Solute transport in dual-permeability porous media



[1] A dual-advection dispersion equation (DADE) is presented and solved to describe solute transport in structured or layered porous media with different nonzero flow rates in two distinct pore domains with linear solute transfer between them. This dual-permeability model constitutes a generalized version of the advection-dispersion equation (ADE) for transport in uniform porous media and the mobile-immobile model (MIM) for transport in media with a mobile and an immobile pore domain. Analytical tools for the DADE have mostly been lacking. An analytical solution has therefore been derived using Laplace transformation with time and modal decomposition based on matrix diagonalization, assuming the same dispersivity for both domains. Temporal moments are derived for the DADE and contrasted with those for the ADE and the MIM. The effective dispersion coefficient for the DADE approaches that of the ADE for a similar velocity in both pore domains and large values for the first-order transfer parameter, and approaches that of the MIM for the opposite conditions. The solution of the DADE is used to illustrate how differences in pore water velocity between the domains and low transfer rates will lead to double peaks in the volume- or flux-averaged concentration profiles versus time or position. The DADE is applied to optimize experimental breakthrough curves for an Andisol with a distinct intra- and interaggregate porosity. The DADE improved the description of the breakthrough data compared to the ADE and the MIM.

1. Introduction

[2] Many porous media exhibit different pore or flow domains due to layering, fracturing, or aggregation. In the following only two pore domains will be considered; domain 1 may represent a fracture or interaggregate pore space with faster flow and domain 2 with slower or no flow in the rock matrix or intraaggregate pore space. The difference in flow and transport of dissolved substances in the two domains may be accounted for by using a physical nonequilibrium concept [Selim and Ma, 1998]. Solute transport in porous media with transient water flow in multiple (unsaturated) domains typically will have to be solved numerically [Gerke and van Genuchten, 1996; Simunek and van Genuchten, 2008]

[3] Analytical solutions for solute transport involving physical nonequilibrium have mostly been restricted to dual-porosity media with a uniform water content and flow in the interaggregate or mobile aqueous domain and no flow in the intraaggregate or immobile domain. Transport in the longitudinal direction, by advection and dispersion, occurs solely in the mobile domain. There is diffusive solute transport between the two domains, which may be quantified by solving Fick's equation if the solid phase of the immobile domain consists of well-defined aggregates [cf. van Genuchten and Dalton, 1986; Rappoldt, 1990] or, more commonly, by assuming a linear first-order exchange process [Lapidus and Amundson, 1952; Coats and Smith, 1964; van Genuchten and Wierenga, 1976]. The solution can be readily extended for additional immobile domains [Veling, 2002]. Bellin et al. [1991] demonstrated that during nonequilibrium transport, the breakthrough curve (BTC) resulting from a Dirac input may have a double peak due to the separate processes of advective transport in the mobile domain and solute transfer between the mobile and immobile domains.

[4] Double peaks are more commonly associated with the concentration distribution versus time or position for transport in media with flow in both pore domains. Less attention has been paid to the analytical solution for transport in dual-velocity or dual-permeability media than for media with flow in just the mobile domain. The former represents a more general scenario. For example, there will be faster water flow in interaggregate pore space or fractures of a porous medium and slower flow in the intraaggregate pore space. The transport mechanism for solutes will be advection and dispersion in the two pore domains and diffusive transfer between the domains. A similar problem is solute transport in a stratified medium with flow parallel to the stratification and the two pore domains are coupled through dispersive solute transfer between the strata. Dykhuizen [1991] resorted to time moments to quantify transport in dual-velocity media noting that solutions for coupled equations and arbitrary boundary and initial conditions are difficult to obtain. In particular, the behavior at larger times could be conveniently approximated with asymptotic solutions. However, the procedure to decouple linear ordinary differential equations is well known in the mathematical literature and can be applied to systems with an arbitrary number of equations [Zwillinger, 1989; Zill and Cullen, 2006]. General solutions were obtained for diffusion in media with two porosities in companion papers by Aifantis and Hill [1980] and Hill and Aifantis [1980]. The approach was extended by Hill and McNabb [1989] for an arbitrary differential operator. The objective of this study is to derive explicit, nontrivial solutions for the solute concentration as a function of time and position for both domains including time moments of the BTC. The solutions will be applied to describe breakthrough curves obtained for solute transport in an aggregated Andisol soil.

2. Mathematical Formulation

2.1. Transport Problem

[5] We consider solute transport during one-dimensional flow in a porous medium with two distinct permeability domains, 1 and 2. A first-order equation is used to approximate the solute exchange between the two domains. The governing equation for transport of a nonreactive solute in either one of the two domains is described with the following dual-advection dispersion equation (DADE):

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where t is time (T), x is distance (L), Ci is the solute concentration expressed per aqueous volume of domain i (M L−3), D is the longitudinal dispersion coefficient (L2 T−1), v is the pore water velocity (L T−1), α is an effective first-order rate coefficient for solute exchange between the aqueous domains (T−1), and θi is the volumetric water content given as the volume of water in domain i per bulk volume (L3 L−3). Purely for mathematical reasons, domains 1 and 2 are designated based on the criterion v1 > v2. Note that θ1 + θ2 = θ is the volumetric water content for the entire medium and θ is equal to the porosity for saturated conditions. A Damköhler number, Dai =αL/qi, may be used to quantify the rate of solute transfer relative to the rate of advective transport at a position x = L, while the inverse of α characterizes the time for mass transfer between the two domains. Two limiting cases of the DADE are the conventional advection-dispersion equation (ADE) and the mobile-immobile model (MIM). For the ADE there is only one domain with α → ∞ whereas for the MIM there is mobile domain 1 with water content θ1 and an immobile domain 2 with water content θ2, where there is no flow (v2 = D2 = 0) and a finite value for α.

[6] The equations for transport in domains 1 and 2 according to equation (1) are coupled because they each contain concentrations for domains 1 and 2 as dependent variables. For mathematical convenience we assume that the dispersion coefficients may be written as Di = κvi with κ as dispersivity (L). This assumption ignores the contribution of molecular diffusion and it assumes that both domains have the same dispersivity to allow the equations to be decoupled [cf. Esfandiari, 2008]. The DADE can now be written as

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with qi = θivi as the Darcy velocity in the i-th domain (L T−1) and the total flow rate given by q = θ1v1 + θ2v2. Figure 1 shows a schematic of transport in dual-permeability media with values for both the Darcy flux and water content that will be used in subsequent examples. Equilibrium partitioning between the aqueous and solid phase can be readily included by multiplying θi with a retardation factor Ri. The mathematical conditions for which the problem will be solved are,

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Note that the first-type inlet condition is applicable to flux-averaged or flowing concentrations. A third-type condition needs to be employed for volume-averaged or resident concentrations. The difference between these concentrations is negligible for a column Péclet number greater than 20 [van Genuchten and Alves, 1982].

Figure 1.

Schematic of the dual-advection dispersion model.

2.2. Solution

[7] The mathematical problem is linear, but both of the two governing equations contain the two dependent variables. Appendix A provides details on how the equations are uncoupled to obtain a nontrivial solution. The Laplace transform is applied to express the problem in matrix form. Subsequently, matrix diagonalization is used to decouple the system of equations. Appendix B briefly reviews the steps pertaining to the solution of the uncoupled system and the Laplace inversion. The following solutions were obtained for the DADE:

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with the following auxiliary variables to represent the difference in flow rate:

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and additional normalized variable for solute transfer:

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Finally, J0 and J1 are the zero- and first-order ordinary Bessel functions of the first kind.

[8] The solutions for both domains are very similar except for the second integral term on the right-hand side of equations (6) and (7). The first term, the double integral, quantifies equilibrium transport; it will vanish for α → ∞. This term constitutes the limiting case of equilibrium transport when the DADE effectively becomes the ADE (v1v2, α → ∞). The second term depends on the difference in pore water velocity for both domains, it will vanish if v1v2. This term is more pertinent for nonequilibrium (α → 0) than for equilibrium (α → ∞) conditions. Finally, the third term is prominent for nonequilibrium conditions and it will vanish if α → ∞. The last two terms yield similar results if θ2 ≠ 0, v2 = 0, and α → 0, and the sum of these two DADE-terms equals the solution according to the MIM.

[9] For effluent samples or other flux-averaged concentrations with flowing fluid sampled across both flow domains, the corresponding effluent concentration is defined as,

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In a similar manner, volume-averaged or resident concentrations may be defined using water content instead of Darcy flux for concentrations obtained for a third-type inlet condition:

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3. Solute Pulse Application

[10] Solute transport will be examined for a pulse input with duration to, i.e.,

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Solutions for a pulse instead of a step input can be readily obtained from equations (6) and (7) using the superposition principle [Toride et al., 1993].

3.1. Moments

[11] Temporal moments of the concentration in the i-th domain at position x are defined as,

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These can be most easily obtained from the solution in the Laplace domain [Leij and van Genuchten, 2002]:

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Moments of the BTC for the DADE were obtained by using a solution in the Laplace domain similar to equation (B4). Time moments of the effluent breakthrough inline image follow directly from the moments for the two domains by weighting with the relative flux-according to equation (11):

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[12] Table 1 lists the results for the first three moments of the concentrations in the two domains. The notation is simplified by dropping the first subscript of m, which denotes the domain. Also included are the moments for the limiting cases of equilibrium transport (ADE), which are obtained by setting θ1 = θ2 = θ/2, q1 = q2 = q/2, and α → ∞, and moments for the MIM by setting q1 = q, and q2 = 0. It is well known that the zeroth moment reflects mass conservation. For the ADE, MIM, and DADE the zeroth moment m0 is in all cases equal to the total applied mass Coto. Because m0 is the same for both domains in the DADE, the net mass exchange between the two domains must be zero. The first moment m1 is used to characterize mean breakthrough time. Note that m1 is the same for the ADE and MIM but for the DADE there is an additional term. This term accounts for the different velocity in the two domains and the transfer rate. Finally, the expression for the second moment m2, which is used to characterize solute spreading, includes a term for a position-dependent dispersion for all three models. For the MIM and the DADE there are additional terms. For the MIM there is dispersion that depends on the rate parameter α and the immobile water content θ2. The additional terms in the expression for m2 are much lengthier for the DADE; they incorporate the rate parameter, as well as water contents and Darcy velocities for both domains. Increased residence times, as indicated by a lower pore water velocity, enhance spreading due to hydrodynamic dispersion or nonequilibrium solute transfer. Differences in the second-order moment will be explored graphically.

Table 1. Time Moments for Breakthrough Curves According to the Advection-Dispersion Equation (CADE), Mobile-Immobile Model (CMIM), and Dual-Advection Dispersion Equation (C1 and C2) at an Arbitrary Position for a Pulse of Duration toa
  • a

    inline image.

CADE inline image inline image inline image
CMIM inline image inline image inline image
C1 inline image inline image inline image
C2 inline image inline image inline image

[13] Moments are normalized with respect to the zero-order moment:

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The normalized first-order moment Mi1 quantifies the mean of the BTC. The variance of the BTC is given by:

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The differentiation of the variance with position allows the definition of an “equivalent” dispersivity:

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[14] First, the behavior of breakthrough time and dispersion for the DADE are illustrated by selecting a particular set of values for x, to, q2, θ1, θ2, and α while varying q1. The ratio of the normalized first-order moment for domains 1 and 2 (M11/M21) as well as the ratio of the variance (σ1/σ2) are plotted as a function of the difference in velocity for the two domains normalized by the average velocity. The value for α was either 0.05 or 0.5 d−1, which corresponds to a Damköhler number for the second domain Da2 of 1.25 or 12.5. Note that the residence time and the variance are always less for domain 1, which has a greater pore water velocity but not necessarily the greater Darcy flux.

[15] The ratio of M1 for the (faster) domain 1 with respect to that for domain 2 is shown for lower (Figure 2a) and greater mass transfer (Figure 2b). This ratio is equal to 1 for v1 = v2, with 2(v1v2)/(v1 + v2) = 0, and transport may be described by the ADE, and becomes zero if v2 = 0 when there is no transport in domain 2, i.e., 2(v1v2)/(v1 + v2) = 2 and transport follows the MIM. For limitations in transfer between domains 1 and 2 (Figure 2a), the residence time and variance remain more similar for both domains for a lower fraction θ1 of the faster domain. The impact of water content on the differences in solute movement and spreading between domains becomes less pronounced for greater α (Figure 2b).

Figure 2.

Ratio of time moments for domains 1 and 2 as a function of 2(v1v2)/(v1 + v2) with q2 = 1 cm d−1, x = 25 cm, κ = 2.5 cm, to = 0.5 d, and θ = 0.5 for three combinations of θ1 and θ2: (a) normalized first-order moment ratio M11/M21 for α = 0.05 d−1, (b) normalized first-order moment ratio M11/M21 for α = 0.5 d−1, (c) variance ratio σ12/σ22 for α = 0.05 d−1, and (d) variance ratio σ12/σ22 for α = 0.5 d−1.

[16] The variance for the fast domain with respect to that of the slow domain behaves in a similar manner as the ratio of the first normalized moments. There will typically be less solute spreading, due to limited transfer between the domains and dispersion, in the faster domain 1 than in domain 2. However, if the faster domain is relatively small (θ1 = 0.05), the variance may be equal to or even greater in the fast domain. The effect of water on the difference in solute spreading between the two domains is more pronounced for smaller α (Figure 2c) because of limited opportunity for solute exchange to smooth out the difference.

[17] Second, the variance σ2 of a hypothetical BTC is plotted as a function of distance in Figure 3 for the DADE and for the limiting cases of the ADE and MIM. Two different combinations of Darcy fluxes are used for domains 1 and 2 for the DADE while the same Darcy flux of 15 cm d−1 is used for the ADE and MIM. The value for α is either 0.05 or 0.5 d−1 in the DADE and MIM. The variance is obtained according to equation (18). For the DADE the flux-weighted variances for the two domains are summed to obtain σ2 for the effluent (compare equation (16)). In all cases the variance increases linearly with position and an estimate for the effective dispersivity, which is estimated according to equation (19) from the slope between x = 15 and 25 cm, is included in the legend of all graphs. The variance for the ADE shows the same dependency on distance for all four cases; it provides the lower limit for the DADE when there is no difference in velocity between the domains and the variance is entirely due to hydrodynamic dispersion. The variance for the MIM constitutes an upper limit for the DADE when there is a maximum difference in velocity between domains, i.e., all water flow occurs in mobile domain 1. The variance for the MIM is greater for α = 0.05 d−1 (see Figures 3a and 3c). Even for α = 0.5 d−1, most of the variance is due to physical nonequilibrium rather than hydrodynamic dispersion judging by the dispersivity values for the ADE and MIM in Figure 3b or 3d. The variance for the DADE is similar to that of the ADE if there is substantial flow in both domains (Figures 3a and 3b). If there is negligible flow in domain 2, the variance for the DADE approaches that of the MIM, especially for higher α (Figure 3d). Because of the minor contribution of hydrodynamic dispersion to the variance when the velocity in one domain is very small compared to the other, the assumption of using the same dispersivity for both domains appears reasonable. If there is an appreciable velocity in both domains, the small impact of dispersion compared to advection on solute transport suggests that differentiating between the velocity is more important than between the dispersivity for the two domains.

Figure 3.

Variance σ2 as a function of distance for the equilibrium ADE, and the physical nonequilibrium MIM and DADE with θ1 = θ2 = 0.25, κ = 2.5 cm, to = 0.5 d: (a) q1 = 10 cm d−1, q2 = 5 cm d−1, and α = 0.05 d−1, (b) q1 = 14.9 cm d−1, q2 = 0.1 cm d−1, and α = 0.05 d−1, (c) q1 = 10 cm d−1, q2 = 5 cm d−1, and α = 0.5 d−1, and (d) q1 = 14.9 cm d−1, q2 = 0.1 cm d−1, and α = 0.5 d−1. Values for effective dispersivity κeff are given in parentheses.

3.2. Effect of Transfer Parameter and Velocity

[18] The solutions will be used to illustrate the impact of the following transport parameters on solute movement within each domain and in the bulk soil: transfer parameter α and pore water velocities v1 and v2. Consider a high-permeability domain 1, which has 20% of the pore space with v1 = 100 cm d−1 and θ1 = 0.1 and a lower permeability domain 2 with v2 = 12.5 cm d−1 associated with the remaining 80% of the pore domain (see Figure 1). The transfer parameter α assumes values of 0.05, 0.5, and 5 d−1 with the following Damköhler numbers at x = 25 cm: Da1 = 0.125 and Da2 = 0.0625, Da1 = 1.25 and Da2 = 0.625, and Da1 = 12.5 and Da2 = 6.25, respectively. Figure 4 shows normalized concentration profiles for the two domains as well as the volume-averaged value after 0.3 d as the result of a 0.1-d pulse input with dispersivity κ = 2.5 cm. Because of the limited solute exchange between the pore domains, the solute pulse moves fairly independently through the two pore domains if there is minimal exchange between the two domains (α = 0.05 d−1). The combined volume-averaged profile Cv(x) has two distinct peaks, which could have been described as the weighted sum of two different solutions of the simple ADE for uniform soils (α → 0). For increased exchange (α = 0.5 d−1), the concentration curves for the higher-velocity domain 1 is skewed to the left, i.e., the outgoing solute front, while for the lower-velocity domain 2 the skew occurs for the incoming solute. Domain 1 contains an appreciable amount of solute near the inlet (x < 15 cm) that originates from domain 2. On the other hand, there is already solute in the lower permeability layer at t = 0.3 d beyond 25 cm that could not have arrived solely by advection in domain 2. The two peaks in the volume-averaged concentration profile become less distinct. Finally, there is considerable solute transfer between the two domains for α = 5 d−1 that allows for only slight differences in concentration between the two domains. Because of the rapid exchange between the two domains, each solute particle experiences the same apparent velocity. Hence, the three concentration profiles are very similar.

Figure 4.

Concentration profiles C1, C2, and Cv at t = 0.3 d resulting from a pulse input (to = 0.1 d) with q1 = 10 cm d−1 and θ1 = 0.1 (v1 = 100 cm d−1), q2 = 5 cm d−1 and θ2 = 0.4 (v2 = 12.5 cm d−1), and dispersivity κ = 2.5 cm for three values of the rate parameter: (a) α = 0.05 d−1, (b) α = 0.5 d−1, and (c) α = 5 d−1.

[19] A comparable trend can be observed for the BTCs as a result of different α (Figure 5). The same parameters were used as for Figure 4 except for a longer duration of the solute pulse (to = 0.5 d), while BTCs are plotted at a distance of 25 cm. For limited transfer (α = 0.05 d−1 or Da2 = 0.25), there is early breakthrough for domain 1 and later breakthrough for domain 2. Because the residence time for solute in domain 1 is relatively short, the shape of its BTC is similar to the Heaviside input. The flux-averaged effluent curve has two distinct peaks associated with the faster and the slower domains. For a value of α = 0.5 d−1 (Da2 = 2.5) it appears that both BTCs are distinct, but more of the solute applied to one domain ends up in the effluent for the other domain. The breakthrough curves for both domains are virtually identical for α = 5 d−1 (Da2 = 25) due to rapid solute exchange in view of the distance of x = 25 cm.

Figure 5.

Breakthrough curves for C1, C2, and Ce at x = 25 cm resulting from a pulse input (to = 0.5 d) with q1 = 10 cm d−1 and θ1 = 0.1 (v1 = 100 cm d−1), q2 = 5 cm d−1 and θ2 = 0.4 (v2 = 12.5 cm d−1), and dispersivity κ = 2.5 cm for three values of the rate parameter: (a) α = 0.05 d−1, (b) α = 0.5 d−1, and (c) α = 5 d−1.

[20] Second, the impact of the pore water velocity v is examined. The value for v in a domain is the ratio of the Darcy flux and the volumetric water content. The Darcy flux dictates the amount of water and dissolved solute that enters, moves through, and exits the domain. It is determined by the permeability and the bulk area of that domain with respect to the total bulk area since Darcy fluxes, and water contents, are expressed with respect to the total bulk area. The speed at which solute moves through the aqueous phase also depends on its volumetric water content. Figure 6 exemplifies the role of Darcy flux and water content in solute breakthrough for a total Darcy flux q = 15 cm d−1 and a total water content θ = 0.5. If the majority of the solute transport (q1 = 10 cm d−1) occurs in the fast domain 1 (θ1 = 0.05), there will be two distinct peaks with a rectangular short peak for the high-velocity domain 1 and a smoothed out peak for the low-velocity domain 2 (Figure 6a). Figure 6b deals with the case where the Darcy fluxes remain as before (q1 = 10 cm d−1, q2 = 5 cm d−1), but the water contents are now the same (θ1 = θ2 = 0.25). The peak for domain 1 appears later and is smoother compared to that in Figure 6a while the opposite is true for domain 2. Finally, for Figure 6c the majority of flow and transport occurs in domain 2 (q1 = 5 cm d−1, q2 = 10 cm d−1) with the solute moving faster through domain 1 than 2 (θ1 = 0.05, θ2 = 0.45). The peak for domain 1 appears a little later, but is otherwise similar to that in Figure 5a because of the relatively small residence times. Conversely, the peak for domain 2 resembles that of Figure 6b. Compared to Figure 6b, the increase in θ2 from 0.25 to 0.45 is approximately offset by an increase in Darcy flux from 5 to 10 cm d−1.

Figure 6.

Breakthrough curves for C1 and C2 and the flux-averaged Ce at x = 25 cm as result of a pulse input (to = 0.5 d) with dispersivity κ = 1 cm and transfer parameter α = 0.05 d−1 and: (a) q1 = 10 cm d−1, θ1 = 0.05 (v1 = 200 cm d−1) and q2 = 5 cm d−1, θ2 = 0.45 (v2 = 11.1 cm d−1); (b) q1 = 10 cm d−1, θ1 = 0.25 (v1 = 40 cm d−1) and q2 = 5 cm d−1, θ2 = 0.25 (v2 = 20 cm d−1); and (c) q1 = 5 cm d−1, θ1 = 0.05 (v1 = 100 cm d−1) and q2 = 10 cm d−1, θ2 = 0.45 (v2 = 22.2 cm d−1).

4. Application to Transport in an Andisol

[21] Andisols are generally developed from volcanic ash consisting of noncrystalline materials such as allophone, imogolite, Al-humus complexes, and ferrihydrite. Andisols comprise 17% of the land in Japan and they are widely used for agriculture. Water flow and solute transport in such soils are of prime interest due to the unique physical and chemical properties of Andisols. The soils are characterized by a low bulk density with a well-developed aggregate structure made up of noncrystalline minerals (Figure 7). There is a distinct interaggregate and intra-aggregate pore space. Solute displacement experiments were conducted for an Andisol from an upland field at the National Institute of Vegetable and Tea Science in Mie, Japan. The Andisol is assumed to have nonvolcanic origin. Figure 8 shows the soil water retention curve (measured with a hanging water column, pressure plate extractors, and dew point meters) for the soil used in the displacement experiment. The bimodal behavior of the pore space can clearly be seen from the two curves [Miyamoto et al., 2003].

Figure 7.

Volcanic ash soil (Andisol) from Mie, Japan.

Figure 8.

Water retention curve, shown as the volumetric water content as a function of soil water pressure head, for an Andisol from Mie, Japan, obtained with hanging water columns, pressure plate extractors, and dew point meters.

[22] The Andisol was packed in a 26-cm-long, 4.4-cm-diameter soil column with a bulk density of 0.85 g cm−3. Four-probe sensors were placed horizontally along the column to monitor the electrical conductivity [Toride et al., 2003]. A steady state saturated flow rate of 1397.5 cm d−1 was established using an aqueous solution with a 0.06 M CaCl2 concentration using a peristaltic pump. After achieving steady flow, the influent concentration was changed from 0.06 M to 0.09 M to create a step input. The experimental data points for BTCs observed at four depths are shown in Figure 9.

Figure 9.

Observed breakthrough curves at positions of 7.7, 10.0, 14.1, and 16.9 cm for Andisol from Mie, Japan, described with the advection-dispersion equation (ADE), mobile-immobile model (MIM), and dual-advection-dispersion equation (DADE).

[23] At the greater depth of 16.9 cm it appears that the BTC has three inflection points and that there is bimodal transport. The experimental data were described with an analytical solution for the ADE, MIM, and the DADE using the CXTFIT optimization routine [Toride et al., 1995]. It is assumed that the solute is nonreactive and that the first-type inlet condition can be applied. The solutions for the ADE and MIM have been reported, among others, by Leij and Sciortino [2012]. For the MIM and the DADE the total concentration was obtained according to equation (12). The optimized curves are also shown in Figure 9. Optimized transport parameters can be found in Table 2 for the four depths and for simultaneously optimizing BTCs by pooling data for all four depths. Also included are the coefficient of determination r2 and the mean square of error (MSE), the ratio of the sum of squared residuals, and the degrees of freedom.

Table 2. Optimization Results for Solute Displacement Experiment
x (cm)7.729.9614.0516.87Pooled
  • a

    ADE: advection-dispersion equation.

  • b

    MIM: mobile-immobile model.

  • c

    DADE: dual-advection dispersion equation.

v (cm min−1)0.9100.9350.9790.8940.925
κ (cm)0.9951.5992.2542.3151.835
MSE × 1042.386.9013.134.720.5
 MIMbm = 0.14, θim = 0.44)
v (cm min−1)0.9591.0861.1391.0171.026
κ (cm)0.7070.3730.4890.6160.884
α (min−1)1.0870.2730.1970.1780.318
MSE × 1041.782.274.8219.314.7
 DADEc1 = 0.14, θ2 = 0.44)
v1 (cm min−1)2.2053.1823.7973.3952.973
v2 (cm min−1)0.5510.5220.4760.5240.610
κ (cm)0.6820.5870.3730.4060.728
α (min−1)0.0470.0290.0340.0120.004
MSE × 1041.690.7240.4381.126.21

[24] Two optimization parameters were used for the ADE: the pore water velocity v and the dispersivity κ. From Table 2 and Figure 9 it can be seen that the description of the data becomes poorer for greater depths while the dispersivity tends to increase with x. The description of the pooled data set is better than that of the BTC at the greatest depth x = 16.9 cm. For both the MIM and DADE the volumetric water contents of domains 1 and 2 were set to 0.14 and 0.44, respectively. These could be optimized, but to minimize the number of fitting parameters they were inferred from the retention curve shown in Figure 8. The following parameters for the MIM were optimized: pore water velocity v, dispersivity κ, and transfer coefficient α. Domains 1 and 2 are typically referred to as the mobile (m) and immobile (im) region with θmvm = θv. The optimization is more accurate than for the ADE. The optimization parameters for the DADE were v1, v2, α, and κ. A considerably better fit of the data is obtained with the DADE than with the ADE and the MIM. Noteworthy is the improved description of the bimodal BTCs observed at the greater depths. Solute transport in the two domains appears to be fairly independent given the small values for the transfer parameter α. Optimized parameters do fluctuate with depth, especially the velocity in the faster pore domain 1. On the other hand, the overall Darcy flux remains fairly constant with depth and the results for the pooled BTCs are considerably more accurate for the DADE than for the other two models.

[25] The pooled variables for the DADE, obtained by simultaneously optimizing BTCs for four depths, were used to predict effluent concentrations from which advective solute fluxes, q1C1 and q2C2 for domains 1 and 2 as well as the sum qCe, were obtained at the four positions (Figure 10). To illustrate the bimodal behavior, the scenario involves the application of a step pulse of duration to = 2 min. The DADE clearly contrasts the solute flux in the faster domains 1 and 2. Especially for the smaller depths, there is high solute flux for a short period in domain 1, whereas the flux is smaller over a longer period for domain 2. The total flux exhibits bimodal behavior more clearly and would be apparent from the experimental BTC for a step input.

Figure 10.

Solute fluxes at positions of 7.7, 10.0, 14.1, and 16.9 cm for to = 2 min predicted with the dual-advection-dispersion equation (DADE) using parameters obtained by simultaneously optimizing BTCs for four depths (pooled variables in Table 2).

5. Concluding Remarks

[26] A mathematical model for solute transport in a dual-permeability medium was presented and solved using Laplace transformation and matrix decomposition as outlined in Appendices A and B. For mathematical convenience the dispersivity was assumed to be the same for both pore domains. This assumption appears reasonable in light of the relatively small contribution of hydrodynamic dispersion to solute spreading when there is a pronounced difference in velocity between the two domains. The analytical solution for the dual-advection dispersion equation (DADE) is slightly lengthier but not more complicated than that for the well-know mobile-immobile model (MIM). Time moments were derived for the DADE, and these were presented in the context of the limiting cases of the advection-dispersion equation (ADE) with just one pore domain and the MIM where there are two pore domains with the greatest contrast in flow.

[27] The analytical solutions and the moments were used to illustrate transport resulting from the application of solute pulse to a dual-velocity medium for different combinations of Darcy flux, water content, and rate parameter. Of particular interest is the difference in travel time for the two pore domains, which leads to double peaks in the volume- and flux-averaged concentration versus position and time, respectively. The analytical solution may be used for similar sensitivity analyses or to verify numerical simulations for simple scenarios. Furthermore, the solution was applied to describe BTCs for a Japanese Andisol with a distinct intra- and interaggregate porosity. The DADE was able to capture most of the bimodal behavior of the breakthrough curve at the greater depth.

Appendix A:: Matrix Decomposition

[28] The following Laplace transform with respect to time (with s as transformation variable)

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is applied to the problem defined by (2) through (5) to obtain

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subject to

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display math

[29] The system of equations given by (A2) may be written in matrix form. Because of the constant dispersivity and the symmetry of the coefficient matrices, it is straightforward to diagonalize the system [Esfandiari, 2008; Kreyszig, 2006; Zwillinger, 1989]. After dividing by qi, the two equations given by (A2) may be represented in matrix form as,

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where the operator L is defined by

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while the matrix A and the transformed concentration vector C are given by:

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The modal matrix to diagonalize equation (a5) is obtained by solving the eigenvalue problem

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where λ denotes one of two possible eigenvalues with corresponding eigenvector v, and I is the identity matrix. Expressions for eigenvalues and eigenvectors were found with the symbolic mathematical software package Mathematica. The results are,

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display math

The variables inline image were already defined by equation (9) while inline image and r are given by:

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display math

[30] The modal matrix P is defined from the eigenvectors {v1, v2} as,

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The modal matrix has the following useful property that allows diagonalization [cf. Esfandiari, 2008]:

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The concentration vector is transformed by using the following equality:

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where Y is the vector with the new dependent variable. Substituting this expression into equation (A5), premultiplying with P−1, and applying (A14) yields:

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Using the definition of the operator given by equation (A6) and applying the same transformation (A15) to the boundary conditions (A3) and (A4) finally gives the decoupled system:

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subject to

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display math

Appendix B:: Solution of Dependent Variable

[31] Equation (A17) is a second-order ordinary differential equation with constant coefficients for which we may readily write down the following general solution [Kreyszig, 2006; Zill and Cullen, 2006]:

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The outlet condition suggests that Ai = 0 and from the inlet condition we find:

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The regular concentration, although still in the Laplace domain, follows from (A15a):

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After substituting the solutions for Yi and the expressions for the eigenvalues λi, the solutions in the Laplace domain may be obtained. These are very similar for domains 1 and 2; only the inversion procedure for domain 1 will therefore be reviewed. The solution in the Laplace domain may be given by:

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with auxiliary variables a, b, and r defined by (9), (A11), and (A12). This solution was used to obtain the moments of the breakthrough curve according to equation (15).

[32] The inverse Laplace transform is carried out by using the shifting and convolution theorems. This leads to the expression

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where the subscript of the Laplace operator inline image denotes the (dummy) temporal variable to which inversion takes place. The following transformation pairs are used [Abramowitz and Stegun, 1970; Polyanin and Manzhirov, 1998]:

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display math

where H is the unit-step or Heaviside function. Differentiating (B7) with respect to k gives the inversion pair:

display math

with δ as the Dirac delta function. The above identities may also be used for inversions with k < 0. In case of complex arguments of the Bessel functions, modified Bessel functions are used based upon the equalities [Abramowitz and Stegun, 1970]:

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with I0 and I1 as zero- and first-order modified Bessel function of the first kind. Substituting the expressions for the inverse Laplace transform in (B5) results in

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and all other variables are as previously defined. The final solution given by (6) is obtained by integrating the product of an arbitrary function f and the Heaviside or Dirac delta functions according to:

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display math

The solution for the second domain is obtained in an identical manner.


[33] The views expressed in this paper are solely those of the authors and do not necessarily reflect the views or policies of the U.S. Environmental Protection Agency. The authors would like to thank the three reviewers and the associate editor for providing constructive comments and suggestions that allowed improvement of an earlier version of the manuscript.