# The 3-D Facies and Geomechanical Modeling of Land Subsidence in the Chaobai Plain, Beijing

## Abstract

The hydrogeologic systems of alluvial fan are characterized by a heterogeneous distribution of various lithological units/facies. The structure (integral scale and volumetric proportion) of the hydrofacies distribution and the values of the hydrogeomechanical parameters of each facies can play a major role on the system response to groundwater withdrawal in term of land subsidence. We propose a novel approach where stochastically simulated hydrofacies distributions are coupled with 3-D finite element groundwater flow and geomechanical simulations to characterize land subsidence and horizontal movements due to groundwater withdrawal under a statistical framework. The integrated approach is applied on the Chaobai alluvial plain, China, an area of about 1,100 km^{2} where the main wellfields supplying water to Beijing are located. Groundwater pumping from the 1960s to now caused a land subsidence larger than 1 m and the present subsidence rate peaks to 70 mm/year. A Monte Carlo simulation with 100 hydrofacies generations is used. The model outcomes highlight how the heterogeneous structure of the hydrofacies fan reflects into the computed displacement fields. The standard deviation associated to the mean displacement field amounts up to 1/10 of the displacement components. The larger coefficient of variation (*CV* up to 0.5) is associated to the zone characterized by longer integral scale and with localized groundwater withdrawals. The computed variability of the subsidence rate, in the range of 1 to 3 mm/year, reflecting the intrinsic heterogeneous nature of an alluvial fan, corresponds to the short-distance variability of land subsidence measured by persistent scatterer interferometry.

## Key Points

- The 3-D hydrofacies coupled to 3-D FE flow and geomechanical models allow characterizing land subsidence from alluvial fans
- Integral scale of 3-D hydrofacies and withdrawal features (localized vs. distributed) control the variability of the land movements
- A first application on the Beijing alluvial fan, with the uncertainty due to facies distribution quantified in 10% of the average subsidence

## 1 Introduction

Despite the scientific research has clearly outlined the consequences of aquifer overexploitation in terms of land subsidence since the beginning of the twentieth century (Gambolati & Teatini, 2015), this is still today one world's major hydrogeological hazard caused by anthropogenic activities. Because of the general trend in economic development, variation in the distribution and timing of precipitation due to climate changes and population growth and concentration in (mega-) cities, the number of countries and people directly or indirectly affected by land subsidence is continuously increasing throughout the world. For example, new, or renewed, exemplifying sites recently affected by land subsidence are as follows: (i) Jakarta, Indonesia, where thousands inhabitants live in suburbs 3- to 4-m below the mean sea level right in front of the Java Sea (Abidin et al., 2015); (ii) Mexico City, where parts of the city are even more frequently flooded by rainwater and others become particularly sensitive to earthquakes because of the huge differential subsidence and related ground ruptures (González-Hernández et al., 2015); (iii) Wuxi, China, where earth fissures associated to land subsidence have made unlivable dozens of houses (Ye et al., 2018); and (iv) Central Valley, California, where large damages to bridges, roads, buried irrigation pipelines, canals, and wells have been exacerbated by land subsidence due to the increased groundwater use to face with the recent drought periods (2007–2009 and 2012–2016) (Faunt et al., 2016).

Since the early 1970s, for example, Gambolati et al. (1974), numerical models have been widely used to quantify the past land subsidence due to groundwater withdrawals and predict the possible future occurrence. Several modeling studies were developed, mainly using a 1-D approximation, as for instance with the SUB package of MODFLOW (e.g., Leake & Galloway, 2010; Minderhoud et al., 2017; Ortiz-Zamora & Ortega-Guerrero, 2010; Wu et al., 2010; Ye et al., 2012) or, more rarely, by means of 3-D approaches (e.g., Burbey, 2006; Ochoa-González et al., 2018; Teatini et al., 2006; Ye et al., 2016). All these regional-scale analyses have been carried out by means of a deterministic approach. The complex hydrogeological setting is generally represented through a layer-cake distribution of hydraulic (e.g., hydraulic conductivity) and geomechanical (e.g., compressibility) properties, which can represent a strong simplification of the actual hydrogeological features.

On the other hand, measurements by Persistent Scatterer Interferometry (PSI), which became available over the last two decades, have revealed quite unexpectedly that land subsidence is highly heterogeneous also in flat sedimentary plains (Figure 1b) (e.g., Dixon et al., 2006; Tosi et al., 2016; Zhu et al., 2015). The displacement patterns provided by persistent scatterers (i.e., the radar reflectors) are characterized by variable rates, significantly far from our understanding of the process when only leveling and GPS measurements were available. This short-scale heterogeneity can be due to measurement noise, superposition of regional, and local consolidation processes but also to the intrinsic heterogeneity of the soil geomechanical properties.

Despite the above evidence and the huge progress made in the geostatistical characterization of complex aquifer systems at the regional scale (e.g., Dai et al., 2007; Harp et al., 2008; Weissmann et al., 1999), the effect of a heterogeneous distribution of soil properties has been accounted for mainly in applications dealing with flow and transport processes (e.g., Dai et al., 2014; González-Nicolás et al., 2015; Soltanian & Ritzi, 2014; Soltanian, Ritzi, Dai, et al., 2015; Soltanian, Ritzi, Huang, et al., 2015). Conversely, very few works have addressed the issue of charactering the heterogeneous distribution of mechanical soil/rock properties (e.g., Darrag & Tawil, 1993; Eivazy et al., 2017; Frias et al., 2004), even less the evaluation of heterogeneity effects on pressure/stress changes and displacement (land subsidence) fields at the regional scale. In Cassiani and Zoccatelli (2000), a Monte Carlo technique was applied with respect to soil compressibility to quantify the risk of land subsidence due to hydrocarbon production. A semianalytical three-dimensional model based on the theory of Geertsma was used as a subsidence model. The study by Ferronato et al. (2006) focused on the vertical uniaxial rock compressibility that is regarded as a random spatial process. An axisymmetric aquifer system is simulated with a stratified hydrogeological setting typical of a sedimentary basin. A basin-scale constitutive law for the rock compressibility, log normally distributed with depth-dependent mean and constant variance, is implemented into a poroelastic stochastic Finite Element (FE) model based on the Monte Carlo simulation of an ensemble of 1000 realizations. Monte Carlo groundwater flow simulations were performed by Teatini et al. (2010) using a FE discretization of an aquifer system based upon a hydraulic conductivity distribution characterized by a lognormal, stationary random process. The pore overpressure resulting from water injection was then implemented into a deterministic FE geomechanical model to investigate the distribution of the land uplift. Zapata-Norberto et al. (2018) investigated the effect of vertical heterogeneity in simulating nonlinear 1-D groundwater flow and consolidation in highly compressible aquitards by means of Monte Carlo numerical simulation.

Here we propose to investigate the impact of sedimentary heterogeneity on land subsidence by a novel integrated approach where 3-D facies, hydrodynamic, and geomechanical models are coupled sequentially. First, the heterogeneity of the sedimentary architecture, or hydrofacies, of an alluvial fan is stochastically simulated through multizone transition probability models (Zhu et al., 2017). Second, the facies model is transferred into a 3-D FE grid populated with facies-dependent hydraulic conductivity and compressibility field. The grid is then used to solve the groundwater flow and equilibrium geomechanical equations with a one-way coupled approach (Teatini et al., 2006). A Monte Carlo procedure is implemented to investigate the effects of the facies heterogeneity.

In this work, we elected to use the Chaobai plain (Figure 1a), North of Beijing, as a test case for the application of the proposed approach. With a population of more than 20 million in 2015, land subsidence and ground rupture are important geo-hazards in Beijing plain. They started to occur in the 1930s and rapidly increased because of the fast economic and urban development requiring larger and larger amounts of good-quality freshwater. From 1955 to 1973, the cumulative land subsidence became larger than 100 mm in some local areas, mainly to the East and North of Beijing downtown, where maximum cumulative subsidence values reached 230 and 130 mm, respectively. Since 1999, land subsidence has become even faster, with a maximum rate of 140 mm/year. Presently, the area with settlement values larger than 50 mm extends for about 4,300 km^{2} and the maximum cumulative subsidence is estimated at about 1.5 m from 1955 to 2013 (Lei et al., 2016; Li et al., 2017). Land subsidence has threatened the safety of people life and properties.

The paper is organized as follows. Initially, the integrated procedure is presented and some details are provided about automatic procedures that have been implemented to properly assign boundary conditions and forcing factors. Indeed, a specific care must be used to avoid nonphysical outcome when different realizations are simulated, for example, the piezometric change obtained if the system is forced to provide water from a low-permeability clay zone. Then, the main hydrogeological setting of the Chaobai plain is introduced and the geological hazards experienced by the area in terms of overexploitation of the groundwater resources and land subsidence are presented. The proposed procedure is applied to the study site and the main results in terms of change of the piezometric level, 3-D displacement field, and deformation in a few representative locations are shown. The simulation time spans the interval between 1965 and 2012. Finally, we close the paper by discussing the major advantages and disadvantages of the proposed approach relative to more traditional deterministic approaches where aquifer systems are represented through layer-cake property distributions, for example, as in Cui et al. (2003) for downtown Beijing.

## 2 Methodological Approach

### 2.1 Multizone Simulations of 3-D Alluvial Fan Structures

Alluvial-fan aquifers often exhibit spatial variations due to the complex depositional and diagenetic processes that occurred during the long-term fan evolution (Zhu et al., 2017). The flooding deposits are usually characterized by spatial zonations along the sediment transport direction, with the volumetric proportions and length distributions of the hydrofacies varying significantly from upstream to downstream (Ritzi et al., 2004). Therefore, the stationarity for a whole alluvial fan is typically tenuous (Weissmann & Fogg, 1999). Here the multizone transition probability method developed by Zhu, Dai, et al. (2016) and Zhu et al. (2017) was used to preserve the local stationarity assumption of the system. Indeed, a large-scale fan can be divided into a number of segments characterized by stationary features, according to the geostatistical analysis of hydrofacies distributions.

*t*

_{ik}(

*h*

_{φ}) is the transition probability from facies

*i*to facies

*k*in the direction of

*φ*with a lag distance

*h*,

*p*

_{k}is the volumetric proportion of facies

*k*,

*δ*

_{ik}is the Kronecker delta,

*λ*

_{φ}is the integral scale in the

*φ*direction, and

*N*is the number of hydrofacies.

The borehole hydrofacies indicator data are processed by the geostatistical tool GEOST, developed by Dai et al. (2014) starting from the Geostatistical Software Library (Deutsch & Journel, 1992) and T-PROGS (Carle & Fogg, 1996), in order to compute the sample transition probabilities. GEOST is used for inversion of the multizone transition probability models. The parameters in equation 1 are optimally inverted through the modified Gauss-Newton-Levenberg–Marquardt method (Dai et al., 2008), with the prior information incorporated into the objective function as “conditional data” and used to define initial values, and minimum and maximum bounds. The 95% confidence intervals are also computed during the optimization process to quantify the uncertainty of estimated facies parameters. According to Zhu, Dai, et al. (2016), the transition probability models for different zones, analyzed through indicator geostatistical simulations, can be used to represent the multizone architectures of the hydrofacies in the alluvial fan.

### 2.2 The 3-D FE Hydrodynamic and Geomechanical Models

*H*is the hydraulic head, ∇ and ∇· are the gradient and divergence operator, respectively,

**is the hydraulic conductivity tensor with principal components**

*K**K*

_{xx},

*K*

_{yy}, and

*K*

_{zz},

*S*

_{s}is the specific storage,

*Q*is the source/sink term,

*t*is time, and

*x*,

*y*,

*z*denote the directions of a standard Cartesian reference.

**and**

*σ***1**are the effective stress and identity tensor, respectively,

*b*is the Biot coefficient,

*P*the pore fluid pressure, and

**the set of external forces. For an isotropic elastic medium with incompressible solid grains, that is,**

*f**b*= 1, equations 3 can be written in terms of displacements (Verruijt, 1969):

*E*and

*ν*are the Young modulus and Poisson ratio, respectively,

*u*

_{i}is the component of the displacement vector

**along the**

*u**i*th coordinate direction,

*ε*is the volumetric strain equal to ∇ ·

**, and ∇**

*u*^{2}is the Laplace operator. Equations 2 and 4 are coupled because

*γ*

_{w}and

*β*are the water specific weight and volumetric compressibility, respectively,

*α*and

*ϕ*are the medium oedometric compressibility and porosity, respectively. The parameter

*α*typically decreases as the effective intergranular vertical stress rises, that is, with both a depth increase and a larger pore pressure drawdown. Moreover, because of the mechanical hysteresis of a porous medium in unloading/reloading conditions (Ferronato et al., 2013; Teatini et al., 2006)

*α*, and, consequently,

*S*

_{s}and

*E*, takes on the elastic values

*α*

_{II},

*S*

_{s,II}and

*E*

_{II}when the effective stress is smaller than the preconsolidation stress or the larger stress ever experienced by the medium; otherwise, the larger plastic values

*α*

_{I},

*S*

_{s,I}and

*E*

_{I}are used. Although relatively simple compared to the state-of-the-science in geomechanics (e.g., Isotton et al., 2019), the constitutive model for aquifer mechanics used in the present analysis is similar to those implemented in widely-used hydrogeological simulators, for example, MODFLOW.

### 2.3 Integration of Static and Dynamic Modules

Understanding the geomechanical response of alluvial fans to groundwater pumping requires integrating the stochastic facies distributions developed through the multizone transition probability model into the FE dynamic simulators. Figure 2 shows the flowchart of the proposed procedure: Data on subsurface lithological distributions acquired through borehole stratigraphy and geophysical investigations are used by the multizone transition probability approach to build up a number *N*_{R} of alluvial fan structures. Each zone is characterized by its own integral scale along the vertical, dip, and strike directions and the volumetric proportions for the various identified hydrofacies. These hydrofacies structures are used as static models in the 3-D groundwater flow and geomechanical models. Using a one-way coupled approach, which is fully warranted for hydrogeomechanical investigation at regional scale (see, e.g., Gambolati et al. 2000), the flow model is applied first and its output in terms of pore pressure change is properly processed as the forcing factor in input to the mechanical module. Pumping rates and precipitation records are the forcing functions of the groundwater flow model, which is calibrated against piezometric records and/or piezometric maps. The geomechanical model is calibrated using subsidence measurements acquired through leveling, GPS, and PSI. Deep compaction recorded in extensometer stations can also be adopted. An iterative approach is required to ensure the use of the same compressibility values in the two modules through equations 6 and 7. Specifically, once the flow model (equation 2) is calibrated on the available piezometric variations for the *i*th time step, the corresponding distribution of *S*_{s} is used to update *α* (equation 6) and *E* (equation 7), and Δ*H* to compute the forcing term *P* for the geomechanical model (equation 5). Then, the geomechanical model (equation 4) is run over the same time interval and the solution compared with the corresponding subsidence measurements. If the match is unsatisfactory, *E* is appropriately varied to reproduce the displacement records. The updated *E* is used to update *α* and *S*_{s}, and the flow model is rerun and rematched on the piezometric records by updating *K* and *S*_{s}. If the new Δ*H* and *S*_{s} differ from the previous values more than a prescribed threshold the *E* distribution is updated and the geomechanical model rerun with the new forcing factor. The procedure is repeated until convergence. After convergence, the computed movements over the *i*th time step is added to the values obtained at the previous (*i*-1)th time step to compute the new cumulative displacement distribution. Each hydrofacies in each zone is potentially characterized by a different hydraulic conductivity, porosity, compressibility, and Poisson ratio.

*:*

*Converting hydrofacies distributions to static FE models (Figure*3*)*. The hydrofacies structures, which are generated on a regular 3-D grid, must be properly transferred to the FE mesh used for the hydrologic and geomechanical simulations. This is carried out by providing each FE with a set of hydrogeological parameters (,*K**α, ν*, and*ϕ*) representative of the closest hydrofacies with respect to its centroid. The parameter values can change for the same hydrofacies according to the different zones identified in the FE mesh. In order to preserve as much as possible the statistical properties of the heterogeneous fields in the numerical models, the characteristic size of the grid elements along the horizontal and vertical directions must be comparable with the geometric properties of the stochastic fields;*Prescribing flow-based hydrologic stresses*. Effective rainfall (*Q*_{er}), single-well discharge/recharge (*Q*_{sw}), well-field discharge/recharge (*Q*_{wf}), and groundwater pumping at county/municipality level (*Q*_{cm}) are the data usually available for the factors forcing pressure evolution and land subsidence in regional aquifer systems, that is, the term*Q*in equation 1.*Q*_{er}represents the portion of the rainfall infiltrating at depth and is quantified by multiplying the measured rainfall by an infiltration coefficient*I*_{er}depending on land use and coverage. Prescribing these conditions is not straightforward when long-term (e.g., decadal) simulations must be run for*N*_{R}alluvial fan structures. The procedure usually adopted in deterministic modeling is to assign the inflow/outflow rates*q*into/from targeted nodes corresponding to the well intake and/or depth of the sandy layers. In our framework, the same node can be located in differently permeable and compressible soils, depending on the specific hydrofacies generation, with the risk of computing unrealistically large pressure changes when withdrawal is prescribed from a low conductivity facies. Therefore, a preprocessor is required to properly quantify the fraction*q*_{i,k}of*Q*(with*Q*corresponding to*Q*_{er},*Q*_{sw},*Q*_{wf}, or*Q*_{cm}) to be prescribed to the*i*th node depending on the*k*th static model:

*w*

_{i, k}takes the form

*n*

_{Q}is the number of nodes where

*Q*takes place (e.g., the nodes on the land surface for

*Q*

_{er}or the nodes in the depth range of the well intake for

*Q*

_{sw}),

*e*

_{i}is the number of elements sharing the

*i*th node, with

*K*

_{j,k}and

*V*

_{j}the hydraulic conductivity and the volume, respectively, of the

*j*th FE;

*Assigning head-dependent and no-flow conditions on the lateral boundary Γ*. Nodes used to simulate boundary conditions are usually grouped into two categories, that is, prescribed-head nodes and no-flow nodes (e.g., Langevin et al., 2017). When*N*_{R}hydrofacies structures are accounted for, these conditions must be assigned to the high-permeability and low-permeability boundary nodes, respectively, by means of a specific preprocessor. Indeed, similarly to the previous item, unrealistically large compaction at depth and land subsidence can be computed along the model boundaries if the pressure head changes, which are generally available from piezometric records in coarse geologic units, are prescribed to nodes located within highly compressible, fine (i.e., typically clay and silty) materials. By assigning the weight*ξ*= 0 to clay and silt and*ξ*= 1 to the other hydrofacies (i.e., from silty sand to gravel), the following boundary conditions are prescribed to the*i*th node, with*i*∈Γ, for the*k*th static model:

_{n}the derivative normal to the lateral boundary, and the given head.

## 3 The Chaobai Alluvial Fan: Hydrogeological Setting and Geohazards

The study area belongs to the Northern Beijing Plain, in the upper and middle Chaobai River alluvial fan (Figure 1a) including the plain of Huairou and Shunyi districts. The ground surface slope is about 2% decreasing southward. The yearly precipitation (Figure 4), which represents one of the main source of the aquifer recharge, is generally larger in the northern part of the basin. It averaged 620 mm from 1959 to 2012, with a maximum of 1,120 mm in Huairou District and a minimum equal to 292 mm in Shunyi District in 1969 and 1965, respectively. About 80% of the yearly precipitation is concentrated between June and September. The average temperature is about 11.8 °C.

The Chaobai River is the main watercourse flowing through the study area from north to south. Since 1981, the riverbed is usually dry, except during the flooding season of some years or when the upstream reservoirs release some water.

### 3.1 Hydrogeological Setting

The alluvial system in the study area was developed by multiple flooding events over the Quaternary. The main stream of Chaobai River remained relatively stable and the Chaobai fan did not transfer much during its formation (Cai et al., 2009). The deposits are mainly composed of coarse-grained sediments in the northern part and gradually change to relatively fine-grained sediments southward (Zhu, Dai, et al., 2016; Zhu, Gong, et al., 2016) (Figure 5a). Water yield also decreases from north to south.

The alluvial fan is characterized by an obvious lithological zonation as highlighted by the available stratigraphic information: a phreatic sandy-gravel aquifer is located in the upper plain, while a complex multiaquifer system characterizes the middle and lower regions. Moreover, the thickness of Quaternary deposits varies significantly, from tens to hundreds of meters from the piedmont to the floodplain. In the southwestern part of the Shunyi district, the Quaternary thickness is larger than 400 m with more than 150-m-thick compressible sediments.

### 3.2 Geohazards

Groundwater resources have been deeply exploited in this area over the last decades. Data about the pumping rates are available since the middle 1960s (Figure 4), with distributed withdrawals provided for each of the 25 towns into which the study area is administratively subdivided.

Because of the huge population growth in Beijing, from about 5 million in 1970 to approximately 20 million inhabitants in 2015, and the drought period experienced in the 2000s, a number of large wellfields have been drilled in the plain to supply water to the capital urban area (Figure 1). A first important groundwater wellfield (belonging to the so-called eighth water factory) operated from 1980 along the Chaobai River, and other three well groups belonging to other “water factories” operated since 1990. An oversized “emergency groundwater resource region” was detected in Huairou District in August 2003. Here a number of 21 groups of wellfields, with a daily pumping volume of about 26 × 10^{4} m^{3}, were drilled along the Huai River, Sha River, and Yanxi River. Another relatively new wellfield was realized in 2009 along the Huai River. The pumping rates for each wellfield are available on a monthly basis.

The long-time overexploitation of groundwater has significantly depressed the piezometric levels and locally inverted the natural north-to-south flow direction (Figure 6a). Because of piezometric lowering, the water provided by eighth water factory decreased largely, from 1.64 × 10^{8} m^{3}/year before 2000 to 0.65 × 10^{8} m^{3}/year in 2009. The piezometric contour lines in 2012 highlight two main depression cones, a first one located along the Chaobai River and the second at southwestern border of the study area centered in Tianzhu (Figure 6b). The south-to-north water transfer project started to provide surficial water to Beijing since December 2014 and decreased the need of groundwater.

The drop of piezometric levels in the aquifer system has been responsible for severe sediment compaction, land subsidence (Zhu et al., 2015), and ground ruptures (Yang et al., 2015). According to historic documents, leveling surveys were carried out in the whole Beijing plain since the 1950s, with the reference benchmark established on the rock outcrops to the West of the downtown Beijing. Figure 6c shows the cumulative land subsidence between 1955 and 2013 obtained by interpolating the available leveling records. A peak value of more than 1-m subsidence was experienced in the southwestern portion of the area. More recently, the PSI technique has been used to monitor land displacements. A total number of 113 images in ascending and descending orbits by ERS, ENVISAT, and TerraSAR satellites from June 1992 to November 2015 were acquired over the study area and processed by Interferometric Point Target Analysis PSI chain (Werner et al., 2003). Multitrack data are combined to obtain time series of land displacement along the line of sight. For example, the line-of-sight displacements obtained over the period 2003–2010 using ENVISAT acquisitions is shown in Figure 6d. The map confirms that also recently, a serious land subsidence affected the southwestern part of the study area. The maximum land subsidence rate increased from ~30 mm/year during the period from 1992 to 1999 (ERS images) to ~50 mm/year between 2003 and 2010 (ENVISAT images) and to ~70 mm/year from 2010 to 2015 (TerraSAR images). Notice that the maximum land subsidence increased over time irrespective of the decrease of the annual withdrawal rates since 2005. This is because the largest subsidence rates are more associated to significant localized drawdown occurring in compressible thick deposits rather than to the total amount of groundwater withdrawn at the regional scale. The distribution of the active wells with respect to the geological setting, the relative location of the wells themselves, and the withdrawal rate from adjacent group of wells (or wellfields) are factors that play the main role.

Compaction at depth has been monitored in the study area through the multidepth extensometer station established at Tianzhu (Figure 1a) operating since 2004. Anchored in the bedrock at ~900-m depth, the extensometer records the deformation of the main fine-layer deposits down to ~300-m depth (Zhu et al., 2015). The cumulative deformation recorded at the Tianzhu extensometer station amounted to 418 mm from April 2004 to the end of 2014.

## 4 Geostatistical Numerical Model of the Chaobai Plain

### 4.1 Modeling Setup

The modeling domain encompasses an aerial extent of 1,155 km^{2} (Figure 1) and extends from the land surface to the bedrock or the 500-m depth below mean sea level along the vertical direction. Indeed, the production wells do not tap aquifers deeper than this value. Because of data availability, the domain is bounded by the physical limit of the alluvial fan in the northern part and by the administrative bound of the Shunyi district southward.

Four lithological categories or hydrofacies, namely, gravel, medium-coarse sand, fine sand, and subclay and clay, were classified on the basis of the interpretation of the cores from 694 boreholes (Figure 5a). In the framework of a Monte Carlo procedure, an ensemble of *N*_{R} = 100 hydrofacies models were generated by GEOST. This *N*_{R} value was selected as a trade-off between a proper investigation on the heterogeneity effect in the geomechanical problem and a reasonable computational burden and disk space occupation. When the sampling number was greater than 80, we found the computed variances and means of groundwater levels starting to converge to some constants. Then, we terminated the Monte Carlo simulations at *N*_{R} = 100 to save computational expenses. Hydrofacies structures were generated on a regular 400 × 400 × 5 m grid. Following the procedure described above (see also Figure 3), the simulated hydrofacies distributions were used to create 3-D FE model structures. The FE mesh is composed of 3′572′194 tetrahedra and 615′726 nodes (i.e., almost 2 million degrees of freedom are use in the solution of equation 4). The 3’D mesh was generated by extruding a 2-D triangular grid along the *z* direction and cropping the elements located above the land surface and below the basement. The element size in the FE mesh is consistent with the GEOST grid, with the characteristic length in the horizontal directions reduced to 200 m around the main wellfields to improve the accuracy of the numerical solution (equations 2 and 3). The hydrological and geomechanical parameters of Table 1 were assigned to each FE according to the hydrofacies distribution. Such values were derived from in situ investigations, evidences of previous models, and a trial-and-error calibration procedure to capture contemporarily the measured water level drawdown and land subsidence. The following additional parameters have also been set: *γ*_{w} = 10^{3} kg/m^{3}, *β* = 4.3 × 10^{−4} MPa^{−1}, and *ν* = 0.3.

Hydrofacies | K_{x} = K_{y} (m/s) |
E_{I} (MPa) |
E_{I}/E_{II} |
ϕ |
---|---|---|---|---|

Gravel | 5.0 × 10^{−3} |
130 | 2 | 0.3 |

Medium-Coarse Sand | 3.8 × 10^{−4} |
20 | 2 | 0.25 |

Fine Sand | 6.5 × 10^{−5} |
12 | 5 | 0.25 |

Clay | 10^{−6} |
4 | 5 | 0.4 |

*Note*. In this work*E*(and therefore α) depends only on the value of the actual vertical effective stress with respect to the preconsolidation stress. The parameters are independent from the fan zones. Moreover,*K*_{x}/*K*_{z}= 10. The initial values were derived from Beijing Institute of Hydrogeology and Engineering Geology (2015) and Zhu et al. (2017).

The simulation period, which spans the years between 1965 and 2012, was covered by an adaptive time step Δ*t* ranging from 1 month to 1 year. Undisturbed (natural) steady state flow regime and a structurally equilibrated condition were assumed in 1965 for the groundwater flow and the geomechanical model, respectively. The following boundary conditions were prescribed in the groundwater flow model. Effective rainfall, which was derived as described in section 2.3 multiplying the recorded precipitation (Figure 4) by an infiltration coefficient *I*_{er}, with 0 ≤ *I*_{er} < 0.6 in the Chaobai plain, was assigned on the land surface and no flow was imposed on the basement. Since quantifying reliably the amount of groundwater exchange through the outer boundary, which is partly an administrative limit as stated above, is practically impossible, Dirichlet conditions are set on the nodes pertaining to permeable (sand to gravel) hydrofacies according to the available regional maps of the piezometric level (see section 2.3). Measurements in the phreatic and confined systems are available on a 10- and 1-year basis before and after 2000, respectively. Linear interpolation in time between these values was used to derive the boundary conditions for each time step. Distributed and wellfield pumping rates were prescribed as described in section 2.3 using the available time and space information provided above and summarized in Figure 4. Notice that because of the lack of specific information, the distributed withdrawals, which differ for each of the county in the study area, were split among the nodes in the depth interval between *z*_{surf} − 0.1(*z*_{surf} − *z*_{bot}) and *z*_{bot} + 0.1(*z*_{surf} − *z*_{bot}), where *z*_{surf} and *z*_{bot} are the elevation of the land surface and bottom at the specific node vertical alignment, according to the weights defined in equations 8 and 9. The bottom is fixed and impervious, the land surface is a traction-free boundary, and the nodes on the outer boundary can move only vertically.

The discrete models produce a sequence of large and sparse symmetric and positive definite linear systems with ill-conditioned matrices due to the huge and abrupt jumps in the material parameters. Here the preconditioned conjugate gradient iterative solver was used accelerated by a state-of-the-art sparse approximate inverse preconditioner. In particular, we use the factorized sparse approximate inverse in its adaptive fashion (Janna et al., 2015; Janna & Ferronato, 2011). A distinct advantage for the Monte Carlo simulation carried out in the present application relies on the fact that the adaptive factorized sparse approximate inverse is breakdown free and perfectly parallelizable, so that shared memory machines were fully exploited.

### 4.2 Results

#### 4.2.1 Hydrofacies Structure

According to the available borehole data, the Chaobai alluvial fan was divided along the flow direction of the river into three zones, namely, the upper fan (zone 1), middle-upper fan (zone 2), and middle-lower fan (zone 3). Local-stationary assumption holds in each zone (Zhu et al., 2015). The zones were identified on the basis of the statistical facies distribution (Zhu et al., 2017). The multizone analytical solution of the transition probability models was adopted to simulate the sedimentary heterogeneity, which incorporates the geologic information on facies proportions, mean lengths, and juxtapositional tendencies into geostatistical simulations (Dai et al., 2005). The parameters of transition probability models were optimally estimated. The results are summarized in Table 2. Notice that the characteristic size of the grid selected to represent the hydrofacies structure is a trade-off between the integral scales provided in Table 2 and the expected computational burden.

Zone | Parameter | Direction/hydrofacies | Estimated value |
---|---|---|---|

1 | integral scale (m) | vertical | 17.1 |

dip | 618 | ||

strike | 309 | ||

volumetric ratio (%) | gravel | 0.53 | |

medium-coarse sand | 0.07 | ||

fine sand | 0.23 | ||

clay | 0.17 | ||

2 | integral scale (m) | vertical | 6.2 |

dip | 360 | ||

strike | 180 | ||

volumetric ratio (%) | gravel | 0.24 | |

medium-coarse sand | 0.07 | ||

fine sand | 0.29 | ||

clay | 0.40 | ||

3 | integral scale (m) | vertical | 5.3 |

dip | 319 | ||

strike | 159 | ||

volumetric ratio (%) | gravel | 0.06 | |

medium-coarse sand | 0.11 | ||

fine sand | 0.33 | ||

clay | 0.50 |

Once the inversed multizone transition probability models were available, the code GEOST was used to simulate the hydrofacies distributions sequentially from zones 1 to 3 using the 694 borehole information as conditional data. As an example, one simulated output is shown in Figure 7. The north-to-south vertical section provided in Figure 5c highlights the hydrofacies transition from the upper to the lower part of the alluvial fan and clearly reveals how the simulated lithological distribution is much more representative of the borehole data (Figure 5a) than deterministic hydrogeological interpretations (Figure 5b).

#### 4.2.2 Piezometric Head

The groundwater flow model was run for each of the *N*_{R} = 100 hydrofacies distributions. A basic statistical postprocessing in terms of mean head
and associated standard deviation *σ*_{H} was carried out for each element of the FE mesh.

The outcomes of the calibrated groundwater flow model are summarized in Figure 8. The figure provides the computed depth-averaged
in 1965 (Figure 8a) and in 2012 (Figure 8b), at the end of the simulation period. Due to the constant grid spacing along the vertical direction, a simple mean of the piezometric changes calculated on the nodes along the same vertical alignment is adopted. The parameter calibration (specifically *K* and *α* and consequently *S*_{s} and *E*) was carried out through a trial and error procedure by simultaneously minimizing the cumulative squared model errors in terms of (1)depth-averaged water level drawdown and (2) land subsidence,over the entire time period spanned by the simulations. The two errors were computed by summing the difference between observation and model outcome in correspondence of surficial node location for each of the *N*_{R} = 100 hydrofacies. The comparison with the measurement-based maps provided in Figures 6a and 6b shows a reasonable match between the model results and the records, as confirmed by Figure 9 where measured and simulated piezometric levels versus times are shown for a few representative wells. What clearly emerges is the significant lowering of the piezometric level because of groundwater withdrawal: The average head decrease is about 40 m, with the formation of two large depression cones. The largest one is located in the Tianzhu and Houshayu regions on the southwestern part of study area. Another cone developed around the Yang town to the southeast. The minimum *H* value in 2012 amounted to about is −15 m below msl. The associated standard deviation *σ*_{H} ranges between 0.5 and 1.5 m.

#### 4.2.3 Displacement Fields

Each groundwater flow simulation was coupled with a 3-D geomechanical run. Figures 8c and 8d provide the simulated cumulative land subsidence between 1965 and 2012 and the average subsidence rate from 2003 to 2010, respectively. The two maps can be compared with the available data (Figures 6c and 6d, respectively). The available records are satisfactorily reproduced in terms of both the pattern and amount of the movements. The most serious occurrence is located in the southwestern part of the study area. The northern zone, where the largest wellfields are located, is relatively stable because of the small thickness of the alluvial deposits and low compressibility of the prevailing coarse hydrofacies.

The horizontal displacements are much smaller than land subsidence (Figure 10), with larger values smaller than 0.1 m, that is, approximately 1/10 of the vertical movements. This is somehow expected because of the scattered distribution of the withdrawals. Notice that the largest values of *u*_{y}, that is, the movement in the south–north direction, are computed along a narrow band crossing from east to west the study area. Figure 11 clearly shows that the largest *u*_{y} develops where the bedrock is steeper.

## 5 Discussion

### 5.1 Statistical Uncertainty Distribution

A major advantage of the proposed approach with respect to the deterministic framework traditionally employed in land subsidence analyses (e.g., Minderhoud et al., 2017; Teatini et al., 2006; Ye et al., 2016) is the possibility of associating “uncertainty quantification” to the simulated occurrence. This range reflects the intrinsic heterogeneous nature of an alluvial fan. Figure 12 provides the maps of the standard deviation of the cumulative displacements between 1965 and 2012 along the east–west (Figure 12a), south–north (Figure 12b), and vertical (Figure 12c) directions, together with that associated to the 2003–2010 subsidence rate, *σ*_{m,2003–2010} (Figure 12d). The values average 0.02, 0.01, and 0.04 m for *u*_{x}, *u*_{y}, and *u*_{z}, respectively, and 2 mm/year for the subsidence rate over the ENVISAT acquisition period. The maximum standard deviation amounts to approximately 1/10 of the maximum expected value, for example, 0.1 m over about 1 m (Figure 8b) for *u*_{z}, and 0.01 m over 0.1 m (Figure 10b) for *u*_{y}. The hydrofacies distribution significantly affects the standard deviation of the computed displacement field, with the largest standard deviation values in the middle-lower fan (zone 3) for *u*_{x} *u*_{y}, and in the upper fan (zone 1) for land subsidence. The influence of the hydrofacies statistical properties is even more evident looking at the coefficient of variation (*CV*), that is, the ratio of the standard deviation to the mean. In fact, the average *CV* decreases from 0.40 in the upper fan to 0.15 in the middle-upper fan and to 0.05 in the middle-lower fan (Figure 13).

- the variability of integral scale and the main features of the pumping distribution affect the variability associated to land subsidence.
*CV*is generally higher where the integral scale takes the longer values (zone 1; see Table 2) and where the aquifer system is exploited through localized well-fields rather than scattered wells causing a uniformly distributed water removal; - the variability associated to the displacement field seems relatively low, as already observed in the literature (e.g., Teatini et al., 2010). The integral nature of land subsidence, which represents the sum of layer deformation, the smoothing effect exerted by the layers above the pumped depth interval, the hyperstatic (or statically indeterminable) nature of the 3-D continuous medium, and the redistribution of the forcing factors (i.e., pumping rates) depending on the location of the facies permeability contribute to this outcome.

### 5.2 Comparison With a Deterministic Approach

It is also interesting to compare the modeling approach here presented with a “traditional” determinist model previously developed by the Beijing Institute of Hydrogeology and Engineering Geology (2015) for the same area. By means of MODFLOW and 1-D SUB package, it was simulated the land subsidence in the Chaobai plain over the period between 2003 and 2010. The model domain was composed of four aquifer units along the vertical direction and a 500 × 500 m grid, comparable with that used in our modeling approach, was used for the horizontal discretization. The same data set on pumping rates was available and similar information on lithological distribution, pumping tests, and data derived from the literature were used to asses an initial distribution of the hydrogeomechanical parameters. The parameter inversion was carried through a trial-and-error approach. The land subsidence distribution (Figure 14c) matches accurately the PSI measurements (Figure 6d), more precisely than the outcome of the integrated facies model provided in Figure 8d, mainly for the zone with the largest subsidence rates. The match required the domain subdivision in a complicate pattern of zones characterized by different hydraulic conductivity (Figure 14a) and storage (Figure 14b) values. However, this parameter distribution is not fully supported by the available hydrogeological information, and the forecasting reliability is questionable.

With the novel integrated approach here proposed, *K* and *α* values are uniquely identified for each hydrofacies, with the lithologic data set used to fully characterize the hydrofacies distribution within the 3-D domain. Recent publications have been focused on directly associating hydraulic conductivity to hydrofacies distribution by combining stochastically simulated hydrofacies with geophysical data (e.g., Zhu, Gong, et al., 2016). No study along this line is known to the authors in relation to the geomechanical parameters, possibly due to the even larger paucity of data available for soil compressibility.

### 5.3 Simulated and Measured Land Subsidence Short-Length Variability

Land subsidence due to groundwater pumping has been traditionally considered as a “smooth,” gentle-gradient process, that is, characterized by a long-scale (a few kilometer at least) variability (e.g., Faunt et al., 2016; Teatini et al., 2006). However, recent measurements carried out by PSI (e.g., Jones et al., 2016) reveal that vertical land displacements in subsiding areas are also characterized by significant differential displacements at a shorter (in the range of tens to hundreds of meters) scale (Figure 1b).

A certain interest is the possibility of comparing the variability obtained by the stochastic modeling approach here proposed and the PSI results. As an example, we focus on the period between 2003 and 2010 when the interferometric outcome is obtained by ENVISAT acquisitions.

Figure 15 shows the average subsidence rate and the associated standard deviation σ_{PSI} obtained by resampling the PSI solution on the same 400 × 400 m regular grid used to generate the hydrofacies structures. σ_{PSI} mainly ranges from 1 to 3 mm/year, which is comparable with σ_{m,2003–2010} (Figure 12d). Values up to 5 mm/year are localized in a few restricted areas. A more in-depth analysis is ongoing,

### 5.4 Horizontal Displacements

The proposed 3-D modeling approach allows investigating the occurrence of possibly large horizontal displacements in correspondence of abrupt variations of the basement depth (Figures 9 and 10). This is an important issue in relation to the possibility of ground rupture generation accompanying land subsidence. Ground ruptures develop in faulted basins (e.g., Ochoa-González et al., 2018) or where a relatively shallow rock basement is characterized by a sudden depth changes (i.e., Holzer & Pampeyan, 1981; Ye et al., 2018).

Also the Beijing plain has been affected by ground ruptures (Figure 15b) generally caused by the reactivation of geologic faults associated to aquifer overexploitation (Hu et al., 2019), which on a hand can explain part of the local variability charactering the PSI outcome, on the other confirm the importance of using a 3-D analysis instead of a traditional 1-D approach. Specific investigations on this topic are worth to be developed.

## 6 Conclusions

An original modeling approach is developed to investigate the effects of hydrofacies distributions typical of alluvial fans on the displacement fields, in particular land subsidence, due to groundwater withdrawal. Available lithological information is used by a multizone transitional probability method to generate possible 3-D facies structures with integral scales (along the vertical, dip, and strike directions) and volumetric proportions of the identified lithologies varying from zone to zone. The static/geologic model is converted into a 3-D FE mesh used by a subsurface flow and geomechanical simulator to compute the piezometric evolution and the displacement field caused by aquifer overexploitation. In the context of a Monte Carlo procedure, with the generation of an ensemble of hydrofacies structures, specific preprocessors must be used to assign Dirichlet and flow (recharge and pumping rates) boundary conditions and to avoid unrealistic piezometric and displacement values.

The modeling approach is applied to the Chaobai alluvial plain, North of Beijing, China, where groundwater pumping caused a significant land subsidence since the 1960s. Hundreds of available borehole stratigraphies were processed to derive the hydrofacies statistical properties for the three zones of the study area is divided into. For each zone, a local stationarity assumption of the system is verified. A number of 100 hydrofacies structures are generated and used as static models for the dynamic simulations of groundwater flow and land subsidence between 1965 and 2012. The model outcomes are postprocessed to provide the first two moments to the pressure head and displacement distributions. A rough trial-and-error calibration of the hydrogeomechanical parameters allows for the average values to well reproduce the available piezometric and land subsidence measurements.

The model results, in terms of standard deviation, significantly improve our possibility of quantifying the uncertainty associated to land subsidence prediction. It is shown how the uncertainty in the hydrofacies structure is reflected on the displacement fields. In particular, the standard deviation is larger for zones with longer integral scales. In the Chaobai plain, the maximum standard deviation amounts to approximately 1/10 of the maximum expected displacement, with a variability coefficient up to 0.5 in the upstream fan where the major well fields are located.

This is a first application of the proposed methodology to a real-world case study. A few updates are under implementation from the numerical point of view to reduce the computational burden and makes the approach more effectively usable to manage water resources in the Chaobai plain and in other sites worldwide that are experiencing large land subsidence because of aquifer overexploitation. A first advancement is aimed at replacing the trial-and-error calibration procedure with an unbiased stochastical approach, for example, using Data Assimilation algorithms. An activity is currently ongoing to include an Ensemble Smoother algorithm (Evensen & van Leeuwen, 2000) within the framework depicted in Figure 2, thus effectively estimating the hydraulic (e.g., Bailey & Baù, 2012) and geomechanical (e.g., Zoccarato et al., 2016) properties of the hydrofacies. Second, the high computational cost and large disk space required to run multiple times the FE simulators and save the outcome for the statistical post-processing, in particular in case of large-scale models, is a rather severe bottleneck. This suggests the use of model reduction methodologies to efficiently approximate the FE solutions. Research is ongoing in the geomechanics context to replace the full ensemble of model runs with polynomial evaluations, using a Markov Chain Monte Carlo sampling technique based on the polynomial chaos expansion surrogate solution (Bottazzi & Della Rossa, 2017; Zoccarato et al., 2018).

## Acknowledgments

This work has been developed within the 2017–2022 MoU between the University of Padova (Italy) and the Capital Normal University (China). The research activities were supported by the Beijing Natural Science Foundation (8202008), Beijing Outstanding Young Scientist Program (BJJWZYJH01201910028032), Young Yanjing Scholar Program, the Capacity Building for Sci-Tech Innovation- Fundamental Scientific Research Funds (025185305000/191), and the University of Padova (2016 International Cooperation Programme). The data used in this study will be freely available at http://www.igcp641.org website.