Tracking Contaminant Transport Backwards with an Operator-Splitting Method

Abstract

Recovering the past movement of a contaminant plume from measurements of its current values is a challenging problem in hydrology. Moreover, modeling the movement of a contaminant plume backwards is an ill-posed problem due to the unstable and non-unique nature of the resulting solution. Therefore, standard numerical methods become unstable, making it impossible to simulate existing contaminant transport models with reversed time. This paper presents two major contributions to solve the backward problem. Firstly, a stable and consistent numerical method based on an operator-splitting concept which is effective in tracking back the contaminant movement, and secondly, an optimal condition for the choice of mesh width that enables the error during computer simulation to stay within a reasonable bound. The numerical method was validated by introducing errors of varied strengths at the starting point and reconstructing the contaminant profiles backwards at any given time.

1. Introduction

While everyday chemicals provide a range of benefits to society, they are often released into the waterbodies and can cause unintended harm. Over the years, water pollution by various contaminants from human activities has been posing a major threat to freshwater resources all over the world. This has gained widespread attention in studying the movement of contaminants by many hydrologists and environmentalists in order to implement remediation processes in the hope of reducing the damage to the environment. A few years ago, one of us studied the transport of contaminants in groundwater under various sorption isotherms and was able to obtain analytical solutions for the contaminant profiles [1,2,3].

One of the potential difficulties faced in tracking back a contaminant is that very little information is known about the movement of a contaminant through waterbodies, so it becomes challenging to implement remediation efforts. If we know how the contaminant has moved through a waterbody and track back its movement, efforts can be made to treat the contaminant and reduce the extent of damage caused. Therefore, it is crucial to track the contaminant transport in reversed time and understand its movement.

Contaminant transport has an inherent dispersive property. Thus, the backward modeling of this phenomenon is an ill-posed problem. Precisely, a problem is called ill-posed if the solution does not satisfy the conditions of a well-posed problem, which are existence, uniqueness, and stability. The stability here refers to the case that the solution depends in a stable manner on the auxiliary conditions and that small changes in the conditions lead to only small changes in the solution. An ill-posed problem causes large errors in the output even for a small error in the observed data. Hence, most of the standard numerical methods perform poorly. Therefore, one needs to develop a stable and consistent numerical method to solve the ill-posed problem.

There have been a number of studies on tracking back contaminant movement to preceding times. Ref. [4] looked at this problem using an optimization approach. However, it was sensitive to measurement errors in contaminant concentrations. Ref. [5] advocated a trial-and-error method to approximate the contaminant movement at preceding times. However, this method was claimed to be sensible only for a selective choice of conditions and hence was declared ineffectual in approximating the contaminant movement generally. In addition, several other techniques have been explored to study the backward contaminant movement and identify the contaminant source; these can be broadly classified as deterministic and/or probabilistic approaches.

Amongst the methods developed for ill-posed problems, the regularization method gained popularity and has been widely used. For example, approaches such as Tikhonov regularization [6,7,8], least-square estimation from analytical approximations [9], non-regularized nonlinear least squares [10], the backward beam equation method [11,12], the method of quasi-reversibility [13,14], and least-square solution as an optimal control problem [15] have been studied. These approaches have been successful in approximating the contaminant source in reversed time from a source of known site but do not provide an estimate for the error associated with the contaminant movement in reversed time. In fact, the performance of these methods is highly influenced by the measurement errors present in the contaminant concentrations recorded at the initial site.

The probabilistic methods explored involve using an optimization model approach [4]; statistical pattern recognition technique [16]; random walk particle tracking [17]; geostatistical techniques [18,19,20,21,22]; adjoint methods [23]; the minimum relative entropy idea [24,25]; and a hybrid Monte Carlo technique [26]. Even though these methods improve some of the shortcomings of the deterministic methods, they are still sensitive to measurement errors. One should note that option pricing models in financial mathematics are also inverse diffusion problems. In a recent article [27], an option pricing model, based on an uncertain fractional-order differential equation, studied the lookback option. Interestingly, unlike an ill-posed problem, where a solution that depends continuously on the initial data does not exist, a lookback option is path-dependent. However, it should be possible to make use of some computational ideas developed for option pricing models for backward contaminant transport problems. Similarly, methods based on renormalization inversion theory in combination with computational fluid dynamics [28], which are used in atmospheric sciences for reconstructing tracer sources, could be adapted for backward contaminant transport problems as well. Recently, Refs. [29,30,31] looked at recovering the contaminant source using a machine learning approach. This approach uses Non-Negative Matrix Factorization (NMF) combined with a custom-made semi-supervised $k$-means clustering algorithm. However, this approach overlooks the physical nature of the problem and thus, at times, might present unrealistic values of contaminant concentrations. In the same vein, a deep neural network model [32] that combines first principles was used to study hydraulic fracturing. The interesting aspect of this work is that the deep neural network was used to estimate any unknown process parameters in hydraulic processing. This is unlike the situation in a contaminant transport model, where, based on laboratory and field experiments, good estimates are already available for the parameters of the model. However, if, in certain contaminant transport situations, one is unable to find good estimates for some parameters, then the deep neural network idea presented in [32] can be valuable.

A numerical method may be consistent and stable for a forward problem, but will perform poorly on the corresponding backward problem. This is due to the ill-posed nature of the backward problem, where any measurement error that may be present in the initial condition recorded at any time $T$ tends to grow exponentially with increasing time steps. The objective of this paper is to develop an innovative and simplistic method to capture the movement of a contaminant travelling backwards in time. We propose a deterministic approach using an operator-splitting method to track back and reconstruct the contaminant movement, that is stable and not significantly impacted by measurement errors. The motivation for using operator-splitting is due to its ability to break down a complex problem to simpler ones. The highlights of this approach are the simplicity of implementation and the optimal mesh width condition that contains the growth of error in a reconstructed contaminant profile.

2. Materials and Methods

2.1. Formulation of the Forward Problem

The transport of a contaminant plume can be described mathematically in terms of dispersion and advection processes. A plume observed at a certain time $t>0$, with a constant flow velocity c and dispersion coefficient $D$, can be formulated by Equation (1) below. If we solve (1) for given initial data at t = 0, we can obtain the solution at any time $t=T$.

The advection–dispersion equation:

$$\begin{array}{c}\begin{array}{c}{u}_t=D{u}_{xx}-c{u}_x\\ -\infty <x<\infty ,0\le t\le T\end{array}\}\end{array}$$

(1)

where the initial contaminant concentration is given by

$$u\left(x,t=0\right)=f\left(x\right).$$

(2)

Here, $u\left(x,t\right)$ is the contaminant concentration, at spatial distance $x$ and time $t$. $f\left(x\right)$ is the contaminant concentration at $t=0$. Now, $-\infty <x<\infty$ is an infinite domain and in order to carry out numerical computations, we need a finite domain $\left[L_1,L_2\right],$ where $L_1$ and $L_2$ are large in magnitudes, to mimic $-\infty <x<\infty$. At the finite boundaries, we impose the periodic boundary condition

$$u\left(L_1,t\right)=u\left(L_2,t\right).$$

(3)

2.2. Operator-Splitting Method

Operator-splitting techniques have been widely adopted for solving complex problems because of their simplicity and efficiency. There has been good evidence of the performance of operator-splitting methods [33], and they have proved to be computationally fast. An operator-splitting method has numerous advantages, mainly dimension reduction, simplifying a complex problem by maintaining the order of the method, and computationally speeding up the performance of the method.

In this paper, we start by applying the operator-splitting method on the forward problem given by (1). The operator-splitting method will split the forward problem into two sub-problems consisting of just the advection part and the dispersion part, respectively. So, the sub-problems are as follows:

$$u_t=c{u}_x$$

(4)

$$u_t=D{u}_{xx},$$

(5)

where (4) is the advection process while (5) is the dispersion process. The procedure of obtaining the numerical solution includes solving the advection process as the first step and the dispersion process as the second step using the results of the first step. The procedure is started by solving the advection process (4) using the initial condition of (1) for time step $\Delta t$. The result obtained by solving the advection process is an intermediate value and is used as the new initial condition for the dispersion process (5). Subsequently, we will solve (4) and (5) sequentially at each time step $\Delta t$ to advance the solution in time. Let $u_{m }^n$ represent the numerical approximation of $u\left(x,t\right)$ at $x=x_m=m\left(\Delta x\right)$ and $t=t_n=n\left(\Delta t\right)$, where $\left(\Delta x\right)$ and $\left(\Delta t\right)$ are the spatial mesh width and time step, respectively, on the computational domain $\left[L_1,L_2\right]$.

Then, our numerical formulation of (4) and (5) is

$$u_m^{n^{*}}=\left[1-p\right]u_m^n+\left[p\right]u_{m-1}^n,$$

(6)

$$u_m^{n+1}=\left[r\right]u_{m-1}^{n^{*}}+\left[1-2r\right]u_m^{n^{*}}+\left[r\right]u_{m+1}^{n^{*}},$$

(7)

where $p=\frac{c\left(\Delta t\right)}{\left(\Delta x\right)}$and $r=\frac{D\left(\Delta t\right)}{{\left(\Delta x\right)}^2}$.

Observe that the solution profile is moving with speed $c$ and so, for every time step $\left(\Delta t\right)$, the solution profile is moving a distance $c\left(\Delta t\right)$. Then, by choosing $\left(\Delta x\right)=c\left(\Delta t\right)$ (i.e., $p=1)$, we can simply shift the solution profile one mesh width at a time, for every time step, in the direction of the positive $x$-axis. Thus, we solve step 1, the advection part, exactly, i.e., there is no error due to the numerical method. In the case of step 2, in order to obtain a reliable solution, one needs to choose the mesh width and time step appropriately within any stability restriction. It is easy to show that the von Neumann stability analysis [34] for the numerical method given by (7) will give the stability condition

$$0<r\le \frac{1}{2}.$$

(8)

In [6], where the Tikhonov regularization method was studied on the advection–dispersion equation, the Gaussian–Hill function given below in (9) was used to validate their results. In the same spirit, we will also use this function

$$u\left(x,t\right)=\frac{5}{\sqrt{25+2t}}{e}^{\left(\frac{-{\left(x-30-ct\right)}^2}{50+4t}\right)}$$

(9)

in our work. In computing the forward-moving contaminant profile using (6) and (7), the initial condition is chosen as $f\left(x\right)=u\left(x,0\right)=e^{\left(\frac{-{\left(x-30\right)}^2}{50}\right)}$ from (9).

Figure 1 presents the numerical solutions obtained using the operator-splitting method for the forward problem (1). Here,

$$c=30,p=\frac{c\left(\Delta t\right)}{\left(\Delta x\right)}\mathrm{and}-250\le x\le 1500,$$

(9a)

and the parameters $\left(\Delta x\right)$ and $\left(\Delta t\right)$ are chosen such that $p=\frac{c\left(\Delta t\right)}{\left(\Delta x\right)}=1$.

Figure 1 assures that the operator-splitting method is able to capture the propagation of the contaminant successfully. Further, it shows that the relative error associated with the numerical solution does not just grow forever but seems to converge to a small and acceptable value with time. Now, let us consider the backward problem and see how we could apply the operator-splitting method to it.

2.3. Formulation of the Backward Problem

If one observes a contaminant profile at time t = $T$ and would like to find out what the profile was at time t = 0, then the question arises of how one can track back the profile to understand its movement at preceding times. In this study, we intend to solve the backward problem when information is available on the contaminant profile at any time $T$; that is, solving the backward problem with the initial condition given at time $T$. In order to comprehend and track the contaminant profile backwards, we obtain the backward problem using the transformation

$$\tilde{t}=T-t$$

(10)

on (1). Then, we obtain

$$u_{\tilde{t}}=-D{u}_{xx}+c{u}_x$$

(11)

subject to the initial contaminant concentration given by

$$u\left(x,\tilde{t}=0\right)=u\left(x,t=T\right).$$

(12)

We also impose the periodic boundary condition given below

$$u\left(L_1,\tilde{t}\right)=u\left(L_2,\tilde{t}\right).$$

(13)

The resulting backward problem given by (11)–(13) describes the movement of the contaminant at preceding times.

As in the case of the forward problem, we consider an operator-splitting method for solving the backward problem. In this case, the sub-problems associated with the advection process and the backward dispersion process are as follows:

$$u_{\tilde{t}}=c{u}_x$$

(14)

$$u_{\tilde{t}}=-D{u}_{xx}$$

(15)

The corresponding numerical formulations for (14) and (15) are given by

$$u_m^{n^{*}}=\left[1-p\right]u_m^n+\left[p\right]u_{m+1}^n,$$

(16)

$$u_m^{n+1}=\left[-r\right]u_{m-1}^{n^{*}}+\left[1+2r\right]u_m^{n^{*}}+\left[-r\right]u_{m+1}^{n^{*}},$$

(17)

where $p=\frac{c\left(\widetilde{\Delta t}\right)}{\left(\Delta x\right)}$and $r=\frac{D\left(\widetilde{\Delta t}\right)}{{\left(\Delta x\right)}^2}$.

Again, we choose the parameters such that $\left(\Delta x\right)=c\left(\widetilde{\Delta t}\right)$ (i.e., $p=$ 1), so the advection part (i.e., (16)) does not involve any computational errors. Now, in order to solve (16), at every time step, we shift the profile one spatial mesh width in the direction of the negative $x$-axis.

However, the von Neumann stability analysis on (17) will quickly show that the method in the form of (17) is unstable. So, the question is, how can one change the form of (17) so that it is a stable method?

We go back to Equation (11) and, following [35], introduce the variable transformation

$$u\left(x,\tilde{t}\right)=e^{a\tilde{t}}v\left(x,\tilde{t}\right),$$

(18)

where $a>0$ is a contraction factor to be determined later. Now, Equation (11) is transformed to

$$v_{\tilde{t}}=-D{v}_{xx}+c{v}_x-av,$$

(19)

subject to the initial contaminant concentration given by

$$v\left(x,\tilde{t}=0\right)=e^{-a\left(0\right)}u\left(x,\tilde{t}=0\right)=u\left(x,t=T\right).$$

(20)

As before, we impose the periodic boundary condition

$$v\left(L_1,\tilde{t}\right)=v\left(L_2,\tilde{t}\right).$$

(21)

The sub-problems that correspond to (14) and (15) in the new variable v are

$$v_{\tilde{t}}=c{v}_x,$$

(22)

$$v_{\tilde{t}}=-D{v}_{xx}-av.$$

(23)

Note that comparing (23) with (15) shows that (23) has an additional term ‘$-av$’.

The new numerical formulations, as opposed to (16) and (17), are

$$v_m^{n^{*}}=\left[1-p\right]v_m^n+\left[p\right]v_{m+1}^n,$$

(24)

$$v_m^{n+1}=\left[-r\right]v_{m-1}^{n^{*}}+\left[1+2r-a\left(\widetilde{\Delta t}\right)\right]v_m^{n^{*}}+\left[-r\right]v_{m+1}^{n^{*}}.$$

(25)

Again, solving (24) at every time step means shifting the contaminant profile by one spatial mesh width. However, now, because of an additional term ‘$-av$’ in (23), the von Neumann analysis of (25) will lead to a stability condition given by

$$\frac{4D}{{\left(\Delta x\right)}^2}\le a\le \frac{2}{\left(\widetilde{\Delta t}\right)}.$$

(26)

This means that if the contraction factor a is chosen according to (26), the method (25) for backward dispersion will be stable. So, once $v$ is obtained numerically using (24) and (25), we can obtain the solution $u$ of (11) by substituting back

$$u=e^{a\tilde{t}}v.$$

At this point, let us consider the situation where the contaminant data collected at time t = T have measurement errors (or noise). The question is, with noise in the initial data for the backward problem, how well can the contaminant profile be tracked back? Due to the ill-posedness of the backward problem, an existing error would amplify at each time step as we track back the contaminant profile. Thus, the tracked contaminant profile at any preceding time may not be acceptable or reliable.

There has been some theoretical work carried out by [36] for the backward dispersion problem. It was shown that given an a priori bound M on the solution u(x,0) at time $t=0$, and given the terminal data g(x) at time $T>0$ to a known accuracy $\delta$ to u(x,T), in the $L^2$ norm, i.e.,

$$||u{\left(x,0\right)||}_2\le M$$

(27)

$$||u\left(x,T\right)-{g||}_2\le \delta$$

(28)

then the optimal $a$ value is given by

$$a^{*}=\left(\frac{1}{T}\right)ln\left(\frac{M}{\delta }\right).$$

(29)

We make use of (29) along with (26) to choose the spatial mesh width and the corresponding time step in order to employ method (25) to solve the backward problem, i.e., we chose $a^{*}\ge \frac{4}{{(\Delta x)}^2}$ and found $\Delta x$ as

$${(\Delta x)}^2\ge \frac{4}{a^{*}}\mathrm{or}$$

(30)

$${(\Delta x)}^2\ge \frac{4}{\left(\frac{1}{T}\right)ln\left(\frac{M}{\delta }\right)}.$$

(31)

Simplifying yields,

$$\Delta x\ge \frac{2\sqrt{T}}{\sqrt{ln\left(\frac{M}{\delta }\right)}}.$$

(32)

In order to simulate the situation with experimental errors, we chose $g\left(x\right)$, the error-prone contaminant data at t = T, such that

$$g\left(x_i\right)=u_{exact}\left(x_i,T\right)+\epsilon *{\left(randomnumber\right)}_i,\forall i\in 1,\dots ,\left(m+1\right),$$

(33)

by generating $\left(m+1\right)$ random values between 0 and 1, where $x_m$ is a grid point. The strength of the error is $\epsilon$.

Let $u\left(x,\tilde{t}\right)$ be the solution of the backward problem given by (11). If appropriate parameters are chosen as discussed above, then for $\epsilon ={10}^{-1}$ we obtain

$$||u{\left(x,\tilde{t}=\mathrm{T}\right)||}_2\le M=2.9770.$$

$$||u\left(x,\tilde{t}=0\right)-{g||}_2\le \delta =0.4805.$$

For simplicity, we fix $M=3$ and $\delta =0.5$.

Then, the choice for $\Delta x$ is such that $\Delta x$ $\ge \frac{2\sqrt{T}}{\sqrt{ln\left(\frac{M}{\delta }\right)}}=4.7249$.

Hence, if we choose $\Delta x\ge 4.7249$, then our resulting numerical solution for the backward problem will be reliable.

3. Results

In this section, we present the results obtained by employing the operator-splitting method to the backward advection–dispersion equation given by (11).

First, in order to obtain the initial data (i.e., the solution profile at $t=T$), for the backward problem, we carried out the numerical simulation of the forward problem using $u\left(x,0\right)$ (from (9)) and obtained the numerical solution profile at $t=T$. Later, we used the numerical solution profile that we obtained for the forward problem at $t=T$ as the initial condition for the backward problem (11). Figure 2 presents the results of the numerical simulations.

It is evident from Figure 2 that our operator-splitting method is effective in tracking the contaminant profile back in time.

Now, let us consider the real scenario where the observed contaminant plume data might have measurement errors and, to mimic this scenario, we introduced different strengths of errors ($\epsilon ={10}^{-3}, {10}^{-2},\mathrm{and} {10}^{-1})$ to the Gaussian–Hill function (9) as described in (33). The numerical simulations are presented in Figure 3, Figure 4 and Figure 5.

Figure 3 and Figure 4 show the robustness of our method for initial errors that are small in strength. In Figure 5, even though the initial error is larger, our method is able to track the backward moving profiles fairly accurately. In order to illuminate the significance of choosing an appropriate $\left(\Delta x\right)$, we chose $\left(\Delta x\right)=2.5$, which violates the condition given by (32). Figure 6 presents the numerical simulations for this choice, and it clearly shows that the numerical solution blows up since the initial error intensifies as the contaminant profile moves backwards. Our numerical observations are summarized in

Table 1. Relative errors corresponding to the initial strength of error.

	Relative Error
$\mathbf{Initial}\mathbf{Error}\mathbf{Strength}\left(\mathit{\epsilon }\right)$	at $\tilde{\mathit{t}}=0$	at $\tilde{\mathit{t}}=2$	at $\tilde{\mathit{t}}=4$	at $\tilde{\mathit{t}}=6$	at $\tilde{\mathit{t}}=8$	at $\tilde{\mathit{t}}=10$	Relative Error Bound
${10}^{-3}$	$2.164\times {10}^{-3}$	$2.801\times {10}^{-3}$	$6.374\times {10}^{-3}$	$1.005\times {10}^{-2}$	$1.367\times {10}^{-2}$	$1.718\times {10}^{-2}$	$2.1\times {10}^{-2}$
${10}^{-2}$	$2.164\times {10}^{-2}$	$1.909\times {10}^{-2}$	$1.748\times {10}^{-2}$	$1.712\times {10}^{-2}$	$1.820\times {10}^{-2}$	$2.064\times {10}^{-2}$	$2.92\times {10}^{-2}$
${10}^{-1}$	$2.164\times {10}^{-1}$	$2.134\times {10}^{-1}$	$2.121\times {10}^{-1}$	$2.135\times {10}^{-1}$	$2.177\times {10}^{-1}$	$2.292\times {10}^{-1}$	$2.918\times {10}^{-1}$

below. Table 1 presents the recorded relative errors at different times for the corresponding initial strength of errors introduced. There is a gradual increase in the relative error as the initial strength of error is increased. However, note that in every case, the relative error stays within the theoretical bound as derived in Section 2.3.

4. Discussion and Conclusions

This paper shows how to construct a stable operator-splitting method to study a backward advection–dispersion problem. The beauty of the method is that there is no computational error for the advection portion of the calculation. In addition, if the spatial mesh width is chosen optimally, the computational error for the dispersion part could be held within an acceptable bound. With respect to the contaminant transport problem, the numerical simulations presented in Figure 2, Figure 3, Figure 4 and Figure 5 provide evidence that our operator-splitting method can track back the contaminant profile successfully and reliably. Additionally, they highlight that the operator-splitting method is effective, and the error of the numerical solution can be reasonably bounded with an appropriate choice of $\left(\Delta x\right)$. Interestingly, the error seems to decrease at the beginning stages before creeping up a little. It should be pointed out that in order to validate the operator-splitting method further, we carried out a number of numerical simulations for $T=20$ and $T=40$ with $c=20, 10, 5, \mathrm{and} 1$, respectively, and found that the method is effective and robust for appropriate choices of mesh width and time step.

The results presented in this work clearly demonstrate two important things. First, the operator-splitting method introduced here is very effective and computationally efficient in tracking back the movement of a contaminant profile. Secondly, the mesh width can be chosen judiciously so that this operator-splitting method is not significantly influenced by any measurement errors one might have in the initial data. At this juncture, we would like to reiterate, as noted in the Section 1, Introduction, that every existing method in the literature for tracking a solution backwards is either very sensitive to measurement errors or does not address errors at all. Though we have presented only a few sample numerical simulations in this manuscript, we were able to validate the robustness of this method by carrying out a large number of simulations for varied parameter values. We feel that our work will help in expanding knowledge in the computational study of backward problems and thus advance the understanding of tracking reliable contaminant plumes backwards. Further, the method proposed here could be easily adapted to other situations in economic, physical, and biological sciences where one might need to back track a solution.