## Abstract

This paper proposes a relationship for the physics and mechanics constants of porous media related to water storage rate and ground settlement under a surface load variation condition. This provides the basis for accurate calculation of ground subsidence. Traditional equations for vertical deformation, groundwater flow and land subsidence due to surface loading were developed using Jacob's assumptions. This paper derives a skeletal elastic specific storage rate. The new deformation and flow equations are similar to the traditional ones based on Jacob's assumptions except that the pore-water head in the traditional equations corresponds with the margin between the pore-water head and the water-column height given in the proposed equations representing the surface load. The analysis show that increasing the surface loading leads to land subsidence, rise in pore-water head and decrease in elastic water storage capacity. The maximum subsidence is equivalent to the subsidence triggered by lowering the water head to the equivalent water column height. The maximum rise of the water head is also equal to the equivalent water column height. The maximum water released to a specific volume of porous medium is close to that resulting from reduction in the water head by the equivalent column height.

## INTRODUCTION

Land subsidence is a geological process of slow ground deformation, found mainly in large- and medium-sized cities worldwide. Locations which are close to inland river deltas and the basin areas of plains are particularly at risk of experiencing land subsidence. Land subsidence is a major cause of geological disasters in China. With the rapid growth in construction, social development and urbanization, the construction of high-level, high-rise buildings causes land subsidence in cities and their surroundings. This can damage underground geological structures and the environment, which in turn would slow down the development of these cities (Burbey 1999; Modoni *et al.* 2013).

Numerous studies have been conducted on land subsidence. These studies observed the distribution of ground subsidence as well as analysed its causes and consequences. Modoni *et al.* (2013) studied the factors influencing the magnitude and distribution of ground settlement observed during the second half of the 20th century in Bologna, Italy, and derived a unified framework for interpreting the observed phenomena and for predicting future scenarios. Pardo *et al.* (2013) investigated the general subsidence occurring in and around Murcia, Spain, using extensometer, piezometer and InSar data. They found that the subsidence caused severe cracking and settlement of buildings. Xu *et al.* (2015) investigated the increased aquifer deformation caused by groundwater pumping from an aquifer in Changzhou, China, and found that land subsidence in the region was becoming a serious geo-hazard. They adopted the Cosserat continuum model to analyse the observed phenomenon.

Some scholars analysed the factors that cause ground subsidence. Chen *et al.* (2013, 2015) analysed the correlation between changes in typical area loading and land subsidence and found a positive correlation between the load density and the homogeneity of the subsidence, especially in areas with high sedimentation rates. Ghazifard *et al.* (2016) investigated the probable causes of earth fissuring and concluded that the main causes of this phenomenon are the reduction in groundwater head, variation in sediment thickness due to bedrock anomalies, and differential vertical compaction. Notti *et al.* (2016) combined detailed geological and hydrogeological data with differential SAR interferometry monitoring to understand the subsidence processes in detrital aquifers with small-scale heterogeneity. Yu *et al.* (2018) quantitatively evaluated parameters for groundwater resources which provide insights for understanding the hydraulic properties of sandstone. Rahman *et al.* (2018) analysed land subsidence of Jakarta and found that heavier buildings and water flow from deeper water wells accelerate land subsidence. Sundell *et al.* (2019) presented a method for risk assessment of groundwater-drilling-induced land subsidence and obtained results of spatial probabilistic risk estimates for each alternative where areas with significant risk are distinguished from low-risk areas.

Several other studies have further established the analysis model of ground subsidence and simulated ground subsidence generated by regional groundwater exploitation. Xu *et al.* (2008) presented various models for predicting subsidence caused by the withdrawal of groundwater in three subsidence regions in China. They found that a three-dimensional (3D) seepage model showed the best results. Wang *et al.* (2011) used a numerical method to examine the sensitivity of parameters, including the relative compressibility of aquifers and aquitards, as well as the permeability in pumped aquifers which control the development of ground subsidence. Panda *et al.* (2015) employed a two-dimensional (2D) coupled seepage and stress–strain finite element framework to model land subsidence and earth fissuring. Their simulation results indicated that the strain exceeded the approximate threshold for fissure formation by 0.02%–0.06% in the area of the identified fissures. Chen *et al.* (2015) established a finite element numerical model for high-rise building loads, groundwater seepage and land subsidence considering the dynamic relationships among the porosity, hydraulic conductivity, soil deformation parameters and effective stress. They showed that land subsidence under the condition of each of groundwater exploitation and high-rise-building load was lower than the sum of the subsidence for the two separate cases. Similarly, Liu *et al.* (2016) established a coupling model that accounted for the effects of both the building load and groundwater exploitation and validated the model's high fitting and prediction accuracy. Mahmoudpour *et al.* (2016) developed a simulation model using PMWIN; the simulation results were in good agreement with the measurement results. They found that land subsidence caused by groundwater pumping is a serious threat to southwest Tehran, Iran. Cui & Jia (2018) studied the deformation of each soil layer considering the dual effects of building load and groundwater withdrawal using a physical model test. They found that compared with the subsidence caused by the building load, the subsidence due to dewatering developed slowly and lasted longer. Zhu *et al.* (2015, 2016, 2017) developed the statistic inversion of transition probability models to characterize aquifer heterogeneity. They increased the accuracy of the simulated hydrofacies architectures. Rajabi (2018) evaluated land subsidence in Qom, Iran, by a numerical model considering aquifer pressure changes and hydrological and geotechnical data. They revealed that land subsidence depended on the characteristics of geological materials.

An exerted surface load does not only cause land subsidence, but also alters the groundwater seepage conditions. The studies reviewed herein were based on the traditional Jacob's assumptions (Jacob 1940, 1950) and have given the relationship between aquifer water storage rate and land subsidence. The stress imposed by porous media leads to one-dimensional vertical elastic deformation; the medium grain itself is incompressible; and the porous medium overburden load remains constant during the deformation process. This paper proposes the relationship formula of physics and mechanics constants of porous media related to water storage rate and ground settlement, which can provide a basis for the accurate calculation of ground subsidence. The groundwater seepage equation and vertical elastic deformation equation under surface loading are derived to analyse the seepage and land subsidence caused by surface loads varying with time. A seepage and consolidation coupling model considering the geological and hydrogeological conditions is established, and ground settlement under the combined action of ground construction load and groundwater seepage is calculated. This study aims to provide a theoretical basis for solving aquifer deformation and seepage problems caused by variable ground load and pumping water in a confined aquifer.

## METHODS

### Deformation equation of surface loading considering the relationship between strain and pore-water head

*E*being Young's modulus, and being Poisson's ratio; and is the elastic constant expressed as . Terzaghi's principle (Terzaghi 1943) of effective stress is written as: where is the vertical stress and

*p*is the pore pressure. Due to the equality of these vertical stresses, Equations (1) and (2) can be combined as:

^{2}) is gravitational acceleration. Therefore, Equation (6) can be written as:

### Seepage equation considering the surface load

*n*is porosity, is the divergence operator, is the volume elastic compressibility of water expressed as , and is the relative flow rate, i.e. the average velocity of flow through a cross-section of porous medium with pore water relative to the solid particles.

When the equivalent water-column height is added to the surface load, the seepage Equation (12) can be analysed for three cases. These cases are as follows:

- (1)
When the groundwater seepage state (in terms of velocity) remains stable or the water storage states (the elastic storage in a unit-volume aquifer) do not change, causing the pore-water head to rise by .

- (2)When the pore-water head does not change, or the pore-water head rises and then returns to its original value (for example, when the surface load elevates the groundwater level and the next groundwater runoff reduces the groundwater table), unsteady seepage will occur and the elastic response of the aquifer leads to the release of water (because ). The drainable volume of the unit-volume aquifer can be calculated as:
Thus, increasing the surface load leads to a drop in aquifer storage capacity per unit volume. The maximum reduction is equivalent to the drainable volume produced by the drop of a unit of water-column height of the underground water table that is equivalent to the surface load.

- (3)In the intermediate state, the drainable volume of a unit-volume aquifer is greater than zero but less than , and the pore-water head thus rises by less than . Assuming that the elastic storage rate of the aquifer in the thickness (vertical) direction of the skeleton is uniform, i.e. , Equation (13) can be rewritten for an aquifer cylinder of unit area and height equal to the aquifer thickness
*b*, such that the elastically drainable volume*W*can be written as:

### Ground subsidence caused by a surface load — application of the deformation equation

*t*can be calculated by integrating Equation (8):

*t*after the beginning of the unstable seepage, and is the change of the equivalent water-column height caused by the exerted surface load. Equation (16) shows that the vertical compressive strain of the aquifer is equal to the product of the elastic storage rate of the skeleton of the aquifer () and the difference between the height change of the equivalent water column by the surface load and the change of the water-head level .

*z*is the coordinate variable for thickness of the aquifer. Assuming that the water storage rate and the change in water head of the aquifer along its thickness is uniformly distributed, Equation (17) can be transformed to: where is the elastic reservoir (release) water coefficient in the

*i*

^{th}layer of the aquifer as , and is the average drop of the water table in the

*i*

^{th}layer. The vertical compression or ground settlement () has a linear relationship with the difference between and when the aquifer skeleton thickness and the elastic reservoir (release) water coefficient are stable. The proportion coefficient is the skeleton elastic storage coefficient of the aquifer. Overall, when considering only the vertical elastic deformation and the changes in surface load, the amount of land subsidence is determined by the variation of water-head, the thickness of the vertical compressive layer and skeleton elastic water storage (release) coefficient of the aquifer.

The deformation Equation (16) and Equation (20) indicate that if the surface load raises the equivalent water-column height, there may be three results:

- (1)
If the deformation is zero ( or ), the maximum added value of the pore-water head can be obtained, which is equal to , i.e. .

- (2)
Thus, an increase in surface load can cause ground settlement, and the maximum settlement magnitude is equal to the amount generated by the equivalent water-column height of the pore-water head.

- (3)
Whereas the previous two cases are at two extreme conditions, case (3) lies in-between. In this case, the ground settlement is greater than zero but less than , and the pore-water head rises by less than . Comparison between the water volume released by a unit-volume aquifer and the maximum ground settlement reveals a consistency with the settlement deformation rule of Jacob's assumptions regarding the principle of volume interchange.

## RESULTS AND DISCUSSION

### Numerical simulation of seepage and consolidation coupling

In this paper, a coupled model of seepage and consolidation considering the engineering geological and hydrogeological conditions was established. In this section, example results are presented from the proposed model using parameters based on field data from the Changchun area in China. The 3D model dimensions are 50 m × 10 m × 50 m (Figure 1). The side boundaries of the model are the water-head type. The bottom of the model is a confining water and no-flow boundary. The aquifer in the study area primarily contains Quaternary loose pore water and includes seven layers (Yu *et al*. 2001). The strata of Changchun city are Songliao wavy plain platform, and the soil layer distribution is stable from top to bottom. The seven layers are: plastic silty clay, soft plastic silty clay, plastic silty clay, hard plastic clay, plastic silty clay, hard clay, and mudstone and sandstone inter-bedded. According to the occurrence conditions, hydrological properties and hydraulic characteristics of the groundwater, the groundwater can be divided into loose rock pore water and plastic rock pore fissure water and structural fissure water. They are characterized by shallow burial and vertical infiltration (Yu *et al*. 2001).

The vertical profile contains seven layers: the unconfined aquifer, confined aquifer I, confined aquifer II, confined aquifer III and cohesive aquitards between two aquifers, shown in Table 1. The bottom boundaries of layers 1–7 are −4 m, −7 m, −23 m, −29 m, −37 m, −45 m and −50 m. The lithology is mostly silty sand and powdery sand. The parameters of each layer are shown in Table 1.

No. . | Strata elevation (m) . | Stratum . | K_{0x} (m/d)
. | K_{0y} (m/d)
. | K_{0z} (m/d)
. | γ (kN/m)
. | φ(°)
. | C (kPa)
. | ν_{0}
. | E_{0} (MPa)
. |
---|---|---|---|---|---|---|---|---|---|---|

1 | −4 m | plastic silty clay | 0.06 | 0.06 | 0.006 | 19.7 | 18 | 21 | 0.48 | 38 |

2 | −7 m | soft plastic silty clay | 0.0005 | 0.0005 | 0.00005 | 18.9 | 20 | 23 | 0.48 | 32 |

3 | −23 m | plastic silty clay | 1.9 | 1.9 | 0.19 | 20.1 | 18 | 21 | 0.49 | 45 |

4 | −29 m | hard plastic clay | 0.000006 | 0.000006 | 0.0000006 | 17.8 | 22 | 23 | 0.48 | 30 |

5 | −37 m | plastic silty clay | 2 | 2 | 0.2 | 20.0 | 20 | 21 | 0.49 | 46 |

6 | −45 m | hard clay | 0.0005 | 0.0005 | 0.00005 | 18.1 | 20 | 20 | 0.48 | 27 |

7 | −50 m | mudstone and sandstone | 2.8 | 2.8 | 0.28 | 20.2 | 20 | 22 | 0.49 | 46 |

No. . | Strata elevation (m) . | Stratum . | K_{0x} (m/d)
. | K_{0y} (m/d)
. | K_{0z} (m/d)
. | γ (kN/m)
. | φ(°)
. | C (kPa)
. | ν_{0}
. | E_{0} (MPa)
. |
---|---|---|---|---|---|---|---|---|---|---|

1 | −4 m | plastic silty clay | 0.06 | 0.06 | 0.006 | 19.7 | 18 | 21 | 0.48 | 38 |

2 | −7 m | soft plastic silty clay | 0.0005 | 0.0005 | 0.00005 | 18.9 | 20 | 23 | 0.48 | 32 |

3 | −23 m | plastic silty clay | 1.9 | 1.9 | 0.19 | 20.1 | 18 | 21 | 0.49 | 45 |

4 | −29 m | hard plastic clay | 0.000006 | 0.000006 | 0.0000006 | 17.8 | 22 | 23 | 0.48 | 30 |

5 | −37 m | plastic silty clay | 2 | 2 | 0.2 | 20.0 | 20 | 21 | 0.49 | 46 |

6 | −45 m | hard clay | 0.0005 | 0.0005 | 0.00005 | 18.1 | 20 | 20 | 0.48 | 27 |

7 | −50 m | mudstone and sandstone | 2.8 | 2.8 | 0.28 | 20.2 | 20 | 22 | 0.49 | 46 |

### Numerical simulation results

Based on the established coupled numerical model of seepage and consolidation, a uniformly distributed load of 7.8 × 10^{4} Pa at the top boundary was used to simulate the construction load. The settlement results of the strata are shown in Figure 2.

The combined effect of loading and seepage led to different responses in the different formations. Compared with the original strata thickness, the stratum subsidence values do not exactly match the formation stratigraphic geometry because the formation thickness and physical and mechanical properties that control the settlement dynamics are different in these strata. Our modelling predicts that the soil settlement, due to such nonlinearities, shows sharp variations in the lowest two layers as compared with the layers above them. The change rule is consistent with those of the simulation and monitoring (Mahmoudpour *et al.* 2016; Zhou *et al.* 2017).

The relationship of ground settlement with time is shown in Figure 3, indicating a general linear increase in land settlement over time. The settlement change ranges from 0.05 to 0.55 m. The results are close to the results of monitoring reported by Chen & Bai (2017). The vertical stress analysis results are shown in Figure 4. The results suggest predominantly uniform vertical stress distribution because the surface load and the vertical stress are uniform. The stress increases gradually with depth. Slight changes in the vertical stress appear across different layers because of the different density of the soil layers.

The results from pore-water pressure analysis shown in Figure 5 indicate that the pore-water pressures in the two bottom layers undergo rapid change and increase from 0 to 2.75 × 10^{5}Pa. The soil pore-water pressure varies only slightly in the top five layers; its distribution pattern is not in accordance with the layer boundaries (similar to the strata settlement profile). These results indicate a strong pore-water flow out of the soil layers which could maintain an almost constant level of pore-water pressure in each layer.

## CONCLUSIONS

This paper presented the derivations for the elastic deformation equation of an aquifer, established the displacement equation coefficient formula under the corresponding conditions, and compared the proposed equation with the traditional equation based on Jacob's assumptions and the corresponding parameters of the formula. The following conclusions are drawn:

- (1)
In a homogeneous isotropic porous medium assuming only vertical elastic deformation, incompressible skeleton particles, homogeneity of the skeleton particles and water density, the relationship between vertical strain and pore-water head under surface load is obtained as well as the equations of seepage control and the relationship between ground settlement and the pore-water head.

- (2)
There is no difference in the elastic storage rate in the skeleton when considering the changes in the surface load and the traditional Jacob's assumptions. The deformation equation and seepage equation were similar to the traditional ones in structure, but the water-head variable in the new equation changed to .

- (3)
Increased surface load can lead to land subsidence, as well as an increase in the pore-water head and a decrease in the elastic storage capacity of the aquifer. The results of analysis and simulation show an increase of surface load will lead to an increase of land subsidence and pore-water head, and an decrease of elastic storage capacity of the aquifer.

- (4)
The maximum ground settlement is equivalent to the change in the water-column height corresponding to a decrease in surface load. The maximum pore-water head is equal to the water-column height corresponding to the surface load. The maximum elastic release of a unit-volume reservoir is equal to the water release of the equivalent water-column height of the pore-water head. The deformation of the aquifer caused by the surface load agrees with the principle of volume interchange.

## ACKNOWLEDGEMENTS

This work has been funded by the Engineering Research Center of Geothermal Resources Development Technology and Equipment of Ministry of Education, Jilin University. Z. Dai thanks Jilin University for the start-up funding and the National Natural Science Foundation of China (grant number: 41772253) supporting this work.

## REFERENCES

*Flow through Porous Media*(R. J. M. De Wiest, ed.), Academic Press, New York, USA, pp. 331–376.

*Geo-Frontiers 2011: Advances in Geotechnical Engineering*(J. Han & D. E. Alzamora, eds), ASCE, Reston, VA, USA, pp.