Transport in Porous Media

, Volume 126, Issue 1, pp 97–114 | Cite as

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

  • Liang Xue
  • Diao Li
  • Tongchao NanEmail author
  • Jichun Wu


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.


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.

Variogram-based methods are based on two-point statistics and have been considered to be weak in characterizing geological patterns. However, variogram models are still indispensable in the stochastic analysis of geological properties. First, variogram models have been widely used at the sites where patterns are not dominating or significant (e.g., Hyun et al. 2002; Guadagnini et al. 2014). Second, new improvements have been continuously made to enhance their ability to capture geologic complexity, e.g., sequential indicator simulations (SIS) (Seifert and Jensen 1999), transition probability-based techniques (Carle and Fogg 1996), fractal models (Painter 2001; Meerschaert et al. 2004; Guadagnini and Neuman 2011), and truncated pluri-Gaussian approaches (Armstrong et al. 2011; Emery 2007; Mariethoz et al. 2009). Third, variogram-based methods seem to be the best choice when simulating long-range correlation in heterogeneous media at the present stage (Riva et al. 2013; Guadagnini et al. 2013; Moslehi and de Barros 2017). This traditional variogram model is essentially built on the statistically homogeneous assumption. The natural logarithm of hydraulic conductivity K(x) can be denoted as:
$$ Y\left( {\mathbf{x}} \right) = { \ln }K\left( {\mathbf{x}} \right), $$
where 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 xi, all the possible realizations constitute an ensemble of the random variable Y(xi). 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:
$$ E(Y({\mathbf{x}})) \equiv E(Y). $$
It indicates that the mean value of the random field is independent of the location; the second moment, i.e., the covariance of the random field, can be denoted as:
$$ {\text{Cov}}(Y({\mathbf{x}}_{1} ),Y({\mathbf{x}}_{2} )) = {\text{Cov}}(Y({\mathbf{s}})). $$
It indicates that the covariance of log hydraulic conductivity between any two points is only dependent on the lag distance s, \( {\mathbf{s}} = {\mathbf{x}}_{1} - {\mathbf{x}}_{2} \), and the corresponding variance is:
$$ {\text{Var}}(Y({\mathbf{x}})) = {\text{Cov}}(Y(0)). $$
The statistically homogeneous assumption implies that the variance (and the integral scale as will be discussed later) of the random field is a finite constant.

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 lnK 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

In a heterogeneous log hydraulic conductivity random field, the spatial variability can be characterized by (semi-)variogram function (Di Federico and Neuman 1997; Hyun et al. 2002; Guadagnini and Neuman 2011):
$$ \gamma \left( s \right) = \frac{1}{2}E\left( {\Delta Y\left( s \right)^{2} } \right) $$
where \( \Delta Y\left( s \right) = Y\left( {{\mathbf{x}} + s} \right) - Y\left( {\mathbf{x}} \right) \) is the spatial increment of log hydraulic conductivity, s is the lag distance and \( E\left( \cdot \right) \) is the ensemble mean operator.
If the log hydraulic conductivity random field is a random fractal field, the variogram can be expressed in the form of power function:
$$ \gamma \left( s \right) = A_{0} s^{2H} $$
where \( A_{0} \) is a constant and H is Hurst coefficient \( \left( {0 < H < 1} \right) \). The combination of Eqs. (5) and (6) shows that the mean square increments of lnK 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.
The random field is self-affine since the power variogram function has the scaling property:
$$ \gamma \left( {rs} \right) = r^{2H} \gamma \left( s \right) $$
where r is any positive constant (r > 0).
The fractal dimension of the self-affine random field is:
$$ D = d + 1 - H $$
where 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

Assume that each statistically homogeneous mode in an infinite hierarchy random field with uncorrelated spatial increments can be characterized by exponential variograms (Deutsch and Journel 1998):
$$ \gamma \left( {s,\lambda } \right) = \sigma^{2} \left( \lambda \right)\left[ {1 - \exp \left( { - \frac{s}{\lambda }} \right)} \right] $$
where \( \sigma^{2} \left( \lambda \right) = A\lambda^{2H} \) is the variance of the natural logarithm hydraulic conductivity random field. Define the mode number \( n = 1/\lambda \) representing the spatial frequency of the random fluctuation, it can be written in terms of mode number as \( \sigma^{2} \left( \lambda \right) = A/n^{2H} \).
Integrate over all the possible modes within the range \( \left[ {n_{\text{l}} ,n_{\text{u}} } \right] \) and weight each mode with a factor of 1/n:
$$ \gamma \left( {s,n_{\text{l}} ,n_{\text{u}} } \right) = \int_{{n_{\text{l}} }}^{{n_{\text{u}} }} {\gamma \left( {s,n} \right)\frac{1}{n}} {\text{d}}n $$
where the lower and upper limits of mode number \( n_{\text{l}} \) and \( n_{\text{u}} \) correspond with the upper and lower limits of integral scale \( \lambda_{\text{u}} \) and \( \lambda_{\text{l}} \) due to the reciprocal relationship.
The upper limit of integral scale \( \lambda_{\text{u}} \) is proportional to the characterization length of window scale, and the lower limit of integral scale \( \lambda_{\text{l}} \) is proportional to the characterization length of data support or measurement scale. Setting these limits to the integral form of Eq. (10) is equivalent to excluding or truncating the modes out of the range. The consequent truncated variogram model is:
$$ \gamma \left( {s,n_{\text{l}} ,n_{\text{u}} } \right) = \gamma \left( {s,\lambda_{\text{u}} } \right) - \gamma \left( {s,\lambda_{\text{l}} } \right) $$
with the analytical form (\( 0 < H \le 0.5 \))
$$ \gamma \left( {s,\lambda_{m} } \right) = \frac{{A\lambda_{m}^{2H} }}{2H}\left[ {1 - \exp \left( { - \frac{s}{{\lambda_{m} }}} \right) + \left( {\frac{s}{{\lambda_{m} }}} \right)^{2H} \varGamma \left( {1 - 2H,\frac{s}{{\lambda_{m} }}} \right)} \right] $$
where \( m = l,u \) and \( \varGamma \left( { \cdot , \cdot } \right) \) is the incomplete gamma function.
If the lower limit \( \lambda_{\text{l}} \) approaches to 0 (representing a point measurement scale or data support) and the upper limit \( \lambda_{\text{u}} \) approaches to infinity (representing an infinitely large window scale or domain), the truncated variogram model reduces to the power variogram model in Eq. (6) with the condition:
$$ A_{0} = \frac{{A\varGamma \left( {1 - 2H} \right)}}{2H} $$
where \( \varGamma \left( \cdot \right) \) is the gamma function. Due to this relationship, variogram model in Eq. (12) is referred to as truncated power variogram model with exponential modes (TpvE).

2.2 Truncated Power Variogram Model with Gaussian Modes

Assume that each statistically homogeneous and independent mode in an infinite hierarchy can be characterized by Gaussian variograms (Deutsch and Journel 1998):
$$ \gamma \left( {s,\lambda } \right) = \sigma^{2} \left( \lambda \right)\left[ {1 - \exp \left( { - \frac{{\pi s^{2} }}{{4\lambda^{2} }}} \right)} \right] $$
Following the similar analysis as discussed above, the truncated variogram model can be expressed as:
$$ \gamma \left( {s,n_{\text{l}} ,n_{\text{u}} } \right) = \gamma \left( {s,\lambda_{\text{u}} } \right) - \gamma \left( {s,\lambda_{\text{l}} } \right) $$
with the analytical form (\( 0 < H \le 1 \))
$$ \gamma \left( {s,\lambda_{m} } \right) = \frac{{A\lambda_{m}^{2H} }}{2H}\left[ {1 - \exp \left( { - \frac{\pi }{4}\frac{{s^{2} }}{{\lambda_{m}^{2} }}} \right) + \left( {\frac{\pi }{4}\frac{{s^{2} }}{{\lambda_{m}^{2} }}} \right)^{H} \varGamma \left( {1 - H,\frac{\pi }{4}\frac{{s^{2} }}{{\lambda_{m}^{2} }}} \right)} \right] $$
If the lower limit \( \lambda_{\text{l}} \) approaches to 0 and the upper limit \( \lambda_{\text{u}} \) approaches to infinity, the truncated variogram model can also reduce to the power variogram model in Eq. (6) but with the condition:
$$ A_{0} = \frac{{A\left( {\pi /4} \right)^{H} \varGamma \left( {1 - H} \right)}}{2H} $$
And variogram model in Eq. (16) is referred to as truncated power variogram model with Gaussian modes (TpvG).
When the truncated variogram models have a finite window scale \( \lambda_{\text{u}} \), they can be used to characterize a stationary field with a finite variance:
$$ \sigma^{2} \left( {\lambda_{\text{l}} ,\lambda_{\text{u}} } \right) = \sigma^{2} \left( {\lambda_{\text{u}} } \right) - \sigma^{2} \left( {\lambda_{\text{l}} } \right) = \frac{{A\left( {\lambda_{\text{u}}^{2H} - \lambda_{\text{l}}^{2H} } \right)}}{2H} $$
And a finite integral scale:
$$ I\left( {\lambda_{l} ,\lambda_{u} } \right) = \frac{2H}{1 + 2H}\frac{{\lambda_{u}^{1 + 2H} - \lambda_{l}^{1 + 2H} }}{{\lambda_{u}^{2H} - \lambda_{l}^{2H} }} $$

3 Generation of Reference LNK and Flow Field

A synthetic case is designed to investigate the performance to describe the characteristics of steady-state saturated groundwater flow through a multiscale porous media. The governing equation of flow field is (Harbaugh et al. 2000):
$$ \nabla \cdot \left[ {K\left( {\mathbf{x}} \right)\nabla h\left( {{\mathbf{x}},t} \right)} \right] = 0 $$
which is subject to the prescribed boundary condition
$$ h({\mathbf{x}}) = \varphi ({\mathbf{x}}),{\mathbf{x}} \in \varGamma_{\text{D}} $$
and the prescribed flow rate boundary condition
$$ K\left( {\mathbf{x}} \right)\nabla h({\mathbf{x}},t) \cdot {\mathbf{n}}({\mathbf{x}}) = Q({\mathbf{x}},t),{\mathbf{x}} \in \varGamma_{\text{N}} $$
where 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.
The medium is assumed to be a random fractal following a fractional Brownian motion (fBm) distribution which can be described using power variogram model as in Eq. (6). The validity of power variogram model to characterize the permeability and porosity data at various sites has been reported in several literatures (Molz and Boman 1993, 1995; Liu and Molz 1996). To generate an fBm realization, there exist several approaches in the literature. These approaches can be classified into three major categories. The first category is based on covariance decomposition, such as Cholesky decomposition methods, Karhunen–Loeve decomposition method and Davies–Harte method (Dieker 2004; O’Malley et al. 2012; Cohen and Istas 2013). The second category is based on sequential simulations, including random midpoint displacement method, sequential Gaussian simulation (SGS), and the Hosking method (Voss 1986; Guadagnini and Neuman 2011; Hosking 1984). The third category relies on spectral simulation, such as fast Fourier transformation method and Fourier–wavelet method (Prykhodko and Attinger 2014). In this study, Fourier–wavelet method listed in Table 1 is chosen to generate a large 2D reference lnK field with the power variogram model parameters as A0 = 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.
Table 1

Methods to generate reference and simulation lnK fields


Variogram type

Conditioning capability


Fourier–wavelet method



Generate the reference lnK field using power variogram model




Generate the simulation lnK field with TpvG variogram type conditioning on 2500 observation data

Traditional GSLIB



Generate the simulation lnK field with Exp variogram type conditioning on 2500 observation data

Fig. 1

Reference lnK and flow fields

4 Synthetic Results and Discussion

To investigate the predictability of TpvG shown in Eq. (16) and the widely used traditional exponential variogram (denoted as “Exp” hereinafter in figures and tables) in groundwater hydrology as shown in Eq. (9) on the characteristics of flow through the multiscale random fractals, the Monte Carlo-based analysis procedures are proposed when the number of measurements is limited. The samples of lnK 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.
Fig. 2

lnK sampling region of the synthetic case

The analysis procedures that is shown in Fig. 3 and the corresponding results to evaluate the predictability of flow characterization through the random fractals by using TpvG and Exp variogram model are presented as follows:
Fig. 3

Flowchart to perform the comparative analysis between TpvG and Exp models

  1. 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 lnK and head fields as the reference fields for the purpose of predictability comparison.

  2. 2.
    Collect 50 × 50 (= 2500) lnK samples in the middle region of the domain as the conditional lnK 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.
    Fig. 4

    Sample variogram and calibration results of TpvG and Exp variogram models

  3. 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. 4.
    Generate lnK field of the entire domain conditional on the 2500 lnK 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 lnK 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 lnK field, the generated lnK fields are essentially uncertain, and an ensemble of conditional lnK realizations with the size of 1000 are used to describe such uncertainty. It is worth noting that the generation of lnK 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 lnK 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 lnK 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 \( (A,\,H,\,\lambda_{\text{l}} ,\,\lambda_{\text{u}} ) = \left( {0.01,\,0.41,\,0,\,10^{5} } \right) \).
    Fig. 5

    Upscaling \( \lambda_{\text{u}} \) parameter from observation scale to window scale

  5. 5.
    Obtain an ensemble of hydraulic head distributions by solving the groundwater governing equations. Here, a thousand conditional lnK 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 lnK 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 lnK 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.
    Fig. 6

    Ensemble mean and variance of hydraulic head based on Exp variogram model: a ensemble mean; b ensemble variance

    Fig. 7

    Ensemble mean and variance of hydraulic head based on TpvG variogram model: a ensemble mean; b ensemble variance

    Table 2

    Summary of statistical values for head distribution based on Exp and TpvG variogram models





    Spatial average of mean




    Spatial average of variance






    Ensemble spreadb



    aObtained through Eq. (23)

    bObtained through Eq. (24)

  6. 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.

RMSE can be computed through (Chen and Zhang 2006):
$$ {\text{RMSE}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {h_{i}^{t} - E(h_{i} )} \right)^{2} } } $$
where 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 ensemble spread can be written as (Chen and Zhang 2006):
$$ {\text{Ensemble Spread}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {{\text{Var}}(h_{i} )} } $$
where \( {\text{Var}}(h_{i} ) \) is the ensemble variance of the hydraulic head. If a predictive model characterizes the uncertainty of the hydraulic head field properly, the ensemble spread of that predictive model should be close to the RMSE.

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 lnK 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 lnK 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 lnK 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.



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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Authors and Affiliations

  1. 1.State Key Laboratory of Petroleum Resources and ProspectingChina University of PetroleumBeijingChina
  2. 2.Department of Oil-Gas Field Development, College of Petroleum EngineeringChina University of PetroleumBeijingChina
  3. 3.Department of Hydrosciences, School of Earth Sciences and EngineeringNanjing UniversityNanjingChina
  4. 4.State Key Laboratory of Pollution Control and Resources ReuseNanjingChina

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