1
Title:
An accurate method for transient particle tracking
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Authors:
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Uli Maier*,
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Claudius M. Bürger,
[email protected]
[email protected]
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Affiliation:
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Center for Applied Geosciences, (ZAG), University of Tübingen, Germany
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Sigwartstr. 10, 72076 Tübingen, Germany
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* corresponding author
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An accurate method for transient particle tracking
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Uli Maier, Claudius M. Bürger
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Center for Applied Geosciences, (ZAG), University of Tübingen, Germany
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Abstract
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Particle tracking methods are widely applied in subsurface hydrology to calculate advective transport
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within flow fields obtained from the numerical solution of the groundwater flow or Richards
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equation. These procedures are used as standard methods to acquire travel time distributions of
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advective transport as well as the delineation of well capture zones. For cell-centered, regular,
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structured grids, analytical solutions for the computation of particle pathlines inside grid cells are
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known for steady state flow fields and transient flow fields under the condition of a constant velocity
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gradient over a full time step. We extend particle pathline computation using an exact semi-
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analytical solution for a fully transient flow field that relaxes this condition and may even treat a
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complete reversal of the velocity gradient from one time step to the next.
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1. Introduction
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Particle tracking methods are a well-established part of the subsurface hydrology toolbox to calculate
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advective transport within flow fields derived from the numerical solution of saturated or variably-
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saturated flow equations (e.g. Richards equation). The delineation of well capture zones as well as
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the calculation of travel time distributions are important and typical applications of these standard
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methods. As particle velocity is the spatial derivative of location with respect to time, pathlines of
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particles can be computed by integration over generally non-uniform flow velocity fields. Usually two
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different approaches are distinguished [Cordes and Kinzelbach, 1992, Bensabat et al, 2000]: (i)
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numerical integration methods [Cheng et al., 1996, Bensabat et al., 2000, Suk and Yeh, 2009] and (ii)
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semi-analytical methods [Pollock, 1988, Lu, 1994].
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One focal point of research during the past two decades has been the improvement of numerical
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integration methods for vertex-centered, unstructured meshes in use with finite element methods
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(FEMs) [Cheng et al., 1996, Bensabat et al., 2000, Suk and Yeh, 2009]. These numerical discretization
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schemes are appealing if irregular, complex geometries need to be very accurately represented.
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However, with respect to pathline computation challenges had to be overcome:
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As noted by Dogrul and Kadir [2006], Yeh [1981] had observed that a simple post-processing of the
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standard Galerkin FEM solution to the groundwater flow equation produced discontinuities in Darcy
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velocities across elemental interfaces with corresponding violations of elemental mass balances. He
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also found errors in the global mass balance. In the same work [Yeh 1981] presented a Galerkin finite
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element scheme that recovered continuous velocity fields and a later comment by Lynch [1984] also
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explained and resolved the global mass balance error. Rather than changing the FEM itself, Cordes
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and Kinzelbach [1992] and Dogrul and Kadir [2006] developed different post-processing methods to
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alleviate the local water balance violation. Based on Yeh’s work particle tracking algorithms for FEMs
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further evolved from Cheng et al. [1996] and Bensabat et al. [2000] to recent numerical pathline
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integration schemes with consecutive refinement of finite elements into sub-elements to improve
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accuracy and applicability of pathline computations, e.g. see [Suk and Yeh, 2009, Suk and Yeh, 2010,
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Suk, 2012].
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In this technical note, however, we focus on particle tracking methods for cell-centered, structured,
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regular grids. Even though less flexible with respect to geometry representation, this type of
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discretization is very appealing due to its ease of implementation and simple topological structure,
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which usually represents a computational advantage, e.g. for domain-decomposition in
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parallelization. Regular grids are also the usual choice for finite difference (FD) or finite volume (FV)
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numerical schemes, the latter of which we want to consider as the basis for our particle tracking
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scheme development. As information in the FV scheme is exclusively passed on from one grid cell to
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another via the flow across their mutual interfaces, local mass conservation is ensured by the
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method itself. Consequently, the combination of regular grids and FV schemes is widely applied in
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fields like reactive transport modelling where local mass conservation and computational efficiency
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are vitally important. In this study we develop an analytical extension of Pollock’s and Lu’s semi-
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analytical methods [Pollock, 1988, Lu, 1994] and investigate numerical treatment of the established
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analytical solution in this context. Applied to the groundwater flow or Richards’ equation, the
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standard FV scheme solves for one primary unknown variable for the interior of each grid cell (i.e.
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the (average) hydraulic head in groundwater applications). The standard FV scheme is inherently
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conserving local mass, as it considers one constant flux vector over the entire interfacial area
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between two adjacent grid cells. Hence, velocity is continuous across an interface, however, these
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calculated velocities are really only given at the interfaces. It is part of the semi-analytical particle
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tracking method to define how the interface velocities (four in 2-D, six in 3-D) determine the velocity
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field inside the grid cell. In the “analytical” part of the method, the particle trajectory is then
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computed by a closed-form solution over the velocity field. In that respect semi-analytical methods
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are accurate within a grid cell and can determine the final particle exit coordinate within a single
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computation step. Nevertheless, bilinear or trilinear interpolation of flow velocities between
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different spatial directions can again result in a violation of the water (mass) balance [Salamon et al.,
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2006].
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Pollock [1988] provided a method for pathline computation within that framework for a steady-state
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flow field, which is the standard particle tracking algorithm widely applied by the groundwater
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community through the use of MODPATH [Pollock, 1994]. Transient simulations are also possible by
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treating each time step as a quasi-steady-state period. Lu [1994] extended that concept for a
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temporal change of the velocity field over one time step. However, Lu [1994] retained the
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assumption that the spatial change of individual velocity components remains constant over the time
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step. With this simplifying assumption the mixed derivative of velocity with respect to space and time
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is neglected (d2vx/dx/dt = 0) and the mean of the spatial velocity changes at the beginning and end of
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the current time step is used instead. This approach still represents an improvement for transient
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flow fields, but it does not adequately reproduce the particle motion for cases with a significant
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change of the spatial derivative of velocity, e.g. if flow drastically changes its direction. In that case
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the approach of Lu [1994] could produce inaccuracies.
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In this technical note we present a method that extends the approach of Lu [1994] with the objective
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to fully account for the changes of flow velocity along each spatial dimension and time in a semi-
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analytical expression. The method is exact within the boundaries of the specific grid cell.
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2. The derivation of the semi-analytical expression
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For cell-centered, structured, regular grids the semi-analytical method considers the velocity field at
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grid cell interfaces assuming a single constant value along each interface. Therefore the particle
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velocity component for one given spatial dimension is independent of its coordinate in the other
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spatial dimension, i.e. dvx/dy = dvy/dx = 0, etc. The basic assumption is a linear velocity gradient along
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each spatial dimension within a grid cell. In our approach, the flow velocity field is allowed to change
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with time according to the interfacial fluxes given by the solution of the governing flow equation,
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independent of further simplifying assumptions. In each spatial dimension, four different velocities
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can be obtained from the flow model for the boundaries of each grid cell of dimension x between
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lower index coordinate 0 and higher index coordinate L, within a timestep t between time 0 and
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time T at the end of the time step. These four velocity components are denoted v0,0, vL,0, v0,T, and vL,T.
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For simplicity, we treat spatial and temporal coordinates in terms of local coordinates within the
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given grid cell. One of the big advantages of regular, structured rectangular grids, is that any
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coordinate can be easily transferred from global domain- to local cell Cartesian coordinates and back
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by a simple subtraction/addition of the global cell edge coordinates without loss of accuracy and
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efficiency. Note that the subscripts 0,L,T given below mark space-time cell interfaces and not particle
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starting locations. The latter are denoted e.g. by x0 and can be freely selected via the initial condition
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x(t0 ) = x0 and for any starting time t0 . The four velocities mentioned above are lumped together
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yielding the cell specific parameters a, b, c and d:
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d = v0,0
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c = (v0,T – v0,0) / T
= v/tx= 0
(2)
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b = (vL,0 – v0,0) / L
= vt= 0
(3)
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a = (vL,T + v0,0 – vL,0 – v0,T) / L / T
= v)/t
(4)
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The velocity field in spatial dimension x within the grid cell and time step t then becomes
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vx 
(1)
dx
 axt + bx + ct + d
dt
(5)
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This scheme represents a bilinear interpolation of the velocity field with respect to each single spatial
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direction and time, which preserves mass of water within each cell and water flow across cell
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boundaries. A sketch of such a velocity field within a rectangular grid cell is provided in fig. 1.
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(Fig. 1)
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Re-arranging (5), the ordinary differential equation (ODE) reads
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dx
 (at  b) x  ct  d , which can be solved if we multiply both sides with the integrating factor
dt
a
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 t
 ( at b ) dt
=e 2
e 
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e
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with K’ being an integration constant of the integral evaluation on the left hand side. The integral on
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the right hand-side is
a
 t 2 bt
2
2
 bt
, yielding:
x  K'  e
a
 t 2 bt
2
a
 t 2 bt
2
(ct  d ) dt
c  t 2 bt

(ct  d ) dt  K  e 2

a
2a
a
b2
 at  b  2 a
bc 

  d    erf 
 e . If we take the initial
a

 2a 
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e
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condition x(t0) = x0 (with 0≤x0≤L, 0≤t0≤T) into account, the lumped integration constant K can be
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determined and we arrive at the solution of particle location x at time t within the interval 0≤x≤L,
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0≤t≤T:
b a
 at  b 
c h
 
bc  2 a  2 t 2 bt   at  b 
x  x0 e  e  1 
d  e
 erf 
  erf  0

a
2a 
a
 2a  
  2a 
2
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h
a
t
2
(6)

t02 b ( t t0 )
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where e h  e 2
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We could further lump the complete equation by taking into account that
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b2 a 2
1
2
 t  bt  at  b   ht
2a 2
2a
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Note that given the definition of ht and h0 the abbreviation eh could also be written as eh  eht  h0 ,
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but we will use h, ht, and h0 as their associated terms are treated different numerically and in order
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to state the formula for travel distance in its most condensed version:
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x  x0 eh 
and
b2 a 2
1
2
 t0  bt0  at0  b   h0
2a 2
2a
  h  erf  h 
c h
 
bc 
e 1 
  d    eht  erf
a
2a 
a


t
0
(7)
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As the limiting case a → 0 raises numerical issues we chose criteria for the selection of an
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appropriate pathline computation method based on ht and h0, of which the latter, in the most simple
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case of t0 = 0, equals b2/2a. As will be shown below the analytical convergence of our solution to Lu’s
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solution is ensured.
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It should be noted that, in case a < 0, the third argument becomes a complex number. However,
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numerical evaluation has shown that imaginary parts of the expression vanish in the final result for x.
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As a matter of fact, we can prove that x is a real number by re-arranging the expression
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that a < 0) to i 2a : (the choice -i 2a does not make a change for the following reasoning):

2a (given

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151
b2
 t 2  bt
c
1  
bc 
x  x0 e  e h  1 
  d    e 2a 2
a
i  2a 
a
h


a
  1 at  b 
 1 at 0  b 
  erf 

 erf 
 i  2a  
  i  2a 
(8)
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It is obvious in (8) that the arguments of the error function have no real components, being solely
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imaginary. Replacing all occurrences of i with (–i) in (8), we obtain the complex conjugate of the
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expression. As the error function is odd, erf(-z) = -erf(z) holds and -1 can be factored out from erf and
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cancels. Hence the right hand side of (8) is equivalent to its complex conjugate, which proves that
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computing x via (8) will yield a real number for any input (provided that a is not equal to 0).
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3. Special cases and accuracy consideration
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Comparing eq. [5] to the solutions of Pollock [1988] and Lu [1994], we see that Pollock’s solution is
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the special case for a = 0 and c = 0 (steady state conditions), whereas Lu’s solution is the special case
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for a = 0. Pathlines can be computed using equation [7], as modern programming languages such as
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Matlab or Fortran 90 can handle complex numbers easily. Functions of complex arguments, e.g. are
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discussed in Zhang et al. [1996] or Jin (available at http://jin.ece.illinois.edu/routines/routines.html).
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Direct calculation of the particle exit time from the cell by solving [7] for time t appears cumbersome,
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if not intractable. Despite of its implicit definition via eq. [7] and the known cell interface
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coordinates, (an approximation of) the exit time can be obtained using an iterative numerical
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approach such as Newton iteration.
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We shall in the following prove mathematically that our solution converges to Lu’s solution for a → 0:
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Solving the basic differential equation (eq. 5) (here shown for the x-coordinate) with the initial
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condition x(t0) = x0 and using the integrating factor method we can obtain the formulation, which is
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the general solution for travel distance before integration
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(9)
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Even though the integral is not explicitly given, we can already at this stage consider the limit of x(t)
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for a → 0 in order to demonstrate the convergence of our formulation to Lu’s method. As a does not
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appear in the denominator, we can set a to 0 in this formulation, resulting in
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(10)
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Equation (10) can be solved by basic integration techniques. The function for the x-coordinate then
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becomes
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(11)
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, which is consistent with Lu’s result. Hence analytically it should be clear (actually mathematically
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proven) that our method converges to Lu’s method when a approaches zero. Correspondingly, the
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solution of Lu is shown to converge to our result by decreasing the timestep (i.e. reducing a), as
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shown in the graphical example of Fig. 3.
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Accuracy of simulation tools is a critical issue, whereas their efficiency is crucial for their practical
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applicability, therefore intensive code testing was performed. A high-order numerical approximation
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of the integral in (9) can be used to evaluate the behavior of our implementation of the analytical
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solution, which remains applicable when a approaches zero. For this purpose we use the formulation
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(9) and approximate the integral using a numerical 7th-order Gauss-Lobatto integration. We call this
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the GL method. Again, as a is not in the denominator, even vanishing of a does not introduce
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numerical problems for the integral approximation, so that the error of the method can be
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approximated by the deviation from the GL solution. Additionally, we perform a – rather time-
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consuming - explicit direct integration of eq. (5) over 103 sub-timesteps (DI method). We re9
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computed our test suits with both the GL and DI method, resulting in maximum deviations of 0.6 %,
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yet in general much smaller. Lu’s method, however, was found to differ remarkably from Gl and DI
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even for small a. Fig. 3., e.g. exemplifies that Lu’s method is not very accurate even for a strongly
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reduced time step size.
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For finite positive a approaching zero (i.e. a  0), the large term exp(b2/2a) is balanced by the
198
difference of the vanishing error functions, which may produce inaccuracies due to truncation in
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floating point operations. This could generally be resolved by switching to the method of Lu [1994]
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for sufficiently small a, or by increased floating point precision (e.g. real*16 type) for that special
201
case. Moreover, a slight reformulation improves the accuracy of our approach until b2/2a > 600. We
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take into account that the difference of two error functions is the accuracy-limiting computation,
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which has to balance the large exponential. Consider, for example, that erfc(x) = 1 – erf(x) converges
204
to zero for large arguments rather than to one as erf(x) does. By actually computing the value as
205
erf(at) – erf(a0) = 1-erfc(at) – (1-erfc(a0)) = erfc(a0) – erfc(at) we can circumvent the problem of large
206
exponents and achieve accuracy until the minimum representable size of numbers (10 300 in double
207
precision, e.g.). Therefore, in our test examples, Lu’s solution needed only to be computed if a = 0. In
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case numerical problems for a approaching zero still occur, we recommend the use of Gauss-Lobatto
209
(GL) integration. A precondition for Newton’s iteration method to be performed is the
210
differentiability of the target function, thus in case a switch between methods is necessary, both
211
functions have to align smoothly around the switch location. However, as iterations take place for
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the flow field of a given grid cell (or, more specifically, its given spatial direction), which has a unique
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parameter set, switching methods is generally not needed for the given framework.
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Results of efficiency testing can be seen in Tab. 2, the iteration scheme usually converges after about
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4 iterations, and never takes more than 16 iterations before success. The same setup run on Lu’s
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solution is only slightly faster, but less accurate. Note that the observed failures of Newton iterations
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displayed in Tab. 2 are associated with target cell faces that are physically impossible to be reached
218
by the particle from the given initial time and location. As eq. 7 cannot be directly solved for time t,
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to our knowledge there is no a priori estimation whether a solution for t exists for a given flow field,
220
t0 and x0. The exit conditions which shall be discussed in § 4, are helpful to exclude some clearly
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impossible incidences, but do not cover all circumstances. Therefore inevitably a number of non-
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convergent Newton iterations have to be performed to find out that a target location is out of reach
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for the particle. Reachable exit faces, however, were found by the iteration procedure in every such
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case. Independent of the numerical method is the question of how many iterations have to be
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performed until the scheme can be considered non-convergent with certainty in case no solution
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exists, i.e. the particle cannot exit though the given cell face within the time step. In that respect our
227
analyses indicated that a pre-defined number considerably larger than 16 may be appropriate.
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A further issue of consideration is the strict conservation of water mass in flow between adjacent
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grid cells within the timestep subject to transient flow conditions. Contrary to bilinear interpolation
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of the flow velocity between two spatial directions, which is known to violate the water (mass)
231
balance [Cordes & Kinzelbach, 1992, Salamon et al., 2006], the spatio-temporal bilinear interpolation
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described above preserves mass. Given that the water balance is closed for each moment in time,
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any linear combination of water balances between two points in time will also preserve mass. A
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further necessary condition for water mass preservation (accounting for stationary as well as
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transient conditions in general) is that the facial flow velocity components have to be computed as
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discharge between cells over the interfacial area and cell-specific volumetric water content  (i.e.
237
porosity in fully saturated media). In other words, the flow velocity has to be discontinuous at the
238
cell face in order to preserve water mass exchange between the cells, in case of heterogeneous
239
porosity. As we do not interpolate  spatially between adjacent grid cells, mass is conserved by our
240
method.
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4. Numerical implementation
11
243
An important task for implementing such analytically derived particle trajectories for larger flow
244
fields is the handling of particle transfers over the boundaries of adjacent grid cells in the framework
245
of finite difference or finite volume methods. Pollock [1988] has discussed the case selections to
246
identify potential particle exit locations extensively. After identification of potential particle exit
247
faces, exit times are computed and the exit face with the shortest exit time is assigned as the exit
248
location. In short, the criteria characterizing a potential exit face can be stated for a steady state flow
249
field as follows:
250

Face velocity points outward of cell (inward for back tracking)
251

Initial particle velocity within the cell points towards given cell face (away from that cell for
252
back tracking)
253
For an actual particle exit, both conditions have to be given at the same time point. For a transient
254
flow field within the grid cell the exit conditions change to a more complex situation:
255

256
257
Cell face velocity at the start or end of the current timestep points outward of cell (actual
outflow occurs)

Particle location at the end of the timestep is outside of cell boundaries; or particle velocity
258
has been changing direction between initial and final particle location. (In the latter case the
259
particle might have left and re-entered the cell, if observed inside after the move.)
260
Thus, all potential exit faces have to be considered, which will increase the number of computations
261
compared to the stationary case. After identification of exit time, an actual location where the
262
particle pathline intersects the cell face must be checked for an outward pointing velocity. This is
263
necessary to exclude the mentioned leaving and re-entering of the cell under transient conditions.
264
A flowchart delineating the computational process of pathline and travel time computation within
265
meshes of a FDM/FVM framework is provided in Fig. 2. The procedure is very similar to the one
12
266
presented by Pollock [1988] with the difference of the iterative solution of exit time to be performed
267
under the preconditions described above.
268
(Fig. 2)
269
270
5. Code demonstration
271
The described method was implemented in a numerical fortran 90 scheme, further referred to as
272
“PTRANS”, which handles transient particle tracking for rectangular FD or FV grids. An example of
273
transient particle movement (computed by our semi-analytical method) within a single rectangular
274
cell is shown in Fig. 3, of which the velocity field in the y-direction corresponds to Fig. 1. The particle
275
coming from the left cell boundary performs a loop and leaves the cell at the top boundary.
276
Information on the flow field within the cell and time step is given in Tab. 1. The accurate analytical
277
solution is compared (i) to a brute force numerical integration of eq. 5 in 103 substeps and (ii) to the
278
method of Lu, in a single step, and increasing numbers of substeps until it converges to the accurate
279
solution. The solution of Lu [1994] becomes quite inaccurate in the given case of ax ~ -49 and ay ~ 96
280
and tends to perform the loop far outside the grid cell. Increasing time step resolution applied to the
281
method of Lu [1994] improves its accuracy, however, it can be seen that about 6000 substeps have
282
to be used to achieve reasonable agreement. Interestingly, the numerical solution (dashed blue line),
283
which shows a good agreement after dividing the whole timestep into 1000 substeps, is more
284
efficient than the method of Lu [1994].
285
(Fig. 3)
286
A second example is shown in Fig. 4 where a coarsely discretized aquifer is experiencing flow
287
reversals by temporally varying infiltration and exfiltration due to river stage fluctuations. The initial
288
water table at z = 5 m is kept constant at the far right boundary at x = 50 m, whereas the water level
289
boundary at x = 0 m rises linearly to z = 9 m after 1.6 days and subsequently drops down linearly to
13
290
an elevation of 3 m until the end of the simulation (4 days). The particles were released at the same
291
locations at different times (between 1.2 days and 2.0 days) and can be seen to shift to the right
292
initially and turning left afterwards. Particle release locations are marked with rectangles. The
293
hydraulic conductivity was 3.162E-04 m/s, the specific storage coefficient was 1.0E-03 m-1 and the
294
porosity 35 %. x = 2 m, z = 1.1 m and a large timestep of 0.2 days was applied. The dataset shown
295
in Fig. 4 was run again using only the method of Lu [1994] for pathline computation, which requires
296
slightly less CPU-time, but somewhat more Newton-iterations, as Lu’s solution also requires
297
iteratively solving for exit time [Lu, 1994]. Results are similar in general, but inaccuracies due to Lu’s
298
simplified flow field are noticeable (green line, long dash). An overview of computational
299
performance of the two model runs is given in Tab. 2.
300
(Fig. 4)
301
302
6. Conclusion
303
In this technical note we extended the particle tracking schemes of Pollock and Lu to fully transient
304
conditions. Our semi-analytical method allows for accurate pathline computation by a closed form
305
expression which can be easily implemented with modern programming languages and incorporated
306
into existing finite difference and finite volume models that use cell-centered, structured, regular
307
grids. Our method closes the gap of existing particle tracking approaches in this context to general,
308
transient, non-stationary flow, which becomes important in situations where flow reversals appear
309
on a short time scale, e.g. river-aquifer interaction in the riparian zone, pulsed well pumping, or wave
310
motion at coastal aquifers. The extension of the presented formula to irregularly shaped finite
311
volume grids using the approach of Cordes and Kinzelbach [1992] could be a promising future task.
312
313
Acknowledgments
14
314
315
316
This work was supported by a grant from the Ministry of Science, Research and Arts of BadenWürttemberg (AZ Zu 33-721.3-2) and the Helmholtz Center for Environmental Research, Leipzig
(UFZ).
317
References
318
Bensabat J., Zhou Q., Bear, J. (2000). An adaptive pathline-based particle tracking algorithm for the
319
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Eulerian-Lagrangian method. Advances in Water Resources 23 (2000) 383-397.
Cheng, H. P., Cheng, J. R. and Yeh, G. T. (1996). "A particle tracking technique for the Lagrangian-
321
Eulerian finite element method in multi-dimensions." International Journal for Numerical Methods in
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Engineering 39(7): 1115-1136.
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Cordes C, Kinzelbach W. Continuous groundwater velocity fields and path lines in linear, bilinear, and
trilinear finite elements. Water Resour Res 1992;28(11):2903-11.
Dogrul, E.C., Kadir, T.N. (2006), Flow Computation and Mass Balance in Galerkin Finite-Element
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Groundwater Models, J. Hydrologic Engineering, 132 (11), doi: 10.1061/_ASCE_0733-
327
9429_2006_132:11_1206_
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Lu, N. (1994): A semianalytical method for pathline computation for transient finite-difference
groundwater flow models. Water Resources Research 30(8), 2449-2459.
Lynch, D.R. (1984): Mass conservation in finite element groundwater models, Adv. Water Resour.
7(2), 67-75.
Park, C-H. and Aral, M. M. (2007). Sensitivity of the Solution of Elders Problem to Density, Velocity
and other Numerical Perturbations, Journal of Contaminant Hydrology, 92: pp.33-49.
Pollock, D.W. (1988): Semianalytical computation of pathlines for Finite-Difference Models. Ground
Water 26(6), 743-750.
Pollock, D.W., 1994, User's Guide for MODPATH/MODPATH-PLOT, Version 3: A particle tracking
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post-processing package for MODFLOW, the U.S. Geological Survey finite-difference ground-water
338
flow model: U.S. Geological Survey Open-File Report 94-464, 6 ch.
339
Salamon, P., Fernandez-Garcia, D., Gomez-Hernandez, J. (2006). A review and numerical
340
assessment of the random walk particle tracking method. Journal of Contaminant Hydrology, 87,
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277-305.
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Suk, H. and Yeh, G. T. (2009). "Multidimensional Finite-Element Particle Tracking Method for Solving
Complex Transient Flow Problems." Journal of Hydrologic Engineering 14(7): 759-766.
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Suk, H. and Yeh, G. T. (2010). "Development of particle tracking algorithms for various types of finite
elements in multi-dimensions." Computers & Geosciences 36(4): 564-568.
Suk, H. (2012). "Practical Implementation of New Particle Tracking Method to the Real Field of
Groundwater Flow and Transport." Environmental Engineering Science 29(1): 70-78.
Yeh, G.T. (1981), On the computation of the Darcian velocity and mass balance in the finite element
modeling of groundwater flow. Wat. Resour. Res. 17, 1529-1534.
Yeh, G.T., Cheng, J.R., Cheng, H.P., and Jones, N.L., 1997, FEMWATER: A three-dimensional
351
density-dependent flow and transport in variably saturated media, US Army, Vicksburg, MS.
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Zhang, S. and Jin, J. M., Computation of Special Functions. New York: John Wiley & Sons, 1996.
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Figure captions
358
Fig. 1. Sketch of a velocity field v(x, t) within a timestep between time 0 and T and spatial dimension
359
x between 0 and L. This graph represents the spatio-temporal velocity distribution shown in the y-
360
dimension of Fig. 2. The red-framed plane which cuts in the middle is the simplified velocity field
361
derived from the method of Lu (1994).
362
Fig. 2 Flowchart for computational particle tracking procedure using regular, rectangular grids in the
363
framework of FD or FVM models.
364
Fig. 3. Particle trajectory within a cell during one timestep using the analytical solution (red line,
365
diamond symbols), and 1000 substep numerical solution (blue dashed line, triangle symbol). Data are
366
listed in Tab. 1. The green dash-dotted line (rectangle symbols) represents the approximation by Lu
367
(1994), for which the loop extends outside the boundaries of the cell. Thin black lines show Lu’s
368
solution divided into increasing numbers of substeps, eventually converging to the exact solution.
369
Fig. 4. Particle trajectories for different release times in a grid of coarse spatial and temporal
370
resolution, and a flow field undergoing a shift from rightward to leftward flow direction.
16
371
372
373
374
375
17
376
Tables
377
Tab. 1. Data used for single grid cell example simulation shown in Fig. 3. Time step
378
ranges from zero to 0.5, with particle release time at time zero. (relative units). The
379
flow field for the y-direction is illustrated in Fig. 1.
x-direction
y-direction
L
1.6
1
x0
0
0.5
v0,0
24
-6
vL,0
17.12
0.3
v0,T
23
0
vL,T
-61.24
31.15
a
-96.7
b
-4.3
6.3
c
-2
12
d
24
-6
49.7
380
381
18
382
Tab. 2. Computational performance and efficiency comparison between the proposed
383
method and the method of Lu (1994) for the dataset shown in Fig. 4, tracking 12
384
particles and calculating 191 particle distribution snapshots in time for each, on a
385
2.40 GHz 32 bit Intel CPU.
Performance
This method
Lu (1994)
Total number of Newton-iteration
142
143
Number of failed calls:
53
67
Total number of Newton-iterations
881
1787
Number of NI involved in success:
391
504
Average number of Newton-
6.2
12.5
4.4
6.6
max. number of NI until success:
16
19
Total used CPU-time
0.39
calls:
(NI)
iterations
Average number of successful
Newton-iterations
sec
0.28
sec
386
19
387
Figures
388
389
390
Fig. 1. Sketch of a velocity field v(x, t) within a timestep between time 0 and T and spatial dimension x
391
between 0 and L. This graph represents the spatio-temporal velocity distribution shown in the y-
392
dimension of Fig. 3. The red-framed plain which cuts in the middle is the simplified velocity field
393
derived from the method of Lu (1994).
394
395
396
20
R ead particle s tarting location
As s ign particle to local grid cell
C ompute cell face velocity components
C ompute particle location after current time-s tep
C ompare particle velocity before and after move
P article might leave current
cell during move?
yes
no
D etermine potential exit faces
(Newton-iteration)
C ompute cell trans it time
and
determine actual exit face
Are locations at intermediate
time required?
yes
C ompute coordinates at
intermediate times
no
Write particle coordinates to file
D etermine new cell location
D is charge point?
no
yes
S top particle
397
Fig. 2 Flowchart for computational particle tracking procedure using regular, rectangular grids in the
398
framework of FD or FVM models.
21
399
400
Fig. 3. Particle trajectory within a cell during one timestep using the analytical solution (red line,
401
diamond symbols), 1000 substep numerical solution (blue dashed line, triangle symbol). Data are
402
listed in Tab. 1, flow field of the y-direction is illustrated in Fig. 1. The green dash-dotted line
403
(rectangle symbols) represents the approximation by Lu (1994), for which the loop comes to perform
404
outside the boundaries of the cell. Thin black lines show Lu’s solution divided into increasing
405
numbers of substeps, eventually converging to the exact solution.
406
22
407
408
Fig. 4. Particle trajectories for different release times in grid of coarse spatial and temporal
409
resolution, and a flow field undergoing a shift from rightward to leftward flow direction.
410
411
23
412
To be added to Fig. 4
413
Plotted area:
9m
Fixed head 5 m
3m
5m
z
x
50 m
24