Published online 20 September 2006
Published in Soil Sci Soc Am J 70:1843-1850 (2006)
DOI: 10.2136/sssaj2005.0166
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
Soil Physics
Specifying Scale-dependent Dispersivity in Numerical Solutions of the Convection–Dispersion Equation
Huaguo Wanga,
Naraine Persaudb,* and
Xiaobo Zhouc
a 2169 McCarty Hall, Soil and Water Science Dep., Univ. of Florida, Gainesville, FL 32611
b Dep. of Crop and Soil Environmental Sciences, Virginia Tech, Blacksburg, VA 24061-0404
c Crop and Soil Sciences Dep., The Pennsylvania State Univ., University Park, PA 16802-3504
* Corresponding author (npers{at}vt.edu)
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ABSTRACT
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Many studies have incorporated various scale-dependent dispersivity functions into the convection–dispersion equation (CDE). For a given function, there are different ways of specifying the dispersivity values in the discretized subdomains of a numerical solution. These ways would depend on how the function's time or space variable is built into the numerical scheme and on how the different ways may be interrelated. No study has addressed how these different ways may affect resulting breakthrough curves (BTCs) or their applicability to solute transport problems. In this study, five ways were specified and designated as local time, average time, apparent time, local distance, or apparent distance dependent dispersivity in the numerical scheme. The main objective was to demonstrate relationships among these ways by assuming that dispersion in the scale-dependent CDE was quasi-Fickian and implicitly based on an estimation of the first moment of the BTC at the given location. How these ways affected calculated BTCs was numerically tested using generalized linear and power scale–dependent dispersivity functions for a one-dimensional problem with initial solute distribution represented as a Dirac delta function. For either type of function, differences were obvious in BTCs obtained for solute transport conditions with small apparent Peclet numbers using the alternative ways of specifying scale-dependent dispersivity in the numerical scheme. The differences decreased with increasing apparent Peclet number. The numerical test results also showed that the derived relationships were not applicable for numerically calculating the spatial solute distribution at a given time. These test results were expected to be generally applicable because any other initial-value solute transport problem can be solved as a superposition of this test problem.
Abbreviations: 1-D, one-dimensional • BTC, breakthrough curve • CDE, convection–dispersion equation • Pe, Peclet number
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INTRODUCTION
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DETERMINISTIC-MECHANISTIC MODELS based on the CDE are most often used for describing and predicting solute transport in porous media. For one-dimensional nonreactive solute transport under steady water flow conditions, the CDE is given as:
 | [1] |
where C is solute concentration (ML–3), D is the hydrodynamic dispersion coefficient (L2T–1), v is average pore velocity (LT–1), x is distance (L), and t is time (T).
The parameter D represents a quasidiffusion coefficient describing solute spreading caused by mechanical dispersion and molecular diffusion, and is defined as (see Leij and van Genuchten, 2000):
 | [2a] |
 | [2b] |
where Dm is the mechanical dispersion coefficient, D0 is the molecular diffusion coefficient (L2 T–1) in porous media, which is the product of molecular diffusion in bulk solution and tortuosity of the porous media, and 
is the dispersivity (L).
Many authors have reported that dispersivity is scale dependent rather than constant, both at the column scales (Han et al., 1985; Porro et al., 1993; Zhang, 1995), and at the field scale (Gelhar et al., 1992;Yasuda et al., 1994; Ellsworth et al., 1996). The scale dependence of the dispersivity over the solute transport domain was explained using fractal theory (Wheatcraft and Tyler, 1988), the fractional advection–dispersion equation (Pachepsky et al., 2002), or using stochastic methods (Gelhar and Axness, 1983; Dagan 1984; Schwarze et al., 2001).
Because most practically applied solute transport models are developed from deterministic models such as Eq. [1], many authors have tried to explicitly incorporate the scale-dependent dispersivity into the CDE. In these studies, the dispersivity in the CDE was no longer constant but was expressed as scale-dependent functions, either as time-dependent dispersivity (Pickens and Grisak, 1981; Basha and El-Habel, 1993; Zou et al., 1996), or as distance-dependent dispersivity (Mishra and Parker, 1990; Yates, 1990; Logan 1996; and Xu and Eckstein, 1995).
The CDE is most often solved numerically such as using finite element methods (e.g., Yeh, 2000), finite difference methods (e.g., Ma and Selim, 1994), or the particle tracking methods developed by Uffink (1985) and Delay et al. (1997). However, when the scale dependent dispersivity is used, the dispersivity value in the numerical scheme for a given scale-dependent function can be specified in many ways depending on how the independent variable in the function is built into the numerical space or time discretization. For example, Pickens and Grisak (1981) specified the scale-dependent dispersivity values as varying spatially from element to element, or as a constant value for the entire domain at a given time step. Mishra and Parker (1990) specified the distance-dependent dispersivity as a local dispersivity or effective dispersivity.
For a given solute transport problem and solute transport conditions, different ways of specifying scale-dependent dispersivity values when numerically solving the CDE may affect the resulting BTCs or spatial concentration distribution and the computational efficiency of the numerical scheme. No reports are available that have: (i) comprehensively detailed all the possible ways for specifying the scale-dependent dispersivity in numerical schemes for solving the CDE, (ii) theoretically clarified definitions and relationships among these various ways, and (iii) quantified how these ways affected the resulting BTCs or spatial concentration distribution, as well as their applicability to different solute transport problems.
The overall purpose of this paper was to address the foregoing three interrelated issues in specifying the scale-dependent dispersivity in numerical schemes. The objectives were to specify the various ways, demonstrate relationships among these ways that are implicitly based on an estimation of the first moment of the BTC at the given location, and assess how such estimation may cause error in the BTC prediction. Testing the effect on predicting BTCs at a given location was emphasized. The test was conducted using the numerical dispersion-free particle tracking method developed by Uffink (1985) and Delay et al. (1997) to generate BTCs for an initial value nonreactive solute transport problem with initial solute distribution represented as a Dirac delta function. This problem was chosen since any other nonreactive initial-value solute transport problem can be solved as a superposition of this test problem, and results would be applicable for any numerical space or time discretization scheme for calculating nonreactive solute BTCs using the CDE with scale-dependent dispersivity.
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SCALE-DEPENDENT DISPERSIVITY IN THE NUMERICAL SOLUTION OF THE CONVECTION DISPERSION EQUATION
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In a numerical solution (i.e., finite difference and finite element solution) of the CDE, the solute transport domain is discretized to subdomains in space, and time is discretized to time steps. Concentrations at these subdomains are sequentially solved at each time step under given initial and boundary conditions. Without loss of generality, the discretized subdomain indexed as i in space and n in time, can be denoted as cell (i, n). Therefore, dispersivity at cell (i, n) has to be specified for calculating the BTC at a given spatial location L. For a given scale-dependent function in the CDE, there may be different ways of specifying the dispersivity at cell (i, n) based on how the time or space variable in scale-dependent dispersivity functions is built into the numerical scheme and also on how these ways may be related with each other.
The following discussion of these ways and their relationships uses terminology that may or may not be related to the basic definitions of dispersion, such as local dispersion and macrodispersion, commonly accepted in the stochastic hydrology literature. It is concerned only with alternative ways of defining the independent variable of the given scale-dependent dispersivity function in the numerical scheme, and not with the correctness of incorporating the scale-dependent dispersivity in the CDE nor with reasons why dispersivity is scale-dependent.
Local Time-Dependent Dispersivity (t)
Even though the assumption that solute transport in porous media is Gaussian or quasi-Fickian was questioned by some authors such as Pachepsky et al. (2002), incorporation of the scale-dependent dispersivity into the CDE still implies that the dispersion is Gaussian or quasi-Fickian. Therefore, the mechanical dispersion coefficient is time dependent, and can be written as:
 | [3] |
where [
m(t)]2 is the variance of the spatial distribution of the solute molecules in the porous medium at time t.
When solute transport in porous media is under steady water flow conditions, the average pore water velocity, v, is constant. From Eq. [2b] Dm(t) = (t)v. Combining this with Eq. [3] gives:
 | [4] |
In the time domain, (t) in Eq. [4] is a local parameter, meaning that it represents the parameter used to quantify the change of the spatial variance of solute distribution in the porous medium only at time t. The corresponding numerical description should be
 | [5] |
Here, C(i, n) represents solute concentration in cell (i, n), that is, the concentration at x = i
x, and t = nt. x is the space length of a discretized cell, and t is the time length of a discretized time step. (•, n) is local time-dependent dispersivity used to specify that the dispersivity values are constant over the entire spatial domain at a given time, but changes with time. A BTC can be generated with this numerical scheme in one computation starting at time = 0.
Average Time-Dependent Dispersivity '(t)
For a given time t, [m(t)]2 is traditionally given by (e.g., Bear, 1988)
 | [6] |
Here '(t) represents a dispersivity parameter used to quantify the spatial variance of the solute distribution in the porous media at time t. This equation exists because from Eq. [1], solute distribution over space for a given time follows a normal distribution with mean displacement of vt and standard deviation of m. According to the Central Limit Theorem, such a normal distribution would exist even for an initial solute distribution that is not a Dirac delta function. Because the dispersivity is no longer a constant, the relationship between the spatial variance and dispersivity becomes time dependent. It is easy to estimate the dispersivity parameter in Eq. [6] from observations of the spatial distribution of the solute at a given time (Lapidus and Amundson, 1952).
Integrating Eq. [4], and then combining it with Eq. [6] gives:
 | [7] |
This result shows that '(t) in Eq. [7] represents a parameter that quantifies the average change of the spatial variance of solute distribution in porous media from time zero to time t. The corresponding numerical scheme becomes
 | [8] |
Because '(t) represents the averaged integral effect of local time-dependent dispersivity from time zero to time t, it is no longer a local time parameter. To generate a concentration point C(L, t) on a BTC, all the dispersivity values in the cells with time index j for which jt 
nt = t, would be set as '(t), in the calculation starting from t = 0. This way to specify the dispersivity in the numerical solution is termed as average time-dependent dispersivity, symbolized as '(•, t = 0 
n) in Eq. [8]. If another concentration point C(L, t') on this BTC needs to be generated, all the dispersivity values in the cells with time index j for which jt nt = t', would have to be reset as '(t'), and the calculation restarted from t = 0.
Using average time-dependent dispersivity makes the numerical scheme computationally inefficient because the time complexity of applying average time-dependent dispersivity '(t) to generate a BTC is O(t2). Consequently, it is impractical to numerically generate the BTC using the CDE with '(t) for large-scale solute transport problems or multidimensional solute transport problems.
Apparent Time-Dependent Dispersivity '(T') or '(T)
One challenge of using (t) in the numerical solution of the CDE is that this function cannot be directly obtained from analysis of an experimental solute BTC. The BTC represents concentration distribution over time at a given location, rather than the concentration distribution over space a given time. Therefore, if the time-dependent dispersivity values (t) can be directly related to the location at which the BTC is observed or generated, analysis of the BTC using the CDE with time-dependent dispersivity will be facilitated. A potential approach to defining this relationship is by defining the mean solute travel time T'as the expected value of the time distribution of the BTC at L. The mean solute travel time T' is given by the first moment of an observed BTC is defined as:
 | [9] |
where f(L, t) is a concentration distribution of the BTC observed at L.
Application of Eq. [9] implies a restriction that the concentration of the BTC and the concentration C(i, n) in the numerical scheme of Eq. [5] and Eq. [8] should have same definition, that is, both are flux concentration or both are resident concentration. This restriction holds for outlet boundary problems when C(i, n) is taken as flux concentration, or when C(i, n) is taken as resident concentration but a zero concentration gradient boundary condition is imposed at the location at which the BTC is generated. For initial value problems, the restriction holds when the BTC is measured as resident concentration. In any case, difference between the two concentrations is minor when solute transport is dominated by convection (Leij and van Genuchten, 2000; Jury and Roth, 1990).
Substituting the definition of T' into Eq.[7] for t = T' gives:
 | [10] |
Because T' is a function of L, '(T') is also a function of L for a BTC generated at L. The value of '(T') is defined to represent (t) or '(t) for generating the BTC at L. This means that the BTC generated by the CDE with a constant value of '(T') will be same as those generated by the CDE with time-dependent dispersivity (t) or '(t) for a velocity under the given initial and boundary conditions. The definition of the T' in Eq. [10] is dependent on f(L, t) which is the concentration distribution function of the BTC generated at L. Therefore '(T'), here termed as apparent time-dependent dispersivity, can be defined from the BTC at a given L. Therefore, it is dependent on the BTC at L, and when L is changed, the corresponding value of '(T') has to be changed.
When the objective of applying scale-dependent dispersivity is to predict the BTC at L, it is redundant and impractical to define a scale-dependent dispersivity, which is dependent on a BTC, to generate this BTC at L. To relate the time-dependent dispersivity to the distance for the practical purposes of prediction, a time defined as a function of L has to be independent on the BTC at L. When apparent time-dependent dispersivity is used for the purposes of prediction, an arbitrary time T has to be defined to represent the expected value of the time distribution of the BTC at L, and the corresponding apparent time-dependent dispersivity is denoted as '(T). In this case, Eq. [10] becomes:
 | [11] |
When the apparent time-dependent dispersivity is used, the numerical scheme using the CDE for generating BTCs at L becomes
 | [12] |
where 
= T', or = T. '(•, ) denotes that the values of the dispersivity are identical for all time steps regardless the value of n. In fact, the numerical solution of Eq. [12] is identical with the numerical solution developed for the CDE with a constant dispersivity. The scale-dependent effect is considered outside of the numerical scheme through calculating a value of '(T') or '(T) from Eq. [10] or Eq. [11].
Apparent Distance-Dependent Dispersivity (L)
As shown in Eq. [9], '(T') or '(T) can also be directly defined as a function of L. In this case, the dispersivity is termed as the apparent distance-dependent dispersivity, denoted as (L). It is straightforward to show that the (L) is related to '(T') [or '(T)] as:
 | [13a] |
or
 | [13b] |
The resulting numerical scheme will be identical as Eq. [12]. Therefore, (L) and '(T') [or '(T)] can all be used to represent apparent scale-dependent dispersivity.
The key implication of applying apparent scale-dependent dispersivity (L) and '(T') [or '(T)] in the numerical scheme of the scale-dependent CDE is that the dispersivity value within the observation scale L is assumed to be constant. When the location for calculating the BTC is given, the dispersivity values in all cells related to the calculation are set to a single value, which is a function only of the location where the BTC is calculated. It is therefore impossible to generate multiple BTCs at different locations in one computation.
Local Distance-Dependent Dispersivity D(x)
Substituting T' = L/v [or T = L/v] and t = x/v, where x is the physical spatial distance in the solute transport domain defined by x = vt, into Eq. [11] and using the relationships of Eq. [13] gives:
 | [14] |
In Eq. [14], (x/v) is an explicit function of distance rather than time. For a given time t, x is uniquely defined by x = vt. Consequently, (x/v) can be directly defined as a function of distance x. In this case, (x/v) is local distance-dependent dispersivity, and denoted as D(x). The relationship of the local time-dependent dispersivity (t) and the local distance-dependent dispersivity D(x) can be expressed as:
 | [15] |
For a BTC observed at L, Eq. [14] can be rewritten as:
 | [16] |
Therefore, the apparent distance-dependent dispersivity (L) represents the averaged integral effect of the local distance-dependent dispersivity D(x) on the BTC, which is observed at L. Equation [16] was also suggested by Mishra and Parker (1990).
As shown in Eq. [15], (t), (x/v), or D(x) is defined from Eq. [14] by using the relationships T' = L/v [or T = L/v] and t = x/v. However, (x/v) or D(x) cannot be defined from Eq. [14] when T' 
L/v [or T L/v]. Therefore, when an observed BTC is analyzed for determining D(x) and (L), Eq. [15] and Eq. [16] can be simultaneously correct if T = T' = L/v. Note that T = L/v when the dispersivity in the CDE is constant (Valocchi, 1985), however, when is scale dependent, such a relationship may not exist.
When the local distance-dependent dispersivity D(x) is applied, the CDE numerical scheme becomes:
 | [17] |
The notation D(i, •) in Eq. [17] is used to denote that the distance-dependent dispersivity values vary spatially, but it is unvarying with time for a given cell. The D(i, •) is specified as a function of ix. Once the spatial location of a cell is given, the dispersivity value in this cell is defined, regardless of where the BTC is calculated. It is therefore possible to calculate multiple BTCs at different locations in one computation.
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SPECIFIYING SCALE-DEPENDENT DISPERSIVITY FOR PREDICTING BREAKTHROUGH CURVES
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Prediction of the BTC using the scale-dependent CDE with apparent scale-dependent dispersivity (L) [or '(T)], and local distance-dependent dispersivity D(x) is implicitly based on an arbitrary time T = L/v, which is expected to be the first moment of the time distribution of the BTC at L. This is true for constant dispersivity (Valocchi, 1985). However, to date, there has been no report for analytically specifying moments of BTCs generated using the scale-dependent CDE. An alternative way for verifying the applicability of foregoing relationships is through numerical tests. BTCs can be generated at L using the numerical solution of the CDE with (t), (L) [or '(T)] and D(x) for a given solute transport problem under given solute transport conditions. If these BTCs were indistinguishable from each other, then these relationships would be applicable for predicting the BTCs at L for the given solute problem under the given solute transport conditions. Otherwise, some of these relationships (or maybe all of them) would be inapplicable. Therefore, the applicability of using (L) [or '(T)], and D(x), or the relationships between them, for a given solute transport problem under differing solute transport conditions needs to be tested.
Solute Transport Problem and Conditions
The solute transport problem used for testing was the CDE with the following auxiliary conditions:
 | [18] |
These conditions define an initially instantaneous and narrow-pulse solute input, which can be represented as a Dirac delta function, that is, 
(x), applied at zero time at an arbitrary location in the infinite domain. While it is generally impossible to specify an infinite boundary condition in a numerical solution, however, it can be approximated as a large enough finite domain. The infinite boundary was approximated as an imaginary outlet boundary at 50 times the effective column length (i.e., distance between pulse and outlet).
Because any nonreactive initial-value solute transport problem can be solved as a superposition of the solute transport problem described in Eq. [18], the test results were expected to be generally applicable to other initial-value solute transport problems. The numerical tests were carried by comparing the BTCs generated numerically using the CDE with (L) [or '(T)] and D(x), to those generated using the CDE with (t). Solute transport conditions were characterized by apparent Peclet numbers (Pe). The apparent Peclet number was defined as
 | [19] |
When the dispersivity is constant, Eq. [19] becomes Pe = L/ or Pe = Lv/D, which is commonly used for characterizing the solute transport conditions (van Genuchten and Alves, 1982). Pe < 50 for dispersion dominated transport and Pe > 100 when convection is dominant.
Scale-Dependent Dispersivity Functions
A power scale-dependent dispersivity function was used in the CDE for numerically generating the BTCs. The power function was proposed by Pickens and Grisak (1981), Wheatcraft and Tyler (1988), Neuman (1990), Basha and El-Habel (1993), and Zou et al, (1996) for describing the scale-dependent dispersivity. The power function (t) can be given as
 | [20a] |
where c and d are constant.
The functions '(T), D(x) and (L) become:
 | [20b] |
 | [20c] |
 | [20d] |
When d = 1 in Eq. [20], the power scale-dependent function becomes the linear function used by Pickens and Grisak (1981), Han et al. (1985), Yates (1990), Jury and Roth (1990, p. 41) and Basha and El-Habel (1993).
Numerical Solution
The one-dimensional particle tracking method (Uffink, 1985; Delay et al., 1997) was used to generate the BTCs using the CDE with the different scale-dependent dispersivity functions. There is no possibility of numerical dispersion when using the particle tracking method, in contrast to numerical approaches using finite differences or finite elements. In the particle tracking algorithm, all solute particles taken as uniformly present in the cell at location zero was used to represent the Dirac delta initial condition in our numerical solution. Because a Dirac delta function is defined as the limit of the normal distribution as the variance approaches zero (Jury and Roth, 1990), this representation of the initial condition is closely (albeit not exactly) Dirac delta. According to the Central Limit Theorem, the solute distribution for this representation will be the same as the solute distribution for the Dirac delta initial value problem after a few time steps. Also, using the analytical solution of the CDE with constant dispersivity, Wang and Persaud (2004) showed that for Pe < 200, BTCs are identical for initial solute distribution specified as Direct delta function at x = 0, or as a uniform distribution centered at x = 0 with width less than 5% of L.
The molecular diffusion coefficient in porous media, that is, D0 in Eq. [2a] was taken as zero in the numerical solution, assuming the negligible effect on solute transport compared to the mechanical dispersion. In all simulations the mass of solute applied was = 1 mmol, porosity = 0.33, cross sectional area = 33.5 cm2, and L = 120 cm. For the linear scale-dependent function, that is, for d = 1 in Eq. [20], the average pore water velocity v was = 1 cm min–1, c = 0.1, 0.04, 0.02, and 0.01, resulting in Pe = 20, 50, 100, and 200, respectively. For the power function, average pore water velocity v was = 1.5 cm min–1 and d = 1.2, c = 0.065, 0.0244, 0.013, and 0.0065, resulting Pe = 21, 56, 106, and 211, respectively.
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RESULTS AND DISCUSSION
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Two sets of BTCs were generated using the CDE with scale-dependent dispersivity. One set of BTCs were from the linear scale-dependent function and the other was from the power scale-dependent function. Because the response of the BTCs to increasing apparent Peclet number was similar for both sets, only the BTCs for the linear function is presented in Fig. 1
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Fig. 1. Breakthrough curves generated using the convection–dispersion equation with scale-dependent dispersivity functions based on the linear assumption.
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These results indicated that BTCs generated using the CDE with (L) [or '(T)] and D(x) were observably different to the BTC generated using the CDE with (t) for solute transport under conditions defined by apparent Pe = 20. For apparent Pe = 50, the BTC generated using the CDE with D(x) were indistinguishable from the BTC generated using the CDE with (t). In this case, the BTCs generated using the CDE with (L) [or '(T)] were different to the BTC generated using the CDE with (t). However, the error may be acceptable. For apparent Peclet numbers of 100 and 200, BTCs generated using the CDE with (L) [or '(T)] and D(x) were indistinguishable from the BTC generated using the CDE with (t). Solute transport is supposed to be dominated by convection when Peclet numbers equal 100 and 200 (van Genuchten and Alves, 1982).
These results indicated that when the apparent Peclet number is larger than 100, application of the dispersivity functions (L) [or '(T)] and D(x) to the numerical solution of the scale-dependent CDE, will not cause observable errors in the BTCs generated for the solute transport problem. Therefore, the arbitrarily defined T = L/v can be used to represent the expected value of the time distribution of the BTC at L for this problem when the Peclet number is larger than 100. Application of (L) [or '(T)] and D(x) will not cause an unacceptable error when the apparent Peclet number is larger than 50. However, when dispersion dominates the solute transport process and the apparent Peclet number is small (<20), the application of (L) [or '(T)] and D(x) will cause observable errors in the generated BTCs. Therefore, the arbitrarily defined T = L/v cannot be used to represent the expected value of the time distribution of the BTC at L when the apparent Peclet number is <20. The results also indicated that the accuracy of applying (L) [or '(T)] and D(x) is determined primarily by the solute transport conditions, rather than by the form (linear or power-law) of the scale-dependent function.
The spatial concentration distribution generated at time T/2 using the CDE with scale-dependent dispersivity functions based on the linear function are presented in Fig. 2
. As shown, the solute distribution curves over the spatial transport domain at t = T/2, generated using the CDE with (L) [or '(T)] are observably different from that generated using the CDE with (t) for all the given solute transport conditions. This result implied that (L) [or '(T)], which were developed to define the average integrated effect of (t) and D(x) on solute dispersion as observed in a BTC at L, cannot be used to analyze the spatial solute distribution at a given time. However, the spatial solute distribution curves generated using D(x) were not observably different from those generated using (t) for the solute transport under conditions specified by apparent Peclet numbers of 100 and 200. They are different for the solute transport under conditions prescribed by Peclet numbers of 20 and 50. Therefore, the CDE with dispersivity function D(x) may be applied to numerically generate the spatial solute distribution curves for the solute transport under conditions specified by high apparent Peclet numbers (>100). It cannot be used for lower apparent Peclet numbers (<50).
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Fig. 2. Concentration distributions over spatial domain at time T/2 generated using the convection–dispersion equation with scale-dependent dispersivity functions based on the linear assumption.
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To calculate the BTC, the functions (L), '(T), '(T'), and D(x) are defined from (t). The practical reasons and implications of defining these functions include:- (i) The BTC represents solute concentration distribution over time at a given location, and it is easy to use scale-dependent dispersivity, which is directly or indirectly defined as a function of distance.
- (ii) The function (t) may be easily identified from the field or laboratory observation of spatial concentration distribution at different times. However, when the observation is a BTC, the apparent distance-dependent dispersivity (L) may be most easily obtained by the analysis of this BTC. At a given L, the BTC generated using the CDE with (L) is identical to the BTC generated using the CDE with constant dispersivity. When the values of (L) at several locations are obtained by analyzing the BTCs at these locations, an explicit function for (L) can be obtained from these values through parameter fitting. Once the explicit function for (L) is obtained, (t) '(T') or D(x) can be obtained using the relationships between them as described above.
- (iii) (t) is difficult to apply when the solute input is not instantaneous. In this case, the convolution has to be used when the BTC is calculated using the CDE with local time-dependent dispersivity. However, calculating the convolution would be a computationally inefficient. Thus application of distance-dependent dispersivity can avoid calculating the convolution. However, the convolution has to be applied for calculating BTCs when (L), '(T), '(T'), and D(x) are used for simulating instantaneous multiple-source solute transport and therefore (t) should be used.
- (iv) Both (t) and D(x) permit calculating BTCs at different locations in one computation. On the other hand, (L), '(T), '(T') are the apparent parameters for a specific location. Therefore, only the BTC corresponding to the apparent scale-dependent dispersivity at this location can be generated in one computation.
In the scale-dependent CDE, the dispersion in porous media is still considered as Gaussian or quasi-Fickian. Hence, the scale-dependent dispersivity should be the local time-dependent dispersivity (t). The other scale-dependent dispersivities (L), '(T), '(T'), and D(x) are derived from (t) for practical purposes as discussed above. The numerical tests showed that BTCs generated from (L), '(T), '(T'), and D(x) are not identical with the BTCs generated from (t) when dispersion is dominant in solute transport (i.e., Pe < 20). Therefore, in this case only (t) should be used in the scale-dependent CDE for numerically calculating BTCs to avoid inaccuracy.
It should be pointed out that all the foregoing definitions and discussion were for one-dimensional solute transport. In field situations solute transport is multidimensional. Nevertheless, results from this research may prove useful for developing numerical solutions to the multidimensional scale-dependent CDE, since the quasi-Fickian assumption of dispersion is also applicable in this case. However, extrapolating the results to the multidimensional problem application would require further study.
Received for publication May 27, 2005.
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INTRODUCTION
