Introduction
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 (K)
distributions in alluvial fans are generally non-stationary (Weissmann et
al., 1999, 2010, 2013; Anderson, 2007; Zhu et al., 2016a).
Hence, the use of these methods may cause large characterization errors and
add significant uncertainty to the predictions achieved by groundwater flow
and contaminant transport models (Eggleston and Rojstaczer, 1998; Irving and
Singha, 2010; Dai et al., 2014a). Zhu et al. (2016a) adopted a
local-stationary assumption by dividing the alluvial fan into three zones
along the flow direction of the Chaobai River, China. The zones were
properly detected based on the statistical facies distribution. Then, the
indicator simulation method was applied to each zone and the simulated
hydrofacies distribution in the three zones was used to guide modeling of the K distribution.
Hydraulic conductivity of granular deposits generally varies with grain
size, porosity and sorting. Traditional methods for K estimate, e.g., well
test, permeability measurements, and grain-size analyses (Niwas et al.,
2011), are very expensive, time-consuming and make it difficult to provide
representative and sufficient field data for addressing spatial variations
of conductivity. Recently, data fusion techniques have been developed for
coupled inversion of multiple-source data to estimate K distributions for
groundwater numerical modeling. Geophysical data (such as surface electric
resistivity and various logging data) are relatively inexpensive and can
provide considerable information for characterizing subsurface heterogeneous
properties (Hubbard et al., 2001; Yeh et al., 2002; Dai et al., 2004a; Morin,
2006; Sikandar et al., 2010; Bevington et al., 2016). Electric resistivity
data have been proven useful to derive sediment porosity distributions
(Niwas and Singhal, 1985; Niwas et al., 2011; Niwas and Celik, 2012; Zhu et
al., 2016b). Zhu et al. (2016b) simulated the spatial distributions of
hydraulic conductivity by combining the interpolated resistivity on the basis of
vertical electrical soundings (VESs) and the stochastic simulated facies through the empirical equation, in which
the hydraulic conductivity was converted from the porosity data calculated
from resistivity measurements and the grain size.
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 K distributions
are local stationary; i.e., the mean and variance of log conductivity are
convergent in each hydrofacies and in each local zone. Therefore, we can
compute the log10(K) semivariogram in each hydrofacies and in each zone.
The spatial structure features of hydraulic conductivity deduced from
semivariograms are used during the geostatistical simulation processes of
the hydraulic conductivity. The Chaobai alluvial fan (or “megafan”
as defined by Leier et al., 2005, and Hartley et al., 2010, for very large
alluvial fans) in the northern Beijing Plain, China, was selected as study
area to test the proposed integrated approach.
Chaobai alluvial fan in the north of Beijing Plain. (a) Location
of the study area and distribution of the field data. (b) Map of the
hydraulic conductivity issued by Beijing Institute of Hydrogeology and Engineering
Geology (2007). The location of the study area is shown in the inset.
Material and methods
Study area
The study area belongs to the Chaobai River alluvial fan (or megafan), in
the northern Beijing Plain (northern latitude 40–40∘30′, eastern
longitude 116∘30′–117∘),
with an area of 1150 km2 (Fig. 1a). The Chaobai River is the second
largest river flowing through the Beijing Plain from north to south. The
ground elevation decreases southward with an average 2 ‰ slope.
Quaternary sediments were mainly deposited by flooding events with
turbulent flow and consist of porous strata containing groundwater. The
aquifer system in the alluvial fan can be divided into three zones according
to the lithological features (Fig. 1): an upper fan zone (or zone 1) with
coarse sediments (e.g., sandy-gravel aquifers); a middle-upper fan zone (or
zone 2), where medium-coarse sediments (e.g., sandy-gravel to sandy-silt
aquifers) were deposited; and a fine sediment (e.g., sand and clay multiple
aquifers) middle-lower fan zone (or zone 3). Four hydrofacies, including
sub-clay and clay (C), fine sand (FS), medium-coarse sand (MS) and gravel (G),
were classified based on the interpretations of the cores and textural
description of almost 700 boreholes (Zhu et al., 2016a).
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 km2 is located in zone 1.
The total groundwater withdrawal amounted to 1.2 × 108 m3
in 2003. Several well fields belonging to the “water supply
factory” were drilled along the Chaobai River in zone 1 and the upper zone 2.
Most of these well fields were built in 1979 with a designed groundwater
pumping volume of 1.6 × 108 m3 per year. The average
thickness of the exploited aquifer system is approximately 300 m. The
long-term over-exploitation of the aquifer system has resulted in a serious
drawdown of water levels, which has reduced the exploitable groundwater
resources and induced geological disasters, mainly land subsidence, fault
reactivation and ground fissures (Cheng et al., 2015; Yang et al., 2015;
Zhu et al., 2015). In 2010, the annual groundwater withdrawal at the EGRR
and the water factory decreased to 0.86 × 108 and
0.65 × 108 m3, respectively.
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.
Methodological approach
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 log10(K) for various facies, including clay, fine sand,
medium-coarse sand and gravel.
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 K variance
in each zone. Finally, the configuration of log10(K) is simulated
through a multiple-zone sequential Gaussian algorithm with estimated
statistic parameters reflecting the K spatial structures in the alluvial fan.
Figure 2 shows the steps involved in the developed approach.
Dataset
Geophysical data
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 Ωm,
and that of C is the smallest with a value of 24 Ωm. Figure 3
compares the outcome of logging data in term of resistivity versus depth and
the corresponding stratigraphy, where the groundwater depth is 12 m. The log
was acquired in the eastern part of zone 2. The average resistivity from
32.4 to 40.5 m depth, where the sediments are mainly G and MS, is
70.8 Ωm. The resistivity curve shows two evident peaks from 97 to
102 m and between 81 and 84.5 m depth, where the MS is located.
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 Ωm is
recorded. Since a hydrofacies with a smaller grain size has a greater total
surface area, the resistivity difference can partially reflect the
distributions of particle sizes and the hydrofacies composition. Since the
obtained resistivity is the apparent resistivity, we used the resistivity
located in the middle of the facies block, where the resistivity is
approximate to the real resistivity. Unfortunately, it is unknown to the
authors if the logs were calibrated in the field and how the salinity of the
formation water, although minimal and almost independent on the site and
depth, has been accounted for. On the other hand, the resistivity
distributions have good correlations with different hydrofacies along the
vertical and horizontal directions. Therefore, in the mathematical framework
that follows, we have assumed that the logs have been calibrated and are
accurate enough for presenting our work as a proof of concept.
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.
Geological and hydrogeological data
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.
Zone
Sub-clay
Fine
Medium-
Gravel
and clay
sand
coarse sand
Zone 1
0.166
0.234
0.067
0.533
Zone 2
0.409
0.286
0.065
0.240
Zone 3
0.503
0.328
0.106
0.063
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-1 at the depth of 180 m, 943 mg L-1 at the depth of 50 m
and 692 mg L-1, respectively. Generally, the TDS is very low with the higher
values measured in the southwestern part of the study area. Because of the
relatively small dataset and the observed low variability, in this paper the
TDS variation in the vertical direction has been neglected. A TDS map was
obtained by interpolating the available records using an ordinary Kriging
method with a spherical semivariogram model.
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.
Hydraulic conductivity estimates from geophysical acquisitions
The hydraulic conductivity K was estimated using the Kozeny–Carman equation:
K(x,y,z)=δgμ×d(x,y,z)2180φ(x,y,z)31-φ(x,y,z)2,
which is widely accepted to derive the hydraulic conductivity from grain
size and porosity (Soupious et al., 2007; Utom et al., 2013; Khalil and Santos,
2013; Zhu et al., 2016b). In Eq. (1), d(x,y,z) is the median grain
diameter (D50, mm) at the location (x, y, z), which was determined according to the
lithology information (or lithological descriptions and grain-size distributions),
g is gravity, μ the kinematic viscosity (kg/(m s-1)), δ the fluid density and
φ(x,y,z) the porosity. φ was estimated using Archie's law
(Eq. 2), which relates the bulk resistivity of granular medium to porosity:
ρ=αρwφ-msw-n,
where ρ is the saturated formation resistivity (Ωm),
α the pore-geometry coefficient associated with the medium
(0.5 ≤ α ≤ 2.5) and m the cementation factor (1.3 ≤ m ≤ 2.5)
(Massoud et al., 2010; Khalil and Santos, 2013). α is set as 1. In
the upper part of alluvial (zone 1 and zone 2), m is set as 1.3 due to the
sand being unconsolidated. In zone 3 m is set as 1.7, which reflects slightly
cemented sandstones (Niwas et al., 2011); sw is the water saturation, and
n the saturation index. The pore-fluid resistivity (Ωm)
ρw is calculated using the following experimental relation:
ρw=5.6(TDS)b1+β(t-18)
with TDS in g L-1, temperature t in ∘C, and b and
β are
constant parameters (Wu et al., 2003). For the most common electrolytes,
b = -0.95 and β = 0.025. Note that the parameters associated with
Eqs. (2) and (3) are site specific and applying these equations to
other sites will mean re-adjusting the related parameters.
Statistical data of logarithm hydraulic conductivity (log10(m day-1))
in the three zones of the Chaobai alluvial fan.
Zone
Parameter
Fine
Medium-
Gravel
sand
coarse sand
Zone 1
Mean
1.07
1.82
2.92
Minimum
-0.94
1.22
2.26
Maximum
1.65
2.45
3.66
Proportion
0.36
0.12
0.32
Zone 2
Mean
0.42
1.17
2.65
Minimum
-2.22
-0.23
0.95
Maximum
1.22
2.07
3.38
Proportion
0.23
0.14
0.31
Zone 3
Mean
0.17
0.81
2.48
Minimum
-2.64
-0.78
0.34
Maximum
0.72
1.43
3.21
Proportion
0.35
0.17
0.12
The logarithmically transformed values of the estimated hydraulic
conductivity (log10(K)) were used for the geostatistical analysis
because of its normal distribution (Neuman, 1990). The histograms of
log10(K) values within each facies are in Fig. 5. There are 102, 2077
and 1716 conductivity samples in zone 1, zone 2 and zone 3, respectively.
Considering that Archie's law can only be used for clay-free granular
sediments, the K values of C were not estimated in this study. Based on the
lithological description information of borehole data, it has been
reasonably assumed that clay fraction is negligible in G, MS, and FS facies.
The statistics of log10(K) for the three facies in three zones are
listed in Table 2. The mean log10(K) values decrease from zone 1 to zone 3,
consistently with the sedimentary transport processes in the alluvial
fan. In the upper region (zone 1), high water-flowing energy made the
deposits consist mainly of larger-grained particles and the coarse-grained
sediments are dominant. In the southern part (zone 3), the deposits change
to relatively fine-grained particles. The mean log10(K) of gravel is
greater than 2.4 (lg(m/d)) and that of fine sand is less than 0.2 (lg(m/d)).
The lithological information at the depth of the conductivity
samples shows that volumetric proportions of FS and MS increase and that of
G decreases from zone 1 to zone 3. The results are consistent with the
statistic outputs deduced from 694 borehole data by Zhu et al. (2016a).
Histograms of log10K for fine sand, medium-coarse sand and gravel.
Statistical methods
Semivariogram of hydraulic conductivity
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
r^khφ=12N(h)∑(o,p)∈N(h)Yzo-Yzp2
can be fitted by an exponential model (e.g., Dai et al., 2014b)
rkhφ=σ21-e-3hλ,
where r^k(hφ) and rk(hφ) are the experimental and model semivarograms of
log conductivity Y for the kth facies at a lag distance h along the
φ direction. In this paper we calculate the semivarograms in the vertical and
dip directions. N(h) is the number of pair measuring points zo and zp
separated by a h lag distance, σ2 is the variance and λ the correlation range.
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 ∂rk∂σ2
and ∂rk∂λ are the partial derivatives of
the semivariogram with respect to variance and range:
∂rk∂σ2=1-e-3hλ,∂rk∂λ=-σ23he-3hλλ-2.
Composite semivariogram of log conductivity
Once the facies semivariograms were obtained in each zone, the composite
semivariogram γ(h) could be calculated through the following equation
(e.g., Ritzi et al., 2004):
γhφ=∑k=1M∑i=1Mrkihφpktkihφ,
where pk and tki(hφ) are the volumetric proportion of
facies k and the transition probability from facies k to facies i in the
φ direction with a h lag distance, respectively. Equation (8)
delineates the composite semivarigoram with respect to the individual facies
semivariogram and transition probability. The general shape function and
range of the composite semivarigoram can be obtained from individual facies
mean length and volumetric proportion with the methods described in Dai et al. (2005).
The transition probability tki(hφ) has an analytical solution
as derived by Dai et al. (2007):
tkihφ=pk+δki-pkexphφλφ,
where δki is the Kronecker delta and λφ is
the integral scale in the direction of φ. A geostatistical modeling
tool GEOST (Dai et al., 2014b) modified from the Geostatistical Software
Library (Deutsch and Journel, 1992) and TPROGS (Carle and Fogg, 1997) were
employed to compute the sample transition probabilities in each zone. The
parameters pk and λφ were optimally estimated
through a modified Gauss–Newton–Levenberg–Marquardt method. More details are
provided by Zhu et al. (2016a). The composite semivariograms for different
zones can help us to understand the heterogeneity variations from the upper
to lower part of the alluvial fan, as well as the stationary property (local
versus regional) of the facies and hydraulic conductivity distributions.
Sequential Gaussian simulation
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 y axis differs for gravel lithology.
With the assumption that the log conductivity distributions are stationary
within each zone, we used the SGSIM simulator implemented into GEOST to model
the log10(K) continuous configuration under a multiple-zone framework.
The conductivity of the FS, MS and G facies in each zone was simulated
sequentially using the structure characteristics of the semivariograms.
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 log10(K) of the various facies in
each zone (Table 2). The stochastic simulated facies was constructed through
the optimized volumetric proportion and mean length of facies in three
directions. The mean length in vertical and dip directions were calculated
through 694 borehole. The mean length in strike direction was assumed as
half as that in the dip direction. During the facies simulation process,
borehole data were used as conditional data (Zhu et al., 2016a). In detail,
since each cell is characterized by specific facies and zone indices, its
conductivity was assigned using the corresponding (in relation to the facies
and the zone) 3-D SGSIM outcome in that position. Note that the hydrofacies
(e.g., C, FS, MS and G) are defined qualitatively based on the sedimentary
structures, borehole lithological descriptions and grain sizes, while the
conductivity samples are then deduced from geophysical measurements for each
facies at each zone. Since the sub-clay and clay contents from zone 1 to
zone 3 are increased due to the changes in the sediment transport
conditions, for the same facies we also found this trend and the overall
hydraulic conductivities are decreased from zone 1 to zone 3. Since sub-clay
and clay are generally characterized by a low hydraulic conductivity value,
a uniform K value equal to 0.0001 m day-1 was set to all the C cells.
Results and discussion
Variation of log10(K) for the various facies
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 log10(K)
semivariogram in vertical direction for the various facies and zones.
Zone
Parameter
Fine sand
Medium-coarse sand
Gravel
Estimated
Confidence
Estimated
Confidence
Estimated
Confidence
value
interval
value
interval
value
interval
(95 %)
(95 %)
(95 %)
Zone 1
Variance
0.23
(0.19, 0.28)
0.32
(0.29, 0.34)
1.60
(1.41, 1. 81)
Range (m)
6.01
(2.01, 20.52)
8.01
(1.53, 14.67)
6.50
(6.50, 12.84)
Zone 2
Variance
0.069
(0.067, 0.070)
0.14
(0.13, 0.15)
1.22
(1.19, 1.24)
Range (m)
3.13
(1.83, 4.42)
8.27
(3.61, 12.93)
15.0
(12.33, 17.67)
Zone 3
Variance
0.05
(0.047, 0.053)
0.126
(0.118, 0.135)
0.62
(0.54, 0.7)
Range (m)
6.52
(2.19, 10.85)
2.72
(0.20, 6.55)
5.98
(0.20, 15.63)
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 log10(K) of different facies along the dip direction
in zone 2 and zone 3.
Zone
Fine sand
Medium-
Gravel
coarse sand
Zone 2
Estimated
0.10
0.15
1.38
value
Confidence
(0.059, 0.141)
(0.071, 0.228)
(1.14, 1.62)
interval
(95 %)
Zone 3
Estimated
0.045
0.068
0.48
value
Confidence
(0.030, 0.0607)
(0.043, 0.093)
(0.22, 0.73)
interval
(95 %)
Due to the small number of conductivity samples in zone 1, the variance of
log10(K) along the dip direction is calculated only in zone 2 and zone 3
(Table 4, Fig. 7), as observed along the vertical direction. This phenomenon
is possibly a result of a sediment transport energy decrease along the flow
direction. Lower energy flow in zone 3 allow for better sediment sorting and
weak heterogeneity (or lower variance) in hydraulic conductivity.
Composited semivariogram of log10(K)
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 y axis differs for gravel lithology.
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
log10(K) in the three-dimensional (3-D) domain representing the Chaobai alluvial
fan: (a) axonometric projection of the 3-D system and
(b) vertical sections along the A–A', B–B', C–C' and D–D' alignments.
The vertical exaggeration is 25. The selected cell size is 300 m in north–south
and east–west directions and 5 m in vertical direction, with a total number
of 747 540 cells. The thickness of the simulated domain is 300 m.
Configuration of log10(K)
The configuration of log10(K) in three dimensions is showed in Fig. 9.
The distribution of conductivity is generally consistent with that of the
facies. Coarse units are more frequently distributed in the upper zone,
which makes the average K much larger in this zone than that in the lower
part of the alluvial fan. The regions with high conductivity (red color in
Fig. 9) in zone 1 are more continuous than that in other parts. The adjacent
cells with the smallest conductivity (blue color in Fig. 9) are obviously
located mainly in zone 3. The mean conductivity is smaller in the southern
part of the study area, where the piezometric drawdowns in the multiple-layer
aquifer system were larger and the surface subsidence more serious (Zhu et
al., 2013, 2015). Note that since we simulated the dip direction along the
main water flow direction and, due to the lack of enough data, the
strike-directional semivariogram is assumed to be similar to that in the dip
direction; the simulated facies in the fan apex did not show a radiating
pattern. More information about simulating the radiating pattern can be
found from Carle and Fogg (1997) and Fogg et al. (1998).
Based on the 3-D K configuration, the average value of K in the
depth range from 0 to 300 m amounts to 194, 25 and 4 m day-1 in zone 1,
zone 2 and zone 3, respectively. These values are comparable with those
provided by the Beijing Institute of Hydrogeology and Engineering Geology (2007)
based on a number of pumping tests carried out over several years in
the study area. In this BIHEG report, the average value of K is > 300 m day-1 in
zone 1, between 30 and 100 m day-1 in zone 2 and < 30 m day-1 in zone 3 (Fig. 1b).
The fact that the arithmetic average K values are gently smaller than the
latter ones is likely due to the fact that the outcome of pumping tests are
generally more representative of coarser sediments.
Investigating the stochastic results along the vertical direction, it is
interesting to notice that the average K in deep units of zone 1 and zone 2
is smaller than that in the shallow strata. For example, in zone 1 the
average K for the cells from 0 to 100 m deep is 295 m day-1, which is three
times as much the value for the depth range between 200 and 300 m.
Conversely, no significant variation of K versus depth is observed in zone 3,
with only a small decrease of the average K from the deeper to the shallower units.
Conclusions
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 log10(K)
configuration is simulated using the sequential Gaussian simulation model
based on statistic parameters of log10(K) and the structure suggested by
a 3-D hydrofacies simulation.
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
log10(K) in the vertical direction shows that the large heterogeneity in
the upper fan (with a value of 0.68) decreases in the lower zone.
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.