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Concentration fluctuations in aquifer transport: A rigorous first-order solution and applications
No full textFlow 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 = ln K, the structure is characterized by the mean [Y], the variance sigma(Y)(2), the horizontal and vertical integral scales I-h and I-v. 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 alpha(dL) and alpha(dT). 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 sigma(C)(2), 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 sigma(Y)(2). 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 sigma(C)(2), is rederived, correcting for an error in the previous work. The general results are applied to determining CVC = sigma(C)/[C] at the center of a small solute body, of initial size much smaller than I-h = I-v, as function of sigma(gamma)(2), t' = tU/I and Pe = UI/D-dT = I/alpha(dT). Though PSD reduces considerably CVC as compared with advective transport (Pe = infinity), 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). (C) 2000 Elsevier Science B.V. All rights reserved
Transport of a passive scalar in a stratified porous medium RID A-2321-2010
No full textA uniform and horizontal head gradient J is applied to a stratified formation whose given random conductivity K is function of the vertical coordinate x(3) only. K is assumed to be stationary and of finite integral scale I. By Darcy's law, the velocity field V-1(x(3)) = JK depicts a fluctuating shear flow. A solute body is injected instantaneously in the formation. In a Lagrangean framework, the second spatial moment of the mean concentration can be related to the one-particle trajectories variance X-11(t, Pe) where Pe=I-v/D-dT and D-dT is the transverse pore-scale dispersion coefficient. X-11 was determined in the past by Matheron and de Marsily (1980). The present study is concerned with determining the local concentration variance sigma(C)(2), that depends on the two-particles trajectories covariance Z(11)(t). The latter is derived exactly and (C) and sigma(C)(2) are determined by assuming normal or lognormal probability distribution of trajectories. The results are illustrated for small and very large (ergodic) solute plumes. For large travel time the concentration coefficient of variation at the center of the plume tends asymptotically to a constant value, unlike formations with finite horizontal correlation length of the hydraulic conductivity. The results may serve for benchmarking of numerical codes and in applications for short travel distances in highly anisotropic formations
Concentration fluctuations in transport by groundwater: Comparison between theory and field experiments RID A-2321-2010
No full textIn a recent work [Dagan and Fiori, 1997] we developed a Lagrangian theoretical framework to compute the concentration variance in transport of a conservative solute in a heterogeneous formation of random stationary structure. This variance, a measure of concentration fluctuations in the plume, depends on the spreading effect of advection and on the mixing effect of pore-scale dispersion. Comparison between theory and the numerical simulations of Graham and McLaughlin [1989] showed very good agreement. Here we compare the same theoretical formulae with the field measurements at Borden Site and Cape God. We rely on the analysis of these experiments carried out by Fitts [1996], who characterized the concentration fluctuations by a global measure. The latter is the standard deviation of the logarithm of the ratio between measured and modeled concentrations over the entire plume and at different times. The theoretical relationships require computing the two particles trajectories' covariances. We simplify the computations considerably by taking advantage of the anisotropy of the heterogeneous structures of the two aquifers. Thus the major effect of advection is in the shearing of the plume in the longitudinal direction, whereas pore-scale dispersion acts mainly through its transverse effect in the vertical direction. We found a good agreement between theory and experiments by using parameter values determined independently in the past, provided that the values of transverse pore-scale dispersivities were 0.5 mm for Cape Cod and 0.44 mm for Borden Site. These values are somewhat smaller than the vertical macrodispersivities inferred from measurements
The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers RID A-2321-2010
No full textTransport 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 D-dij. 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 D-dij = alpha(dij)U are constant. The relative effect of the two mechanisms is quantified by the Peclet numbers Pe(ij) = UI/D-dij = I/alpha(dij), 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 sigma(C)(2), the concentration mean and variance, respectively. The solution, derived in the past, for Pe = infinity is reconsidered first. By assuming a normal X probability density function (p.d.f.), closed form solutions are obtained for [C] and sigma(C)(2). 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 X-d 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 sigma(C)(2) 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 sigma(C)(2). Simple results are obtained for a small V-0, 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
Time-dependent transport in heterogeneous formations of bimodal structures: 2. Results
No full textThe theoretical results of part 1 [Dagan and Fiori, 2003] for modeling time-dependent, advective transport of a conservative solute in porous formations of bimodal structure are applied to illustrate the behavior of a few trajectory statistical moments as function of time, of the permeability contrast kappa, and of the inclusions volume density n. The computations are carried out for circular (2D) and spherical (3D) inclusions to represent isotropic media. Advective transport is solved by studying the distortion of a thin plume, linear (2D) or planar (3D), normal to the mean velocity U and moving through a single inclusion. The deformation of the plume is determined from the residual trajectories of solute particles that are derived numerically by a quadrature. The longitudinal macrodispersivity is defined by alpha(L)(t; n, kappa)=(2U)(-1) dX(11)/dt, where X-11 is the trajectories second moment in the mean flow direction. The general behavior of the time-dependent longitudinal dispersivity alpha(L)(t; n, kappa) and, in particular, its constant, large time limit are examined. The tendency of alpha(L) to the "Fickian'' limit with time depends strongly on the conductivity contrast; in particular, for low permeable inclusions (kappamuch less than1) it may be extremely slow. It is shown that the first-order approximation in the conductivity contrast kappa is of limited validity (0.3<κ<2). The transverse moment X-22 tends asymptotically to a constant value. The analysis of the trajectory high order moments shows that the probability density function (pdf) of the solute trajectories tends to normality at large time. Similar to the "Fickian'' limit the normal distribution may be reached at very large time in presence of low conductivity inclusions, with the pdf characterized by significant tailing for the trailing part of the pdf
The impact of Concentration measurements upon estimation of flow and transport parameters: the lagrangian approach
Full text linkTransport of a conservative solute takes place in a heterogeneous formation of spatially variable conductivity. The latter is modeled as a random space function of stationary lognormal distribution. As a result the velocity field and the concentration are also random. Assuming that measurements of concentration of an existing plume are available, the problem addressed here is to assess their effect upon identification of log conductivity and flow transport variables. The solution is sought in a Lagrangian framework in which transport is represented in terms of the random trajectories of particles originating from the initial plume. A concentration measurement is equivalent to the passage of a trajectory through the measurement point at a given time. The impact of measurements is achieved by conditioning any variable of interest on realizations for which at least one trajectory satisfies the requirement. It is shown that cokriging of concentration and another flow or transport variable leads to the correct conditioned mean of the latter. In contrast, the conditional variance based on cokriging is erroneous. The procedure is illustrated for two-dimensional flow under a few simplifying assumptions. The effect of a concentration measurement upon the expected value and variance of log transmissivity and plume centroid are examined in a few particular cases. The procedure may improve the solution of the inverse problem and the prediction of transport of existing plume
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