# A Bayesian Assessment of an Approximate Model for Unconfined Water Flow in Sloping Layered Porous Media

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## Abstract

The prediction of water table height in unconfined layered porous media is a difficult modelling problem that typically requires numerical simulation. This paper proposes an analytical model to approximate the exact solution based on a steady-state Dupuit–Forchheimer analysis. The key contribution in relation to a similar model in the literature relies in the ability of the proposed model to consider more than two layers with different thicknesses and slopes, so that the existing model becomes a special case of the proposed model herein. In addition, a model assessment methodology based on the Bayesian inverse problem is proposed to efficiently identify the values of the physical parameters for which the proposed model is accurate when compared against a reference model given by MODFLOW-NWT, the open-source finite-difference code by the U.S. Geological Survey. Based on numerical results for a representative case study, the ratio of vertical recharge rate to hydraulic conductivity emerges as a key parameter in terms of model accuracy so that, when appropriately bounded, both the proposed model and MODFLOW-NWT provide almost identical results.

## Keywords

Dupuit–Forchheimer analysis Layered porous media Bayesian hypothesis testing Railway track drainage## 1 Introduction

The modelling of unconfined water flow in layered porous media is a challenging problem with important applications in Earth sciences and engineering. Relevant examples of such applications are found in the drainage of agricultural lands (Schmid and Luthin 1964), or the internal drainage of ballasted railway tracks (Rushton and Ghataora 2009), among others. An exact analytical solution to the problem is virtually impossible due to the nonlinearity of the unconfined boundary condition and the fact that the location of this boundary is unknown (Bear 1972). This modelling complexity is accentuated when dealing with sloping layered porous media with recharge (Rushton and Youngs 2010), which typically requires the use of numerical methods such as finite-difference (FD) (Wang and Anderson 1982; Todsen 1971; Lee and Leap 1997) or finite element (FE) models (Shamsai and Narasimhan 1991; Rulon et al. 1985; Chen et al. 2008; Zheng et al. 2009) to approximate the exact solution. However, these numerical methods become computationally demanding and hence non-feasible for several activities such as model calibration, parameter estimation and optimization, since such analyses require a great number of model evaluations. Groundwater numerical models, especially those considering unconfined flow with recharge (e.g. precipitation), may require significant CPU time to complete a single forward run. Approximate models can be used to address the computational complexity of the numerical models; however, they require a number of simplifying assumptions so that they can be solved analytically.

The Dupuit–Forchheimer (D–F) theory is perhaps the most powerful and widely accepted simplifying theory for treating unconfined flows (Bear 1972), although most of the available solutions are restricted to homogeneous isotropic porous media (Schmid and Luthin 1964; Wooding and Chapman 1966; Towner 1975; Chapman 1980; Yates et al. 1985; Knight 2005; Castro-Orgaz and Giráldez 2012). Despite its practical relevance, very few references can be found in the literature dealing with some form of approximate model to efficiently approach the problem of unconfined water flow in layered porous media with recharge. Youngs (1965, 1966) provided an analytical formulation of the unconfined seepage flow problem in soils with hydraulic conductivity varying with depth that was further extended in Youngs (1971) for sloping lands; notwithstanding, these works do not consider solutions for water table profiles. More recently, Youngs and Rushton (2009b) have provided an approximate model for steady-state water table prediction in two-layered undulating soils with recharge in the context of a railway track drainage problem, although it is restricted to systems with two parallel layers, which significantly bounds the practical scope of the proposed solution. Moreover, there are known limitations of the D–F theory based on the assumed simplifying hypotheses about seepage flow. Most authors agree that solutions must be restricted to problems where flow is essentially horizontal with a small inclination of the water table (Bear 1972; Lee and Leap 1997; Castro-Orgaz and Giráldez 2012). Others, in contrast, have shown that D–F solutions are sufficiently accurate even when water table slope is considerable and there is a significant vertical velocity component (Towner 1975; Youngs and Rushton 2009a, b). However, as evident from the results in Youngs and Rushton (2009a, b), the accuracy of the proposed D–F approximations greatly depends on the adopted values of some model parameters. These model parameters are not fitting parameters that need to be tuned nor estimated by comparing model predictions against observed data. Rather, these are physical parameters that represent the actual properties of the porous medium. Hence, it is important to identify the values of the model parameters that makes the D–F approximation accurate when compared against a system response taken as a benchmark.

*null hypothesis*in this inference problem corresponds to the event that the hypothesized model is equivalent to the reference MODFLOW model. Relative probabilities are used to quantify the degree of belief that the null hypothesis is true, conditioned on the values of model parameters. Next, an inverse problem is formulated based on Bayes’ theorem where the probability distributions of the model parameters are estimated conditioned on the event that the null hypothesis holds.

Third, since the identification of the model parameters can be computationally challenging when dealing with significantly large search spaces, this paper proposes a novel two-stage inverse problem implementation methodology. A schematic view of this methodology is shown in Fig. 2. By this methodology, the Bayesian assessment of the null hypothesis is first performed across several partitions of the parameter space into various parameter subspaces, and then the problem of parameter identification, which is computationally more demanding, is run over those subspaces with higher relative plausibilities. Consequently, the approach has the advantage of being able to identify (1) the values of the model parameters within a given subspace and also (2) the subspaces within the overall parameter space where the null hypothesis is more likely to hold, with quantified uncertainty. Upon the implementation of the proposed methodology to a representative case study, new evidence is obtained regarding the suitability of the D–F theory for non-horizontal flow in unconfined layered systems.

The paper is organized as follows: Sect. 2 presents the formulation of the proposed model. In Sect. 3, the proposed Bayesian framework for model assessment is presented. Section 4 is devoted to providing results and discussing the proposed Bayesian framework for model assessment, along with computational details about implementation. In Sect. 5, a practical engineering example about railway track drainage is provided to illustrate the applicability of the model in a real-life scenario. Finally, some concluding remarks are provided in Sect. 6.

## 2 Proposed Model

### 2.1 Governing Equations

Let us consider a physical system represented by a two-dimensional *n*-layered sloping porous medium. This system may represent in practice a layered soil overlying an impermeable bed that rises to a peak between drains at the downstream boundaries of the system with water head \(H_{B}\), as depicted in Fig. 1. A uniform steady-state vertical recharge flow rate *q* (e.g. precipitation intensity) is considered as an input to the system.

*x*as follows:

*n*is the number of

*wet layers*, i.e. those laying totally or partially under the water table, and \(Q_{\ell }\) is the water flow through the \(\ell \)th layer at section

*x*. This flow can be expressed as \(Q_{\ell }=v_{\ell }S_{\ell }\), with \(v_{\ell }\) being the velocity of flow (averaged through the thickness) within layer \(\ell \) and \(S_{\ell }\) the section perpendicular to the flow in that layer. Therefore, Eq. (1) becomes:

*h*(

*x*) is the water table height as a function of the distance

*x*, \(t_\ell (x)\) is the vertical thickness of the \(\ell \)th layer at a distance

*x*, \(L_{x}\) is the horizontal length of the system, and \(\tan \alpha _0\) is the slope of the impervious base, as shown in Fig. 1. From Darcy’s law, the velocity of flow within the \(\ell \)th layer, \(\ell =1,\dots ,n\), can be expressed as

*n*-layered sloping system is obtained as:

*x*and

*s*, where

*s*is a coordinate measured along the sloping bed with \(s = 0\) corresponding to the water table height at \(x = 0\), and \(s=s_{B}\) at \(x=L_{x}\). These independent variables can be shown to be geometrically related as (Youngs and Rushton 2009b):

*streamtubes*parallel to the base. In addition, no-flow and constant head boundary conditions are assumed at the left-hand and downstream boundaries, respectively; thus:

### 2.2 Solution Method

*h*(

*x*) with no closed-form explicit solution. An implicit parametric solution specified by \(\mathscr {W} \ni w \mapsto \left( x(w),h(w)\right) \in \mathbb {R}^{2}\), can be obtained by the variable change:

*x*(

*w*) can be obtained by differentiation w.r.t

*w*based on Eq. (12), resulting after some algebraic manipulation in:

*b*,

*c*are constants defined as follows:

*w*is defined within the subspace \(\mathscr {W}=[w_B,\infty )\subset \mathbb {R}^{+}\). By taking values \(w \in \mathscr {W}\), (e.g. by defining a grid within \(\mathscr {W}\)), values for

*x*(

*w*) can be readily obtained from Eq. (19), which are subsequently used to obtain values for the water table height

*h*(

*w*) from Eq. (10), as:

*n*th layer, i.e. \(\sum _{\ell =1}^{n-1}t_{\ell }(x)\leqslant h(x)<\sum _{\ell =1}^{n}t_{\ell }(x),~\forall x \in (0,x_{B}]\subset \mathbb {R}^{+}\). However, this is a particular case of a more general one where the water table may cross the boundary between layers with different hydraulic conductivities at an unknown point \(x_C\in (0,x_{B}]\). In this case, the complete solution for the water table will be given by a piecewise continuous function where each sub-function is defined in the generic interval \((x_C,x_B]\), with \(x_B\) being the abscissa of the known boundary condition, and \(x_C\) the abscissa of the crossing point, which becomes the known boundary condition for the subsequent sub-function. The determination of \(x_C\) may be challenging especially when considering layers of contrasting hydraulic conductivities (Youngs and Rushton 2009b). A generic trial and error method might be adopted to approximate \(x_C\), although this method may lead to error propagation that is hard to control. To overcome this drawback, the Newton–Raphson method (Carnahan 1969) is applied here to systematically obtain a parametric approximation to the abscissa \(x_{C}\) with a controlled level of accuracy. To this end, let us define the function

*h*(given by Eq. (20)) and the vertical height of the boundaries of layer

*n*, \(z_n\), which is defined by:

*n*th layer, and \(n^*=n\) otherwise. Thus, \(x_C=x(w_C)\) can be obtained as the point where \(\delta (w_C)=0\) holds. By Newton–Raphson’s formula, an estimation of \(w_C\) can be obtained as follows:

*i*th iterating approximation to \(w_C\). The term \(\delta (w_{C}^{(i)})\) in Eq. (23) is obtained by Eqs. (20) and (22) as:

*b*is given by Eq. (16a). Finally, by substituting expressions (24) and (26) into Eq. (23), an iterative approximation to \(w_C\) is obtained starting with an initial value \(w_{C}^{(i=0)}\) and repeating the process for increasing values of \(i \in \mathbb {N}\) until \(|x(w_{C}^{(i)})-x(w_{C}^{(i-1)})|<\epsilon \), with \(\epsilon \) being a sufficiently small error tolerance. An algorithmic description of the proposed piecewise prediction of the steady-state unconfined water table in layered porous media is provided in Algorithm 1.

## 3 Bayesian Model Assessment

The model proposed in Sect. 2 is just an idealization of reality based on a set of modelling assumptions. For a particular system output (e.g. water table height), the validity of such simplifying assumptions depends on the adopted values of certain model parameters, such as hydraulic conductivities or slope of layers. A Bayesian inverse problem framework is proposed in this section to efficiently identify the value of the model parameters that better suit the hypothesis that both the proposed model and a reference numerical model given by MODFLOW-NWT (Niswonger et al. 2011) render identical outputs. MODFLOW-NWT is a MODFLOW variant that uses the Newton–Krylov method (Knoll and Keyes 2004) and unstructured, asymmetric matrix solvers to numerically solve the exact formulation of the two-dimensional groundwater flow problem. MODFLOW-NWT is shown to be particularly suitable for unconfined layered systems like the one considered here where the water table crosses the interface between layers with contrasting hydraulic conductivities (Painter et al. 2008; Keating and Zyvoloski 2009). To avoid repetition of the literature, the interested reader is referred to Harbaugh (2005) and Niswonger et al. (2011) for specific details about MODFLOW modelling.

### 3.1 General Settings

Let \(\varvec{f}=\varvec{f}(\varvec{x},\varvec{u};\varvec{\theta })\) be the water table height as given by the proposed model in Sect. 2 for a particular system configuration, where \(\varvec{x}=(x_1,\dots ,x_i,\dots ,x_{n_{\varvec{x}}})\in \mathbb {R}^{n_{\varvec{x}}}\) are the abscissa values where \(\varvec{f}\) is evaluated, \(\varvec{u}\in \mathbb {R}^{n_{\varvec{u}}}\) is a vector containing known model inputs (e.g. geometry inputs), and \(\varvec{\theta }\in \varvec{\Theta }\subset \mathbb {R}^{n_{\varvec{\theta }}}\) are model parameters (e.g. hydraulic conductivities) defined over parameter space \(\varvec{\Theta }\subset \mathbb {R}^{n_{\varvec{\theta }}}\). Let us also consider a *reference model* for unconfined water table prediction denoted by \(\varvec{g}=\varvec{g}(\varvec{x},\varvec{v};\varvec{\theta })\), which, in the absence of experimental data, constitutes our best available knowledge about the system being represented. In this study, \(\varvec{g}=\varvec{g}(\varvec{x},\varvec{v};\varvec{\theta })\) will be represented by the solution given by the FD model MODFLOW-NWT, with \(\varvec{v}\in \mathbb {R}^{n_{\varvec{v}}}\) being particular model inputs defining the geometry and the configuration of the numerical model.

*degree of belief*of hypothesis \(\mathscr {H}\) for a specific set of model parameters \(\varvec{\theta }\), by assuming that \(J(\varvec{\theta })\) is uncertain and that it follows a probability model denoted by \(p(\mathscr {H}|\varvec{\theta })\). To this end, \(J(\varvec{\theta })\) is conservatively assumed to be modelled as a zero-mean Gaussian distribution, i.e. \(J(\varvec{\theta })\sim \mathscr {N}(0,\sigma )\), following the principle of maximum information entropy (Jaynes 1957). This principle enables a rational way to establish a probability model for the discrepancy function such that it produces the largest uncertainty (largest Shannon entropy) in the degree of belief of hypothesis \(\mathscr {H}\); the selection of any other probability model would lead to an unjustified reduction in such uncertainty (Beck 2010). Thus, the degree of belief of hypothesis \(\mathscr {H}\) can be described through the probability model

### 3.2 Assessment of Model Parameters

*likelihood function*for hypothesis \(\mathscr {H}\) given \(\varvec{\theta }\). However, our interest precisely lies in the reciprocal information, i.e. to determine the values of \(\varvec{\theta }\) among the set of values in \(\varvec{\Theta }\subset \mathbb {R}^{n_{\varvec{\theta }}}\) that lead to models that more likely satisfy hypothesis \(\mathscr {H}\). This

*inverse problem*can be formulated by Bayes’ theorem (Tarantola 2005; Rus et al. 2016), as:

*prior*degree of belief about the models specified by \(\varvec{\theta }\) in regards to the fulfilment of hypothesis \(\mathscr {H}\). In this work, the uniform PDF is conservatively adopted for \(p(\varvec{\theta })\) as a way of representing our prior state of ignorance about the values \(\varvec{\theta }\in \varvec{\Theta }\) satisfying hypothesis \(\mathscr {H}\). Note that Bayes’ theorem takes this initial degree of belief and updates it by using the information given by the likelihood function in Eq. (28). The resulting information \(p(\varvec{\theta }|\mathscr {H})\) is formally referred to as the

*posterior*PDF of model parameters.

*proposal distribution*\(\pi (\varvec{\theta }^{'}|\varvec{\theta }^{(\zeta )})\), the M–H algorithm obtains the state of the chain at \(\zeta +1\), given the state at \(\zeta \), specified by \(\varvec{\theta }^{(\zeta )}\). The candidate parameter \(\varvec{\theta }^{'}\) is accepted (i.e. \(\varvec{\theta }^{(\zeta +1)}=\varvec{\theta }^{'}\)) with probability \(\text {min}\{1,r\}\) and rejected (i.e. \(\varvec{\theta }^{(\zeta +1)}=\varvec{\theta }^{(\zeta )}\)) with the remaining probability \(1-\text {min}\{1,r\}\), where:

### 3.3 Assessment of Parameter Subspaces

*posterior plausibility*of the

*j*th subspace in \(\varvec{\Theta }\), i.e.: \(P(\Theta _{j}|\mathscr {H},\varvec{\Theta })\), Bayes’ theorem is extended at the level of the subspaces as follows:

^{1}

*j*th subspace in \(\varvec{\Theta }\), so that \(\sum _{j=1}^{n_{s}}P(\Theta _{j}|\varvec{\Theta })=1\). This prior plausibility expresses the initial relative degree of belief of the models evaluated in \(\Theta _{j}\) within \(\varvec{\Theta }\) in regard to the fulfilment of hypothesis \(\mathscr {H}\). The factor \(p(\mathscr {H}|\Theta _{j})\) is the

*evidence*for model subspace \(\Theta _{j}\in \varvec{\Theta }\) and expresses how likely hypothesis \(\mathscr {H}\) is satisfied if model parameters in subspace \(\Theta _{j}\) are adopted. This evidence can be obtained by using the total probability theorem:

## 4 Numerical Validation by Bayesian Assessment

### 4.1 System Configuration

Range of values for model parameters defining each subspace

Parameter | \(\Theta _{1}\) | \(\Theta _{2}\) | \(\Theta _{3}\) | \(\Theta _{4}\) |
---|---|---|---|---|

\(q/K_\ell \) | \(\left[ 0.01,0.015 \right) \) | \(\left[ 0.015,0.15 \right) \) | \(\left[ 0.15,1 \right) \) | \(\left[ 1,1.5 \right] \) |

\(\tan \alpha _0\) | \(\left[ 0 ,0.5 \right] \) | \(\left[ 0 ,0.5 \right] \) | \(\left[ 0 ,0.5 \right] \) | \(\left[ 0 ,0.5 \right] \) |

From this standpoint, both the proposed model and the numerical reference model given by MODFLOW-NWT are repetitively run for different values of model parameters \(\varvec{\theta }\in \Theta _{j}\subset \varvec{\Theta },~j=1,\dots ,4\), driven by the M–H algorithm, so as to obtain the required posterior probabilities in Eq. (29). For the MODFLOW-NWT simulation, a spatial grid is considered by discretizing each model layer into one row and 200 columns. Such discretization is appropriately selected after a grid convergence study such that the solution given by MODFLOW-NWT is independent of the grid size for each subspace of parameters \(\Theta _j \in \varvec{\Theta },~j=1,\ldots ,4\). The orthomin/stabilized conjugate-gradient solver, also called \(\chi \)MD (Niswonger et al. 2011), is chosen as matrix solver for MODFLOW-NWT. Default solver input values are scaled as suggested by Niswonger et al. (2011) under the “complex” solver option. No-flow boundaries are set at the left-hand boundary (\(x=0)\) and the impermeable base, while a specified flow boundary is applied to the top of the upper layer using the “Recharge” package (McDonald and Harbaugh 1988). For this example, a constant value \(q=1\times 10^{-5}\) m/s is adopted for the recharge flux rate; therefore, the variability in \(q/K_\ell ,~\ell = 1,2,3\), is achieved by correspondingly varying the values for \(K_\ell \). A head-dependent boundary condition is considered at the right-hand boundary using the “General Head Boundary” package, allowing for *seepage face* formation (Rushton and Youngs 2010; Bear 1972). Water flow across this boundary is obtained from Darcy’s law using a gradient calculated as the difference between the specified head outside the boundary (\(H_\mathrm{B}\)) and the head computed by MODFLOW-NWT on the boundary. To assess the accuracy of the numerical solution, the water budget error (i.e. the difference between water inflow and outflow) is computed for each simulation so that the solution is accepted only if water budget error is less than \(0.5\%\) (Anderson et al. 2015).

### 4.2 M–H Algorithm Implementation

As stated in Sect. 3.2, the prior PDF associated with the set of model parameters \(\varvec{\theta }\) for a particular subspace \(\Theta _j\) is modelled as a uniform distribution defined within the interval of definition of such parameters; i.e. for the *i*th component \(\theta _i \in \varvec{\theta }\in \Theta _j\), \(p(\theta _{i}|\Theta _j)=\mathscr {U}(\theta _{i,\min },\theta _{i,\max }),~j=1,\dots ,4\), where \(\theta _{i,\min }\) and \(\theta _{i,\max }\) are given in Table 1. It should be noted that each component \(\theta _i \in \varvec{\theta },~i=1,\dots ,n_{\varvec{\theta }}\), is conservatively assumed to be stochastically independent (Chiachío et al. 2015); thus, \(p(\varvec{\theta }|\Theta _j)\) is defined as the unconditional product of the individual priors, i.e. \(p(\varvec{\theta }|\Theta _j)=\prod _{i=1}^{n_{\varvec{\theta }}}p(\theta _{i}|\Theta _j),~n_{\varvec{\theta }} = 4,~j=1,\dots ,4\). At the level of the subspaces of parameters, a discrete uniform distribution function is adopted for the prior probabilities of such subspaces, i.e. \(P(\Theta _{j}|\varvec{\Theta })={1}/{4}, j = 1,\ldots ,4\), representing our initial state of ignorance about the subspaces where hypothesis \(\mathscr {H}\) is more likely to hold. To obtain the required posterior PDF of parameters \(p(\varvec{\theta }|\mathscr {H},\Theta _j)\) for the *j*th subspace, the M–H algorithm is applied with a multivariate Gaussian for the proposal PDF, i.e. \(\pi (\cdot |\varvec{\theta }^{(\zeta )})=\mathscr {N}(\varvec{\theta }^{(\zeta )},\varvec{\varSigma }_{\pi })\) in Eq. (31), where \(\varvec{\varSigma }_{\pi } \in \mathbb {R}^{n_{\varvec{\theta }}\times n_{\varvec{\theta }}}\) is the covariance matrix of the random walk. Note that since model parameters \(\varvec{\theta }\) are assumed to be stochastically independent, \(\varvec{\varSigma }_{\pi }\) is a diagonal matrix, i.e. \(\varvec{\varSigma }_{\pi }=\hbox {diag}(\sigma _{\pi ,1}^{2},\ldots ,\sigma _{\pi ,n_{\varvec{\theta }}}^{2})\); hence, each component parameter in \(\varvec{\theta }\) performs an independent random walk within \(\Theta _j\). The diagonal elements of \(\varvec{\varSigma }_{\pi }\) are appropriately selected through initial test runs such that the monitored acceptance rate (ratio between accepted M–H samples over total amount of samples) is within the suggested range \(\left[ 0.2,0.4\right] \) for the M–H algorithm (Roberts and Rosenthal 2001). For the definition of Eq. (28), the standard deviation of the discrepancy function is set to \(\sigma =0.05\), taking it as known. This parameter has been shown to have a relatively low influence on the model parameters identification.

### 4.3 Results and Discussion

*q*/

*K*) formally emerges as a critical parameter for model assessment so that when conveniently bounded, the assumption of unidimensional flow parallel to the impervious base can be safely adopted. For subspaces \(\Theta _3\) and \(\Theta _4\), the posterior range of plausible values for the slope parameter is significantly reduced with respect to the prior around the value \(\tan \alpha _0=0\), as evident from Fig. 5c, d. This is also manifested in the lower values for the relative plausibilities for those subspaces, as shown in Fig. 4. These low values for the plausibilities can be explained based on the likelihood function, which is evaluated using prior samples from a region of the parameter subspace far from the narrow region of high likelihood (recall Eq. 35), which requires values of the slope parameter close to \(\tan \alpha _0=0\) (horizontal flow).

*ad hoc*solutions have been suggested in the literature to emulate the effect of the seepage face on the D–F solution by introducing an artificial boundary condition (Mizumura 2009; Rushton and Youngs 2010; Dan et al. 2012). These solutions could be conveniently incorporated into the proposed model to better fit the reality in such region.

## 5 Engineering Case Study

## 6 Conclusions

An approximate model based on the Dupuit–Forchheimer theory has been presented to efficiently predict the steady-state water table height in sloping layered porous media with recharge. The model was developed as an alternative to the numerical modelling version of the same problem, which becomes computationally intractable in a number of practical problems requiring multiple model evaluations. To verify and validate the proposed model, a novel approach based on Bayesian hypothesis testing has been developed to evaluate the accuracy of the new model against the numerical model MODFLOW-NWT, an open-source finite-difference code by the U.S. Geological Survey for unconfined groundwater flow, considered here as reference model. The assessment is carried out through probabilities that measure the relative extent of agreement between both models for the many possible values of model parameters while accounting for the underlying modelling uncertainties. The numerical implementation of this Bayesian methodology is facilitated by considering multiple subspaces within the overall parameter space, and the probabilities across such multiple subspaces are integrated using principles of conditional probability and total probability. The ratio of vertical recharge to hydraulic conductivity formally emerges as a critical parameter for model accuracy so that when conveniently bounded, both the proposed model and MODFLOW-NWT provide almost identical results.

Building on this work, a future research direction is the application of the proposed Bayesian framework to infer an approximate model for unconfined flow in large-scale heterogeneous porous media taking as reference model a stochastic numerical modelling approach (Mantoglou 1992; Mousavi Nezhad et al. 2011), which would allow the consideration of spatially variable hydraulic properties in the assessment. Another desirable further work in the context of model development is the assessment of a sound approach to improve the lack of fitting accuracy of the proposed model in the vicinity of the seepage face.

## Footnotes

- 1.
Note that, while \(p(\cdot )\) is used to denote a probability density function (PDF), \(P(\cdot )\) is used to denote probability.

## Notes

### Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/M023028/1], “Whole-life Cost Assessment of Novel Material Railway Drainage Systems” along with RSSB who jointly and equally support it, and also the Lloyd’s Register Foundation which partially provides support to this work. RSSB is a rail industry body. Through research, analysis and insight, RSSB supports its members and stakeholders to deliver a safer, more efficient and sustainable rail system. The Lloyd’s Register Foundation is a charitable foundation in the UK helping to protect life and property by supporting engineering-related education, public engagement and the application of research. Work on this paper has been benefited from the open-source Python package Flopy (Bakker et al. 2016) to preprocess and run MODFLOW-NWT using Python.

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