous structure, is derived in Appendix A along the lines of Indelman and Abramovich [1994].

Effective Conductivity of an Isotropic Heterogeneous Medium of Lognormal Conductivity Distribution

The study aims at deriving the effective conductivity Kef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function of lognormal distribution and finite integral scale IY. We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K 0 within a sphere

Correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations

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Impact of Climate Change on the Hydrology of Upper Tiber River Basin Using Bias Corrected Regional Climate Model

, of radius R 0, and they occupy a volume fraction n. Uniform flow of flux

Ergodic transport through aquifers of non‐Gaussian log conductivity distribution and occurrence of anomalous behavior

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The Lagrangian concentration approach for determining dilution in aquifer transport: Theoretical analysis and comparison with field experiments

prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere

Hydraulic structures subject to bivariate hydrological loads: Return period, design, and risk assessment

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