Steady plumes in heterogeneous porous formations: A stochastic Lagrangian approach


Corresponding author: A. Fiori, Dipartimento di Scienze dell'Ingegneria Civile, Universita' di Roma Tre, Via Vito Volterra 62, Rome I-00146, Italy. (


[1] We study three-dimensional steady plumes in heterogeneous porous formation by a stochastic analytical approach. Flow is uniform in the mean and the hydraulic conductivity K is a space random function, resulting in a random local concentration C. The primary concern of our work is to characterize C in terms of its mean value and its fluctuation; this is achieved, for low/medium heterogeneity, by a rigorous Lagrangian model based on the first-order approximation. Large-scale advection and pore-scale dispersion (PSD) govern the movement of solute particles through the medium of log conductivity Y = lnK. Transport is mainly governed by vertical and lateral spreading, different from the extensively studied transient transport, which is in turn governed by longitudinal dispersion. The concentration field depends on the Peclet number Pe, which controls dilution; the latter results from the intertwining of large-scale advection and PSD. The mean concentration inline image recovers the well-known solution for homogeneous media with the obvious differences in the dispersion/diffusion parameters. The standard deviation σC, evaluated here for the first time, fully analytically within the Lagrangian approach for steady state plumes, exhibits a maximum far from the plume centerline; the coefficient of variation can be very large in the plume fringe. The rigorous analytical evaluation of σC requires a huge computational effort in order to calculate the trajectory moments and perform the numerical integrations needed. We propose two simplified solutions with different degrees of approximation, that can be used in applications and preliminary studies: (i) the first one is obtained neglecting the longitudinal component of the trajectory fluctuations; the solution brings considerable simplifications although numerical integrations are still required in order to evaluate σC; this approximation is quite accurate for the mean concentration while slightly overestimates the concentration variance; (ii) the second one leads to a closed-form expression for σC; although somewhat conservative, this simple approximated solution works very well in a broad range of distances from the source, and may be used for a preliminary assessment of C in applications.

1. Introduction

[2] Solute transport in porous formations is ruled by the spatial distribution of the hydraulic properties, the most relevant being the hydraulic conductivity K. The last four decades have seen a tremendous effort by the scientific community in order to adequately capture the relevant structural features which govern solute transport in complex subsurface environments (see e.g., the monographs by Dagan [1989] and Rubin [2003]). A common assumption is to model K as a space random function, following experimental evidence, Y = lnK is modeled as a second-order stationary normal random variable, with geometric mean KG, variance inline image, and two-point covariance CY.

[3] A successful and widely employed approach has been the first-order one, which is formally valid for small/moderate heterogeneity, i.e., inline image. Despite the latter limitation, the first-order method has led to useful relations between the statistical moments of the concentration field C and the main flow/structure parameters, e.g., the mean velocity, the statistical moments of K, and the injection conditions (for a comprehensive review on the subject the reader is addressed to the previously cited monographs). The derivation of simple relations is generally not possible by numerical methods, which although are virtually free from the assumption inline image often require a huge computational burden. Instead, the development of analytical or semianalytical solutions is important for applications, especially for a quick estimate of the concentration field, for their simplicity in the input parameters, easy software implementation and computational effort (see e.g., the popular codes BIOSCREEN [Newell et al., 1996] and BIOCHLOR [Aziz et al., 2000]).

[4] This paper focuses on approximate, semianalytical, and analytical solutions for 3-D steady transport of conservative solutes under mean uniform flow. Steady plumes, i.e., in which the concentration C does not depend on time, may occur for instance from continuous releases of a contaminant from a source area. Field investigations on contaminated sites, where the plume derives from old contamination events, often show that the plume is at steady state. Models for the prediction of the local concentration of steady plumes are important for many applications, e.g., those related to biodegradation of organic compounds [e.g., Liedl et al., 2005; Maier and Grathwohl, 2006; Cirpka and Valocchi, 2007; Gutierrez-Neri et al., 2009; Cirpka et al., 2011]; also, models for conservative solute constitute the fundamental prerequisite for building more elaborated models which embed different complex biochemical processes.

[5] Analytical or semianalytical solutions are available for the simple case of homogeneous formations [e.g., Domenico, 1987; Wexler, 1992]. Our interest here is in transport through heterogeneous porous formations, with spatially variable and random K. As a consequence, the concentration C is also random, and our main target is the calculation of its mean inline image and variance inline image as function of the relevant flow/structure parameters. Models of transport which explicitly consider heterogeneity are important for a few reasons. First, they provide a measure of both the mean of C and its uncertainty, measured by σC; the uncertainty analysis is quite useful in applications, and even more so in engineering applications which require a description of C in probabilistic terms (e.g., risk assessment). Second, the analysis of transport in heterogeneous formations helps in elucidating fundamental issues, like e.g., the interplay between large-scale advection and local-scale dispersion/diffusion, their impact on the local dilution features (which are important in presence of additional biochemical processes), and allows distinguishing between macrodispersion and the actual decay of concentration. We are not aware of analytical models specifically designed for steady 3-D transport in heterogeneous formations; conversely, different stochastic approaches (e.g., Eulerian or Lagrangian) have been adopted in the last decades for the development of models for transient transport after instantaneous solute injection.

[6] In the present work, we shall derive the first two statistical moments of concentration for steady 3-D transport by adopting the first-order Lagrangian approach [see e.g., Dagan, 1989; Fiori and Dagan, 2000]. The plan of the paper is as follows. The general mathematical framework is introduced first; then, the first-order analysis is formally adopted and introduced in the principal expressions. Semianalytical and analytical expressions are then explicitly derived, with different levels of approximation. A set of illustration examples serves as vehicle for discussion. Then, the conclusions section ends the paper.

2. Mathematical Framework

[7] Steady, mean uniform flow takes place in a three-dimensional, unbounded domain; the Cartesian system is inline image, where the longitudinal direction x1 is aligned with the mean velocity inline image (see Figure 1). The medium is of random and stationary hydraulic conductivity inline image, with Y = lnK normal, of two-point covariance inline image, with inline image. Following experimental evidence [see Rubin, 2003, Table 2.1], Y is assumed as axisymmetric, with inline image and inline image the horizontal and vertical log-conductivity integral scales, respectively. Thus, at second-order analysis the stationary structure is characterized entirely by the four parameters KG (the geometric mean), inline image, inline image, and f, where inline image is the anisotropy ratio (generally f < 1).

Figure 1.

Schematic of the domain.

[8] Conservative solute is continuously released at x1 = 0 transverse to the mean flow, over an injection plane (IP) with dimensions L2, L3, and area A0 = L2L3. The solute concentration at the IP is C = C0; the concentration in the porous formation at the initial time t = 0, when the solute injection begins, is null. In the limit t a steady plume establishes in the medium, with constant concentration inline image. In what follows, we analyze the statistical moments of the concentration field inline image by employing a Lagrangian formulation of transport. The basic approach builds on the framework developed by Fiori and Dagan [2000] and Fiori et al. [2002], although the derivations are different.

[9] We follow a generic solute subparticle, defined at the pore scale, along its movement in the porous medium, after its release from the IP. The “total” trajectory inline image of the subparticle, which was released at time t0 and position inline image within the IP, is written as the sum of two independent components: (i) an advective one inline image, which is related to the Darcy-scale advective flow, and (ii) a pore-scale dispersion (PSD) component inline image, which represents the dispersion/diffusion effects acting at scales smaller than the Darcy one [see Fiori and Dagan, 2000]. The PSD trajectories are described by a Wiener process and are characterized by the (local) dispersion coefficients inline image; in the following, we shall adopt for simplicity an isotropic PSD tensor inline image. Since the transverse local dispersivity plays a crucial role for the mixing processes to take place, the definition of Peclet which should be employed in applications involves the transverse dispersivity αT, i.e., inline image. Since Ih varies between a few meters and αT is typically of the order of fraction of millimeters at the field scale, typical Peclet values one might expect are around inline image; lower and higher values are also possible, though.

[10] From each elementary area da within the IP, a continuous string of solute particles originates, the front of which moves with time. The front reaches a generic control plane (CP) located at distance x1 at time inline image, which is the travel time of the subparticle released at a at t = 0; the relation between the travel time τ and the total trajectory inline image follows the identity inline image. The transport process at this scale is governed by advection and the local concentration inline image, defined at the subparticle level (i.e., the pore scale), can be written as

display math(1)

with H the Heaviside function and Xt2, Xt3 the transverse and vertical total trajectories of the subparticle originated at a.

[11] The concentration inline image, which is defined at the Darcy scale, is obtained by averaging (1) over the PSD component inline image. This intermediate step was performed by Fiori and Dagan [2000] before the further averaging procedure which is required for the calculation of the ensemble moments of C; however, it is more convenient to perform the two steps simultaneously, along the following lines.

[12] We start from the mean steady concentration inline image, which is obtained by averaging of (1) over the random variables inline image and taking the limit inline image. The averaging procedure is simplified along the first-order approximation due to the lack of correlation of the quantities inline image [Dagan, 1989]. Thus, we average (1) first over Xt2, Xt3 conditioned on τ, and then on τ itself, to get

display math(2)

where inline image are the probability density functions (pdfs) of the related random variables.

[13] The concentration variance inline image is obtained in a similar fashion by squaring (1), taking the expectation and subtracting the average (2). The procedure yields the result

display math(3)

with inline image the joint pdf of the trajectories inline image of two particles originated from inline image and inline image and inline image, the joint pdf of the related travel times.

3. First-Order Analysis

[14] The general Lagrangian analysis carried out in the previous section is cast here in the framework of first-order analysis. The latter assumes that the Y field is low/mild heterogeneous, i.e., inline image [see e.g., Dagan, 1989; Rubin, 2003]. First-order analysis allows the analytical determination of the pdfs which are required to obtain the concentration moments (2) and (3). When inline image, the trajectory pdfs inline image can be assumed as normal, because of the linear relation between X and Y [Dagan, 1989] and the normality of inline image; similarly, the joint pdfs inline image are bivariate normal [Fiori and Dagan, 2000]. Conversely, the travel time pdf fτ is the one pertaining to a the first-passage distribution of a particle trajectory, which is normally distributed through the CP [see e.g., Redner, 2001], which is the well-known inverse-Gaussian distribution

display math(4)

where Dt11 is the sum of the macrodispersion coefficient D11 and the PSD one Dd. Expression (4) implicitly assumes Fickianity, i.e., inline image; we are not aware of extensions of the inverse-Gaussian distribution to time-variable Dt11. Although other forms of fτ have been suggested in the past, like e.g., the lognormal distribution [Russo, 1993; Cvetkovic et al. 1992], we adhere here to the more rigorous inverse-Gaussian one as it allows recovering the classic solutions for homogeneous media, as will be discussed later.

[15] The selection of the joint pdf inline image is more problematic, as the existing literature does not indicate a clear and unique definition for the bivariate inverse-Gaussian distribution [e.g., Balakrishnan and Lai, 2008]. In this study, we shall therefore employ a Copula distribution [e.g., Salvadori et al., 2007], built on the univariate pdf (4). The copula-based joint travel time distribution is introduced and discussed in Appendix The Travel Time Joint pdf fτ,τ′.

[16] Once the shape of the distribution is set, the first- and second-statistical moments of trajectories are needed in order to carry out the calculations. The moments of the bivariate normal distribution inline image (i = 2, 3) are

display math(5)

where inline image and Zii indicate the trajectories variances and covariance, respectively. The bivariate pdf inline image (see Appendix The Travel Time Joint pdf fτ,τ′) requires the statistical moments of the longitudinal trajectories, as follows [Fiori et al., 2002]

display math(6)

with inline image. The statistical moments appearing in the previous expressions have been extensively studied in the last decades in the context of the first-order theory; Appendix Trajectory and Travel Time Statistical Moments reproduces the principal formulas for the derivation of moments Xtii, D11, and Zii.

[17] The numerical burden required for the calculation of the various quantities may be significant. It can be alleviated by assuming inline image, as done by Fiori and Dagan [2000], which is rigorously valid for small plumes inline image; Tonina and Bellin [2008] have recently shown that this approximation may be a valid one also for large plumes. The derivation of the C moments is carried out in the following section.

4. Concentration Moments and Simplified Expressions

[18] After the above preliminary steps, we can now derive the expressions for the mean and variance of the concentration. Starting from the mean inline image, integration of (2), with fτ given by (4), leads to

display math(7)


display math(8)

[19] The solution (7) is identical to the well-known and classic formula which was developed for deterministic formations [e.g., Wexler, 1992], with inline image instead of 2Diit (i = 2, 3).

[20] The concentration variance is obtained by introducing the joint pdfs, with the trajectory moments in (3) and performing the integrations; the final result is

display math(9)


display math(10)

[21] The above formulas for inline image and inline image require some numerical quadratures; in particular, the calculation of (9) is not straightforward: it requires (4) quadratures and it involves the complex joint distribution inline image, which in turn needs the two-times, two-points cross-correlation definition inline image. The derivations can be considerably simplified by neglecting the longitudinal transport inline image; the latter is a common approximation in steady transport in homogeneous formations [Rolle et al., 2009; Cirpka and Valocchi, 2007; Gutierrez-Neri et al., 2009; Liedl et al. 2005], and it was found that it has a limited impact on the predicted concentration in homogeneous aquifers, especially when inline image [Wexler, 1992]. Introducing inline image in both (7) and (9), it yields

display math(11)
display math(12)


display math

[22] The above solution shall be denoted in the following as “DL0.” The previous expressions constitute a significant simplification over their counterparts (7) and (9), and shall be checked later.

[23] Equation (12) can be further simplified by the procedure adopted by de Barros et al. [2011], in which the approximation inline image was introduced; the previous is consistent with the assumption of small source. The procedure leads to the fully analytical result

display math(13)


display math(14)

[24] We shall check in the following section the validity of both the DL0 approximation (11,12) and the simplified, closed-form solution (13), for both moments inline image.

[25] Finally, it is of particular interest the solution for a point source inline image. In such case, the previously discussed approximation inline image is exact. Also, the point source solution is valid for any plume size at the large distance limit, i.e., inline image. The point source solution is easily obtained by performing the expansion of (11) and (12) for inline image, leading to the simple results

display math(15)
display math(16)

with inline image the correlation coefficient of the trajectory inline image.

5. Illustration Example and Discussion

[26] We illustrate here some results for inline image and inline image for the following set of parameters: inline image. The chosen anisotropy ratio reflects the typical values encountered in the field [see Rubin, 2003, Table 2.1], while the relatively large value of inline image, beyond the presumed validity of the first-order approach, is adopted in order to emphasize differences which may emerge between the solutions. Analysis performed on alternative sets of parameters has indicated a substantial similarity with the results discussed in the sequel, with the obvious differences (e.g., a general slower/faster overall dynamics for larger/smaller injection areas A0).

[27] Figure 2 shows inline image as function of the CP distance x1, along the plume mean centerline x2 = x3 = 0 (Figure 2a) and along the transverse direction x2 for a fixed inline image (Figure 2b), for a few values of inline image. Two solutions are compared in the figures: (i) the “full” one (equation (7), solid lines), and (ii) the DL0 solution (equation (11), dashed lines). First, it is seen that, after an initial decay, the purely advective solution ( inline image) tends to a constant value for inline image. This unique feature is related to the convergence to zero of the transverse and vertical macrodispersion coefficients D22,D33 for inline image. Although this feature is predicted exactly by the first-order theory [see e.g., Dagan, 1989], recent studies have shown that D22 and D33 may be nonzero in highly heterogeneous formations [Attinger et al., 2004; Janković et al., 2009]. Hence, the constant asymptotic value of inline image observed in Figure 1 may not be valid for any inline image. The behavior of inline image reflects the complete lack of mixing implicit in the inline image solution.

Figure 2.

The mean concentration inline image (a) at the mean centerline and (b) as function of x2 ( inline image), for inline image and for a few values of the Peclet number; full solution (equation (7), solid lines) and DL0 solution (equation (11), dashed lines).

[28] The presence of PSD (finite Peclet) leads to a monotonous decrease of inline image down to 0, with distance x1. The lowering of the inline image is faster when the PSD is the dominant phenomena (low Pe). Such feature, which is well illustrated in Figure 2a, differs from the instantaneous injection case, for which inline image is practically insensitive of Pe [Fiori and Dagan, 2000]. A similar property characterizes in the transverse direction (Figure 2b) and in the vertical direction (not shown). It is seen that the solution DL0 is generally an acceptable approximation for the mean C, as already known from the existing literature for homogeneous media (the issue was discussed in the previous section); since the solution for inline image is the same as the one for homogeneous media, with the macrodispersion coefficients replacing the PSD ones, we defer further discussion to previous work, and concentrate in the following on the more interesting standard deviation σC, which is introduced and discussed here for the first time.

[29] Figure 3 illustrates the standard deviation σC along the plume mean centerline x2 = x3 = 0 (Figure 3a) and transverse to it (Figure 3b), for a few values of Peclet; the parameter values are the same as Figure 2.

Figure 3.

The concentration standard deviation σC (a) at the mean centerline and (b) as function of x2 ( inline image); the style and parameter values are the same as Figure 2.

[30] The distribution of C is binary in absence of PSD ( inline image) furthermore σC is equal to inline image [Dagan, 1989]; the latter, together with the asymptotic results of Figure 2a, explain the asymptotic behavior of the standard deviation for inline image for inline image.

[31] We emphasize that for inline image reflects the uncertainty in the location of the plume parcels originating from the IP, rather than a clear and definite dilution process. The latter is indeed triggered by PSD, i.e., it only occurs when Peclet is finite. Figure 3a shows that the presence of PSD (finite Peclet) determines a peak of σC followed by a decrease with distance x1, both inverse proportional to Pe. At very large distances inline image, meaning that the solute has fully mixed in the medium, and transport has become a deterministic diffusive process; the feature will be better illustrated in the sequel. As shown in Figure 3, the complete mixing may occur at very large distances from the IP for a typical Peclet number inline image.

[32] For a fixed x1, σC displays a peak in a lateral region which expands and moves far from the centerline with x1 and inline image(Figure 3b); the behavior is very similar in the vertical direction x3 (not shown). A similar behavior was found by de Barros and Nowak [2010] and Nowak et al. [2010] through different methods; the feature is related to the expansion of the plume fringe with distance, which is mainly influenced by PSD and its impact on transverse/vertical macrodispersion.

[33] Figures 3a and 3b also show that the DL0 solution provides a good estimate of σC for high and low values of the Peclet number. For intermediate Pe, the DL0 solution generally overestimates σC, in a region around the mean centerline x2 = x3 = 0. At any rate, the differences are bound to vanish for increasing x (say when inline image), as visible in Figures3a and 3b, for which the two solutions converge. We have implemented a different expression for the copula function, as well as a completely different bivariate travel time pdf (the bivariate lognormal one), and we got identical results. We conclude that the approximation which neglects longitudinal transport may in some cases (e.g., those discussed above, like for small/intermediate values of x and intermediate Peclet) work less well for σC than inline image. The variance is overestimated by DL0 because it neglects the longitudinal component of the local dispersion tensor, which helps in enhancing mixing and therefore the reduction of the concentration standard deviation. Still, because of its simplicity and its convergence to the full solution for inline image, the DL0 solution may provide a reasonable prediction of the concentration uncertainty in important engineering problems, like e.g., risk assessment or the design of remediation strategies, where the overestimation of uncertainty is in favor of safety. We remark again that the calculation of full solution (9) is generally complex, especially when x1 is large, and the DL0 represents a major simplification computation wise.

[34] The impact of the anisotropy ratio f is examined next. Figures 4a and 4b depict the mean and the standard deviation of the concentration evaluated by (11) and (12) for two different anisotropy ratios, f = 0.1 and f = 0.5, and for the isotropic case (f = 1). The behavior of the mean value is similar to that observed for f = 0.1. The effect of increasing anisotropy is to increase, for a fixed location, the concentration mean value; the result derives from the decreasing vertical and transverse dispersions with f. The concentration standard deviation generally decreases with the anisotropy ratio but few inversions are possible when Peclet is large. Due to the involved analytical form of (11) the behavior of the standard deviation cannot be easily explained. Analysis of the case inline image indicates the value σC = 0.5 as the upper bound of the standard deviation, which is reached when the mean concentration is equal to 0.5. Finally, we emphasize that the uncertainty evaluated by the standard deviation can be rather large for high values of Peclet.

Figure 4.

(a) The mean concentration inline image and (b) the concentration standard deviation σC evaluated by the DL0 solution (equations (11) and (12)) at the mean centerline, for inline image, for a few value of inline image and a few values of the Peclet number.

[35] We analyze now the concentration coefficient of variation inline image, up to large distances, and compare it with the approximate/asymptotic solutions presented in the previous section. We shall not discuss again the full solution (equations (7)-(9)), which has been described before (Figure 3), and which by the way requires cumbersome (and sometimes unfeasible) calculations for large values of x1. Figure 5 shows CV as function of x1 (Figure 5a) and x2 (Figures 5b and 5c) for the same parameters examined before. The figures depict DL0, the simplified analytical solution (equations (11)-(13)), denoted as “Analytical,” and the point-source one (equations (15) and (16)). Starting from the behavior in the longitudinal direction (Figure 5a), the CV displays the same features which have been observed for σC; the crucial impact of Peclet is here more visible. The asymptotic CV for larger distances is, derived from equations (15) and (16), as follows

display math(17)
Figure 5.

The concentration coefficient of variation inline image (a) at the mean centerline and (b) as function of x2 at inline image and (c) inline image; parameter values are the same as Figure 2. DL0 solution (equation (12), solid lines), approximated analytical formula (equation (13), dashed-dotted lines) and large distance, asymptotic behavior (equation (16), dashed lines).

[36] Figure 5a (dashed lines) shows that the above formula is able to reproduce the tail of CV, and it provides an useful description of the concentration uncertainty when inline image. Following ((17)), the drop of CV2 with inline image along the mean centerline inline image is inline image, where the asymptotic relation inline image have been used [Fiori and Dagan, 2000; equation (23)]. The decay with distance of the coefficient of variation clearly indicates the mixing effects associated with finite Peclet at large distances, with consequent reduction of uncertainty.

[37] Conversely, when inline image it is inline image, as previously discussed; hence, the absence of PSD does not allow mixing and the consequent decrease of uncertainty.

[38] The lateral behavior of CV is observed in Figures 5b and 5c, for inline image and 1000, respectively. It is seen that CV grows far from the mean centerline, as also predicted by (17), the behavior being similar to that observed for transient plumes [e.g., Fiori and Dagan, 2000]. The rate of growth with x2, x3 is proportional to Peclet, and the uncertainty reduction at the fringe for decreasing Pe again reflects the smearing effect of PSD. The increasing validity of the asymptotic solution (equation (17)) with x1 and inline image is clearly visible.

[39] The results represented in Figure 5 show that the analytical, closed-form solution (equation (13)) works reasonably well in the entire range of possible values for x. In any case, it provides the correct small- and large-distance limits, the convergence being faster for decreasing Peclet. Because of the simple analytical form, and considered all the sources of uncertainty, expression ((13)) seems rather appealing for applications and overall preferable to the more complex expression ((9)).

6. Conclusions

[40] We have analyzed steady plumes in heterogeneous porous formations by a stochastic approach, after assuming that hydraulic conductivity K is a space random function. Thus, K is modeled as a second-order stationary random field, lognormally distributed, fully characterized by the four parameters inline image. A rigorous Lagrangian first-order approximation, formally valid for low/medium heterogeneity, was employed, after assuming mean uniform flow. Hence, transport of a passive tracer is studied by analyzing the trajectory of a solute particle, undergoing large-scale advection and local-scale dispersion/diffusion. The modeling approach is aimed to a first, preliminary assessment of the first two statistical moments of the concentration (mean and variance) as function of distance from the injection area, as an alternative to the common solutions available for homogeneous media.

[41] Because of the rather complex mathematical structure of the solution for the variance inline image, two simplified expressions have been developed and tested, as well as an asymptotic, large distance solution for both inline image and inline image, which is valid also for point-like injections. Of particular interest is the solution which neglects the longitudinal dispersion (coined as DL0), which lead to considerable simplifications.

[42] The main results of the present study can be summarized as follows.

[43] (1) The mean concentration inline image recovers the classic, widely employed solution for steady transport in homogeneous formations [e.g., Wexler, 1992], with the “total” macrodispersion coefficients (sum of macroadvection and pore-scale dispersivities) replacing those for the homogeneous formations. It is seen that the plume shape and the concentration field is mainly influenced by the transverse and vertical macrodispersion and pore-scale components, while longitudinal spreading/diffusion plays a marginal role. The analysis of variance emphasizes however that the mean concentration inline image is by no means indicative of mixing in the system, and the concentration coefficient of variation CV can be rather large in absence of local dispersion mechanisms.

[44] (2) The concentration variance inline image typically exhibits a peak at a distance from the plume centerline, which moves away from it with increasing distance x1 from the source. inline image also exhibits a peak along the centerline, the magnitude and distance being proportional to Peclet. In contrast, the coefficient of variation inline image typically grows with lateral/vertical distance from the source, i.e., along the plume fringe, proportional to Peclet, while it tends to be relatively small in a region around the centerline, of size similar to the IP. We remark that the concentration uncertainty displayed by steady plumes is generally smaller than that pertaining to transient plumes, e.g., after an instantaneous injection.

[45] (3) The Peclet number exerts a significant influence on both moments and its correct estimation plays a fundamental role in the steady plume analysis. Contrary to the instantaneous injection case [e.g., Dagan, 1989], the mean concentration is highly affected by Pe; the reason is that the transverse and vertical trajectory moments, which rule the behavior of inline image, strongly depend on Pe. The variance inline image is also profoundly affected by Peclet; in fact, pore-scale and molecular mechanisms are those which rule mixing in the system, leading to a general decrease of the concentration uncertainty. It is seen that in all cases, PSD (i.e., finite Peclet) leads to a decrease of inline image; in turn, the latter tend to a constant value when inline image, indicating lack of local dilution/mixing.

[46] (4) The anisotropy exerts its effect by means of the trajectory covariances. The effect of the anisotropy is to decrease the concentration mean value. Generally, the anisotropy decreases the variance but, for large Pe, its behavior is not monotonous. When inline image the value 0.25 constitutes the upper bound of the variance.

[47] (5) The DL0 solution works very well for inline image, while the differences with complete solutions can be larger for inline image and intermediate values of Pe. Still, the predictions made by DL0 are quite accurate in all cases, leading to a somewhat overestimation of inline image for intermediate Peclet and at distances not far from the source. Thus, the DL0 solution is relatively simple to use and applicable in most of the cases of interest.

[48] (6) A fully analytical solution was also derived, based on DL0. It has the advantage of simplicity and it can be related with existing analytical solutions. The analytical, closed-form solution works reasonably well in a broad range of distances from the source, and it anyway provides the correct small- and large-distance limits. Because of the simple analytical form, and considered all the sources of uncertainty, the closed-form solution may be a useful tool for applications.

[49] Although the present work is focused on the first and second moment of the local concentration, an assessment of the concentration probability density function (pdf) can be performed by employing the Beta distribution. The latter was suggested by Fiori [2001], and tested by Bellin and Tonina [2007] through accurate numerical simulations and analysis of field experiments. The effectiveness of the beta distribution is also supported by recent studies based on numerical simulations and field test [e.g., Schwede et al., 2008, Cirpka et al., 2011]. The analytical expressions proposed in this work allow estimating the two parameters of the beta distribution, hence leading to a more effective characterization of the local concentration through its statistical distribution.

[50] The derived solutions can be used for a preliminary contamination assessment, all the approximations notwithstanding, may also serve as a basis to build more solutions dealing with more complex reactive solute processes.

Appendix A: The Travel Time Joint pdf inline image

[51] Copulas provide an easy and general structure to define multivariate distributions. This technique, commonly employed in many disciplines such as financial applications, is widely studied and applied also in the hydrological field. Salvadori et al. [2007] emphasize that all the bivariate distributions currently used can be obtained through Copulas. In this section, we recall the derivation of bivariate travel time distribution inline image; for a more general description of Copula techniques the reader can refer to the work by Salvadori et al. [2007].

[52] The starting point to define inline image are the marginal distribution (4) and the travel time correlation inline image quantified at first order through (B2) and (B9). We employed the Gumbel-Hougaard Copula, which is a particular expression of the Archimedean family. The cumulative density function (CDF) inline image is related to the marginal CDF Fτ by the

display math(A1)


display math(A2)

[53] The constant parameter θ accounts for the correlation between the two subparticles travel times inline image. Under the first-order approximation, at the CP x1 the travel time correlation coefficient of two subparticles ρ11, released within the IP at the same time t0, depends on both τ and inline image; it was approximated as the value evaluated for the mean travel times inline image. Substituting inline image in (A1) [Balakrishnan and Lai, 2009] the closed form of the bivariate density function can be obtained trough the following relation

display math(A3)

[54] The resulting analytical formula is rather long and involved and shall not be reproduced here.

Appendix B: Trajectory and Travel Time Statistical Moments

[55] Under the first-order approximation, the evaluation of the trajectory covariance starts from the well-known equation that relates, in the Fourier space inline image, the velocity field covariance and the hydraulic conductivity covariance [e.g., Dagan, 1989]

display math(B1)

where δii is the Kroneker's delta, inline image and inline image are the Fourier transforms respectively of the velocity field and the hydraulic conductivity field covariance. The following calculations are carried on assuming, for inline image an axial-symmetric anisotropic covariance structure inline image and an isotropic PSD tensor ( inline image). Introducing the dimensionless variables inline image the one particle covariance is obtained along the outlines traced by Fiori [1996] as follows

display math(B2)

[56] We have employed the Gaussian covariance CY, as it allows to reduce the numerical quadratures required and the overall computational effort

display math(B3)

where inline image.

[57] After few manipulations, recasting the problem in cylindrical coordinates inline image and performing three analytical integrations, we obtain the trajectory covariance

display math(B4)


display math(B5)
display math(B6)
display math(B7)

where I0 and I1 are modified Bessel function of the first kind. The asymptotic longitudinal macrodispersion coefficient D11 is obtained through derivation of (B4) and taking the limit inline image

display math(B8)

[58] The trajectory moments of two particles originated at a and b at time t0 = 0 are expressed as function of their travel times τ and inline image

display math(B9)

[59] Neglecting the dependence on ab and using the cylindrical coordinates, we obtain the final expressions

display math(B10)


display math(B11)