Modified mixed Lagrangian-Eulerian method based on Numerical
framework of MT3DMS on Cauchy boundary
Appendix 2: Example 2
The second example deals with the scenarios motivating this work, where contaminants
enter the subsurface in recharge zones with low-permeability, where the velocity may be
close to zero. In this scenario, the low-permeability material is assumed to be clay material. It
is assumed that above the clay surface, a ponding condition of H o is formed due to intense
rainfall, and clay is initially saturated with a hydraulic head of H i . Accordingly, the onedimensional governing equation describing the subsurface flow within the clay can be written
as:
H
2H
Ss
K
t
x 2
(S7)
where H is hydraulic head, S s is specific storage, and K is hydraulic conductivity.
Initial and boundary conditions are as follows:
H x,0  H i
(S8)
H 0, t   H o
(S9)
H , t 
0
x
(S10)
The analytical solution of Equation S7 with initial condition S8 and boundary conditions
S9–S10 can be written by replacing R  S s , D  K , and V  0 (Lapidus and Amundson
1952; Ogata and Banks 1961), as follows:
H x, t   H i  H o  H i Ax, t 
(S11)


Ss x 

where Ax, t   erfc
. Once the hydraulic head distribution is analytically
 2KS t 12 
s


obtained, seepage velocities can be obtained with Equation S12 by differentiating Equation
S11 with x .
1
S x2
 S 2  s
V x, t   H o  H i  s  e 4 Kt

 Kt 
K
(S12)
Since seepage velocities can be analytically obtained, a numerical simulation of solute
transport using all numerical schemes can be performed using analytical velocities from
Equation S12 at each time step of the simulation t  t, 2t, , T  . To determine a semianalytical solution of the contaminant concentration for comparison, the Generalized Integral
Transformation Technique (GITT) (Liu 2000) is used. GITT uses an arbitrary initial
concentration F x  , arbitrary inlet concentration f t  , and analytical seepage velocity at
each time step. The solute concentration distribution using GITT can be calculated at every
time step by using the previous concentration distribution, the analytical velocity distribution
of Equation S12 at current time, and the dispersion coefficient distribution calculated from
the analytical velocity distribution. It should be noted that previous concentration
distributions at each time step are obtained from the concentration calculated at the previous
time step using GITT. Those should be defined in the form of a spatial function for
calculation of GITT, which is always achieved by an infinite series of eigenfunctions (Liu
2000). Accordingly, here GITT cannot provide solute concentration distributions for all
times, but it can give semi-analytical solutions for pre-determined times, t  t , 2t ,  , T .
The input conditions and parameters of the numerical simulation, and the material
properties of the clay are listed in Table S1. For the implementation of this example problem,
spatial and temporal discretizations of all numerical schemes, such as the Neuman method,
the LEZOOMPC, the proposed method, the MOC, MMOC, HMOC, upstream FDM and
TVD, are assigned with uniform grid sizes and constant time step sizes, respectively, in Table
S1. Also, refinement strategies of Neuman and LEZOOMPC approaches and particle
emplacement strategies of MOC, MMOC, and HMOC are the same as Example 1. All
schemes in MT3DMS use the GCG solver.
Similar to Case 1D, corresponding to the dispersion-dominated problems in Example 1,
the results of the FDM, MOC, MMOC, HMOC, TVD, and the proposed methods show good
agreement with the accurate solution, as shown in Figure S1. But, the solutions of the
Neuman and LEZOOMPC approaches greatly deviate from the accurate solution because of
the inappropriate numerical treatment; i.e., the Lagrangian formulation calculating the
Lagrangian concentration at the Cauchy boundary.
Table S1
List of input parameters for solute transport simulation of Example 2
Parameters
H o (m)
Parameters for flow
Parameters for
transport
Initial and
boundary
conditions for
transport
Parameters
for transport
simulation
Values
Notes
0.5a
a
is used by author
b
is used by author
H i (m)
0.0b
S s (1/m)
2.0Χ10-4,c
K (m/day)
3.5Χ10-4,d

0.34e
V x, t  (m/day)
6.9Χ10-5 ~ 4.4Χ10-4,f
L
(m)
3.0Χ10-2,g
D * (m2/day)
1.7Χ10-4,h
Dx, t  (m2/day)
1.8Χ10-4 ~ 1.9Χ10-4,i
F x  (mg/l)
D   L V  D*
0.0
Initial condition
f t  (mg/l)
1.0
Inlet concentration for Cauchy
boundary condition
t (day)
0.25
Constant time step size
x (m)
0.01
Uniform grid size
L (m)
0.3
Domain length
T (day)
10.0
Simulation time
c
is chosen from Batu (1998) for
clay
d
is chosen from Domenico and
Schwartz (1990) for clay
e
is chosen from Morris and
Johnson (1967) for clay
f
is calculated from Eq. (S12)
g
is used by author
h
is chosen from Cussler (1997)
i
is calculated from
j
Mesh Peclet number, Pe
-3
3.9Χ10 ~ 2.4Χ10
-2,j
Pe  V
k
Mesh Courant number, C
1.7Χ10-3 ~ 1.1Χ10-2,k
is calculated from
x D
is calculated from
C=
V t x
Figure S1. Results of concentration profiles of Example 2 at t = 10 days.
References
Batu, V. 1998. Aquifer Hydraulics: A Comprehensive Guide to Hydrogeologic Data Analysis,
John Wiley & Sons, New York.
Cussler, E. L. 1997. Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York:
Cambridge University Press.
Domenico, P.A. and F.W. Schwartz. 1990. Physical and Chemical Hydrogeology, John
Wiley & Sons, New York.
Lapidus, L., and N.R. Amundson. 1952. Mathematics of adsorption in beds. VI. The effects
of longitudinal diffusion in ion exchange and chromatographic columns. Journal of
Physical Chemistry 56: 984-988.
Morris, D.A. and A.I. Johnson. 1967. Summary of hydrologic and physical properties of rock
and soil materials as analyzed by the Hydrologic Laboratory of the U.S. Geological
Survey, U.S. Geological Survey Water-Supply Paper 1839-D.
Ogata, A. and R.B. Banks. 1961. A solution of the differential equation of longitudinal
dispersion in porous media. U.S. Geological Survey Professional Paper 411-A, A1-A9.