Ó Springer-Verlag 2004
Stochastic Environmental Resea (2004) 18: 22–30 DOI 10.1007/s00477-003-0161-5
ORIGINAL PAPER
B. X. Hu Æ J. Wu Æ D. Zhang
A numerical method of moments for solute transport in physically and chemically nonstationary formations: linear equilibrium sorption with random Kd
Abstract A Lagrangian perturbation method is applied to develop a method of moments for solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity stems from medium nonstationarity and internal and external boundaries of the study domain. The solute flux is described as a space-time process where time refers to the solute flux breakthrough through a control plane (CP) at some distance downstream of the solute source and space refers to the transverse displacement distribution at the CP. The analytically derived moment equations for solute transport in a nonstationarity flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. The developed method is applied to study the effects of heterogeneity and nonstationarity of the hydraulic conductivity and chemical sorption coefficient on solute transport. The study results indicate all these factors will significantly influence the mean and variance of solute flux. Keywords Groundwater Æ Stochastic method Æ Heterogeneity Æ Nonstationarity Æ Method of moment
B. X. Hu (&) Desert Research Institute, Division of Hydrologic Sciences, University and Community College System of Nevada, Las Vegas, Nevada, USA e-mail:
[email protected] J. Wu Nanjing University, Department of Earth Sciences, China D. Zhang Hydrology, Geochemistry and Geology Group, Earth and Environmental Sciences, Los Alamos National Laboratory, New Mexico, USA
1 Introduction It has been widely recognized that natural media are generally heterogeneous, which is shown in the dramatically spatial variations of their physical and chemical parameters, such as the hydraulic conductivity and chemical sorption coefficient. The heterogeneity of the media will significantly influence the flow and solute transport processes through them. Owing to spatial variability of the parameters in the natural media and the scarcity of the available field data, these parameters are generally treated as spatial random variables. Stochastic theories have been developed to study the effects of the random parameters on flow and transport processes. However, most stochastic transport theories (e.g., Dagan, 1982, 1984; Gelhar and Axness, 1983; Gelhar, 1986; Winter et al., 1984; Naff, 1990; Neuman and Zhang, 1990; Deng et al., 1993; Cushman, 1997) are based upon the following assumptions: (1) steady-state flow with no boundaries; (2) constant mean flow velocity in space; (3) stationary hydraulic conductive field; and (4) small variance of log hydraulic conductivity. Though not very realistic, these assumptions significantly simplify the calculations. However, these strict assumptions limit the application of the stochastic theories to solute transport in complicated natural media, such as the media with multi-scale heterogeneity and with internal and external boundaries, which poses a critical question for the development of stochastic theory ‘‘how can stochastic theory be applied to predict groundwater flow and solute transport for practical environmental projects?’’ Another issue related to most stochastic theories is the lack of study on prediction uncertainty. The ensemble mean concentration, or its spatial moments, is the simplest property of a concentration field resulting from the selected ensemble and indisputably represents the most important one. However, most research has focused on the mean concentration or its spatial moments alone. Probably the most unfortunate result is the
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fact that even today, many regulatory guidelines are still based on the mean calculation results alone, and very little attention has been paid to the importance of other statistical properties of the concentration field, primarily the variance about the mean. At the same time, uncertainty in estimating cleanup costs is associated with the lack of detailed knowledge related to the spatial and temporal distribution of contaminants, as well as the lack of knowledge on the effectiveness of the remediation. These difficulties highlight the need for the development of new modeling approaches for subsurface contaminant plume delineation. Recently, a numerical method of moment for solute transport in a nonstationary, heterogeneous medium has been developed (Zhang et al., 2000; Wu et al., 2003), where a Lagrangian perturbation method is applied to develop the framework for solute transport in a nonstationary flow field and a numerical method is implemented to obtain the solution. For this method, the first three assumptions listed above for most stochastic theories are not required. Even though the small variance of log-conductivity (smaller than 1) is still required in the theory development, the first-order method may be more relaxed since accounting for nonstationary features in a hydraulic conductivity field reduces the underlying variance. The predicted quantities will be the mean of solute mass flux through a control plane and the variance about the mean. The method is designed for solute transport in complicated natural media. In Zhang et al.’s (2000) study, the chemical sorption process is linear and non-equilibrium with constant sorption coefficient. Although the general framework is for two- or three-dimensional flows, it has only been implemented for a two-dimensional flow case. In this study, the chemical sorption is considered to be linear and equilibrium, but the sorption coefficient is random and nonstationary. Zhang et al.’s general framework for solute flux in a nonstationary flow field is adopted, but a new and more general method is developed for the moments of solute transverse displacements for threedimensional flow fields, where the expressions are given in the differential equation form, rather than in the integral form (Zhang et al., 2000). The analytical moment equations are solved numerically through a finite difference method. The developed method is then applied to study the effects of heterogeneity and nonstationarity of hydraulic conductivity and chemical sorption coefficient distributions on solute transport process.
2 Physical and chemical nonstationarity There are many different types of nonstationarity in natural media, which are summarized by Zhang (2002). In this study, we consider a common one, zonal stationarity. It has been frequently observed in natural fields that a geologic formation includes several distinct geologic units (e.g., layers, zones), which may be results of different geologic processes. Although the hydraulic
conductivity and/or sorption coefficient of each geologic unit are stationary processes (fields), each unit has different statistical moments than do its neighboring units. Therefore, the hydraulic conductivity or sorption coefficient field of the entire formation is not stationary. Such a random field is called a zonally stationary field. The nonstationarity of a log hydraulic conductivity field, Y ðxÞ ¼ ln KðxÞ, may manifest in such ways; its mean hY ðxÞi and variance r2Y ðxÞ vary spatially, and its two-point covariance CY ðx; vÞ depends on the actual locations of x and v. For a zonally stationary field, hY ðxÞi and variance r2Y ðxÞ are fixed within each zone, but vary from one zone to another. CY ðx; vÞ ¼ CY ðx vÞ is stationary if x and v are in same zone, and is zero if x and v are in different zones. Although CY ðx vÞ can take any form, here we consider, for simplicity, only the exponential form 8 " # 9 < ðx v Þ2 1=2 = i i CY ðx vÞ ¼ r2Y ðxÞ exp ð1Þ : ; k2i ðxÞ where i is the coordinate direction, ki ði ¼ 1; 2; 3Þ is the correlation length, and r2Y is the variance. Einstein summation is applied in (1). In this study, the chemical sorption is assumed to be linear and in equilibrium, so the influence of the chemical sorption process on solute transport is described by the retardation factor, R, which is elated to the sorption coefficient as, R ¼ 1 þ qnb Kd0 , where qb is the medium’s bulk density (g/cm3 ), n is the medium porosity (cm3 / cm3 ) and Kd0 is the sorption coefficient (cm3 /g). For simplification, we introduce a new parameter, Kd ¼ qnb Kd0 . Generally speaking, the spatial variation of Kd0 is much larger than those of qb and n. Therefore, we assume qb and n be deterministically constant in one zone (their values may vary from one zone to another), and the randomness of Kd or R is only caused by Kd0 . Similar to hydraulic conductivity, dramatically spatial variation of Kd ðxÞ or RðxÞ, has also been observed in field measurement (e.g., Brusseu, 1994). The distribution of RðxÞ is also assumed to be zonally stationary, which means that the mean hRðxÞi and variance r2W ðxÞ are fixed within a zone, but vary zonally, and CR ðx; vÞ ¼ CR ðx vÞ if x and v are in the same region (medium), otherwise CR ðx; vÞ ¼ 0. The correlation between Y ðxÞ and RðxÞorKd ðxÞ, has been studied for many years, however, there is no general rule for the correlation. For inorganic compounds, the medium sorption capacity is proportional to the sorbing surface area, which, in turn, is inversely proportional to grain size. This relationship indicates that the RðxÞ is negatively correlated with Kd ðxÞ. However, for organic compounds, the medium sorption capacity mainly depends on its organic carbon content, not the particle size. Field study results at Borden site, Canada (Robin et al., 1991) indicate that even for inorganic compounds, the correlation between sorption coefficient and hydraulic conductivity is not obvious and may change with scale. Based on these facts, the uncorrelated
24
model between the two parameters proposed by Bellin 0 et al. (1993) is used in this study, RðxÞ ¼ 1 þ KdG ðxÞeW ðxÞ , where KdG ðxÞ is the geometric mean of Kd ðxÞ and W 0 ðxÞ is a normally distributed random space function with zero mean, variance r2W ðxÞ, covariance function is and h i1=2 , where x CW ðx vÞ ¼ r2W ðxÞ exp ðxi vi Þ2 =k2w;i and v are in the same zone, and kw;i is the correlation length along the xi axis. By expanding the exponential terms in the above modals in Taylor series, we can obtain mean of the retardation factor in the first order of 2 r2w as, h RðxÞi ¼ 1 þ KdG ðxÞe½rW ðxÞ=2 . Similarly, the covariance is given by: CR ðx; vÞ ¼ CR ðx vÞ i 2 2 h ¼ KdG ðxÞ erW ðxÞ eCW ðxvÞ 1 ðx and v are in the same zone) ¼ 0 ðx and v are not in the same zone)
ð2Þ
and CYR ðx; vÞ ¼ 0.
4 A method of moments for reactive solute flux in a nonstationary medium A solute of total mass M is released into the flow field at time t ¼ 0, over the injection area A0 located at x ¼ 0, either instantaneously or with a known release rate quantified by a rate function, /ðtÞ½T 1 . We denote with q0 ðaÞ½M=L2 an areal density of injected solute mass at the location a 2 A0 . With Da denoting an elementary area at a, the parcel with mass q0 Da is advected by the random, spatially nonstationary retarded velocity UR ðxÞ. For t > 0, a solute plume is formed and advected downstream by the flow field toward a (y, z)-plane, located at some distance from the source, through which the solute mass flux is to be predicted or measured. The plane is referred to as the control plane (CP). The expected value of the solute mass flux component orthogonal to the CP, q, in the case of point sampling can be expressed as (Andricevic and Cvetkovic, 1998; Zhang et al., 2000; Wu et al. 2003), Z hqðt; xÞi ¼ q0 ðaÞ/ðt sÞf1 ½sðx; aÞ ¼ t; gðx; aÞ ¼ yds da
3 Nonstationary velocity field
A0
We consider incompressible groundwater flow in a heterogeneous aquifer with spatially variable hydraulic conductivity. Groundwater seepage velocity, U(x), satisfies the continuity equation and Darcy’s law, r ohðxÞ UðxÞ ¼ 0 and Ui ðxÞ ¼ KðxÞ n oxi , subject to boundary conditions hðxÞ ¼ H ðxÞ x 2 CD and UðxÞ cðxÞ ¼ XðxÞ x 2 CN , where hðxÞ is hydraulic head, KðxÞ is hydraulic conductivity (assumed to be isotropic locally), H ðxÞ is prescribed head on Dirichlet boundary segments CD , XðxÞ is prescribed water flux across Neumann boundary segments CN , and cðxÞ is an outward unit vector normal to the boundary. Zhang and Winter (1999) developed a numerical moment approach for groundwater flow in a stationary conductivity field in a bounded domain. In this study, we consider a chemical transport under linear equilibrium sorption. The chemical sorption process will retard movement velocity of the solute. The retarded velocity field, UR ðxÞ, is related to the groundwater velocity, UðxÞ, as, UR ðxÞ ¼ UðxÞ=RðxÞ. We decompose UðxÞ and RðxÞ into their means and perturbations, UðxÞ ¼ hUðxÞiþ uðxÞ and RðxÞ ¼ hRðxÞi þ R0 ðxÞ, then obtain the mean and variance of UR ðxÞ in the first-order accuracy of r2u and/or r2R as R hUðxÞi ð3Þ U ðxÞ ¼ h RðxÞi CuRij ðx; vÞ ¼
Cuij ðx; vÞ hUi ðxÞihUj ðvÞi þ CR ðx; vÞ hRðxÞihRðvÞi hRðxÞi2 hRðvÞi2
ð4Þ
Equation (4) is the expression of the retarded velocity covariance with nonstationary fields of log-hydraulic conductivity and chemical sorption coefficient.
ð5Þ where f1 ½sðx; aÞ; gðx; aÞ denotes the joint probability density function (PDF) of travel time s for a parcel from a to reach x and the corresponding transverse displacement gðg ðg; nÞÞ, where g; n are displacements in the y- and z- coordinates, respectively. The variance of the solute flux for the point sampling is evaluated as r2q ðt; xÞ ¼ hq2 i hqi2 with 2
hq ðt; xÞi ¼
Z Z
ð6Þ
q0 ðaÞq0 ðbÞ/ðt s1 Þ/ðt s2 Þ
A0 A0
f2 ½s1 ðx; aÞ ¼ t; g1 ðx; aÞ ¼ y; s2 ðx; bÞ ¼ t; g2 ðx; bÞ ¼ yda db
ð7Þ
where f2 ½s1 ðx; aÞ; g1 ðx; aÞ; s2 ðx; bÞ; g2 ðx; bÞ is the twoparcel joint PDF of travel time and transverse displacement. The solute discharge is another quantity of interest defined as the total solute mass flux over the entire CP at x, Z Z Qðt; xÞ ¼ q0 ðaÞ/ðt sÞdðy gÞda dy ð8Þ A0 CP
where y is a point in the CP. Its mean and variance are given as Z Z1 q0 ðaÞ/ðt sÞf1 ½sðx; aÞda ð9Þ hQðt; xÞi ¼ A0
0
25
r2Q ðt; xÞ ¼ hQ2 i hQi2 ð10Þ Z Z q0 ðaÞq0 ðbÞ/ðt s1 Þ/ðt s2 Þf2 ½s1 ðx; aÞ hQ2 ðt; xÞi ¼
dg0 ðx; aÞ ¼ A1 ðx; hgi; hniÞuR1 ðx; hgi; hniÞ þ Bðx; hgi; hniÞ dx uR2 ðx; hgi; hniÞ þ C1 ðx; hgi; hniÞg0 ðx; aÞ þ D1 ðx; hgi; hniÞn0 ðx; aÞ
A0 A0
¼ t; s2 ðx; bÞ ¼ tda db
ð11Þ
where f1 ½sðx; aÞ is the marginal PDF of f1 ½sðx; aÞ; gðx; aÞ, and f2 ½s1 ðx; aÞ; s2 ðx; bÞ is the marginal PDF of f2 ½s1 ðx; aÞ; g1 ðx; aÞ; s2 ðx; bÞ; g2 ðx; bÞ. To evaluate the statistical moments of solute flux, one needs to know the one- and two-parcel PDFs f1 and f2 , or an infinite number of statistical moments. The approach used in this study is to evaluate the first two statistical moments and assume certain functions for f1 and f2 . The travel time sðL; aÞ, the time required for a parcel originated at a to cross the plane x ¼ L is determined by the Lagrangian retarded velocity, which is related to the Eulerian retarded velocity. The mean, hsi, and variance, r2s , are obtained, to first order, as hsðL; aÞi ¼
ZL ax
dx hU1R ðx; hgi; hniÞi
ð12Þ
and r2s ðL; aÞ
¼
ZL ZL ax ax
dx1 dx2 hU1R ðx1 ; hg1 i; hn1 iÞi2 hU1R ðx2 ; hg2 i; hn2 iÞi2
huR1 ðx1 ; hg1 i; hn1 iÞuR1 ðx2 ; hg2 i; hn2 iÞi
ð16Þ
and dg0 ðx; aÞ ¼ A2 ðx; hgi; hniÞuR1 ðx; hgi; hniÞ þ Bðx; hgi; hniÞ dx uR3 ðx; hgi; hniÞ þ C2 ðx; hgi; hniÞg0 ðx; aÞ þ D2 ðx; hgi; hniÞn0 ðx; aÞ where A1 ðx; hgi; hniÞ ¼
hU R ðx;hgi;hniÞi hU R2ðx;hgi;hniÞi2 . 1
ð17Þ The other coeffi-
cients, A2 , B, C1 , C2 , D1 and D2 are also the functions of the mean velocity like A1 . For brevity, their expressions are not given in this article. We use (16) and (17) to obtain the expressions of autocovariances of g and n, and the covariances among g; n; uR1 , uR2 and uR3 . Zhang et al. (2000) used the integral form to obtain the integral expression for g’ in a twodimensional case. However, for three-dimensional case, it is very difficult to obtain this kind of expression from (16) and (17). Therefore, we use differential equation forms to obtain the expressions of the various correlation functions. Multiplying (16) by g0 ðx2 ; bÞ, we obtain dhg0 ðx1 ; aÞg0 ðx2 ; bÞi dx1 ¼ A1 ðx1 ; hg1 i; hn1 iÞhuR1 ðx1 ; hg1 i; hn1 iÞ g0 ðx2 ; bÞi þ B1 ðx1 ; hg1 i; hn1 iÞhuR2 ðx1 ; hg1 i; hn1 iÞ
þ g1 ðx1 ; hg1 i; hn1 iÞg1 ðx2 ; hg2 i; n2 iÞhg01 g02 i
g0 ðx2 ; bÞi þ C1 ðx1 ; hg1 i; hn1 iÞhg0 ðx1 ; aÞg0 ðx2 ; bÞi
þ g2 ðx1 ; hg1 i; hn1 iÞg2 ðx2 ; hg2 i; hn2 iÞhn01 n02 i
þ D1 ðx1 ; hg1 i; hn1 iÞhn0 ðx1 ; aÞg0 ðx2 ; bÞi
þ 2g1 ðx2 ; hg2 i; hn2 iÞhuR1 ðx1 ; hg1 i; hn1 iÞg02 i
ð18Þ
the first order and therefore require the variation of velocity to be (formally, much) smaller than 1. This condition may be satisfied for many practical subsurface flows where the variance of log-transformed hydraulic conductivity is moderately large. The statistical moments of g and n to the first order of r2ui , velocity variance, are obtained as,
Similarly, we can also obtain the expressions of hg0 n0 i, hxi0 n0 i, huR1 g0 i, huR1 n0 i, huR2 g0 i, huR2 n0 i, huR3 g0 i, huR3 n0 i, which are not shown here for the sake of brevity. The velocity correlations, huRi uRj iði ¼ 1; 2; 3; j ¼ 1; 2; 3Þ, can be obtained from the velocity field and are treated as known quantities or input data here. Owing to the complexity of these equations, we solve them numerically. The joint moments hs01 ðx1 ; aÞg02 ðx2 ; bÞi and hs01 ðx1 ; aÞn02 ðx2 ; bÞi can be obtained by multiplying (17) by g0 ðx2 ; bÞ and n0 ðx2 ; bÞ, respectively, then taking expectation. In this study, we assume the travel time, s, obeys lognormal distribution and transverse displacements, g and n, satisfy the normal distributions. These assumptions are consistent with the previous studies (Bellin et al., 1994; Cvetkovic et al., 1996; Zhang et al., 2000). Under these assumptions, the one- and two-parcel joint PDF of s and g can be obtained.
dhgðx; aÞi hU2R ðx; hgi; hniÞi ¼ R dx hU1 ðx; hgi; hniÞi
ð14Þ
5 Illustrative Examples
hU3R ðx; hgi; hniÞi hU1R ðx; hgi; hniÞi
ð15Þ
þ 2g2 ðx2 ; hg2 i; hn2 iÞhuR1 ðx1 ; hg1 i; hn1 iÞn02 i þ 2g1 ðx1 ; hg1 i; hn1 iÞg2 ðx2 ; hg2 i; hn2 iÞhg01 n02 i
ð13Þ
gj ¼ gðxj ; aÞ and nj ¼ nðxj ; aÞðj ¼ 1; 2Þ, R o½hU1 ðx;g;nÞi g ¼ hgi and g2 ðx; hgi; hniÞ ¼ g1 ðx; hgi; hniÞ ¼ n ¼ hni : og R o½hU1 ðx;g;nÞ> g ¼ hgi . These expressions are derived up to on
where
n ¼ hni
dhnðx; aÞi ¼ dx
In this section, we illustrate our approach through several synthetic case studies. We focus on the combined
26 Fig. 1 a Sketch of the research domain, the locations of source, CP, and a small cube A, and b the mean flow field of case 1
effects of the physical and chemical heterogeneity and nonstationarity on the prediction of the mean solute flux and the associated uncertainty. Figure 1a is a sketch of the study domain, a small cube called cube A with the size, 5k 5k 5k, located at the center of a larger cube with the size, 10k 10k 10k, where k is the log-conductivity correlation length and the two cubes are parallel with each
other. The spatial distributions of hydraulic and chemical properties in one of the two subdomains (the small cube, subdomain 2, and the rest part of the study domain, subdomain 1) are statistically homogeneous, but the distributions in the two subdomains are statistically different. A 1k 1k areal solute source with uniform density and unit mass is located at the yz-plane centered at the point (0:5k, 5k, 5k). The source is represented by
27
Table 1 Computation cases for nonstationary flow field
Cases
Y1
Y2
r2Y1
r2Y2
kY1
kY2
KdG1
KdG2
r2W1
r2W2
kW1
kW2
1 2 3 4 5 6 7 8 9 10 11 12
1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
)1.0 )1.0 0.0 )1.0 )1.0 )1.0 )1.0 )1.0 )1.0 )1.0 )1.0 )1.0
0.8 0.8 0.8 0.8 0.8 0.8 0.2 0.2 0.2 0.2 0.2 0.2
0.8 0.8 0.8 0.8 0.8 0.8 0.2 0.8 0.8 0.8 0.8 0.8
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.2 0.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.2 1.0
2.0 4.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
2.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
0.8 0.8 0.8 0.8 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.8 0.8 0.8 0.8 0.2 0.8 0.8 0.8 0.8 0.8 0.8 0.8
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.2 0.2 0.2 0.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.2 1.0 1.0 1.0
Fig. 2a,b Breakthrough curves of solute flux Q, through the CP at x ¼ 9:1k with various mean sorption coefficient and hydraulic conductivity for nonstationary flow field: a expected value hQi, and b standard deviation value rQ
5 5 uniformly distributed parcels. The control plane is located at x ¼ 9:1k. The hydraulic boundary conditions of the box domain are specified as follows: constant hydraulic head (h ¼ 10k) for the left side (x ¼ 0), constant hydraulic head (h ¼ 0) for the right side (x ¼ 10k),
and no-flow for the other four sides. Since several factors will influence the solute transport process, such as the means, variances and correlation lengths of hydraulic conductivity and sorption coefficient, 12 specific cases with different combinations of parameter
28
values, shown in Table I, are proposed to study the influences of these factors on transport process. The groundwater flow fields for all cases are calculated. For the purpose of illustration and brevity, the results of case 1 shown in Fig. 1b are used to exhibit the nonstationarity of the flow fields. One can see from the figure that the existence of cube A results in a significant change in the mean flow in both direction and magnitude. Cases 1–4 are grouped to study the effects of mean sorption coefficient and medium nonstationarity on the mean and variance of solute flux. The calculation results of the four cases are shown Figs. 2a and 2b. In cases 1 and 2, the distribution of log-hydraulic conductivity is nonstationary, but the distribution of chemical sorption coefficient is stationary. It is shown from the mean breakthrough results of the two cases that longitudinal plume dispersion increases with the increase of KdG , which is consistent with Bosma et al’s (1993) results for a stationary medium. This phenomenon can be explained from Eq. (2) that the variance of sorption coefficient increases with the increase of KdG . In cases 3 and 4, the Fig. 3a,b Breakthrough curves of solute flux Q, through the CP at x ¼ 9:1k with various variance values of the sorption coefficient and hydraulic conductivity for nonstationary flow field. a expected value hQi, and b standard deviation value rQ
distributions of sorption coefficient are the same and nonstationary. The distribution of log-conductivity for case 4 is the same as that in case 1 or 2; the distribution for case 3 is stationary in all the study domain. The mean breakthrough results of the two cases show that the plume longitudinal dispersion is enhanced by the physical nonstationarity. In comparison with the results of cases 1, 2 and 4, the breakthrough in case 4 is between the two chemically stationary cases, which shows KdG ’s influence on the transport process. The results in Figs 2a and 2b also show that the shape of rQ ’s breakthrough curve is similar to hQi0 s, but with larger value. The variances of the sorption coefficient and log hydraulic conductivity are used to describe the degrees of chemical and physical heterogeneities. Fig. 3 is used to present the influences of the variances on breakthrough curves of solute flux. The results of cases 4, 5 and 6 show that the solute dispersion increases with the increase of variance of sorption coefficient, r2w , and the results of cases 6, 7 and 8 indicate the solute dispersion also increases with the increase of r2f . The influence of r2w on the break-
29 Fig. 4a,b Breakthrough curves of solute flux Q, through the CP at x ¼ 9:1k with various correlation length values of the sorption coefficient and hydraulic conductivity for nonstationary flow field. a expected value hQi, and b standard deviation value rQ
through curves is more obvious than that of r2f . Interestingly, variation of r2W or r2f in the rest of the study domain except cube A has a minor impact on spreading in the longitudinal direction (comparing the results of case 4 with 6, and case 6 with 8), which implies that the medium hydraulic and chemical properties in cube A have a much larger influence on solute transport process than those in the rest of the study domain. The correlation length is an important parameter to describe the spatial distribution of a random variable. Fig. 4 presents the effects of correlation lengths of log hydraulic conductivity and sorption coefficients on breakthrough curves of solute flux. The results of cases 8 and 9 show that the solute dispersion decreases with the decrease of the kW , and the results of cases 10 and 11 indicate the dispersion also decreases with the decrease of kY . Similar to our findings in Fig. 3, variation of kW or kY in the rest of the study domain except cube A has little impact on the spreading of solute flux in the longitudinal direction (the solute flux distributions in
cases 8, 10 and 12 are almost identical), which indicates again that the property of cube A dominates the spreading process in the case studies.
6 Summary In this study, we developed a solute transport theory for solute flux through a three-dimensional, spatially nonstationary flow field. The flow nonstationarity results from the nonstationary conductivity field and hydraulic boundary conditions. For solute under chemical sorption process, the nonstationarity of sorption coefficient distribution will also lead to the nonstationarity of the retarded velocity field. These nonstationarity cases are not within the scope of the classical stochastic theory (Gelhar and Axness, 1983; Dagan, 1982, 1984), but widely exist in natural fields. We applied a Lagrangian perturbation method to develop a stochastic framework analytically for the mean and variance of solute flux in a physically and
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chemically nonstationary medium. The theory development is based on the assumption that distribution of each parcel during its transport process satisfies log-normal distribution in the longitudinal direction, and normal distribution in transverse directions. Under this assumption, the solute flux distribution can be calculated with the means and variances of the all the parcels’ travel times through the CP and transverse locations on the CP. The means and variances are related to the mean and covariance of the retarded velocity field, which, in turn, are related to the conductivity field through Darcy’s law. Owing to the nonlinearity of the stochastic governing equations for flow and transport and the complex boundary conditions, numerical methods are implemented to obtain the solutions. The general approach was illustrated with some synthetic cases of solute transport in nonstationary media. The purpose of these case studies is to investigate the influences of various factors on solute transport process. These factors include mean values, variances and correlation lengths of log hydraulic conductivity and sorption coefficients in the two subdomains. The sensitivity studies shows that all these factors will significantly influence the solute transport process. The variance of solute flux is generally larger than its mean value, which means that the solute transport process may significantly deviate from the mean prediction. This result is consistent with previous studies on solute transport in stationary media through moment equation method or Monte Carlo simulation method. Conditional study on field measurements is required to reduce the uncertainty. Acknowledgements This work was partially funded by Desert Research Institute VPR, DOE Yucca Mountain project under contract between DOE and the University and Community College System of Nevada, NSFC (project number 40272106), and the Teaching and Research Award Program for Outstanding Young Teachers (TRAPOYT) of MOE, China.
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