Alluvial fans are highly heterogeneous in hydraulic properties due
to complex depositional processes, which make it difficult to characterize
the spatial distribution of the hydraulic conductivity (

Alluvial fans usually house valuable groundwater resources because of significant water storage and favorable recharge conditions. Sedimentary processes forming alluvial fans are responsible for their complex long-term evolution. Usually, the coarsest material (gravel) is deposited in the upper fan, with the gravel passing into sand in the middle of the fan and then into silt and clay in the tail. A high heterogeneity characterizes the deposit distribution because of the shifting over time of the sediment-transporting streams (Zappa et al., 2006; Weissmann et al., 1999).

Hydraulic conductivity distributions in alluvial fans can be assigned
according to the various hydrofacies simulated by conditional indicator
geostatistical methods (Eggleston and Rojstaczer, 1998; Fogg et al., 1998;
Weissmann and Fogg, 1999; Weissmann et al., 2002a, b; Ritzi et al.,
2004, 2006; Proce et al., 2004; Dai et al., 2005; Harp et al., 2008; Hinnell
et al., 2010; Maghrebi et al., 2015; Soltanian et al., 2015; Zhu et al.,
2016a). However, the geostatistical methods require the stationary
assumption; i.e., the distribution of the volumetric proportions and
correlation lengths of hydrofacies converge to their mean values in the
simulation domain. The hydrofacies and hydraulic conductivity (

Hydraulic conductivity of granular deposits generally varies with grain
size, porosity and sorting. Traditional methods for

This study proposes a novel approach to reconstruct the three-dimensional (3-D)
configuration of conductivity in alluvial fans by combining the hydrofacies
spatial heterogeneity provided by a multiple-zone transition probability model
with hydrogeological and hydrogeophysical measurements, in particular
inexpensive VESs properly calibrated through
resistivity logs acquired in a few wellbores. We assume the

Chaobai alluvial fan in the north of Beijing Plain.

The study area belongs to the Chaobai River alluvial fan (or megafan), in
the northern Beijing Plain (northern latitude 40–40

The study area is one of the most important regions for the supply of
groundwater resource to Beijing. The Huairou emergency groundwater resource
region (hereafter EGRR) with an area of 54 km

The largest cumulative land subsidence from June 2003 to January 2010 was quantified in approximately 340 mm by Zhu et al. (2013, 2015) in Tianzhu County to the south. The characterization of the distribution and spatial variability of the hydraulic conductivity is vital for an optimal use of the limited water resources in this area.

Presently, a large set of hydraulic conductivity samples can be derived by
integrating appropriate relations of various geological data, including
hydrogeophysical measurements, borehole lithostratigraphies and
hydrogeological information (total dissolved solid (TDS) and groundwater
level). These databases can be statistically processed to derive the spatial
variation of

In this paper, the statistical assessment is separately carried out for
separated zones, building-up experimental semivariograms that are fitted
with exponential models. The optimal parameters of the latter are
estimated through a generalized output least-squares criterion. Then,
the composite semivariograms are computed using a hierarchical sedimentary
architecture (Ritzi et al., 2004; Dai et al., 2005) to obtain the

Geophysical data include resistivity logging and vertical electrical soundings. There are six well-electric logs continuously recording the formation resistivity versus depth. Five logs were collected in zone 2 and one in zone 3. Each well log has a lithological description, which helps to relate the resistivity values to the corresponding facies.

Flowchart of the geostatistical methodology.

The average resistivity of G is the largest, with a value of 198

The C resistivity is relatively low due to the good intrinsic electrical
conductivity of this facies. For example from 16.5 to 23.5 m depth, where
C is the prevalent facies, a low resistivity equal to 27.2

Typical depth behaviors of resistivity and corresponding stratigraphy in the eastern part of zone 2.

Inversed resistivity and corresponding stratigraphy in zone 1.

VESs using the Schlumberger electrode configuration were carried out by the Beijing Institute of Hydrogeology and Engineering Geology (BIHEG). A number of 113 detecting positions were selected, with a maximum half current electrode space equal to 340 m and the potential electrode space ranging from 1 to 30 m. All the sounding data (1356 VES measurements) recorded the apparent resistivity of the porous medium. These data were inverted to real resistivity using the nonlinear Occam inversion method (Constable et al., 1987), with a low root mean square relative error of 2 %. Figure 4 shows the layered structure fitting model of resistivity and the borehole lithologic observations. The inversed resistivity generally reflects the difference of facies; the thick gravel layer has larger resistivity, whereas the fine-sand and clay layers have relatively smaller resistivity.

Almost 700 borehole lithologic logs were collected in the study area. The sedimentary deposits show large heterogeneity from the upper to the lower fan zone. In zone 1, the dominant facies is G with a volumetric proportion of 53 %. The volumetric proportion of C is 16 %. In zone 2, the volumetric proportion of C increases to 40%, while that of G decreases sharply to 24 %. In zone 3, the proportion of G decreases further to 6 % and that of C increases to 50 % (Table 1). More detailed information is given in Zhu et al. (2016a). The lithological information in a buffer zone of 200 m around the VES locations has been used to represent the actual facies distribution in the area surrounding the sites of the geophysical acquisitions.

Values of the volumetric proportion for the various facies in three zones.

A total of 35 hydrochemistry measurements with a depth from 20 to 270 m
were obtained throughout the area. The minimum, maximum and average TDS
values are 423 mg L

A large number of depth of water level measurements were also collected to map the thickness of the unsaturated unit. The TDS and groundwater level at each VES and resistivity log location were derived from the interpolated surfaces.

The hydraulic conductivity

Statistical data of logarithm hydraulic conductivity (

The logarithmically transformed values of the estimated hydraulic
conductivity (

Histograms of

Semivariogram describes the degree of spatial dependence of a spatial random
field or stochastic process. It is a concise and unbiased characterization
of the spatial structure of regionalized variables, which is important in
Kriging interpolations and conditional simulations. The experimental semivariogram

The variance and range were optimized using the least-squares criterion,
which was solved by the modified Gauss–Newton–Levenberg–Marquardt method
(Clifton and Neuman, 1982; Dai et al., 2012). The sensitivity equation
method was derived to compute the Jacobian matrix for iteratively solving
the gradient-based optimization problem (Samper and Neuman, 1986; Carrera and
Neuman, 1986; Dai and Samper, 2004; Samper et al., 2006; Yang et al., 2014;
Zhu et al., 2016a). The two sensitivity coefficients

Once the facies semivariograms were obtained in each zone, the composite
semivariogram

The transition probability

The sequential Gaussian simulation (SGSIM) is a widely used stochastic simulation method to create numerical model of continuous variables based on the Gaussian probability density function. The process is assumed to be a stationary and ergodic random process (Deutsch and Journel, 1992; Dimitrakopoulos and Luo, 2004). This method can preserve the variance and correlation range observed in spatial samples. SGSIM provides a standardized normal continuous distribution of the simulated variable.

Experimental (circle symbol) and model (solid line) semivariogram along
the vertical direction for the various hydrofacies in the three zones. Notice
that the range in the

With the assumption that the log conductivity distributions are stationary
within each zone, we used the SGSIM simulator implemented into GEOST to model
the

Finally, the 3-D conductivity configuration was derived by
combining the stochastic simulated facies (Zhu et al., 2016a) with the SGSIM
conductivity distribution and the mean

The optimized vertical correlation range and variance of the log conductivity semivariogram (Eq. 5) are listed in Table 3, along with their 95 % confidence intervals. The fitting between the experimental and the model semivariograms is the best in zone 2 because of the abundant samples, whereas the fitting in zone 1 is the worst (Fig. 6). The fitting result of the semivariogram for the G facies is the worst in zone 1. There are two reasons for this: the first is the high variance of the log conductivity of gravel in this zone, and the second is the limited number of samples (102 samples), which makes quite small the pair numbers within each lag spacing. Hence, the computed semivariogram is highly uncertain.

Optimized parameters in the fitting exponential function of

The variance of FS, MS and G in the vertical direction decreases from zone 1 to zone 3. In the upper alluvial fan, sediments were deposited under multiple water-flowing events and with poor sorting. The deposits consist of wide ranges of sediment categories and grain sizes. The variance of G is larger than 1.5, which reflects the high heterogeneity of hydraulic conductivity in coarse deposits. The variances of FS and MS are smaller with values equal to 0.23 and 0.32, respectively. In zone 3, these values decrease to 0.05 and 0.13, respectively, with that of G sharply decreasing to 0.62. In the middle-lower fan zone, the conductivity variation within each facies reduces gradually because the ground surface slope becomes smaller or flat, the sediment transport energy decreases and the deposits within the three facies are well sorted.

Note that the ranges are correlated with the facies structure parameters such as the indicator correlation scale, mean thickness (or length) and volumetric proportion (Dai et al., 2004b, 2007). The estimated correlation ranges of FS, MS and G along the vertical direction in zone 1 do not show big difference with values equal to 6.0, 8.0 and 6.5 m, respectively. Zone 2 was extended from the fan apex zone (zone 1) with much larger area, which allows for greater preservation potential of finer sediments – such as MS, FS and C – than the more proximal zone 1. Therefore, in zone 2 the volumetric proportions for these three facies increase while that of gravel decreases. The estimated ranges of G and MS are increased. In zone 3, the range difference among the three facies decreases gradually. The range of FS is about 6.0 m, which is twice as much as that of MS. The spatial variation of the structure parameters of three facies causes the large changes of the correlation ranges from zone 1 to zone 3.

Variances of

Due to the small number of conductivity samples in zone 1, the variance of

The composite semivariogram in the vertical direction at each zone is calculated by Eq. (8), using the volume proportions (Table 1) and transition probability (Eq. 9) with the same values of the lag distance used to compute the facies semivariograms (Fig. 8). The values of the optimized variance are 0.68, 0.11 and 0.03 in zone 1, zone 2 and zone 3, respectively. The high-flow energy and the large number of flooding events contributing to sediment deposition are the main causes of the high heterogeneity (largest variance) of the deposits in the upper part of the alluvial fan. The changes of variance between the three zones support the utilization of the local-stationary assumption and simulation of multiple-zone-based conductivity distributions for the Chaobai alluvial fan.

Experimental (circle symbol) and model (solid line) semivariogram
along the dip direction for the various hydrofacies in zone 2 and zone 3.
Notice that the range in the

Experimental (circle symbol) and model (solid line) composited semivariogram along the vertical direction for the three zones.

Distribution of hydrofacies (after Zhu et al., 2016a) and

The configuration of

Based on the 3-D

Investigating the stochastic results along the vertical direction, it is
interesting to notice that the average

This paper proposes a geostatistical method under a multiple-zone framework,
properly supported by a large number of geophysical investigations, to
detect the distribution and the related variance of the hydraulic
conductivity in 3-D domains. In particular, the optimized
statistical parameters (e.g., log conductivity variance and correlation
range) of semivariograms are estimated using the modified
Gauss–Newton–Levenberg–Marquardt method. The Chaobai alluvial fan is used as
a case study area. Multiple data, including downhole resistivity logging
data, vertical electric soundings, well-bore lithologic logs, TDS
measurements and depths to the water table, are integrated to derive a
dataset of conductivity values in a 3-D setting. Log
conductivity semivariograms fitted with exponential functions were
constructed for three facies, including fine sand, medium-coarse sand and
gravel, in each of the three zones into which the Chaobai fan is divided to
guarantee local stationarity of the statistical process. The composite
semivariogram of the three facies has been derived for the two zones where a
sufficiently large number of samples are available. The

For the specific test case, the variance along the vertical direction of
fine sand, medium-coarse sand, and gravel decreases from the upper part of
the alluvial fan, where the values amount to 0.23, 0.32 and 1.60, to the
lower portion of the Chaobai plan with values of 0.05, 0.126 and 0.62,
respectively. This behavior reflects the higher transport energy in the
upper alluvial fan that causes a poor sediment sorting. In the middle
alluvial fan, the transport energy decreases and the sediments tend to be
relatively well sorted. The variance of the gravel is larger than that of
other lithologies. The different flow energy significantly affected the
coarse sediments in the vertical direction. Along the dip direction, the
variance of three facies (gravel, medium-coarse sand and fine sand) in the
middle fan is larger than that in the lower fan. The composite variance of

The distribution of hydraulic conductivity is consistent with that of the facies. Hydraulic conductivity is much larger in the upper zone than that in the lower part of the alluvial fan. This result provides valuable insights for understanding the spatial variations of hydraulic conductivity and setting-up groundwater flow, transport and land subsidence models in alluvial fans.

Concluding, it is worth highlighting that we depicted an original method to detect the variance and configuration of conductivity by fusing multiple-source data in 3-D domains. The proposed approach can be easily used to statistically characterize the hydraulic conductivity of the various alluvial fans, which worldwide are strongly developed to provide high-quality water resources. We are aware of some restrictions in the dataset available at the date for the Chaobai alluvial fan, for example the assumed uniform distribution of TDS versus depth and the relatively small number of the conductivity samples in the upper fan zone. A more accurate description of the semivarigrams in the dip and lateral directions will be included in our future study to improve the developed 3-D permeability field. Moreover, our assumption that the logs are well calibrated might be another source of uncertainty that can be reduced in our forthcoming work. Nonetheless, the proposed methodology will be re-applied in the near feature as soon as new information will become available, thus allowing one to improve the estimation accuracy of spatial statistics parameters and the configuration of hydraulic conductivity in this Quaternary system so important for the Beijing water supply.

The geophysical measurements, borehole lithostratigraphies and hydrogeological information in the northern part of the Beijing Plain can be partly accessible by contacting Beijing Institute of Hydrogeology and Engineering Geology.

Lin Zhu, Huili Gong and Zhenxue Dai derived the method of spatial variance and 3-D configuration of conductivity, performed data analysis and wrote the draft manuscript. Gaoxuan Guo collected the geological and geophysical data, and discussed the results. Pietro Teatini discussed the results, and reviewed and revised the manuscript.

The authors declare that they have no conflict of interest.

This work was supported by the National Natural Science Foundation (nos. 41201420, 41130744) and Beijing Nova Program (no. Z111106054511097). Pietro Teatini was partially supported by the University of Padova, Italy, within the 2016 International Cooperation Program. Edited by: G. Fogg Reviewed by: G. Weissmann and two anonymous referees