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1 23
Stochastic Environmental Research
and Risk Assessment
ISSN 1436-3240
Volume 30
Number 3
Stoch Environ Res Risk Assess (2016)
30:1005-1016
DOI 10.1007/s00477-015-1089-2
Statistic inversion of multi-zone
transition probability models for aquifer
characterization in alluvial fans
Lin Zhu, Zhenxue Dai, Huili Gong, Carl
Gable & Pietro Teatini
1 23
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ORIGINAL PAPER
Statistic inversion of multi-zone transition probability models
for aquifer characterization in alluvial fans
Lin Zhu
1,2
•Zhenxue Dai
2
•Huili Gong
1
•Carl Gable
2
•Pietro Teatini
3
Published online: 12 June 2015
ÓSpringer-Verlag Berlin Heidelberg 2015
Abstract Understanding the heterogeneity arising from
the complex architecture of sedimentary sequences in allu-
vial fans is challenging. This paper develops a statistical
inverse framework in a multi-zone transition probability
approach for characterizing the heterogeneity in alluvial
fans. An analytical solution of the transition probability
matrix is used to define the statistical relationships among
different hydrofacies and their mean lengths, integral scales,
and volumetric proportions. A statistical inversion is con-
ducted to identify the multi-zone transition probability
models and estimate the optimal statistical parameters using
the modified Gauss–Newton–Levenberg–Marquardt
method. The Jacobian matrix is computed by the sensitivity
equation method, which results in an accurate inverse solu-
tion with quantification of parameter uncertainty. We use the
Chaobai River alluvial fan in the Beijing Plain, China, as an
example for elucidating the methodology of alluvial fan
characterization. The alluvial fan is divided into three sedi-
ment zones. In each zone, the explicit mathematical formu-
lations of the transition probability models are constructed
with optimized different integral scales and volumetric
proportions. The hydrofacies distributions in the three zones
are simulated sequentially by the multi-zone transition
probability-based indicator simulations. The result of this
study provides the heterogeneous structure of the alluvial fan
for further study of flow and transport simulations.
Keywords Multi-zone transition probability Alluvial
fan Sediment heterogeneity Structure parameter
uncertainty Statistic inversion Indicator simulation
1 Introduction
The heterogeneous architecture of aquifers has a great impact
on groundwater flow and solute transport in subsurface sys-
tems. To characterize aquifer heterogeneity is challenging
since in most cases aquifer spatial structures vary greatly while
available observation data are limited or not well-distributed in
the three-dimensional domain (Zhu et al. 2015a). The sparsely-
distributed borehole, outcrop, and geophysical data can only
provide partial information to roughly define the aquifer
hydrogeologic properties (e.g., the hydrofacies, hydraulic
conductivity, and porosity distributions). Therefore, deter-
mining the large-scale heterogeneous structures from the above
mentioned small-scale data has been recognized as one of the
critical unresolved problems in groundwater hydrology
(Weissmann and Fogg 1999; Anderson 2007;Harpetal.2008).
Geostatistical and stochastic approaches are common
methods to address aquifer characterization and the corre-
sponding uncertainty when borehole or geophysical data do
not exist or are sparsely distributed. The Markov chain
method provides a conceptually simple and theoretically
powerful stochastic model for simulating geological
structures with different materials (Carle and Fogg 1997;
Weissmann et al. 1999; Rubin 2003; Ritzi et al. 2004;
&Zhenxue Dai
daiz@lanl.gov
&Huili Gong
gonghl@263.net
1
College of Resources Environment and Tourism, Capital
Normal University, Laboratory Cultivation Base of
Environment Process and Digital Simulation,
Beijing 100048, China
2
Earth and Environmental Sciences Division, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA
3
Department of Civil, Environmental and Architectural
Engineering, University of Padova, Padua, Italy
123
Stoch Environ Res Risk Assess (2016) 30:1005–1016
DOI 10.1007/s00477-015-1089-2
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Zhang et al. 2006; Dai et al. 2007a; Ye and Khaleel 2008).
A continuous Markov chain is mathematically a transition
probability model described by a matrix of exponential
functions (Agterberg 1974; Dai et al. 2007b). The Markov
chain model has been numerically solved by computing the
eigenvalues of the transition rate matrix (Carle and Fogg
1997). By using the numerical approach of the transition
probability models, Proce et al. (2004) developed a two-
scale Markov chain model to study the heterogeneity of
facies assemblages in a regional glacio-fluvial stratigraphy.
Sun et al. (2008) modeled the sedimentary architecture of
the alluvial deposits with a hierarchical structure. However,
sometimes there are too many unknown parameters in the
transition rate matrix and it is difficult to calculate the
eigenvalues and the corresponding spectral component
matrices when using the numerical approach (Dai et al.
2007b). Furthermore, without an explicit mathematical
formulation, the numerical approach of the Markov chain
model cannot reveal the relationships among different
hydrofacies and their mean lengths and volumetric pro-
portions. By using two assumptions that the cross-transi-
tion probabilities are dictated by facies proportions only
and the juxtapositional tendencies of the facies are sym-
metric, Dai et al. (2007b) derived an analytical solution of
the transition probability matrix, which was successfully
applied to aquifer characterization and upscaling transport
parameters (Ritzi and Allen-King 2007; Harp et al. 2008;
Dai et al. 2009; Deng et al. 2010,2013; Soltanian et al.
2015a,b,c).
This paper develops a statistical inverse framework of a
multi-zone transition probability approach for characteriz-
ing sedimentary heterogeneity in alluvial fans. Alluvial
fans, generally consisting of stream and flooding deposits,
can serve as groundwater supply fields since they have
abundant water storage and favorable conditions for
receiving recharge from precipitation and river leakage.
Alluvial-fan aquifers often exhibit spatial variations due to
the complicated depositional and diagenetic processes
which occurred during long-term fan evolution. Investi-
gation of the heterogeneous structures and construction of
the stratification sequences of alluvial fans help to improve
our understanding of the original sediment transport pro-
cesses and current hydrogeological properties in fluvial
systems (Ritzi et al. 1995; Zappa et al. 2006). Weissmann
and Fogg (1999) used transition probability geostatistics in
a sequence stratigraphic framework to quantify the facies
distributions in an alluvial fan. The flooding deposits of
alluvial fans may present spatial zonation along the original
sediment transport direction. The volumetric proportions
and length distributions of the hydrofacies vary dramati-
cally from the upstream to downstream. Based on the
stratigraphic character, geologists usually divide an alluvial
fan into three zones: upper fan, middle fan, and lower fan
(Miall 1997). As Weissmann and Fogg (1999) stated, the
assumption of stationarity for a whole alluvial fan is ten-
uous. Here, we apply the stationarity assumption in each
alluvial fan zone since the sediment properties are similar
within each zone. The Chaobai alluvial fan in the Beijing
Plain, China, is used as an example for developing the
multi-zone transition probability approach to characterize
the alluvial fan heterogeneity. The sediments of the
Chaobai alluvial fan are classified into four categories or
hydrofacies: sub-clay and clay, fine sand, medium-coarse
sand, and gravel. Multi-zone transition probability models
are developed to simulate the sedimentary heterogeneity
with different integral scales, volumetric proportions, and
mean lengths of the hydrofacies.
We adopt an analytical solution of the transition prob-
ability models which incorporates the geologic information
on facies proportions, mean lengths and juxtapositional
tendencies into geostatistical simulations (Dai et al.
2007b). During this process, the most important step is the
inversion of transition probability models to optimally
determine the multi-zone aquifer statistical parameters.
The statistic inversion of transition probability models is
conducted using the generalized output least squares (OLS)
criterion to fit the sample transition probability matrix with
the analytical solution. The inverse problem is resolved by
a modified Gauss–Newton–Levenberg–Marquardt method
(Clifton and Neuman 1982; Dai et al. 2012). The sensitivity
equation method is 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). Based on
the estimated statistical parameters for the multi-zone
transition probability models, the three dimensional
hydrofacies distributions in different zones are modeled by
the transition probability-based indicator simulations. This
study will provide the heterogonous structure of the allu-
vial fan for the further study of flow and transport
simulations.
2 Study area
The study area (Fig. 1) is located in the upper and middle
Chaobai River alluvial fan formed in Late Pleistocene in
the Beijing Plain, which covers an area of 1155 km
2
. The
mean elevation is about 40 meters above the sea level. The
ground surface slope is about 3 %towards the southeast.
The Chaobai River is one of the major rivers located at the
central axis of the fan, flowing through the Miyun Reser-
voir at north of the study area. The average precipitation is
624 mm per year (data from 1959 to 2010). About 80 %
the rainfall concentrates in the period from July to
September.
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In this study area, the alluvial fan deposits are mainly
coarse-grained sediments in the north and gradually change
to relatively fine-grained sediments in the south. The
Chaobai River alluvial fan is divided into three zones: the
upper fan zone, middle fan zone, and lower fan zone. The
upper fan zone (also called Zone 1) mainly consists of
coarser sands and gravels with a wide range of sediment
sizes. Because of the high conductivity in the sand and
gravel, the hydrodynamic conditions in the upper fan zone
are favorable for receiving recharge from precipitation and
river leakage. In the middle fan zone, the strata structure
consists of alternate layers of gravels, sands, fine sands,
sub-clay and clay. The hydrodynamic inter-connections
between different layers are relatively weak. Generally, the
cumulative thicknesses of the strata layers and the com-
pressible sediments (sub-clay and clay) in the middle fan
Fig. 1 Location of the study area, distribution of the boreholes, 35 borehole logs (5 logs in Zone 1, 15 logs in Zone 2, and 15 logs Zone 3)
Stoch Environ Res Risk Assess (2016) 30:1005–1016 1007
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zone are greater than those in the upper fan zone. In the
western part of the middle fan zone, the total thickness of
sediments is generally more than 400 m (Zhu et al. 2013).
Since the middle fan zone has a large variation in sediment
content and thickness, this zone is divided into two sub-
zones based on the lithology described in Zhu et al.
(2015b): middle-upper zone and middle-lower zone (or
Zone 2 and Zone 3). The lower fan zone mainly consists of
the compressible fine sediments (e.g., clay, sub-clay, loam
clay, and silt clay). With a very low porosity and hydraulic
conductivity, this zone is insignificant for water supply and
is not included in this study.
The statistics of hydrofacies properties are listed in
Table 1for the above defined three zones (upper fan zone,
middle-upper fan zone, and middle-lower fan zone). There
are 52 boreholes in Zone 1, 279 boreholes in Zone 2, and
363 boreholes in Zone 3 for characterizing the sedimentary
structures. Among all of these boreholes, the maximum
exposure depth is about 400 m and the smallest distance
between boreholes is about 160 m. Four hydrofacies (e.g.,
sub-clay and clay, fine sand, medium-coarse sand, and
gravel) are classified based on the interpretations of the
cores and textural descriptions of the 694 boreholes. In
Zone 1 the gravel sediments are dominant with a measured
volumetric proportion of 0.56 while the measured propor-
tion of sub-clay and clay deposits in this zone is 0.15. In
Zone 2, the proportion of gravel decreases to 0.27 and that
of sub-clay and clay increases to 0.42. In Zone 3, the
dominant material is sub-clay and clay with a measured
proportion of 0.5, and the proportion of gravel decreases to
0.08 (see Fig. 2; Table 1). The maximum cumulative
thickness of sub-clay and clay in Zone 1 is much less than
that in Zone 3, where the value reaches to 253 m while it is
99 m in Zone 1. The cumulative thickness of gravel in
Zone 1 is 336 m and much larger than that in Zone 3. The
mean thicknesses of different facies in three zones show
the similar features as the maximum cumulative thick-
nesses (see Table 1).
3 Statistic inversion of transition probability
models
3.1 Transition probability model
The transition probability is the probability of sedimentary
facies transitions at different lag distances within a three
dimensional domain (Agterberg 1974). By incorporating
facies spatial correlations, volumetric proportions, juxtapo-
sitional tendencies into a spatial continuity model, Carle and
Fogg (1996) and Ritzi (2000) developed transition proba-
bility models for aquifer characterization. These models can
improve the implementation of geostatistical simulations of
permeability or hydraulic conductivity by taking into
Table 1 Statistical data of three alluvial fan zones observed from 694 boreholes
Zone Borehole
number
Area
(km
2
)
Parameters Sub-clay and
clay
Fine
sand
Medium-coarse
sand
Gravel
Zone 1 52 159.85 Proportion 0.148 0.215 0.075 0.562
Mean thickness (m) 7.63 9.08 8.78 17.3
Maximum cumulative thickness
(m)
99.7 147.8 95.8 336
Zone 2 279 360.92 Proportion 0.423 0.220 0.087 0.270
Mean thickness (m) 7.31 4.77 3.87 7.97
Maximum cumulative thickness
(m)
200 130.1 86.8 214
Zone 3 363 633.81 Proportion 0.502 0.293 0.121 0.084
Mean thickness (m) 8.53 6.20 4.57 5.78
Maximum cumulative thickness
(m)
252.8 200.1 120.3 101.3
Whole
area
694 1154.58 Proportion 0.447 0.262 0.106 0.185
Mean thickness (m) 8.07 5.82 4.46 8.21
0
3
6
9
12
15
18
0
0.1
0.2
0.3
0.4
0.5
0.6
Zone 1 Zone 2 Zone 3
Propotion
Sub-clay and clay Fine sand
Medium-coarse sand Gravel
Integral scale
Integral scale (m)
Fig. 2 Volumetric proportions of four hydrofacies and integral scales
in three zones
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account geological and sedimentary information, especially
when permeability or hydraulic conductivity measurements
are not sufficiently abundant to support the computation of
the statistical parameters of permeability or conductivity
(Ross 1988; Deutsch and Journel 1992). Transition proba-
bility models have been used by geologists and hydrologists
to describe the heterogeneity of sedimentary facies for a few
decades (e.g., Carle and Fogg 1997; Harp et al. 2008).
Recently, Ritzi et al. (2004) and Dai et al. (2005) incorpo-
rated the work of Carle and Fogg (1997) to relate the sta-
tistical parameters of the indicator random variables to
distributions, geometry, and patterns of the hydrofacies.
With two assumptions, an analytical solution for the transi-
tion probability model was derived by Dai et al. (2007b) with
the auto- and cross-transition probability model (Eq. 1).
tik hu
¼pkþdik pk
ðÞ
exp hu
ku
ði¼1;2;...;N;k¼1;2;...;NÞ;ð1Þ
where tik hu
is the transition probability from facies ito
facies kin the direction of uwith a lag distance h,p
k
is the
volumetric proportion of facies k,dik is the Kronecker delta,
ku is the integral scale in the direction of u,andNis the
number of hydrofacies. Los Alamos National Laboratory
developed a geostatistical modeling tool GEOST (Dai et al.
2014) modified from the Geostatistical Software Library
(Deutsch and Journel 1992) and TPROGS (Carle and Fogg
1997). The geostatistical tool is employed here to compute
sample transition probabilities from the borehole hydrofacies
indicator data. The sample transition probabilities will be used
for inversion of the multi-zone transition probability models.
3.2 Statistical inversion
Two types of statistical parameters, volumetric proportion
and integral scale, are included in the analytical solutions
of the transition probability models (Eq. 1). By estimating
these statistical parameters, we are able to fit the computed
analytical solutions to the corresponding sample transition
probabilities. Let the statistical parameter vector x¼
p1;p2;p3;...;pN;ku
;then, the least-squares criterion
ExðÞand the corresponding constraints can be expressed as
ExðÞ¼Minimize X
L
l¼1
ðUlxðÞFlÞ2
;
X
N
k¼1
pk¼1and X
N
k¼1
tik hu
¼1;
0pk1and 0tikðhuÞ1
ði¼1;2;...;N;k¼1;2;...;NÞ
ð2Þ
where Lis the number of sample transition probability
data, UlxðÞis the output of the analytical solution (1), and
Flis the sample transition probabilities. The constraint
equations require that the sum of the volumetric propor-
tions of the Nfacies is one and the sum of the transition
probabilities in one row of the transition matrix is also one.
The least squares Eq. (2) can be solved by the modified
Gauss–Newton–Levenberg–Marquardt method as (Dai
et al. 2008)
Xjþ1¼XjJTxJþaI
1JTxEðxÞð3Þ
where Xjis the vector of parameter values at the jth iter-
ation, xis a diagonal weighting matrix, Iis the identity
matrix and ais the Marquardt parameter, and Jis the
Jacobian matrix reflecting the sensitivity of the output
variable to the statistical parameters. Generally, there are
three methods to compute the Jacobian matrix: the finite
difference method (Doherty and Hunt 2009), the varia-
tional method (Sun and Yeh 1990) and the sensitivity
equation method (Samper and Neuman 1989). Here we use
the sensitivity equation method to analytically derive the
sensitivity coefficients. In Eq. 4,ot
opand ot
okuare the sensi-
tivity coefficients (or partial derivations) of the transition
probability to volumetric proportion and integral scale,
respectively.
J¼
ot1
op1
ot1
op2
ot1
opN
ot1
oku
ot2
op1
ot2
op2
ot2
opN
ot2
oku
.
.
..
.
...
..
.
..
.
.
otL
op1
otL
op2
otL
opN
otL
oku
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
ot
opk
¼1exp hu
ku
ðk¼1;2;...;NÞ
ot
oku
¼dik pk
ðÞðhuÞexp hu
ku
k2
u
ði¼1;2;...;N;k¼1;2;...;NÞ
ð4Þ
Prior parameter information is incorporated into the
objective function as additional ‘‘observation data’’ and it
is also used to define parameter initial values, minimum
and maximum bounds, which can help deciding a range of
acceptable values that parameters can take during the
optimization process. Incorporating prior information into
the objective function can also alleviate the ill-posedness
and non-uniqueness of inverse problems (Dai and Samper
2004; Samper et al. 2006). At the beginning of each iter-
ation of the Gauss–Newton–Levenberg–Marquardt
method, ais reduced by a factor bin an attempt to push the
algorithm closer to the Gauss–Newton method. If this fails
to give a reduction in the objective function, ais repeatedly
increased by factor buntil a reduction is obtained. By
Stoch Environ Res Risk Assess (2016) 30:1005–1016 1009
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comparing parameter changes and objective function
improvement achieved in the current iteration with those
achieved in the previous iteration, the algorithm can tell
whether it is worth undertaking another optimization iter-
ation. If so, the whole process is repeated. Finally, the
algorithm stops when a convergent solution is met or the
maximum number of iterations is attained.
3.3 Parameter uncertainty analysis
Analyzing inversion error and quantifying parameter
uncertainty are even more important than finding the best-
fit parameters because models never exactly fit the data
even when the model structures are correct. During the
optimization processes, we compute the variances of the
estimated parameters, as well as the corresponding
covariance matrix CðpÞ,
CpðÞffis2ðJTxJÞ1
;ð5Þ
where s2is the total variance of estimated parameters, Cii is
the diagonal element of the covariance matrix which rep-
resents the variance of parameter i, and xis the diagonal
weighting matrix. Parameter uncertainty is quantified by
confidence intervals, which are computed from the
covariance matrix (5) as expressed by Mishra and Parker
(1989)as
Prxix
i
ffiffiffiffiffi
v2
a
qffiffiffiffiffiffi
Cii
p
¼1að6Þ
where x
iis the estimated value of the parameter, xiis the
true parameter value, and v2
ais the Chi square statistics
corresponding to Mdegrees of freedom and a confidence
interval of 100ð1aÞ(Carrera and Neuman 1986).
3.4 Multi-zone indicator simulation models
Generally, the indicator cross-variogram or indicator
covariance is used in geostatistics to determine the spatial
variability of indicator variables (Deutsch and Journel
1992). Here, we incorporate the multi-zone transition
probabilities into indicator geostatistical models to simu-
late the multi-zone architectures of the hydrofacies in the
alluvial fan. According to the equations derived by Carle
and Fogg (1996) and Ritzi et al. (2004), the relationships of
the indicator cross-variogram cik hu
and covariance
Cik hu
with the transition probability can be expressed as:
cik hu
¼pidik tik hu
þtik hu
2
Cik hu
¼pi½tik hu
pk
ð7Þ
Equation (7) indicates that incorporating the transition
probability models into the indicator simulations is
equivalent to using the indicator cross-variogram or
covariance models. The multi-zone transition probability
matrices in vertical, dip, and strike directions are computed
by the using analytical solutions with the estimated multi-
zone statistical parameters from the statistical inversion in
the three zones. When the multi-zone indicator simulations
are conducted, the corresponding borehole indicator data
are used as the conditional data, which means that the
simulations honor the known hydrofacies distributions
observed in the boreholes.
4 Results and discussion
4.1 Statistic inverse results
By using the derived statistical inverse methodology, we
identified three sets of transition probability models which
correspond to the three zones in the alluvial fan. Three sets
of statistical parameters are estimated including indicator
integral scales for each zone in the vertical, dip, and strike
directions, and facies proportions for each hydrofacies. The
95 % confidence intervals are also estimated for quantify-
ing the uncertainty of estimated parameters (Table 2). The
estimated integral scales and facies proportions vary from
Zone 1 to Zone 3, which is consistent with the fact that the
sedimentary deposits in the alluvial fan originate from
multiple flooding events. The dominant hydrofacies chan-
ges spatially and their average grain sizes reduce from
Zone 1 to Zone 3 (Fig. 2). In Zone 1, the hydrofacies of the
gravel and medium-coarse sand are inter-bedded with
minor sub-clay and clay facies. The gravel is predominant
with an estimated volumetric proportion of 53.3 %. The
volumetric proportions of the sub-clay and clay are much
lower (with an estimated value of 16.7 %). This zone has
integral scales of about 17 and 618 m in the vertical and
dip directions, respectively. This result indicates that with
high transport energy in this zone, the average grain sizes,
the mean thickness, and length of the sedimentary facies
are relatively larger than those in the two down-stream
zones. Figure 3shows the fitting results between the
sample and computed transition probabilities in the vertical
and dip directions. The vertical analytical transition prob-
abilities computed with the estimated optimal statistical
parameters fit the well-defined sample transition probabil-
ities reasonably well (Fig. 3a).The sample transition
probabilities in the dip direction are not well defined
because the well spacing in that direction is too large,
relative to the mean length of the hydrofacies, to fully
capture the variations of the hydrofacies. The plotted
sample transition probabilities are sparse with high uncer-
tainty (Fig. 3b). Therefore, we used a trial and error
method to estimate the integral scale and fix the volumetric
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proportions to be the same as those estimated from the
vertical transition probability. Although the fit is not as
definitive as in the vertical direction, it is still reasonably
matched with the volumetric proportions and integral scale
in the dip direction. The sample transition probability in
strike direction is even more sparsely-distributed with
many zero values. We did not conduct inverse modeling
for the strike transition probability and assumed that the
integral scale in this direction is half of the integral scale in
the dip direction for all three zones, and the volumetric
proportions of the hydrofacies are the same as those in the
vertical and dip directions.
Table 2 The estimated statistical parameters for hydrofacies in three zones
Zone Parameters Categories Estimated parameter Confidence interval (95 %)
Zone 1 Integral scale (m) Vertical 17.05 (11.08, 23.03)
Dip 618 By trial and error
Strike 309 By assumption
Volumetric proportion Sub-clay and clay 0.1657 (0.1229, 0.2085)
Fine sand 0.2346 (0.1919, 0.2773)
Medium-coarse sand 0.0669 (0.0236, 0.1101)
Gravel 0.5328 (0.4889, 0.5768)
Zone 2 Integral scale (m) Vertical 6.245 (2.664, 9.826)
Dip 360 By trial and error
Strike 180 By assumption
Volumetric proportion Sub-clay and clay 0.4093 (0.3717, 0.4469)
Fine sand 0.2854 (0.2479, 0.3229)
Medium-coarse sand 0.0653 (0.0277, 0.1030)
Gravel 0.2400 (0.2025, 0.2775)
Zone 3 Integral scale (m) Vertical 5.348 (3.740, 6.957)
Dip 319 By trial and error
Strike 159.5 By assumption
Volumetric proportion Sub-clay and clay 0.5028 (0.4839, 0.5217)
Fine sand 0.3277 (0.3088, 0.3465)
Medium-coarse sand 0.1066 (0.0877, 0.1255)
Gravel 0.0629 (0.0441, 0.0818)
0.0
0.5
1.0
080
t
11
(a)
Lag(m)
t
12
t
13
t
14
t
21
t
22
t
23
t
24
t
31
t
32
t
33
t
34
t
41
t
42
t
43
t
44
0.0
0.5
1.0
06500
t11
(b)
t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
Fig. 3 Sample (circle symbol) and computed (solid line) transition probabilities in vertical (a) and dip (b) directions in Zone 1
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Zone 2 and Zone 3 both belong to the middle zone of the
alluvial fan, but the estimated volumetric proportions of the
hydrofacies and the integral scales in these two zones show
significant differences. In Zone 2, the multiple strata of the
hydrofacies mainly consist of gravel, fine sand, sub-clay
and clay. The estimated volumetric proportion of the sub-
clay and clay is significantly greater than that in Zone 1 and
its value increases to 40.9 %. The volumetric proportion of
fine-grained sand also shows a slight increase, however, the
volumetric proportion of the gravel decreases sharply to
24 % in comparison with that in Zone 1. The integral
scales in Zone 2 are 6.2 and 360 m in the vertical and dip
directions, respectively, much smaller than those in Zone 1.
The reduced grain sizes of the hydrofacies and the
increased numbers of the sedimentary layers correspond to
the fact that the sedimentary transport energy level
decreases from Zone 1 to Zone 2. The transition probability
fitting results for the Zone 2 are shown in Fig. 4.InZone3,
the estimated volumetric proportion of the sub-clay and
clay increases to 50.2 % and that of the fine sand to about
33 %, while that of the gravel reduces to only 6.3 %. The
integral scales in Zone 3 are 5.3 and 319 m in the vertical
and dip directions, respectively, much smaller than those in
Zones 1 and 2 (Fig. 5). Borehole data shows a higher
occurrence frequency (or more layers) of the sub-clay and
clay facies in Zone 3 than in Zone 2, which represents the
low-energy water-laid sediments.
If an averaged single-zone transition probability model
was adopted for the whole alluvial fan, we found that the
transition probability cannot describe the sedimentary
spatial variations and the statistical properties of the sedi-
mentary features (e.g., the volumetric proportions of the
four hydorfacies and the integral scales). The single-zone
transition probability model does not fit the sample
transition probabilities well, especially for the facies of
gravel (Fig. 6). In the averaged single-zone transition
probability model, the estimated vertical integral scale is
8.23 m, which is larger than the integral scales in Zone 2
and Zone 3 while much smaller than that in Zone 1. The
dominant facies is sub-clay and clay with a volumetric
proportion of 42.23 %, the fine sand has a proportion of
30.33 %, the gravel has a proportion of 18.58 %, and the
medium-coarse sand has a proportion of 8.86 % (Table 3).
This volumetric proportion pattern is similar to the average
of the sedimentary features in Zone 2 and Zone 3. How-
ever, the hydrofacies features in Zone 1 with the dominant
gravel material are not reflected. A single-zone transition
probability model implies that the hydrofacies distributions
in the whole alluvial fan are stationary, which is not true in
this example. Therefore, this result demonstrates that an
averaged single-zone transition probability model cannot
reflect the sedimentary structure variations in the alluvial
fan, and the multi-zone transition probability models are
needed to honor the observed transition relationships of the
hydrofacies and to represent the sedimentary architectures
in this study area.
4.2 Multi-zone simulations of the alluvial fan
structure
With the identified multi-zone transition probability mod-
els, we use the code GEOST to simulate the hydrofacies
distributions sequentially from Zone 1 to Zone 3. The
statistical attributes of the stratigraphy and the borehole
indicator data are honored during the indicator simulation
processes. The borehole indicator data are described with a
vertical interval of 1 m and are used as the hard conditional
data. The estimated volumetric proportions and integral
0.0
0.5
1.0
080
Lag (m)
t11
(a)
t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
0.0
0.5
1.0
010000
Lag (m)
t11
(b)
t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
Fig. 4 Sample (circle symbol) and computed (solid line) transition probabilities in vertical (a) and dip (b) directions in Zone 2
1012 Stoch Environ Res Risk Assess (2016) 30:1005–1016
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0.0
0.5
1.0
080
Lag (m)
t11
(a)
t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
0.0
0.5
1.0
0 10000
Lag (m)
t11
(b)
t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
Fig. 5 Sample (circle symbol) and computed (solid line) transition probabilities in vertical (a) and dip (b) directions in Zone 3
0.0
0.5
1.0
080
Lag (m)
t11
(a)
t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
0.0
0.5
1.0
0 10000
Lag (m)
t11
(b)
t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
Fig. 6 The averaged single-zone sample (circle symbol) and computed (solid line) transition probabilities in vertical (a) and dip (b) directions
with all of the borehole data
Table 3 The estimated
statistical parameters for
hydrofacies with a single-zone
transition probability model in
the study area
Parameters Categories Estimated parameter Confidence interval (95 %)
Integral scale (m) Vertical
Dip
Strike
8.233
469
234.5
(3.77, 12.69)
By trial and error
By assumption
Volumetric proportion Sub-clay and clay 0.4223 (0.3807, 0.4639)
Fine sand 0.3033 (0.2619, 0.3447)
Medium-coarse sand 0.0886 (0.0470, 0.1302)
Gravel 0.1858 (0.1444, 0.2272)
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scales in the vertical, dip, and strike directions for the three
zones are applied as the major statistical parameters for
simulating four hydrofacies distributions in the alluvial fan.
The simulated hydrofacies distributions in the three
dimensional domain are shown in Fig. 7.
There is an obvious lithological change from the upper
fan zone to middle fan zones. The gravel deposits, repre-
sented by red color, are dominant in the upper fan zone
(Fig. 7) and they are well connected. The sub-clay and clay
deposits, marked by blue color, are discontinuously dis-
tributed with a smaller volumetric proportion.
From the upper fan zone to middle fan zones (Zone 2
and Zone 3), the volumetric proportions of the gravel
deposits decrease gradually. The connections between the
gravel deposits become worse. The simulated multiple
layers of the sub-clay and clay, fine, and medium-coarse
sand deposits are distributed alternately in Zone 2 and Zone
3, which are consistent with what we observed from the
boreholes. Multiple aquifers, consisting of fine and med-
ium-coarse sands, are inter-bedded with sub-clay and clay
which have thicknesses around tens of meters and act as
aquitard or confining layers. The increased volumetric
proportion of the sub-clay and clay from Zone 1 to Zone 3
decreases the hydraulic inter-connection in the vertical, dip
and strike directions. Although there is some degree of
uncertainty in the estimated statistical parameters, the
simulated heterogeneous structures are similar to the pat-
terns observed from the boreholes in this area. They will be
incorporated into numerical models for next-step study of
groundwater flow and transport simulations.
5 Summary and conclusions
Sedimentary architectures in the Chaobai alluvial fan show
obvious heterogeneity from the upper fan to lower fan. This
paper developed a statistical inverse framework to identify
multi-zone transition probability models and used them to
characterize and model the spatial heterogeneity. The
optimized statistical parameters were estimated using a
nonlinear optimization technique. The sensitivity equation
method was used to calculate the Jacobian matrix to
increase accuracy of the statistical inversion. The uncer-
tainties of the statistical parameters were quantified with
95 % confidence intervals.
On the basis of the borehole geological descriptions and
log data collected from the alluvial fan, four categories of
hydrofacies were identified, including gravel, medium-
coarse sand, fine sand, and sub-clay and clay. These
hydrofacies vary dramatically in their integral scales and
volumetric proportions from the upstream to the down-
stream of the alluvial fan. In the upper fan zone (Zone 1),
the dominant hydrofacies is gravel with a volumetric pro-
portion of 53 % which is inter-bedded with relatively thin
layers of sand and clay. The content of the material of sub-
clay and clay is minor with a volumetric proportion of
16 %.
In the middle-upper fan zone, the volumetric proportion
of the sub-clay and clay increases to 40 %, while that of the
gravel decreases sharply to 24 %. In the middle-lower fan
zone, the proportion of gravel decreases further to 6 % and
that of sub-clay and clay increases to 50 %, which is
alternatively inter-bedded with multiple thin layers of fine
sand and medium-coarse sand. The fine sand and medium-
coarse sand have proportions of about 33 and 11 %,
respectively. The estimated vertical integral scale is the
largest in upper fan zone (17 m), which corresponds to the
largest mean thickness of the gravel in this zone. The
vertical integral scales decrease to 6.2 and 5.3 m, in Zone 2
and Zone 3, respectively.
The averaged single-zone transition probability model
does not fit well the sample transition probability, which
demonstrates that the multi-zone transition probability
models are needed to characterize the complex heteroge-
neous structures of alluvial fans. The simulated hydrofacies
distributions with multi-zone transition probability models
Fig. 7 Simulated three dimensional hydrofacies structure and cross sections in the study area (with 15 times of vertical exaggeration)
1014 Stoch Environ Res Risk Assess (2016) 30:1005–1016
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represent the sedimentary structures of the alluvial fan
reasonably well. Since the well spacing is too large to fully
catch the facies variations in the dip and strike directions,
the estimated integral scales in these two directions contain
some degree of uncertainty. Further geological and geo-
physical work may increase the accuracy of the identified
transition probability models in the dip and strike direc-
tions, as well as the simulated hydrofacies architectures.
Acknowledgments This work was supported by the National Nat-
ural Science Foundation (Nos. 41201420, 41130744), Beijing Nova
Program (No. Z111106054511097) and Beijing Young Talent Pro-
gram. We benefited from discussions with Robert W. Ritzi of the
Wright State University and his comments and suggestions greatly
improve this paper.
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