Convolution-based particle tracking method for ... - Springer Link

Comput Geosci (2012) 16:551–563 DOI 10.1007/s10596-011-9265-z

ORIGINAL PAPER

Convolution-based particle tracking method for transient flow Gowri Srinivasan · Elizabeth Keating · John David Moulton · Zora V. Dash · Bruce A. Robinson

Received: 3 May 2011 / Accepted: 8 November 2011 / Published online: 7 December 2011 © Springer Science+Business Media B.V. 2011

Abstract A convolution-based particle tracking (CBPT) method was recently developed for calculating solute concentrations (Robinson et al., Comput Geosci 14(4): 779–792, 2010). This method is highly efficient but limited to steady-state flow conditions. Here, we present an extension of this method to transient flow conditions. This extension requires a single-particle tracking process model run, with a pulse of particles introduced at a sequence of times for each source location. The number and interval of particle releases depends upon the transients in the flow. Numerical convolution of particle paths obtained at each release time and location with a time-varying source term is performed to yield the shape of the plume. Many factors controlling transport such as variation in source terms, radioactive decay, and in some cases linear processes such as sorption and diffusion into dead-end pores can be simulated in the convolution step for Monte Carlo-based analysis of transport uncertainty. We demonstrate the efficiency of the transient CBPT method, by showing that it requires fewer particles than traditional random walk particle tracking methods to achieve the same levels of accuracy, especially as the source term increases in duration or is uncertain. Since flow calculations under transient conditions are often very expensive, this is a computationally efficient yet accurate method. G. Srinivasan (B) · J. D. Moulton Theoretical Division, Los Alamos National Laboratory, PO Box 1663, Los Alamos, NM 87545, USA e-mail: gowri@lanl.gov E. Keating · Z. V. Dash · B. A. Robinson Earth and Environmental Sciences Division, Los Alamos National Laboratory, PO Box 1663, Los Alamos, NM 87545, USA

Keywords Solute transport · Particle tacking · Convolution · Transient flow

1 Introduction Transport in porous media is a subject of immense interest because it plays a central role in many physical, biological, and biochemical phenomena [2–5]. The process involves a combination of three passive transport mechanisms: advection, molecular diffusion, and hydrodynamic dispersion. Standard mathematical descriptions of such phenomena are based on a continuum approximation of these mechanisms in a representative element volume in combination with the conservation of solute mass. This approximation takes the form of a partial differential equation for the concentration and is commonly referred to as the advection–dispersion equation (ADE). Most numerical solutions of the ADE based on Eulerian discretization schemes introduce “numerical dispersion,” which manifests itself in excessive smearing of concentration profiles at large Péclet numbers. Since high degrees of anisotropy are often characteristic of hydrodynamic dispersion in porous media, this often results in unrealistic dilution and hence lower values of peak concentration. Numerical dispersion can be controlled by employing Eulerian–Lagrangian algorithms [6, 7], the method of characteristics [8], dynamic mesh refinement [9], total variation diminishing integration schemes (e.g., [10]), and other [11] algorithms, although many of these methods are computationally expensive. Particle tracking methods [12, 13] offer several advantages, most importantly the ability to maintain sharp

552

fronts in systems which are advection dominated, or where transverse dispersivities are two orders of magnitude lower than the longitudinal dispersivity [14]. Implementations of the particle tracking algorithm lend themselves naturally to parallel computing since particle trajectories are independent. Simulations using large numbers of particles can be broken down into subsets of smaller numbers of particles and run in parallel. There are also a few drawbacks of the particle tracking technique. It was shown in [15] that systems with abrupt changes in transport velocities or porosities may exhibit local mass conservation errors unless bi-linear or tri-linear interpolation schemes are used. Particle tracking methods can require a very large number of particles [16] especially when source terms are spread out spatially and/or temporally. The convolution-based particle tracking (CBPT) method that was recently developed in [1] is a computationally efficient approach that provides an accurate solution of plume concentrations using the particle tracking method. The CBPT is restricted to steady-state flow conditions and is valid for linear transport processes. It employs the principle of superposition in space and time to take advantage of efficiencies that can be gained under the assumption of linearity. In reality, however, steady-state flow conditions are rarely seen in the field. Transient flow conditions can be the result of several natural [17] and man-made [18] conditions. Examples include seasonal variations such as rainfall rates, the presence of water supply wells with varying pumping rates, etc. Neglecting transient effects in these cases could introduce substantial error in transport predictions. In the present study, we develop an extension to the CBPT method that removes the highly restrictive assumption of steady-state flow. The CBPT method requires a process model run, which consists of the flow calculations, and generation of particle trajectories for an initial pulse input. The transient CBPT method proposed here generally requires fewer particles in the underlying particle tracking process model run to resolve the transients in the flow with similar or greater accuracy than random walk particle tracking (RWPT) methods which require a finer resolution of temporal release of particles to simulate the source terms. Both the CBPT and RWPT methods depend on a flow simulation/particle tracking run, and so the computational demands are, for some applications, quite similar. In fact, in the case of single transport simulation considering a pulse source, the computational demands of the two methods are virtually identical. The efficiency advantage of CBPT only emerges in two specific (but common) types of applications: (1) where the source term is time varying and (2) in sensitivity or uncertainty

Comput Geosci (2012) 16:551–563

analyses requiring a large number of calculations, each with different transport parameters, when flow conditions remain unchanged. In the first type of application, the RWPT requires a time-varying release of particles to represent the time-varying source, whereas the steady-state CBPT method requires only one particle to represent any time-varying source. For applications with complex, time-varying source terms, this can be a very important advantage. In transient flow applications, this advantage is somewhat diminished but can still be significant. As will be discussed below, this is because in transient flow applications, the CBPT method requires time-varying particle releases to capture flow transients. In the second class of applications, uncertainty or sensitivity calculations, enormous computational efficiency can be gained by only having to conduct the CPU-intensive step (calculating particle paths) once and repeating the fast convolution a large number of times. Many factors controlling transport such as variation in source terms and radioactive decay can be simulated as part of the numerical convolution. In cases where the temporal variation in flow is sufficiently slow, sorption and diffusion into deadend pores can also be simulated in the convolution step. This advantage would apply to both steady-state and transient applications. Transport under transient conditions, which is the norm rather than the exception, can be solved in a computationally efficient manner using the transient CBPT method. The article is organized as follows: We first present the theory of convolution for transport under steadystate and transient flow conditions in Section 2. Section 3 outlines the numerical implementation details of the transient CBPT method including the error analysis due to the assumption of steady-state over sub-intervals. Section 4 documents the ability of this method to accurately and efficiently simulate plume concentrations for several test problems of increasing complexity with varying flow and transport parameters. The key results and conclusions are summarized in Section 5.

2 Convolution theory 2.1 Steady-state flow Solute transport in porous media can be described by the conventional form of the ADE for the rate of change of concentration as follows: ∂(θC) + ∇ · (θCv) − ∇ · (θ D · ∇C) = 0 ∂t

(1)

Comput Geosci (2012) 16:551–563

553

Here C is the resident concentration of the solute, defined as the mass of the solute per unit volume of fluid. θ is the porosity of the medium, v is the Darcy velocity vector, and D is the dispersion tensor. Let g(x, t − τ |x0 ) denote the Green’s function for Eq. 1 subject to the initial condition g(x, τ = t|x0 ) = δ(x − x0 ) denoting a pulse source located at x = x0 and appropriate homogeneous boundary conditions. The Green’s function is now thought of as the response to this pulse source. Concentration c(x, t) resulting from spatiotempo˙ 0 , t) (units of mass/time) rally distributed sources m(x can now be computed as t ˙ 0 , τ )g(x, t − τ |x0 )dx0 dτ, m(x

c(x, t) = 0

(2)

where is the computational domain. In other words, the response to any time-varying source can be computed as a convolution of the source term with the response to a pulse source. The Green’s function depends only on the time difference between the time at which concentrations are computed and the time at which the pulse source is activated. Hence, the Green’s function can be shifted in time if the time of release of the pulse source is not at t = 0. The Green’s function g(x, t|x0 ) can also be expressed in terms of particle trajectories as [20] g(x, t|x0 ) = δ [x − X(t)]|X(0)=x0 ,

(3)

where the angular bracket denotes the ensemble average over the random vector ξ . A standard RWPT [19] solves the ADE (1) by recognizing its equivalence to the Fokker–Planck equation and, hence, to Ito’s interpretation of the Langevin equation [20] √ X(t + t) = X(t) + A[X(t)]t + B[X(t)] · ξ t. (4) Here X(t) is a particle’s location at time t, and ξ is the transition-length vector which is uncorrelated following a Gaussian distribution with zero mean and unit variance. The background drift vector A and the second-order displacement tensor B are related to the coefficients in Eq. 1 by A = v + ∇D,

BB = 2D. T

(5)

Due to the advantages that particle tracking methods mentioned already, we use the particle tracking algorithm 3 and 4 to compute the Green’s function g(x, t − τ |x0 ) in this work. The discussion on adapting the method for transient flow is presented next, in the context of a particle tracking base process model run.

The concepts presented are general and remain true regardless of the method used to generate the process model run, i.e., the particle trajectories. 2.2 Transient flow Transport under steady-state flow conditions can be fully described by a collection of particle trajectories, each released at a single time, say time t = 0. However, in the case of transport in a transient flow field, solute particles entering the flow field at different times experience different flow fields. Hence, in order to accurately describe the temporal nature of the transport, particles must be released at each source location throughout the time domain of transport, such that one may approximate steady-state flow conditions over each interval of particle release. In other words, particles are released at a sequence of times tn ∈ [0, T], and we compute a separate Green’s function gn (x, t|x0 ) for each particle release. The interval of particle release depends upon the transients present in the flow; the number of particles released at each time depends on transport parameters such as dispersivity. Thus, we divide time-varying source into N sub-sources, to coincide with the time of particle release such that ˙ m(x, τ) tn < τ < tn+1 , ˙ n (x, τ ) = m (6) 0 otherwise. ˙ The original source term m(x, τ ) can now be considered ˙n a superposition of the individual sources m ˙ m(x, τ) =

N

˙ n (x, τ ). m

(7)

n=0

Each of the sub-sources now follows the set of particle trajectories corresponding to those particles released at the start time of the source. The concentration cn due to each sub-source mn can be written as a convolution integral t ˙ n (x0 , τ )gn (x, t − τ |x0 )dx0 dτ. m

cn (x, t) = 0

(8)

The total concentration is simply the sum of the individual concentrations, obtained by the principle of superposition and c(x, t) =

N

cn (x, t).

(9)

n=0

The same principle of superposition applies for multiple sources, each of which can be sub-divided in time

554

Comput Geosci (2012) 16:551–563

into multiple sub-sources. Once the process model run is completed with a sufficient number of particles to represent the spatial distribution of the source terms as well as the temporal variations in the flow, the convolution technique can be applied for any number of sources with the same spatial distribution, but with different time-varying forms, without having to rerun the original process model. Similar to the original CBPT method for steady-state flow, the assumption of linearity must be made in order for the superposition technique to be valid. Hence, we can model processes such as radioactive decay with a linear rate constant. t c(x, t) = 0

˙ 0 , τ )g(x, t − τ |x0 )e−k(t−τ ) dx0 dτ. (10) m(x

However, the transients in the field introduce additional limitations which were not present in the case of steady-state flow. Although time-dependent processes such as sorption or diffusion into the matrix that essentially create an additional delay or retention in the cell could be modeled in the original steady-state formulation, these processes cannot generally be handled in the extension for transient flow, unless the transients in the flow are much slower in comparison with the time delay due to diffusion or sorption. The discussion so far has focused on the calculation of resident concentrations. Calculating flux-averaged concentrations involves approximating the liquid fluxes at various times of interest. This is theoretically possible but beyond the scope of the current research. It must be noted that the derivation presented above is general and that any numerical method can be employed for solving the transport equation. We choose a particle tracking method in our study due to the advantages of such methods mentioned already.

3 Numerical implementation details The complete details of the numerical implementation for steady-state flow conditions are described in detail in [1], and we summarize a few key points here briefly. The resident concentration is computed from the Green’s function g(x, t|x0 ) obtained from particle tracking results. The most efficient approach is to store information regarding the time history of the location of each particle and perform the contribution to the convolution integral on a particle-by-particle basis. For each particle and cell combination,we have tin , the time at which a particle enters a given cell, and tout , the

time at which a particle exits that cell. The resident concentration (please see [1] for details) is given by τin

c(x, t) =

p∈Np

τout

˙ m(x, t)e−k(τ −t) dt

Np (xo )θ (x)V(x)

(11)

where Np is the total number of particles associated with the given source location, k is the radioactive decay constant, θ(x) is the volumetric water content at this location, V(x) is the volume of the cell, τout = max(0, τ − tout ), τin = max(0, τ − tin ), and the summation in the numerator occurs over all particles spending time in the cell. The limits of integration define the time interval during which the input mass flux values must be determined in order to translate input mass into the cell at time τ , and the max functions ensure that only particles that have spent time in the cell at or before time are included. Simulation of particle tracking results in a time history of spatial locations and residence time distributions. For the model developed in the present study, we used FEHM is a finite volume heat and mass transfer code which uses the Burnett–Frind dispersion tensor and Pollock’s method for velocity interpolation within a cell. The particle tracking algorithm in the flow and transport code FEHM [21] was modified to output a keyword indicating transient flow conditions in the header of the file containing particle locations and times. Information is recorded at each time and cell number when a particle shifts from one cell to the next. Pre-processing scripts were developed to split each source term into sub-sources based on the timing of particle release in the process model run. The particle tracks generated by FEHM and the sub-sources form the input to the auxiliary module PLUMECALC which implements the CBPT method by performing the numerical convolution [1]. PLUMECALC was enhanced as part of this study to implement the CBPT method for transient flow. In addition to the mass flux and particle tracking information, the PLUMECALC code needs basic grid geometry such as cell volumes and the rock and transport properties of the original flow model on a cell-by-cell basis. Resident concentrations are computed at specified times at each cell for given time-dependent mass flux functions. The code performs the simulation through a numerical implementation of Eq. 11 for the convolution part of the model. Care must be taken to ensure that the particle trajectories generated by the process model run has sufficient temporal resolution to ensure that a particle’s stay in each cell that it passes through is documented. In this case, FEHM controls the time-step size in each cell

Comput Geosci (2012) 16:551–563

555

based on flow and medium properties to ensure that it is sufficiently small to allow for multiple steps taken within a cell. This is an additional level of precaution similar to choosing a sufficient number of particles to accurately describe a spatially varying source or choosing the appropriate grid resolution. The accuracy of the CBPT solution is dependent upon the input, i.e., particle trajectories, from the process model run.

For the more specific case of one-dimensional flow with velocity linearly varying with time considered in the examples presented here, the error involved in this approximation can be estimated as follows: For a known velocity in the system u at time t1 and acceleration a, the CBPT method assumes that the distance traveled Scbpt at any future time T by a particle released at time τ ∈ [t1 , t2 ] is

3.1 Error analysis

1 Scbpt = u(T − τ ) + a(T − τ )2 . 2

The transient CBPT method assumes steady-state conditions in the interval between consecutive particle releases. The error in concentration due to this approximation can be computed from Eq. 10. Assuming steady-state flow in the time interval [t1 , t2 ], particles are released at time t1 in the process model run. Let the corresponding Green’s function be given by g. The exact concentration at time T due to a source located at x0 which is non-zero in the interval [t1 , t2 ] and zero elsewhere is given by t2 ˙ 0 , τ )g(x, T − τ |x0 )dx0 dτ. m(x

cex (x, T) =

(12)

t1

Since the source function in the interval [t1 , t2 ] is assumed to follow the same pathline as the particles released at t1 , the concentrations computed by the CBPT method are given instead by t2 ˙ 0 , τ )g(x, T − t1 |x0 )dx0 dτ. m(x

ccbpt (x, T) =

(13)

t1

Suppressing the location of the source for convenience, the error in concentration is given by cex (x, T) − ccbpt (x, T) =

t2

˙ )g(x, T − τ ) m(τ

t1

− g(x, T − t1 )dx0 dτ.

(14)

The error in concentration is directly linked to the error in the Green’s function, which is the error in particle position. In the general case of a flow field with smoothly varying velocity, the velocity at time τ can be expressed in terms of the velocity and its time derivatives at t1 via the following Taylor expansion: u(τ ) = u(t1 ) +

∂u 1 ∂ 2 u (τ − t ) + (τ − t1 )2 + .... 1 ∂t t1 2 ∂t2 t1 (15)

(16)

The exact distance travel-led SEx is given by 1 Sex = (u + a(τ − t1 ))(T − τ ) + a(T − τ )2 . 2

(17)

The error in the distance travel-led is S = a(τ − t1 )(T − τ ).

(18)

The maximum error due to the approximation occurs at τ=

T − t1 . 2

(19)

Since we are most likely interested in the error due to this temporal discretization at large times, for T t1 , t2 , τ , this occurs when τ = t2 . In other words, the maximum error in position of a particle at any future time T is given by Smax = a(t2 − t1 )(T − τ ) ≈ at(T − t1 )

(20)

where t is the interval over which steady-state flow conditions have been assumed. Since the error in Green’s function or position is O(t), the error in concentration according to Eq. 14 is O(t2 ). From this estimate of maximum error in position, it is possible to arrive at a bound for t. The quadrature rule used in the calculation of the convolution integral 10 is the Riemann left sum, where the value at the left end point of the time interval is used to compute the integral. If we were to instead use the trapezoidal rule and approximate the integral with the average of the values at the two end points, t2 ctrap (x, T) = t1

1 ˙ ) (g(T − t1 ) + g(T − t2 )) dx0 dτ, m(τ 2 (21)

the error in concentration can be reduced to O(t3 ). Since the pathline for a particle released at t2 , which is used in the next time interval is already known, this method would be an improvement in the concentration estimates without having to release additional particles.

556

context to the steady-state examples and have been demonstrated in [1]. The transient CBPT algorithm was verified first for the steady-state flow scenario. A constant source of 33.333 mol/day (such as the one shown in Fig. 1) is subdivided into three sub-sources to coincide with particle release times and convolved with the particle trajectories to obtain the shape of the plume at all times. This test was performed to ensure that releasing particles at multiple times and sub-dividing a source into multiple sub-sources as would be the case for transient flow still yielded the same results as the original steady-state simulation. We consider two different flow scenarios. The first test scenario uses a transient flow condition with flow rate increasing linearly with time, where the transport

a

Source (mol / day)

Once the flow parameters such as the flow velocity and its rate of change are known, the choice of timestep is directly related to the maximum acceptable level of error in concentration. In realistic scenarios where flow is spatially and temporally varying, it is not possible to determine a priori the ideal timestep required. One good rule of thumb is to always choose a timestep no smaller than the timestep used in flow calculations. If the temporal flow profile is well understood, in general, it would be advisable to use more particle releases in regions of significantly varying flow. Fewer timesteps are required at times when transient effects are not significant. However, sensitivity analysis must be performed to determine the timesteps required for the desired accuracy in the concentration solution. As with choosing grid resolution or number of particles to be used in the base process model run, some trial and error is required to arrive at the optimum choice of userdetermined parameters. It is worth noting here that the error due to the steady-state approximation is unlike the error due to numerical dispersion. Errors due to the steady-state approximation over small time intervals results merely in packets of solutes traveling at an average or “group” velocity. This results in jagged profiles for the plumes rather than continuous one, if a sufficient number of particles are not used. There is no artificial spreading, as is the case with errors due to numerical dispersion, thus maintaining the sharp fronts characteristic of advection dominated transport.

Comput Geosci (2012) 16:551–563

15

10

5

0 0

50

100

4 Application to transport in 3D porous media

200

250

300

Time (days)

b

Source (mol / day)

We apply the new CBPT technique developed for transient flow to transport in macroscopically homogeneous media. The test scenarios have been designed to showcase various types of transient flows and the performance of the CBPT algorithm in such cases. The flow fields considered here vary linearly with time, but are spatially homogeneous in nature. Spatially heterogeneous flow fields are the result of storage in the medium or heterogeneities in the media, such as varying porosity and permeability. The pathlines of particles generated by the process model run are functions of the flow field and its derivatives, and hence, the effects of storage and heterogeneities are already factored into the particle pathlines. The convolution integral faithfully follows the particle pathlines and hence also reflects the effects of storage or heterogeneities. Application of the CBPT algorithm to heterogeneous media as well as radioactive decay are identical in the transient

150

15

10

5

0 50

0

50

100

150

200

250

300

Time (days) Fig. 1 Source functions: a constant continuous source and b pulse source at t = 0 days

Comput Geosci (2012) 16:551–563

is due to advection only. The second scenario involves a transient flow field which has a flow rate decreasing linearly with time up to a certain time and later remains a constant. In this scenario, we first start with advection only transport. We then gradually introduce more complexity in our test cases by including longitudinal and transverse dispersion terms. The test cases documented here use a simple threedimensional model with flow velocity aligned in the X-direction. The domain is discretized uniformly into 4,949 nodes, 101 in the X-direction (10 km model length), 7 nodes in the Y-direction (600 m model width), and 7 nodes in the Z direction (600 m model thickness). Within the region through which the simulated plume travels, the grid spacings are 100 m in all three dimensions. The flux at the inlet is prescribed, as a distributed water source. A constant head boundary is applied on the downstream (x = 10 km) plane, and no flow conditions are assumed on the other sides. The solute mass is input into the domain at the upstream end as a point source, centered in the middle of the Y–Z plane. Uniform permeability of 10−12 m2 and porosity of 0.25 are assigned throughout the domain. Since the flow field is unidirectional, all the plumes are one-dimensional in nature, except for the last case tested, namely that with non-zero transverse dispersion parameters. The goal is to verify the accuracy of the CBPT method for transient flow conditions and demonstrate the efficiency of the algorithm with respect to the RWPT method. 4.1 Verification techniques Two separate techniques are used to verify the results of the numerical simulations. The first is a semianalytical particle-based method, which can be used in scenarios where the transport is due to advection only and where the rate of change of velocity can be given as a simple function of time. Since for the sake of simplicity, the domain considered is a homogeneous medium with no storage, for known velocity v(0) at time t = 0 and the rate of change of velocity a(t), the velocity v(t) at any time t is given by t a(τ )dτ (22) v(t) = v(0) + 0

Since this method, like the CBPT, is also particlebased, in the case of a continuous constant source term, particles are released from t = 0 up to the final time t = t f . Since we are only interested in the time at which the concentrations need to be calculated, t = T, we set t f = T. Suppose N + 1 particles are released at equal intervals in t ∈ [0, T] with the time of releases

557 nT being [0, δt, 2δt, ...., tn , ...T] where tn = N+1 . Since the flow is a well-defined function of time v(t), a particle released at time tn starts with a velocity v(tn ). At the time of interest T, the distance traveled by a particle, Sn , released at time tn is given by t T T Sn = v(0) + v(t) · tdt = a(τ )dτ · tdt (23) tn

tn

0

Now that we have an analytical solution for calculating the distance traveled by each particle released in the interval [0, t f ], the concentrations are computed numerically, by releasing a certain number of particles over the time interval of interest. For a given T, we first count the number of particles in each cell based on the distance traveled by each particle. N(x, T) is the number of particles in the control volume associated with x at time T and is calculated according to N(x, T) = 1|Sp ∈ x (24) p

The resident concentration C(x, t) which is defined as the total mass in a cell divided by fluid volume is now given by C(x, T) =

N(x, T)m θ V(x)

(25)

where m is the mass associated with each particle and θ and V(x) are as defined previously. This method is used in verifying the steady state and linearly increasing flow scenarios. In cases where the velocity field is a welldefined function of space and time, this is an accurate, yet computationally very efficient technique, where as many as 107 particles can be used to compute the concentration profile. It must be noted here that since the typical change in porosity due to transient effects is less than 1%, our algorithm ignores changing volume in transient conditions as they are not significant enough to affect concentration calculations. The algorithm can also be translated into a computer program or script with relative ease. This technique is not capable of handling more complex flow scenarios, or non-zero diffusion, or heterogeneities in the medium. It is, however, a powerful tool in the validation of both the CBPT and the RWPT verification technique described next for the first two test scenarios. The second verification method is a RWPT method, which is a numerical technique (e.g., [12]). The source is simulated in the process model run, by releasing particles continuously throughout the length of the source duration. We assume a certain mass associated with each particle and the number of particles released depends upon the strength of the source. Additionally, when dispersion is considered in the transport model,

558

multiple particles may be released at every release time and the mass of the particle scaled accordingly. The output of the process model run only specifies particle locations at different times, so in order to compute concentrations, a post-processing step was added to count the number of particles residing in each cell at user specified times, to calculate the total mass in each cell and then to divide by fluid volume according to Eq. 25. As with any particle tracking method, the optimum number of particles needed to achieve a desired level of accuracy is not known beforehand, but only by trial and error. Often the number of particles used must be gradually increased until a further increase in the number of particles does not produce an increase in levels of accuracy. This method can be used with any flow scenario, even when diffusion or heterogeneities in the medium are present, and is used in the verification of all three test scenarios. However, it will be shown that it is computationally less efficient than the CBPT, i.e., requires many more particles to achieve an accuracy identical to that of the CBPT method.

Comput Geosci (2012) 16:551–563

at uniform intervals over T = 50 days and four times as many particles over T = 200 days. Since the flow is linearly increasing, given the initial velocity u and the acceleration a, a particle released at time tn starts with a velocity u + atn . In this method, velocity and acceleration are assumed to be functions of time only, and not space. At the time of interest T, the distance traveled by a particle, Sn , released at time tn is given by 1 Sn = (u + atn )(T − tn ) + a(T − tn )2 2

(26)

The RWPT (numerical) verification method uses a total of 10, 000 particles released uniformly over t = 300 days in the process model run. The concentration in both these cases is calculated according to Eq. 25.

a 5

x 10 Semi-Analytical RWPT CBPT

We first concern ourselves with a simple case of flow velocity increasing linearly with time from t = 0 to t = 300 days. Again, the simplest transport mechanism of advection is simulated both with the CBPT as well the RWPT algorithm for streamline particle tracking (SPTR). Two separate test cases with different source terms are used to demonstrate the efficiency of the CBPT algorithm. Since the convolution with the source function is handled by the CBPT algorithm, it is only necessary to run the underlying process model once using the SPTR module. The pathlines obtained as a result can be used to convolve with any number of timevarying source terms, in this case, a constant and pulse source term (similar to those shown in Fig. 1). In order to capture the temporal variations in the flow, 1,200 particles are released over the course of the simulation, uniformly spaced at the rate of one particle every 0.25 days (four in a day). The sources are divided into sub-sources for the constant source term, and each subsource is matched to the appropriate particle trajectory, where the source start time is the same as the time of particle release. In the case of the pulse source which is activated at time = 100 days, the trajectory followed is the same as that of particle number 401, released at time 100 days. The results for the concentrations obtained from the CBPT method are verified by the semi-analytical and numerical methods described in Section 4.1. In the semi-analytical method, we release 107 particles

3

2

1

0 0

2000

4000

6000

8000

10000

12000

distance along X (m)

b 5

x 10 Semi-Analytical RWPT CBPT

4

concentration (mol/l)

4.2 Linearly increasing flow


4

3

2

1

0 0

2000

4000

6000

8000

10000

12000

distance along X (m) Fig. 2 One-dimensional plume for linearly increasing flow with constant continuous source for advection only at a t = 50 days and b t = 200 days

Comput Geosci (2012) 16:551–563

559

a 1.4

30 particles


300 particles 600 particles 900 particles

0.8 0.6 0.4 0.2 0 0

2000

4000

6000

8000

10000

12000


b 6

x 10 1200 particles

5.5


5

10000 particles 4.5 4 3.5 3 0

2000

4000

6000

8000

distance along X (m) 6

x 10 30 particles

x 10

4

3

2

5.5


RWPT CBPT


100 particles

1

c 5

x 10

1.2


The plume is plotted at two different times for the constant (Fig. 2) and pulse (Fig. 3) sources. It can be seen in the case of the constant source that fewer particles are required with the CBPT method in order to obtain the same level of accuracy as the RWPT method (Fig. 4). In Fig. 4a, b, we show the concentration profile for the constant source at time 200 days using the RWPT technique. It can be seen in Fig. 4a that even for 900 particles, the concentration profile does not monotonically increase with distance. It takes as many as 10,000 particles as can be seen in Fig. 4b to capture the concentration profile accurately. With 1,200 particles, we obtain a similar accuracy with CBPT as that achieved by using 10,000 particles in the RWPT case. In fact, the CBPT solution for as few as 100 particles gets much closer to the actual solution than the RWPT solution using the same number of particles. Using the concentration computed with the semi-analytical method as a reference, the maximum error in CBPT concentration at any time is 9.6 × 10−9 mol/l (less than 2%). Similarly, L2 errors in concentration for both methods were computed and are listed in Table 1. This result demonstrates the conclusion that fewer particles are needed for the CBPT method to achieve similar accuracy as the RWPT method. While CBPT has been shown to require far fewer particles when the duration of the source term is long, in the case of the pulse source, only one particle is required in both cases. The CBPT method hence provides the same or better accuracy with the same number of particles. As the duration of the source term increases,

100 particles 300 particles

5


4.5 4 3.5

1

0 0

3 0

2000

4000

6000

8000

distance along X (m) 2000

4000

6000

8000

10000

distance along X (m) Fig. 3 One-dimensional plume for linearly increasing flow with pulse source released at t = 100 days for advection only at t = 200 days

Fig. 4 One-dimensional plume for the same linearly increasing flow with constant continuous source for advection only at t = 200 days for different number of particles in the case of a RWPT, b RWPT, and c CBPT

560

Comput Geosci (2012) 16:551–563

Number of particles

RWPT method

CBPT method

30 100 300 600 900 1,200 2,000 3,000 5,000 7,500 10,000

5.0431e−06 8.6617e−07 9.2213e−07 8.4980e−07 8.3123e−07 1.6924e−07 1.4068e−07 1.3800e−07 1.3615e−07 1.2558e−07 1.2471e−07

5.2693e−07 1.6542e−07 1.5358e−07 1.4911e−07 1.3598e−07 1.2467e−07

a 4

x 10

Semi−Analytical RWPT CBPT

3.5


Table 1 Error in concentration (L2 norm, units moles per liter) for RWPT and CBPT methods for a continuous source for increasing flow at time = 200 days

3 2.5 2 1.5 1 0.5 0 0

2000

4000

6000

8000

10000


b the CBPT method requires fewer particles and is more efficient than its counterpart.

5

x 10 Semi−Analytical RWPT CBPT

The next type of flow we consider is one that decreases linearly up to a certain time and then remains constant. Having demonstrated the use of the CBPT algorithm for various types of sources in the previous case, additional complexity in the transport is now introduced. We start with advection-only transport, add longitudinal dispersion, and, finally, add transverse dispersion, both horizontal and vertical. The longitudinal dispersivity is set to be α L = 100 m, and the transverse dispersivities are one tenth the value of the longitudinal dispersivity, αTH = αTV = 10 m. Whereas in the case of advection-only transport, one particle released at any time is sufficient to capture the transport-related characteristics, modeling dispersion requires more particles to accurately represent the distribution of transport path lines. A constant continuous source term is used. The source is sub-divided into as many time intervals as that of particle release, with each sub-source following a set of particles path lines instead of a single path line. The number of particles released was varied in order to gain better understanding of the performance metrics such as runtime and accuracy. For the case with advection, 1,200 particles were used in the CBPT method versus 10,000 for the RWPT, and the case with longitudinal dispersion used 6,000 and 60,000 particles, respectively. In these two instances, since transverse dispersion is neglected, the plume remains one-dimensional in nature. The plumes are shown in Figs. 5 and 6 for two different times. Figure 5 shows the results using CBPT and both verification techniques for the advection-only scenario. Figures 7 and 8 show the XY cross section of the three-


4

4.3 Decreasing flow

3

2

1

0 0

2000

4000

6000

8000

10000

distance along X (m) Fig. 5 One-dimensional plume for decreasing flow with constant continuous source for advection only at a t = 50 days and b t = 200 days

dimensional plume obtained by both methods, CBPT and RWPT. Comparable results were achieved by using 20,000 particles for the CBPT method as opposed to 120,000 for the RWPT method. In each case, the number of particles used in both methods was systematically increased until the desired accuracy was achieved. The CBPT results are in excellent agreement with the RWPT results for the advection-dominated scenarios. When the effects of dispersion are considered, the concentration profiles do not match exactly. Studies of the effects of dispersion in transient flow fields [22] show that lager values of macrodispersivity are observed in both longitudinal and transverse directions. This could possibly lead to more variations in flow paths when steady-state approximations are made over the small time intervals.

Comput Geosci (2012) 16:551–563

a 1.2

561

a

x 10 RWPT CBPT

0.8 0.6 0.4

2.5

x 10

200

distance along Y (m)


1

300

2 100 1.5 0 1 100 0.5

200

0.2 0 0

300 0

2000

4000

6000

8000


b 2

4000

6000

b

300

2

100 1.5

0 1

100

300

0 0

0.5

200

0.5

2000

4000

6000

8000

Fig. 6 One-dimensional plume for decreasing flow with constant continuous source for advection and longitudinal dispersion at a t = 50 days and b t = 200 days

Table 2 lists the run times and number of particles used for the test cases documented in this manuscript. All the test cases were run on a HP laptop with a 64bit AMD processor, 384 MB RAM, and 787 MHz CPU clockspeed. The number of particles used and runtimes for the steady-state scenario and the transient test cases

Table 2 Total computational time for the CBPT and RWPT methods for a continuous source

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000


10000


x 10

200


1

10000

2.5

x 10

1.5

8000


RWPT CBPT


2000

10000

Fig. 7 XY section of three-dimensional plume at z = 0 for decreasing flow with constant continuous source t = 50 days for a CBPT and b RWPT methods

shown in Figs. 2, 6, and 7 are documented. The number of particles needed for advection-only transport with decreasing flow (Fig. 5) is not shown, since the statistics were similar to those in Fig. 2. For each method, the time taken by the process model run (FEHM), the time taken by each method that calculated concentration (CBPT or the particle counting algorithm in case of RWPT), as well as the total time are shown. Since the

Test

Particles

CPU time CBPT (s)

Particles

CPU time

case

CBPT

FEHM

CBPT

Total

RWPT

RWPT (s)

Steady-state Figure 2 Figure 6 Figure 7

3 1,200 6,000 20,000

0.39 73.51 140.72 322.89

0.56 0.86 6.45 36.33

0.95 74.37 147.17 359.22

10,000 10,000 60,000 120,000

13.73 100.68 746.04 1,546.03

562

a

Comput Geosci (2012) 16:551–563

300 11

x 10

10

200


9 8

100 7 6

0

5 4

100

3 2

200

1

300 0

1000

2000

3000

4000

5000

6000

7000

8000

So not only are fewer particles required for the process model run, but once the underlying process model is run for a given flow and transport scenario, multiple cases of varying source terms can be simulated very rapidly. This is one example of where sensitivity studies can be performed much faster using CBPT. The test scenarios described here are merely illustrative and that the number of particles used will be higher in more complex scenarios involving spatially varying source terms or where the transients in flow are significant. The efficiency gained by using CBPT in realistic scenarios will vary depending upon complexities in the flow and source terms.

9000 10000


b

300

11

x 10

10

200


9 8

100 7 6

0 5 4

100

3 2

200

1

300 0

1000

2000

3000

4000

5000

6000

7000

8000

9000 10000


Fig. 8 XY section of three-dimensional plume at z = 0 for decreasing flow with constant continuous source t = 200 days for a CBPT and b RWPT methods

runtime is proportional to the number of particles, it can be seen that the CBPT method provides a computationally efficient way to obtain higher accuracies while using fewer particles. It can also be seen that the time taken by the CBPT algorithm is a small fraction of the total time taken for each full run (process model + CBPT). In the case of Figs. 2 and 3 where flow and transport scenarios are identical and the only variation is in the time varying source term, only one process model run is required. Hence, there was no separate process model run required for the case corresponding to Fig. 3. For the pulse source shown in Fig. 3, the only additional CPU time required by the CBPT model was for the convolution step, which took 0.41 s, whereas for the RWPT case, both flow and concentration computations had to be performed, needing a total of 70.66 s.

5 Conclusions We have presented an extension to the recently developed CBPT method for transient flow conditions. The transient CBPT method has retained several advantages of the original method over traditional RWPT techniques. In addition, it is capable of dealing with transient flow conditions which are present in typical transport scenarios in the field. The first is that it generally requires fewer particles to resolve the transients in the flow and use convolution subsequently than to simulate the source terms with a continuous release of particles for similar or greater accuracy. This is especially true with increase in duration of the source terms. Additionally, as with any particle tracking algorithm, the underlying process model used in this case lends itself to parallel computing since particle trajectories are independent. Hence, simulations using large number of particles can be broken down into subsets of smaller number of particles and run in parallel. Another advantage, as with the steady-state version, is that fewer simulations are required to explore the parameter space in a sensitivity study by allowing a variety of factors to be simulated from a single particle tracking run. Many factors controlling transport such as variation in source terms, radioactive decay, and in cases where the temporal variation in flow is sufficiently slow, sorption, and diffusion into dead-end pores can be simulated as part of the numerical convolution which is typically much faster, provided the underlying flow parameters remain identical. Since flow calculations under transient conditions often form the bulk of the computational expense, using just a single process model run where possible is of enormous computational advantage for Monte Carlo-based analysis of transport uncertainty. However, if any of the flow or dispersion parameters are varied, a new process model run would be required.

Comput Geosci (2012) 16:551–563

The verification simulations illustrated the accuracy of the CBPT method compared to more conventional RWPT approaches. It can be seen from the results documented here that as the duration of the source term increases, the number of particles required to resolve the transients in the flow are far less than the number of particles necessary to represent the timevarying source. While the original CBPT method [1] was restricted to steady-state flow conditions, the extension proposed here gives us an accurate yet highly efficient implementation for the more general transient flow condition. Acknowledgements This work was partially supported by the U.S. Department of Energy Underground Test Area Project and the U.S. Department of Energy Office of Science Advanced Computing Research (ASCR) program in Applied Mathematical Sciences. The authors would also like to thank Edward Kwicklis, Scott Painter and two anonymous reviewers for valuable discussions and comments.

References 1. Robinson, B.A., Dash, Z.V., Srinivasan, G.: A particle tracking transport method for the simulation of resident and flux-averaged concentration of solute plumes in groundwater models. Comput. Geosci. 14(4), 779–792 (2010) 2. Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972) 3. Stein, D., van der Heyden, F.H.J., Koopmans, W.J.A., Dekker, C.: Pressure-driven transport of confined DNA polymers in fluidic channels. Proc. Natl. Acad. Sci. 103(43):15853– 15858 (2006) 4. Koennings, S., Tessmar, J., Blunk, T., Göpferich, A.: Confocal microscopy for the elucidation of mass transport mechanisms involved in protein release from lipid-based matrices. Pharm. Res. 24(7), 1325–1335 (2007) 5. Rueda, F.J., Schladow, S.G., Clark, J.F.: Mechanisms of contaminant transport in a multi-basin lake. Ecol. Appl. 18(8), A72–A88 (2008) 6. Neuman, S.P.: Adaptive Eulerian–Lagrangian finite element method for advection–dispersion. Int. J. Numer. Methods Eng. 20(2), 321–337 (1984) 7. Younes, A., Fahs, M., Ackerer, P.: A new approach to avoid excessive numerical diffusion in Eulerian–Lagrangian methods. Commun. Numer. Methods Eng. 24(11), 897–910 (2008) 8. Chiang, C.Y., Wheeler, M.F., Bedient, P.D.: A modified method of characteristics technique and mixed finite ele-

563

9.

10.

11.

12.

13.

14.

15.

16.

17.

18. 19.

20.

21.

22.

ments method for simulation of groundwater solute transport. Water Resour. Res. 25(7), 1541–1549 (1989) Wolfsberg, A.V., Freyberg, D.L.: Efficient simulation of single species and multispecies transport in groundwater with local adaptive grid refinement. Water Resour. Res. 30(11), 2979–2991 (1994) Cox, R.A., Nishikawa, T.: A new total variation diminishing scheme for the solution of advective-dominant solute transport. Water Resour. Res. 27(10), 2645–2654 (1991) Ewing, E.R., Wang, H.: A summary of numerical methods for time-dependent advection-dominated partial differential equations. J. Comput. Appl. Math. 128(1–2), 423–445 (2001) Tompson, A.F.B., Vomvoris, E.G., Gelhar, L.W.: Numerical Simulation of Solute Transport in Randomly Heterogeneous Porous Media: Motivation, Model Development, and Application. UCID 21281. Lawrence Livermore National Laboratory, Livermore (1987) Tompson, A.F.B., Gelhar, L.W.: Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media. Water Resour. Res. 26(10), 2541–2562 (1990) Gelhar, L.W.: Perspectives on field-scale application of stochastic subsurface hydrology. In: Dagan, G., Neuman, S.P. (eds.) Subsurface Flow and Transport: A Stochastic Approach, pp. 157–176. Cambridge University Press, New York (1997) Labolle, E.M., Fogg, G.E., Tompson, A.F.B.: Random-walk simulation of transport in heterogeneous porous media: local mass-conservation problem and implementation methods. Water Resour. Res. 32(3), 583–593 (1996) A.E.Hassan, Mohamed, M.M.: On using particle tracking methods to simulate transport in single-continuum and dual continua porous media. J. Hydrol. 275, 242–260 (2003) Cirpka, O.A., Attinger, S.: Effective dispersion in heterogeneous media under random transient flow conditions. Water Resour. Res. 39(9), 1257–1271 (2003) Reilly, T., Pollock, D.W.: Sources of water to wells for transient cyclic systems. Ground Water 34(6), 979–988 (1996) Delay, F., Ackerer, P., Danquigny, C.: Simulating solute transport in porous or fractured formations using random walk particle tracking: a review. Vadose Zone J. 4, 360–379 (2005) Risken, H.: The Fokker-Planck Equation: Methods of Solution and Applications, 2nd edn. Springer Series in Synergetics, vol. 18. Springer, Berlin (1996) Zyvoloski, G.A., Robinson, B.A., Dash, Z.V., Trease, L.L.: Summary of the Models and Methods for the FEHM Application—a Finite-Element Heat- and Mass-Transfer Code. LA-13307-MS. Los Alamos National Laboratory, Los Alamos (1997) Rehfeldt, K.R., Gelhar, L.W.: Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers. Water Resour. Res. 28(8), 2085–2099 (1992)