# Efficient Uncertainty Quantification for Unconfined Flow in Heterogeneous Media with the Sparse Polynomial Chaos Expansion

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## Abstract

In this study, we explore an efficient stochastic approach for uncertainty quantification of unconfined groundwater flow in heterogeneous media, where a sparse polynomial chaos expansion (PCE) surrogate model is constructed with the aid of the feature selection method. The feature selection method is introduced to construct a sparse PCE surrogate model with a reduced number of basis functions, which is accomplished by the least absolute shrinkage and selection operator-modified least angle regression and cross-validation. The training samples are enriched sequentially with the quasi-optimal samples until the results are satisfactory. In this study, we test the performance of the sparse PCE method for unconfined flow with the presence of random hydraulic conductivity and recharge, as well as pumping well. Numerical experiments reveal that, even with large spatial variability and high random dimensionality, the sparse PCE approach is able to accurately estimate the flow statistics with greatly reduced computational efforts compared to Monte Carlo simulations.

## Keywords

Uncertainty quantification Unconfined flow Sparse PCE Feature selection Quasi-optimal sampling## 1 Introduction

In realistic subsurface problems, description of flow and transport in porous media often involves some degree of uncertainty, due to the strong heterogeneity and incomplete knowledge of the formations. In order to quantify the uncertainties, stochastic partial differential equations need to be solved with some stochastic approaches. The most common one is the Monte Carlo (MC) method, where the model inputs are randomly sampled and the statistical properties of the outputs can be obtained from the simulated realizations. However in the MC, a large number of realizations are usually required to reduce the sampling errors for obtaining high-order moments and the probability density function (PDF), which leads to unaffordable computational efforts (Ballio and Guadagnini 2004). Besides, perturbation method and moment equation method (Zhang 2001; Zhang and Lu 2004) are other stochastic approaches to uncertainty quantification. Due to the need of manipulating the governing flow equations, the solving procedures could be complicated and cumbersome.

As an efficient alternative, the polynomial chaos expansion (PCE) method has drawn more and more attentions over the past decades (Ghanem 1998; Xiu and Karniadakis 2002a, b; Oladyshkin et al. 2012; Dai et al. 2014). In the PCE method, the random output is represented by a series of orthogonal polynomial basis functions (Ghanem and Spanos 2003). The orthogonal polynomials are associated with the distribution of the input random variables. For instance, Hermite, Laguerre, and Jacobi polynomials are optimal for Gaussian, Gamma, and Beta random variables, respectively, and the generalized PCE is developed for arbitrary variables (Xiu and Karniadakis 2002a, b). If the random inputs are correlated or characterized by random field, parameterization methods such as Karhunen–Loève (K–L) expansion can be used to transform them to independent random variables (Li and Zhang 2007, 2013; Li 2014). Once the PCE coefficients are evaluated, the polynomial surrogate model can be used for uncertainty quantification (UQ) and global sensitivity analysis (GSA) (Oladyshkin et al. 2012; Dai et al. 2014).

The PCE method can be classified as intrusive and non-intrusive approaches. The intrusive approach requires manipulation of governing equations. The best known is the stochastic finite element method, which utilizes the Galerkin scheme and requires solving a set of coupled equations (Maître et al. 2002; Ghanem and Spanos 2003). In contrast, the non-intrusive approach allows using the existing deterministic simulator, which can be treated as a black box. In this category, the probabilistic collocation method (PCM) and stochastic collocation method (SCM) are widely used for UQ (Webster et al. 1996; Li and Zhang 2007; Babuška et al. 2007; Chang and Zhang 2009). In these methods, a set of decoupled equations are derived on some collocation points, which are associated with Lagrange interpolation or Galerkin integral (Li and Zhang 2007; Chang and Zhang 2009). The PCE coefficients can be obtained by directly solving the equation set or by regression (Blatman and Sudret 2010, 2011). These approaches are found to be accurate and computationally efficient for problems with relatively low random dimensionality (Li and Zhang 2007; Chang and Zhang 2009; Liao and Zhang 2015).

Stochastic modeling of flow in heterogeneous porous media has been widely studied (e.g., Polmann et al. 1991; Zhang 2001; Li and Zhang 2007; Nezhad et al. 2011). For unconfined groundwater flow, strong spatial nonstationarity exists due to the presence of nonlinearity, recharge/discharge, and injection/pumping wells. Larger random dimensionality and higher degree of the PCE are required to better describe such complex process, where the number of basis functions will explode (Xiu and Hesthaven 2005). Shi et al. (2009) and Li et al. (2009) utilized the PCM for unconfined and unsaturated flow problems but they are restricted to relatively low random dimensionality. To overcome this “curse of dimension” problem, some dimension reduction methods need to be applied for UQ. Some researchers introduced the idea of compressive sensing to guarantee sparsity with the L1-norm restriction (Doostan and Owhadi 2011; Yan et al. 2012; Hampton and Doostan 2015). Fajraoui et al. (2012) reduced the dimensions by retaining terms that make significant contributions to the variance of the model. Blatman and Sudret (2010, 2011) made use of the stepwise regression and least angle regression (LAR) to construct the PCE model with a reduced number of basis functions. Elsheikh et al. (2014) further applied LAR for Bayesian inference of subsurface flow models. In fact, both the stepwise regression and LAR are common algorithms in the field of feature selection (Tibshirani 1996; Guyon and Elisseeff 2003). Feature selection was first proposed in the area of statistical learning for removing redundant and irrelevant features. A mature system for feature selection has been established, where extensive algorithms and successful applications have been published in the last two decades (Tibshirani 1996; Efron et al. 2004; Hastie et al. 2009). Meng and Li (2017) proposed the sparse PCE for UQ of confined groundwater flow, where the least absolute shrinkage and selection operator (LASSO) modified least angle regression (LASSO-LAR) algorithm is used for feature selection.

This present paper is an extension of our earlier work for confined flow (Meng and Li 2017) to unconfined flow problems, where higher random dimensionality is involved. In the process of constructing the sparse PCE, the LASSO-LAR is applied for feature selection. Different factors are studied in this paper, e.g., coupling of random hydraulic conductivity and recharge, as well as pumping well. In contrast to confined flow, quantifying uncertainty for unconfined flow with these factors is of more difficulties. As a result, the *S*-value constrained quasi-optimal sampling method is adopted to improve the optimality of the samples in constructing the sparse PCE, in contrast to Meng and Li (2017), where only Latin hypercube sampling was used. In all the studied cases, comparisons are made between the sparse PCE and MC in terms of both computational accuracy and efficiency. The numerical results show that the sparse PCE is accurate and efficient for UQ of unconfined flow.

## 2 Methodology

### 2.1 Governing Equations

### 2.2 Karhunen–Loève (K–L) Expansion

*x*direction) can be expressed analytically (Zhang and Lu 2004):

### 2.3 Polynomial Chaos Expansion (PCE)

*i*-th dimension. If \(\xi _i \) obey a certain basic distribution, e.g., Gaussian, Gamma or Beta distribution, \(\varphi _{\alpha _i }^{(i)} \) is Hermite, Laguerre or Jacobi polynomials, respectively (Xiu and Karniadakis 2002a, b). For arbitrary distribution, \(\varphi _{\alpha _i }^{(i)} \) is derived by generalized PCE technique (Xiu and Karniadakis 2002a, b; Li et al. 2011).

*d*, so that a finite number of terms are retained in PCE. The truncated PCE is:

*P*terms have to be determined. Once the coefficients are evaluated, then the mean and variance of the random output can be written as:

### 2.4 Coefficients Evaluation

### 2.5 Feature Selection

The feature selection methods originate from statistical learning area and have been widely used for model constructions in many fields (Guyon and Elisseeff 2003; Saeys et al. 2007). The redundant or irrelevant features are discarded to improve prediction accuracy and provide a better understanding for datasets. A number of algorithms have been proposed for feature selection, e.g., Fisher Score, sequential forward selection, regularization model, etc. (Dash and Liu 1997; Guyon and Elisseeff 2003). Some of them have been applied for UQ problems in previous works (Blatman and Sudret 2010, 2011; Elsheikh et al. 2014). In this study, we utilize the LASSO-LAR algorithm, to construct the sparse PCE surrogate model, which is highly efficient for basis reduction (Meng and Li 2017).

*t*sufficiently small, some of the coefficients will be exactly zero and the correlated features will be discarded.

LASSO-LAR algorithm: | |
---|---|

1. | In preparation, each column vector of features in the training dataset is normalized and the target values are centered, i.e., \(\frac{1}{N}\sum \nolimits _{i=1}^N {{\varvec{\Phi }} _{ij} } =0, \frac{1}{N}\sum \nolimits _{i=1}^N {{\varvec{\Phi }}_{ij}^2 } =1, j=1,\ldots ,P\) and \(\frac{1}{N}\sum \nolimits _{i=1}^N {h_i} =0\). Initialize with the residual \(\mathbf{r}=\mathbf{h}\) and set \(a_1 ,\ldots ,a_P=0\) |

2. | Find the column vector \(\Phi _j \) most correlated with \(\mathbf{r}\) |

3. | Move \(a_j \) from 0 toward its least square coefficient \(\left\langle {\Phi _j ,\mathbf{r}} \right\rangle \), until some other competitor \(\Phi _k \) has as much correlation with the current residual as does \(\Phi _j \) |

4. | Move \(\{a_j ,a_k \}\) in the direction defined by their joint least square coefficient of the current residual on \((\Phi _j ,\Phi _k )\), until some other competitor \(\Phi _l \) has as much correlation with the current residual |

5. | If a nonzero coefficient hits zero, drop its feature from the active set of features and recompute the current joint least square direction |

6. | Continue in this way until \(\min (N-1,P)\) features have been entered |

*k*steps of LASSO-LAR, the hybrid LAR coefficients are generated for higher accuracy, which is defined as:

*N*is the number of samples. When \(\mathrm{Err}_\mathrm{GCV} \) reaches a certain stopping criterion, the surrogate model constructed by current feature subset is adopted for further uncertainty analysis.

### 2.6 Sequential Quasi-Optimal Sampling

The surrogate model is also highly influenced by the sampling method to generate the sampling (collocation) points. With higher-quality samples, the predictions can be more accurate. Space-filling samples are commonly used for experimental design, such as Latin hypercube samples and quasi Monte Carlo samples. In the SCM and PCM, specific interpolation points and Gaussian integral points are directly utilized as training dataset (Li and Zhang 2007; Chang and Zhang 2009). Besides, some new methods have been proposed in recent years such as coherence-optimal sampling (Hampton and Doostan 2015), subsampling with QR column pivoting (Seshadri et al. 2016), etc. Since in the sparse PCE, the basis functions are selected from a large number of candidates, we have no idea how many samples are sufficiently enough for coefficients evaluation beforehand. Therefore, we may start from a conservatively small size of samples at first and gradually enrich the experiment design to improve the surrogate model. A sequential sampling method, the quasi-optimal sampling with *S*-value constraint (Shin and Xiu 2016), is adopted in this study for this purpose.

*S*-value criterion is defined as:

*j*-th column of \({\varvec{\Phi }}_{{{\mathcal {X}}}\cup \xi _i}\). The

*S*-value is calculated for each sample in the candidate set. The sample corresponding to the maximum

*S*-value, i.e., \({\varvec{\xi }} _{\max } \), is added to current samples at last. The process is repeated with the updated \({\varvec{\mathcal {X}}}={{\mathcal {X}}}\cup {\varvec{\xi }} _{\max } \) and \({{\mathcal {X}}}_L ={{\mathcal {X}}}_L \backslash {\varvec{\xi }} _{\max } \), until the expected number of new samples are generated. Then the new surrogate model will be constructed and the sample set will be enriched iteratively until satisfactory results are obtained.

In fact the *S*-value is a criterion representing the differences between the estimations and targets. The computation of *S*-value requires repetitively calculating the determinant which is an expensive procedure. Such unnecessary cost can be avoided by analytical derivations of the *S*-value. The readers could refer to Shin and Xiu (2016)) for more details of mathematical proofs and implementations. Comparisons of the quasi-optimal sampling with other sampling methods are also discussed in Fajraoui et al. (2017).

## 3 Case Studies

### 3.1 Problem Statement

In this section, we test the performance of the sparse PCE for unconfined groundwater flow in heterogeneous aquifers. Some synthetic numerical experiments are performed and comparisons are made, with consideration of the effects of different spatial variability of hydraulic conductivity, random recharge and the presence of pumping well. The flow domain is a square of size \(L_x =L_y =20\) m, uniformly discretized into \(40\times 40\) square elements. The hydraulic head is prescribed constantly at the boundary \(\varGamma _1 \) (along \(x=0, 0\le y\le 15\) m) and \(\varGamma _2 (\hbox {along } x=20\,\hbox {m}, 5\le y\le 20\,\hbox {m})\) as 7.0 and \(5.0\,\hbox {m}\), respectively. Other borders are defined as no-flow boundaries.

In the following studied cases, the log hydraulic conductivity and recharge are assumed to be second-stationary Gaussian random fields with separable exponential covariance, where the correlation length in each direction is \(\eta _x =\eta _y =5.0\,\hbox {m}\). Each random field can be represented by the truncated K–L expansion with independent variables, which is the source of uncertainty. In each case, we train the surrogate model with a relatively small size of samples initially and gradually enrich the experiment design with the quasi-optimal sampling points. The results are compared with Monte Carlo solutions (with 10,000 simulations), which are regarded as the benchmark. The simulations are accomplished with the existing groundwater flow simulator, MODFLOW. The deterministic solutions from the flow simulator are deemed as reliable.

### 3.2 Effects of Spatial Variability

In this section, we explore the accuracy and efficiency of the sparse PCE with different spatial variabilities of the random field. For simplicity, a constant recharge over the domain \(R=4\,\hbox {cm/day}\) is given and the random log-conductivity field \(Y=\ln K\) is studied. At first, we set the field of zero mean and unit variance, i.e., \(\left\langle Y \right\rangle =0,\; \sigma _Y^2 {=}1.0\), and the coefficient of variation of the hydraulic conductivity equals to \(Cv_K =\sigma _K /\left\langle K \right\rangle =131\% \). In the MC benchmark, we retain the first 624 terms to represent the field where \(\sum _{i=1}^{624} {\lambda _i } /D\sigma _Y^2 =96\% \) energy is preserved in K–L expansion, which is found to be adequate to reproduce the ensemble statistics of the random field. More energy is unnecessary since the effect of small scale fluctuations in the conductivity field on the flow responses can be neglected (Zhang and Lu 2004). However for constructing the surrogate models, 624 dimensions would lead to enormous computational efforts. Thus for the sparse PCE, we preserve 70% energy and retain 23 terms, with which some unnecessary details are ignored for convenience.

*S*-value constraint. The mean and variance of hydraulic head obtained from the sparse PCE trained by 140 samples are shown in Fig. 1. The mean head can be well estimated owing to the particular setting of the field in our examples; hence, we mainly focus on the variance for comparison. In this case, the PCE up to \(3{\mathrm{rd}}\) degree are provided for model construction and there are \(P=\left( {\begin{array}{l} 23+3 \\ 3 \\ \end{array}} \right) =2600\) basis functions in total. With the traditional PCE, 2600 or more samples are required to estimate the basis coefficients. However with the sparse PCE, around 80 basis functions are selected as the major stochastic features and the statistical moments obtained with 140 samples are close to the benchmark.

*y*:

### 3.3 Effects of Random Recharge

In this case, random recharge is imposed to the unconfined flow domain, which increases the nonlinearity in the probability space. The log-conductivity is Gaussian with moderate variance, i.e., \(\sigma _Y^2 {=}1.0\) and \(Cv_K =131\% \). We set the spatial recharge with \(\left\langle R \right\rangle =4\,\hbox {cm/day}\) and \(\sigma _R^2 =27.49\,\mathrm{cm}^{2}/\mathrm{day}^{2}\) so that the coefficient of variation also equals to \(Cv_R =\sigma _R /\left\langle R \right\rangle =131\% \). With the K–L expansion, the recharge field is also represented by 23 independent random variables. The conductivity and recharge are assumed to be independent to each other; thus, they are governed by two set of variables \({\varvec{\xi }} _K \) and \({\varvec{\xi }} _R \). Each of them has 23 random dimensions and the problem is involved with 46 random dimensions in total.

The coefficient distribution of the basis functions is inspected to study how the surrogate model is constructed at the center of the flow domain. Coefficients for the basis functions derived from \({\varvec{\xi }} _K \) and \({\varvec{\xi }} _R \) are shown in Fig. 6 separately. Although polynomials up to \(3{\mathrm{rd}}\) degree are provided, only a small part of the \(3{\mathrm{rd}}\) degree polynomials are given very limited coefficients. Therefore, only the first 2 degree polynomials are shown in Fig. 6. The \(1^{\mathrm{st}}\) degree polynomials are given larger coefficients for both \({\varvec{\xi }} _K\) and \({\varvec{\xi }}_R \). Since these basis functions strongly affect the random input by the K–L expansion, they are selected as the major stochastic features. Some basis functions are derived by \({\varvec{\xi }}_K \) and \({\varvec{\xi }}_R \) interactively and most of these basis functions are given zero coefficients, indicating that the random field is affected by the conductivity and recharge independently. Thus these interactive polynomials are identified as redundant features and discarded in the selection process. Among all 18,424 basis functions, around 180 terms are presented with nonzero coefficients and others are discarded, and the surrogate model can be constructed with merely 260 samples.

### 3.4 Effects of Pumping Well

## 4 Conclusions

- (1)
The non-intrusiveness of the sparse PCE method allows one to employ any existing deterministic flow simulators.

- (2)
The sparse PCE method is feasible for quantifying uncertainty associated with unconfined flow in random porous media. With relatively small computational efforts, the random process can be accurately and efficiently represented. A much faster convergence rate is observed compared to MC simulation. The number of flow simulations is significantly reduced, which is the most time-consuming step in uncertainty quantification. In comparison, the extra computational efforts caused by the feature selection process and sequential sample generation can be ignored. The method is highly efficient for high-dimensional problems where redundant and irrelevant features are discarded.

- (3)
The robustness of the sparse PCE method has been tested with respect to different spatial variability, independent random recharge, and the existence of pumping well. In these numerical cases, different distributions of the statistical moments and PDFs of hydraulic head are well produced. For those complex problems, more basis functions and more samples are required to construct a satisfactory surrogate model. With the sequential quasi-optimal sampling method, one can train the model with a small size of samples at first and gradually enrich the experiment design to improve the results. The method is self-adaptive and the best use of the computational resources can be made.

## Notes

### Acknowledgements

This work is partially funded by the National Science and Technology Major Project of China ( Grant nos. 2017ZX05039-005 and 2016ZX05014-004-006).

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