# Predictive Assessment of Groundwater Flow Uncertainty in Multiscale Porous Media by Using Truncated Power Variogram Model

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## Abstract

The spatial distribution of hydrogeological properties is essentially heterogeneous. Heterogeneity can be characterized quantitatively using geostatistics, which conventionally assumes that the stochastic process is stationary. However, growing evidence indicates that the spatial variability has the multiscale self-similarity characteristics and can be better characterized using non-stationary model but with statistically homogeneous increments. A general framework is developed in this work to conduct the uncertainty quantification analysis by using truncated power variogram model, which can explicitly account for measurement scale, observation scale, and window scale. The effect of the multiscale characteristics of the hydrogeological properties on the uncertainty and the consequential risk associated with the groundwater flow process is investigated. A synthetic two-dimensional saturated steady-state groundwater flow problem is used to evaluate the performance to predict the flow field distribution. For comparative purposes, the evaluation is based on both the truncated power and the traditional variogram models when the underlying porous medium is a random fractal field. The results show that the truncated power variogram model can perform the uncertainty quantification more accurately, and the adoption of traditional variogram model tends to result in a smoother estimation on the flow field and underestimate the uncertainty associated with the hydraulic head prediction. Upscaling is generally inevitable to avoid predictive uncertainty underestimation when the underlying random field exhibits multiscale characteristics.

## Keywords

Multiscale Random fractals Observation scale Truncated power variogram Geostatistics## 1 Introduction

Groundwater flow and contaminant transport problems have attracted intensive attention in recent years since groundwater resources are vital to sustain the human society and the environment. The resolution of both problems requires an accurate characterization of the groundwater flow through the subsurface porous media. The characteristics of the groundwater flow are largely determined by the hydrogeological properties of the porous medium. The spatial distribution of hydrogeological variables, such as hydraulic conductivity, is essentially heterogeneous. Due to insufficient measurements on the hydrogeological properties and lack of definite knowledge to describe the spatial variability associated with these properties, the hydrogeological variables are usually characterized stochastically. Therefore, the governing equation of the groundwater flow is treated as stochastic partial differential equation. The corresponding state variables derived from this equation, such as hydraulic head and flow velocity, are associated with uncertainties.

Many methods have been developed to characterize heterogeneity of the hydrogeological properties. Since the concept of Geostatistics was introduced to hydrogeology (Matheron 1963), geostatistical analysis has been widely used to deal with the flow and solute transport in heterogeneous porous media (Delhomme 1979; Hoeksema and Kitanidis 1989; Gelhar 1993; Carlson and Osiensky 2010; Gaus et al. 2003; Ahmadi and Sedghamiz 2007). Geostatistical inverse modeling methods are also adopted in model calibration (Sun and Yeh 2010; Kitanidis 1996; Zimmerman et al. 1998; Yeh and Liu 2000). To characterize more complex features of curvilinearity and connectivity, such as channeling, crevasse splays, clay lens, deltaic fans, karstic caving, geologically based models have been proposed, including geologically realistic models (Jung and Aigner 2011), geologic origin-based models (Goncalvès et al. 2004; Gabrovšek and Dreybrodt 2010), object-based models (Lopez et al. 2009; Michael et al. 2010; Borghi et al. 2012), models based on multiple-point statistics (MPS) (Strebelle 2002; Hu and Chugunova 2008). These approaches are able to reflect certain geologic complexity of the porous medium. However, the geological characteristics of natural formation are too complicated to be completely captured by a single method. No model at the current stage is compatible with all the geological observations, such as channeling, layering, self-affinity, long correlation, and continuous-categorical compound. Every method has its own limitations. Geologically realistic models generally lack quantitative indicators for objective criteria (Jung and Aigner 2011). Object-based models require sufficient databases from analog sites to determine a large number of parameters (Eschard et al. 2002). Geologic origin-based models are helpful to understand the underlying mechanism but “poorly suited for conditioning to direct and indirect data” (Linde et al. 2015). MPS-based methods learn repeating structures in categorical variables from training images which are difficult or expensive to obtain in fields. Furthermore, MPS-based modeling relies on pattern reconstruction on a small grid (e.g., with hundreds of nodes or less in two dimensions), which is difficult for continuous variables and cannot account for long correlation (Meerschaert et al. 2004; Sahimi 2011). Though some attempts have been made to take into account non-stationary and continuous variables (e.g., Strebelle and Zhang 2004; Wu et al. 2008; Mariethoz et al. 2009), there still exist problems (e.g., discontinuity) in applying MPS-based modeling to regional non-stationary fields.

*K*(

**x**) can be denoted as:

**x**is the spatial coordinate. Since the hydraulic conductivity field is heterogeneous and random, the random field

*Y*(

**x**) takes different values on each spatial location. For a given spatial location

**x**

_{i}, all the possible realizations constitute an ensemble of the random variable

*Y*(

**x**

_{i}). The random field of log hydraulic conductivity is considered as stationary or statistically homogeneous if the joint probability of

*Y*(

**x**) is only dependent on the relative locations of these points but not on the exact locations, especially for the first two statistical moments of

*Y*(

**x**). Specifically, the first moment, i.e., the mean, can be denoted as:

**s**, \( {\mathbf{s}} = {\mathbf{x}}_{1} - {\mathbf{x}}_{2} \), and the corresponding variance is:

However, growing evidence shows that the stationary or statistically homogeneous field may fail to characterize the true natural system accurately (Neuman 1990; Gelhar et al. 1992; Molz and Boman 1995; Painter 1996a; Tennekoon et al. 2003; Riva et al. 2013). If the sample variograms of some documented data are plotted on a logarithmic paper, they tend to fall on a straight line (e.g., Desbarats and Bachu 1994; Hyun et al. 2002), which suggests the random field of log hydraulic conductivity is statistically heterogeneous but with homogeneous increments (Di Federico and Neuman 1997; Siena et al. 2011). The focus of the stochastic characterization on log hydraulic conductivity field has shifted from the statistical distribution of the ln*K* value at each location to the distribution of the increment between two locations. Stochastic fractal models have been introduced to characterize such random field, including Gaussian-based fractional Brownian motion and fractional Gaussian noise (Neuman 1990; Molz and Boman 1993,1995; Liu and Molz 1996; Eggleston and Rojstaczer 1998), multifractals (Liu and Molz 1997; Boufadel et al. 2000), non-Gaussian-based fractional Levy motion or fractional Levy noise (Painter 1996a, b), and Gaussian scale-mixings (Painter 2001; Meerschaert et al. 2004; Guadagnini and Neuman 2011; Guadagnini et al. 2013; Riva et al. 2015).

Different models to characterize the multiscale phenomenon observed in the subsurface porous medium have been developed progressively. However, it remains unclear how the multiscale characterization of the porous medium affects the predictive uncertainty during the stochastic analysis of groundwater flow processes. The understanding of this issue is crucial to the risk analysis and the decision-making in the groundwater-related environmental problems. To address this issue, a multiscale analysis method proposed by Di Federico and Neuman (1997) is adopted in this study. Essentially, this method derives a multiscale mode-superposition model, in which the power law variogram behavior of log hydraulic conductivity random field can be considered as a weighted integral of infinite hierarchies of mutually uncorrelated stationary fields. Each hierarchy is referred to as a mode in the spectra of such random field. Mathematically, it has been shown that each mode can be characterized by exponential or Gaussian variogram. It renders the so-called truncated variogram models when cutoff values are imposed on the upper and lower limits of the integral. The leading-order statistical characteristics of flow and transport with truncated variogram models have been investigated analytically by using perturbation method (Di Federico and Neuman 1998). A hydrogeological rationale behind this multiscale mode-superposition model is demonstrated by Neuman (2003) using indicator Kriging. It demonstrates that the variogram of an attribute sampled over overlapping categories is the summation of the variograms sampled over individual categories weighted by their volumetric proportions.

Even though the multiscale characteristics of the porous medium have been observed for decades, the truncated power variogram model has been rarely used in the field application. The first reason may be that it is not intuitive to interpret the truncated variogram model parameters in terms of different scales, especially observation scale. For aquifer properties exhibiting fractal behavior, it is not able to characterize the complete structure and uncertainty of the aquifer by analyzing sampled data in finite observation range using traditional stationary variograms. The second reason may be that the performance to characterize the flow field in a fractal medium by using truncated power variogram model over traditional stationary variogram model has not been fully investigated. To address these issues, a general analysis framework is first provided in this paper to conduct the stochastic analysis of groundwater flow through a random fractal medium, when finite measurements of hydrogeological properties are available in observation scale, and evaluate the uncertainty associated with such a groundwater system. Then the capabilities of truncated power and traditional stationary variogram models to describe the flow behavior through a multiscale porous medium have been compared numerically using Monte Carlo simulation.

## 2 Theoretical Background

*s*is the lag distance and \( E\left( \cdot \right) \) is the ensemble mean operator.

*H*is Hurst coefficient \( \left( {0 < H < 1} \right) \). The combination of Eqs. (5) and (6) shows that the mean square increments of ln

*K*field depend only on the lag distance

*s*but not the actual location

**x**, and thus the random field is heterogeneous but with statistical homogeneous or stationary increments.

*r*is any positive constant (

*r*> 0).

*d*is the Euclidean or topologic dimension (Voss 1986).

If the random field is further Gaussian, the field can be characterized by fractional Brownian motion (fBm). When the Hurst coefficient \( 0.5 < H < 1 \), the spatial increments are positively correlated, which is referred to as persistence; when \( 0 < H < 0.5 \), the spatial increments are negatively correlated, which is referred to as antipersistence; when \( H = 0.5 \), the spatial increments are independent of each other, and the random field reduces to Brownian motion.

### 2.1 Truncated Power Variogram Model with Exponential Modes

*n*:

### 2.2 Truncated Power Variogram Model with Gaussian Modes

## 3 Generation of Reference LNK and Flow Field

*h*is pressure head; \( \varphi \) is the prescribed head on Dirichlet boundary \( \varGamma_{\text{D}} \);

*Q*is the prescribed flux on Neumann boundary \( \varGamma_{\text{N}} \); \( {\mathbf{n}}({\mathbf{x}}) \) is the outward vector normal to \( \varGamma_{\text{N}} \). In the designed synthetic case with two-dimensional rectangle domain, the numerical grid is uniformly discretized into 1000 × 1000 cells. The top and bottom sides of the domain are no-flow boundaries, and the left and right sides of the domain are prescribed head boundaries with the value of 10 on the left side and the value of 5 on the right side.

*K*field with the power variogram model parameters as

*A*

_{0}= 0.046 and

*H*= 0.25, as shown in Fig. 1a. The steady-state groundwater flow equation is solved by using the groundwater flow simulation code, MODFLOW (Harbaugh et al. 2000), and the reference flow field is given in Fig. 1b.

Methods to generate reference and simulation ln*K* fields

Tool | Variogram type | Conditioning capability | Function |
---|---|---|---|

Fourier–wavelet method | Power | No | Generate the reference ln |

GSLIB-TPV | TpvG | Yes | Generate the simulation ln |

Traditional GSLIB | Exp | Yes | Generate the simulation ln |

## 4 Synthetic Results and Discussion

*K*data are distributed in a rectangular region which is enclosed in a larger domain, as shown in Fig. 2. The number of samples is 50 × 50 (= 2500), which takes up 0.25% of the domain grid size (1000 × 1000 = 1,000,000). Therefore, the information collected to characterize the field can be considered as being rare. Figure 2 also demonstrates three levels of scale involved in this work, i.e., the measurement scale or data support that is considered to be a point scale here with \( \lambda_{\text{l}} = 0 \) (this assumption is valid in most practical cases since \( \lambda_{\text{l}} \ll \lambda_{\text{u}} \)), the observation scale that characterizes the region of measurements, and the window scale that represents the domain under investigation.

- 1.
Generate the random log hydraulic conductivity fractal field characterized by fractional Brownian motion using the Fourier–wavelet method proposed by (Heße et al. 2014), and obtain the corresponding hydraulic head distribution by MODFLOW. Use these ln

*K*and head fields as the reference fields for the purpose of predictability comparison. - 2.Collect 50 × 50 (= 2500) ln
*K*samples in the middle region of the domain as the conditional ln*K*data, and obtain a sample variogram by using sample variogram calculation GAMV program in Geostatistical Software Library (GSLIB) package (Deutsch and Journel 1998). The sample variogram is depicted as the scattered dots in Fig. 4. - 3.
Use TpvG and Exp variogram models as the simulation models, and obtain the parameter values in these models by fitting the simulation variogram models with the sample variogram obtained in step 2. Here, the Levenberg–Marquardt algorithm is used to obtain the maximum likelihood estimation of the parameter values. The calibrated model parameter values are: \( (A,H,\lambda_{\text{l}} ,\lambda_{\text{u}} ) = \left( {0.01,\,0.41,\,0,\,151.6} \right) \) for TpvG model and \( \left( {\sigma^{2} ,\lambda } \right) = \left( {0.98,23.85} \right) \) for Exp model. The sample variogram and the calibrated simulation variogram models are depicted in Fig. 4.

- 4.Generate ln
*K*field of the entire domain conditional on the 2500 ln*K*measurement data. The Fourier–wavelet method at this stage cannot generate the conditional random field, as noted in Table 1. To generate the conditional random ln*K*field in the simulation case, the sequential simulation SGSIM program in GSLIB has been modified (and denoted as GSLIB-TPV) to add TpvG variogram model in order to make it support both TpvG and Exp variogram models. These 2500 measurements are the only available information in the simulation case. Due to the limited information to characterize the entire ln*K*field, the generated ln*K*fields are essentially uncertain, and an ensemble of conditional ln*K*realizations with the size of 1000 are used to describe such uncertainty. It is worth noting that the generation of ln*K*random field based on Exp variogram can use the calibrated parameter values directly, i.e., \( \left( {\sigma^{2} ,\lambda } \right) = \left( {0.98,23.85} \right) \). However, the calibrated \( \lambda_{\text{u}} \) parameter value is consistent with the observation scale as shown in Fig. 2. Since the TpvG model has the capability to describe the multiscale characteristics of the random field, particularly the observation scale and the window scale in this case. When it is required to generate the realizations of ln*K*field on the window scale or the entire domain, we need to upscale the value of this parameter from the calibrated observation scale to a value that is consistent with the stochastic characteristics of the field within the window scale. If the ln*K*field is a multiscale random fractal field and its spatial variability can be characterized by power variogram model, the upper limit of the integral scale, \( \lambda_{\text{u}} \), can be ideally set as positive infinity. In this case, it has been tested that upcaling the \( \lambda_{\text{u}} \) value from 151.60 to \( 10^{5} \) can be sufficiently large to represent the multiscale random fractal characteristics as shown in Fig. 5. Therefore, the actual TpvG parameter set used to generate the conditional*lnK*realizations should be \( (A,\,H,\,\lambda_{\text{l}} ,\,\lambda_{\text{u}} ) = \left( {0.01,\,0.41,\,0,\,10^{5} } \right) \). - 5.Obtain an ensemble of hydraulic head distributions by solving the groundwater governing equations. Here, a thousand conditional ln
*K*realizations are generated in the terms of either TpvG variogram model or Exp variogram model to ensure the stability of the obtained statistics. Summarize the statistics of the hydraulic head fields to obtain the ensemble mean and variance averaged over these 1000 realizations. They are plotted in Figs. 6 and 7, respectively. By comparing the hydraulic head results of ensemble mean based on Exp model (as shown in Fig. 6a) and TpvG model (as shown in Fig. 7a) with the reference hydraulic head flow field (as shown in Fig. 1b), it can be found that result obtained by TpvG model is much more consistent with the reference flow field. The flow field obtained by traditional Exp model tends to be more uniform, which is indicated by the fact that the contour lines seems to be parallel with each other. It demonstrates that the Exp model may result in a less variable ln*K*field than what TpvG model does, even though both Exp and TpvG models fit the sample variogram almost equally well as shown in Fig. 4. In addition to the visual comparison, the spatially averaged scalar statistics listed in Table 2, such as spatial average of the ensemble mean and variance, can also give similar analysis results. By comparing the spatial average of the ensemble mean hydraulic head values based on Exp and TpvG variogram models with that of reference hydraulic head field, it can be found that the relative error for TpvG model is 0.038% and the relative error for Exp model is 0.667%. The spatial average of the ensemble mean value based on TpvG model is more close to reference value than that based on Exp model. This indicates that TpvG model results in a more accurate estimation on the hydraulic head distribution. The reason is that the TpvG model can explicitly account for scaling effect of the random fractal ln*K*field, particularly the observation scale and window scale here, but Exp model has no such capability. The scaling characteristics of the random field determine the performance of the simulation model. By comparing the spatial average of the ensemble variance values based on Exp variogram model with that based on TpvG model, it can be found that the spatial average of the ensemble variance value based on TpvG model is more than 80 times larger than that based on Exp model. This indicates that the Exp variogram model may remarkably underestimate the predictive uncertainty of the hydraulic head field, when the underlying porous media are multiscale random fractals. It clearly shows the necessity of proper upscaling when dealing with such multiscale fields.Table 2Summary of statistical values for head distribution based on Exp and TpvG variogram models

Statistics

Reference

Exp

TpvG

Spatial average of mean

7.5547

7.5043

7.5576

Spatial average of variance

–

0.0005

0.0398

RMSE

^{a}–

0.2244

0.1972

Ensemble spread

^{b}–

0.0228

0.1996

- 6.
Evaluate the predictability of the hydraulic head distribution obtained through the TpvG and Exp variogram models by using two representative statistics. The selected statistics are RMSE (root-mean-square error) and ensemble spread.

*N*is the number of the grid blocks, \( h_{i}^{t} \) is the reference hydraulic head value on a given grid block (it is known in synthetic case), and \( E\left( {h_{i} } \right) \) is the ensemble mean of hydraulic head on a given grid block. RMSE represents the consistency level of the predicted hydraulic head field with the reference field.

The values of RMSE and ensemble spread for Exp and TpvG variogram models are listed in Table 2. It can be observed that the ensemble spread value is close to RMSE for hydraulic head results based on TpvG model. But the ensemble spread value greatly underestimates RMSE for hydraulic head results based on Exp model almost by an order of magnitude. This indicates that the predictability on hydraulic head distribution based on TpvG model is superior to that based on Exp model when the underlying porous media are multiscale random fractals. Circumspection is greatly required when assessing the system uncertainty in a random field exhibiting multiscale characteristics.

## 5 Conclusion

The multiscale porous media can be better characterized using random fractals, which assumes that the heterogeneous hydrogeological fields are non-stationary but with stationary increments. Power variogram model for a Gaussian random field follows the stochastic characteristics of fractional Brownian motion. When the high- and low-frequency modes are excluded from the infinite hierarchy set, it renders the so-called truncated power variogram model with either exponential or Gaussian modes. The merits of the truncated power variogram model over the traditional variogram model have been investigated here, especially the performance to predict the flow distribution in a multiscale random fractal porous medium.

Truncated power variogram model allows one to upscale spatial variation of the media, i.e., to infer information of the whole fractals based on measurements in the finite observation ranges. In this work, the predictabilities of flow distribution through a multiscale random fractal field based on the TpvG and the traditional Exp variogram model are evaluated by using Monte Carlo simulation analysis in a synthetic case. The reference hydraulic head field is obtained through a multiscale reference ln*K* field described using non-stationary power variogram model. The TpvG and Exp variogram are, respectively, used as the simulation models to characterize such field with limited ln*K* measurements, and the statistical prediction performance of flow distribution are evaluated. The analysis results show that the traditional Exp model tends to predict the flow characteristics worse than TpvG model does and tends to underestimate the spatial variability of the flow field. The latter is particularly important in the consequential risk assessment analysis. Furthermore, the evaluation of predictability based on these two different types of variogram models is conducted through the consistency of two statistical measurers: RMSE and ensemble spread. The results show that the values of RMSE and ensemble spread are closer for TpvG-based model, indicating a better predictability for this model. It confirms the superiority of TpvG model over Exp model to characterize the flow distribution when the underlying ln*K* field is a multiscale random fractal medium. More importantly, great prudence is needed to avoid uncertainty underestimation when analyzing flow models in a random field exhibiting multiscale characteristics.

## Notes

### Acknowledgements

This work is funded by the Science Foundation of China University of Petroleum - Beijing (Grant No. 2462014YJRC038), the National Science and Technology Major Project (Grant No. 2016ZX05037003), National Natural Science Foundation of China (Grant No. 41602250) and China Geological Survey (Grant No. DD20160293).

### Compliance with Ethical Standards

### Conflict of interest

The authors declare no conflicts of interest or financial disclosures to report.

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