Numerical Investigation on Stability of Lanxi’s Ancient City Wall during a Major Flood Propagation Process

Abstract

Major flood propagation processes often cause instability and damage to the ancient waterfront city walls. To quantitatively reveal the impact of major floods on the stability of ancient city walls, this paper takes Lanxi’s ancient city wall as a study object and constructs a numerical model to investigate the influence of the major flood process in 2017 on the wall stability and reveals the varying laws of its seepage, displacements, maximal shear stresses and safety factors with flood propagation time on the basis of flood level data, combining indoor experiments and field observations. The results show that flood level variations significantly affect the PWPs (pore water pressures) of the fillings behind the wall. During the flood period, the maximal horizontal and vertical displacements are mainly induced by soil extrusion and deformation, and the maximal shear stresses of the outer and inner wall also significantly increase. The changing rates of the wall’s safety factors measurably exceed that of the flood level. The flood level variation range dramatically affects the safety factors when it changes near and above the wall foot. The minimum of the safety factors decreases with the increasing flood level falling rate when it drops near the wall foot at different rates. The ancient city wall usually does not experience serious instability under a single major flood. This study can provide a theoretical basis for the selection of reinforcement measures for flood control ancient city walls and the protection of ancient waterfront buildings.

1. Introduction

The major flood propagation processes have a measurable impact on the stability of soil bank slopes and their protective structures, which involves the complex transient flow of saturated–unsaturated soils [1,2,3]. The instabilities and disasters of the reservoir bank slopes triggered by water level changes often occur in flood seasons every year [4,5]. Generally, the rising water level has little adverse influence on slope stability, while the retreat process of floods, especially the sudden drop, easily results in instability [6,7,8]. In the process, the initial water level variation, flood level decline rate and its decline amplitude, soil’s seepage coefficient, water table, etc. all have an impact on the slope stability [9,10,11,12,13]. The seepage causes fluctuations in soil’s pore water pressure (PWP) under the varying processes of flood levels, altering its physical and mechanical properties, leading to deformation and instability of soil bank slopes [14,15,16]. The periodic rise and fall of the flood levels also prompt the reduction of soil engineering performance [17,18]. The soil moisture content increase leads to the decrease of matrix suction and internal friction angle, inducing the shear strength reduction [19]. The bank slope deformation increases in the process of reservoir impoundment, and even tension cracks appear, the direct weakening manifestation of soil mechanical properties [20].

In Southeast China, there is abundant rainfall and numerous rivers, and flood hazards occur year after year. The local people are accustomed to building city walls along rivers outside the cities and towns, which were used as military defense facilities or for flood control in ancient times. They have been eroded by floods for thousands of years and repeatedly built and reinforced many times [21]. In modern times, the ancient city walls still meet the flood control requirements in many cities and towns. Meanwhile, they have become the business cards and landmark buildings of many locales [22,23,24]. However, with increasing rainfall and frequent floods, the topic of damage to ancient city walls has been widely reported over recent years, and the damage mechanism urgently needs to be studied [25,26].

The ancient city walls that have been around for thousands of years are seriously weathered [27], and the climatic conditions, engineering geology, hydrogeology and living environment are relatively complex. They are often threatened by floods, waterlogging and water level fluctuations [28]. This also makes it difficult to clarify the damage mechanism. Many engineering reinforcement measures are not targeted, resulting in the frequent destruction of ancient city walls, which not only brings largely hidden dangers to urban flood control and people’s safety but also has a serious impact on the protection of ancient buildings and urban cultural relics [29,30,31].

Flood level fluctuations can trigger filling deformation and wall instability, posing a threat to the ancient city wall’s safety [32]. The stability of the filling behind the wall is dramatically affected by its moisture content. The changing moisture content can directly alter the matrix suction, pore pressure and other soil properties, which affect the soil strength and leads to its stability change [33,34]. The physical and mechanical properties of the fillings at the back of the wall are constantly varying under flood level fluctuations [35], making its potential sliding surface show a dynamic change process. The slope of the ancient city wall is often quite large, and the filling at its back forms a high and steep slope. The filling deforms and squeezes the wall during flood level process. In view of that the ancient city wall has been in disrepair for years, the strength of the wall material is generally not high, making it prone to displacement, bulging deformation or instability damage.

The ancient city wall has experienced hundreds of years of historical changes. Its failure involves many factors and the failure mechanism is also complex. On the basis of flood level data, combining indoor experiments and field observations, this paper takes Lanxi’s ancient city wall as a study object and constructs a numerical model to investigate the influence of the major flood process on the wall stability, quantitatively revealing the varying laws of its safety factors. This study can provide theoretical basis for the selection of reinforcement measures for flood control ancient city walls and the protection of ancient waterfront buildings.

2. Overview of Study Project

Lanxi City is situated between 119°13′30″–119°53′50″ east longitude and 29°1′20″–29°27′30″ north latitude in Zhejiang Province, China (Figure 1a,b), 67.5 km long from east to west, and 38.5 km wide from north to south, with a total area of 1310 km². The ancient city wall to be studied is located in the city and seated by Lan River (Figure 1c); it was originally built more than 1000 years ago, and about 600 m of it still remains. The wall was rebuilt on a large scale in 1512 AD. According to the records of Lanxi City [36], the maximal length of the city wall in Ming Dynasty was 2.77 km, which was very spectacular. The city wall has a long history and has been weathered by wind and rain for hundreds of years. It has been destroyed and repaired multiple times. On 18 May 2014, a collapse accident occurred in the ancient city wall near the West Gate Tower. Subsequently, the local water resources management department carried out maintenance and reinforcement design on the wall within a range of about 200 m there. The repair construction of this section was completed in 2015. During the flood on 25 June2017, multiple areas of the wall along the river experienced water seepage, pipe surges and partial collapse, making the situation critical. Floods flooded Lanxi City, causing the urban area to be submerged and affecting about 100,000 people. At that time, the flood control standard of the wall was less than once every 20 years, which posed a safety hazard during the flood season. Around 2020, the ancient city walls were reinforced and their flood control standards were raised to once every 50 years. Up to now, the ancient city wall is still an important flood control facility of the entire Lanxi City (Figure 1d), which brings enormous flood control benefits in flood seasons.

Lanxi City is in a subtropical monsoon climate region with abundant rainfall, with an annual average precipitation of 1476.5 mm. From mid to late June to early July every year, affected by the quiet front, there is continuous cloudy and rainy weather, accompanied by rainstorms, commonly known as plum rain, which can very easily cause watershed flood disasters. There is a rainwater collection area of 18,259 km² above the intersection of the three rivers, with an average annual transit water volume of 16.65 billion m³. In rainy seasons, the Qu River and the Jinhua River converge to the Lan River, causing severe flood level fluctuations in the Lan River. At 20:15 on 25 June 2017, the peak flood level elevation of the Lan River (Lanxi section) achieved 32.04 m, and since records began, the only larger flood level (33.49 m) than this one occurred on 21 June 1955. The Lan River’s daily average water levels in the most recent five years are shown in Figure 2.

In Figure 2, the Lan River’s water levels show a significant sudden rise and fall in the flood seasons every year, and the ancient city wall bears major flood control pressure and sometimes collapses (Figure 3a,b). Many problems have also appeared in the wall under repeated hydraulic erosion, such as wall deformation, leakage (Figure 3c), cracking (Figure 3d), and wall brick damage (Figure 3e), which seriously threaten its stability and safety and gradually evolve into a potential safety hazard for urban development. Therefore, the stability of the ancient city wall must be evaluated.

3. Methods and Materials of Numerical Investigation

The numerical model was carried out by Finite element software GeoStudio (2018 R2) in the geotechnical engineering field. The main theoretical introduction of relevant calculations is as follows.

3.1. Theory of Calculation

3.1.1. Theory of Seepage Calculation

The filling behind the ancient city walls often transitions in a saturated–unsaturated states during flood propagation processes [37,38]. The following equation can be used to describe the seepage process of different soil states in typical cross-sections of ancient city walls [39,40,41,42,43].

$${\displaystyle \frac{\partial }{\partial x}}\left(k_w{\displaystyle \frac{\partial H}{\partial x}} \right)+{\displaystyle \frac{\partial }{\partial y}}\left(k_w{\displaystyle \frac{\partial H}{\partial y}} \right)+Q_w={\displaystyle \frac{\partial {\theta }_w}{\partial t}}$$

(1)

where $Q_w$ is unit flow, 1/s; $H$ is flood level, m; $t$ is time, s; ${\theta }_w$ is moisture content, %, ${\theta }_w={\theta }_r+({\theta }_s-{\theta }_r)/\left({[1+{\left(\alpha \psi \right)}^n]}^m\right)$, in which ${\theta }_s$ and ${\theta }_r$ are saturated and residual moisture contents, %, respectively, $\psi$ is matrix suction, kPa, and $\alpha$, $n$ and $m$ are parameters, wherein,$\alpha$ is related to intake value, 1/kPa, and $m=1-1/n$; $k_w$ is seepage coefficient of unsaturated soil, cm/s, $k_w=k_s{S}_e^{0.5}{[1-{\left(1-S_e^{1/m}\right)}^m]}^2$, in which $k_s$ is seepage coefficient of saturated soil, cm/s, and $S_e$ is relative saturation, %,${ S}_e=1/{[1+{\left(\alpha \psi \right)}^n]}^m$.

3.1.2. Theory of Stability Calculation

The change in groundwater level alters the matrix suction, and its influence on soil properties cannot be ignored [44,45]. The matrix suction can significantly affect the soil’s physical and mechanical properties that are directly related to the ancient city wall’s stability [46,47]. The soil strength can be calculated by the following equation [48,49,50].

$${\tau }_s=c^{\prime }+\left({\sigma }_n-u_a\right)tan{\varphi }^{\prime }+(u_a-u_w)tan{\varphi }^b$$

(2)

where ${\tau }_s$, ${\sigma }_n$, $c^{\prime }$, $u_w$ and $u_a$ are shear strength, normal stress, effective cohesion, PWP and PGP (pore gas pressure), respectively, kPa; and ${\varphi }^{\prime }$ and ${\varphi }^b$ are effective internal friction angle and internal friction angle, °,$tan{\varphi }^b={ S}_{e}tan{\varphi }^{\prime }$.

The safety factors of the ancient city wall during the flood process were calculated using finite element method (FEM). The application of FEM in calculating safety factors is mainly reflected in slope stability analysis and the calculation of safety factors for geotechnical structures. This method does not require any assumptions and can automatically determine the critical slip surface of any shape and the corresponding minimum safety factor, while also accurately reflecting the process of slope instability and plastic zone development [51]. The calculation equation for the safety factor of slope engineering is usually expressed as follows [52,53].

$$F_s={\displaystyle \frac{\sum S_r}{\sum S_m}}$$

(3)

where $F_s$ is the safety factor; $S_m$ and $S_r$ are sliding and anti-sliding shear forces, kN, among them, $S_m=d{\tau }_m$ and $S_r=d{\tau }_s$, in which, ${\tau }_m$ is sliding shear stress, kPa, and $d$ is each strip bottom length, m.

3.2. Numerical Calculation Model

3.2.1. Calculation Parameters

(1) Calculation cross-section and geometric parameters

The typical cross-section of the ancient city wall is shown in Figure 4. Meanwhile, the Figure is also a geological cross-section schematic diagram.

In Figure 4, according to different geological conditions, the cross-section is divided into four layers along the vertical direction, in which the first to second layers (I₁, I₂) are both artificial fillings, the third layer is gravel pebble and the fourth layer is argillaceous silty sand. The wall is almost vertical, with an angle of 88° from the horizontal direction and constructed with red sandstone (8 m high, including a 1 m parapet). On the basis of the typical cross-section in Figure 4, the numerical model with the prototype was constructed, as shown in Figure 5. Due to the relatively regular nature of the rock and soil layers, they were all divided into quadrilateral grids, while the thickness of the ancient city wall is relatively small. To ensure that the calculation nodes can cover the analyzed parts of the wall as much as possible, triangular grid division was used, and the grid division was generated by the software’s built-in grid division tool. To assure calculation accuracy and efficiency, the grid size was set to 0.5 m, and gradient transition processing was used at the size changes [54]. The entire model has 6193 elements and 6352 nodes.

(2) Property parameters of rocks and soils

The performance parameters of the rocks and soils used for the calculation model were obtained according to the engineering investigation and laboratory experiments, as shown in

Table 1. Performance parameters of rocks and soils.

Material Names	Dry Density ${\mathit{\rho }}_{\mathit{d}}$ (g/cm3)	Seepage Coefficient of Saturated Soil ${\mathit{k}}_{\mathit{s}}$ (m/s)	Compressive Modulus ${\mathit{E}}_{\mathit{s}}$ (MPa)	Effective Cohesion ${\mathit{c}}^{\mathit{\prime }}$ (kPa)	Effective Internal Friction Angle ${\mathit{\varphi }}^{\mathbf{\prime }}$ (°)	Poisson’s Ratio $\mathit{v}$
Artificial filling (I1)	1.57	9.60 × 10−4	2520.0	2.5	25	0.33
Artificial filling (I2)	1.68	5.79 × 10−4	25.0	2.0	30	0.33
Sandy gravel (ΙΙ)	1.92	1.74 × 10−3	35.0	0.0	35	0.30
Pelitic siltstone (ΙΙΙ)	2.05	5.50 × 10−6	40.0	5.0	37	0.25
Red sandstone	2.10	2.32 × 10−3	10,000.0	200.0	45	0.21

.

The SWCCs (soil–water characteristic curves) of the fillings can be measured by the pressure plate method, and used to fit and predict the parameters of the van Genuchten empirical model, as shown in Figure 6a [55]. Using matric suction test data to calculate unsaturated soil seepage coefficients, the relation curves between the fillings’ seepage coefficients and matrix suctions were fitted as shown in Figure 6b [56].

The pelitic siltstone (ΙΙΙ) is always below the water surface, thus its seepage coefficient is taken as a saturated one. While the gravel pebble layer (ΙΙ) contains fewer soil particles and is also below the water surface for a long time, its seepage coefficient can also be taken as a saturated one. The wall is constructed with red sandstone bricks, there are some macro cracks between the bricks and its seepage is generated from the brick joint. The seepage coefficient varies little in saturated–unsaturated states, and in the calculation, the saturated seepage coefficient was selected.

3.2.2. Boundary Conditions of Numerical Model

The wall is situated at the intersection of three rivers, and the flood level varies sharply. To explore the influence of the flood propagation process on the wall stability, the flood fluctuation process curve from 21 June to 2 July 2017 was selected based on Figure 2 as the hydraulic boundary condition and hydrostatic pressure load conditions, as shown in Figure 7. Due to the rock layer at the bottom of the model, there is little deformation during the flood process, and the bottom can be considered as a semi wireless boundary condition [57]. The distance between the two sides of the cross-section and the earth city wall is relatively far. In the calculation process, the numerical model‘s horizontal displacement is much smaller compared to its vertical displacement, and it can be considered that no horizontal displacement occurs at the boundary. Thus, the strain boundary conditions are applied that the displacements of left and right ends of the model in horizontal direction are confined, and the bottom displacements in both horizontal and vertical directions are restricted [58].

3.2.3. Calculation Duration

The time it takes for the water level to start to rise and then fall back to the initial level is 270 h. Afterwards, the water level changes very little. Thus, the calculation duration is taken as 270 h.

4. Results and Discussions of Numerical Investigation

4.1. Seepage Analyses

The relation curves of the PWPs of the fillings at Points F1, F2, F3, F4 and F5 behind the wall vary with time, as shown in Figure 8.

According to Figure 8, in the process of flooding, the PWPs of the filling behind the wall vary accordingly when the flood level continues to rise or fall, and the changing law is consistent with the water level fluctuations. The PWP variation delays slightly, which is due to a certain amount of time required for seepage in the soil, and its changing rate is less than that of the flood level. In space, the variation range of the PWPs decreases with the increasing depth. The PWP at Point F5 exceeds 60 kPa at the highest water level and at Point F4 is also above 30 kPa, while the value at Point F1 is always negative during the flood. The increase of the PWP at Point F2 is not obvious with the increasing soil depth, but that at Point F3 is large, and the maximal PWP is about 10 kPa. Thus, the impact of the flood level fluctuation on the PWP variations of the lower fillings is more significant, the PWPs variation with the flood propagation process is more intense and the variation range is also large.

Due to the significant changes in PWPs in the lower fillings of the ancient city wall, and the fact that Point F5 is located at the bottom of the wall, it cannot reflect the changes in PWPs in the fillings. To investigate the changing laws of the horizontal PWPs of the fillings, Point F4 in Figure 5 and three characteristic points located 10 m, 20 m and 30 m away from Point F4 in the horizontal distance (represented by F4-10, F4-20 and F4-30) were taken to study the PWP variations, as shown in Figure 9.

In Figure 9, the variation range of the PWPs decreases with the increasing distance between the wall and soils. The influence of the flood level fluctuation on the change of the PWPs of the fillings (Point F4) behind the wall is obvious. The PWPs there reach more than 30 kPa at the highest flood level. The PWPs at Point F4-20 and F4-30 are both small (around 5 kPa), and there is still a long time delay.

To verify the correctness of the seepage calculation results, the measured data in mid-March and early April 2021 are selected to calculate and analyze the corresponding sections of the ancient city wall. The comparison between the calculated and measured results is shown in Figure 10.

According to Figure 10, the calculated results are basically consistent with the measured data, and their changing laws are also consistent with the flood level fluctuations. This indicates the seepage calculation method and the selection of calculation parameters is appropriate.

Based on the above analyses, it can be concluded that a major flood process significantly affects the PWPs of lower fillings behind the wall. The drastic changing PWPs trigger filling deformations and displacements, which leads to the extrusion of the fillings against the wall and the problem that the wall deforms with flood level fluctuations.

4.2. Displacement and Stress Analyses

The changing laws of horizontal direction displacement of the wall and fillings in the flood propagation process are shown in Figure 11.

According to Figure 11, the horizontal displacement variations of the ancient city wall during the flood are consistent with that of the fillings. The maximal horizontal displacement at Point W1 reaches 2.75 cm, and that at Point W5 is only 0.43 cm. The ancient city wall is made of red sandstones that have stronger strength than that of the fillings behind the wall. The PWPs of the fillings vary when the maximal flood peak comes, and the soil deforms and compresses the wall to make it move. The wall foundation has a large thickness, good stiffness and is constrained by soil and water pressures. It has a strong compressive capacity, and its displacement is relatively small, which can also better limit soil displacement. The wall can be regarded as a cantilever beam in the small deformation range because the red sandstone bricks are firmly bonded. The horizontal displacement increases from bottom to top under soil pressures.

The flood propagation process not only causes horizontal displacements of the ancient city wall and the soils behind, but also vertical displacements. The curves of their vertical displacements varying with time are as shown in Figure 12.

In Figure 12, the vertical displacement of the soil behind the wall back is relatively large in the flooding process. The maximal vertical displacements are 2.76 cm (at Point F1), and followed by 2.06 cm (at Point F5). This difference is due to the increasing PWPs in the seepage process, and it raises the soil volume and results in horizontal and vertical displacement increase [59]. The maximal vertical displacements of Point W1 and F5 are 1.83 cm and 1.75 cm. Their difference is smaller. As the ancient city wall is made of red sandstones, its stiffness and density are large, and the deformation after immersion is small. Although the joint filling between red sandstone bricks is not considered in the calculation, its width is only 1–2 cm. Under the constraint of bricks, the vertical displacements of the joint filling materials increase very little, resulting in little change in the overall vertical displacement of the wall. The phenomena also indicate that the horizontal displacements of the wall are primarily initiated by soil’s horizontal deformations and extrusions in seepage process, while the vertical displacement is caused by soil deformation.

The influence of flood level rise and fall on the wall displacements in horizontal direction that is caused by the fillings behind the wall sensibly exceed the displacements in vertical direction. Therefore, the horizontal displacement of the fillings is extremely important to the wall’s stability. The relation curves of horizontal displacements at Points W2, F2, S1 and S2 in Figure 5 vary with time in the flood propagation process, as shown in Figure 13.

In Figure 13, the maximal displacement in vertical direction is close to 2.5 cm when the filling is about 1.0 m away from the wall. It does not exceed 1.0 cm when the soil is 8.0 m away from the wall, and is about 0.25 cm when the distance increases to 20 m. Thus, the impact of flood propagation process on the backfill behind the wall is more significant. In the subsequent discussion, the effect of the filling deformation near the wall back on the wall stability can be directly analyzed.

To verify the correctness of the results, the displacement data measured during the construction in the middle and late March 2021 are collected to be compared with the calculation values, as shown in Figure 14 and Figure 15.

From Figure 14 and Figure 15, it can be seen that the measured displacement data are in good agreement with the calculation values. This indicates that the displacement calculation method and parameters were chosen correctly.

The maximal stress should appear at the inner and outer walls under the water and soil pressures because the wall is regarded as a whole in the calculation. Due to the large thickness of the wall, there are large differences in the shear stresses among the outer, inner and middle of the wall. Points WI1, WI2, WI3, WI4, WI5 and Points WM1, WM2, WM3, WM4, WM5 are taken to represent the characteristic points of the inner and middle of the wall horizontally corresponding to Points W1, W2, W3, W4, W5 on the outer wall in Figure 5 and their maximal shear stress variation in the flood propagation process is analyzed. The changing laws of the maximal shear stresses in the wall and filling during the flood are shown in Figure 16, Figure 17 and Figure 18.

In Figure 16, the flood level elevation reached 28.0 m (near Point W4) when the flood peak appeared on 23 June 2017, while the shear stress at Points W4 and W3 increased and decreased slightly at Point W5. The flood peak had little impact on the maximal shear stresses at Points W1 and W2. According to Figure 17, the maximal shear stresses at Points W4, W3 and W2 on the inner wall decrease, increase at Point W1, and vary slightly at Point W5. As can be seen from Figure 18, the maximal shear stresses at Points F1, F2 and F3 of the fillings at the wall back gradually increase with the variations of the flood peak level, while the maximal shear stresses at Points F4 and F5 below the flood peak level decrease. The maximal shear stresses at Points WM3 and WM4 in the middle of the wall are sensitive to the flood level fluctuation. The maximal shear stress is reduced by more than 20 kPa. The influence of flood level fluctuations below the flood peak on the wall and the fillings behind the wall is obvious when the Lan River flood level rises, but the effect of small flood level fluctuations on the maximal shear stresses is limited, and the values at each characteristic point of the wall do not exceed 30 kPa.

When the major flood peak level (32.04 m) of the Lan River appeared on 25 June, obvious changes of the stresses occurred at points on the inner and outer walls and fillings, especially that of the characteristic points below the flood level were more significant. After the arrival of the maximal flood peak, the flood level reached near Point W2, and the maximal shear stresses at Points W3 and W4 increased significantly, reaching 138.25 kPa and 145.99 kPa, respectively. They decreased sharply to the stress level before the flood peak as it receded. The maximal shear stresses decrease significantly at Points WI3 and WI4 on the inner wall when the flood peak arrives. They increase significantly to about 100 kPa when the flood level drops, then gradually increase to more than 110 kPa with the decreasing flood level. The variation law of the maximal shear stresses at the points in the middle wall is consistent with that of the corresponding characteristic points on the inner wall, but their values decrease. The variation amplitude of the maximal shear stresses does not exceed 50 kPa in the flood process. The maximal shear stresses of the fillings at the wall back in different depths decrease to varying degrees in the flood process, and their reduction ranges are more significant, close to 20 kPa. As the flood level drops, the maximal shear stresses at the fillings quickly return to their size before the flood peak.

The above analysis shows that the maximal shear stresses at the wall, especially its middle and lower parts, are measurably affected by the flood. The stresses at the outer wall are mainly affected by hydrostatic pressures, while at middle characteristic points of the inner wall, they are mainly affected by the soil pressures generated by seepage deformation. The maximal shear stresses at the selected characteristic points of the wall are less than the shear strength of the red sandstone in the flood process, thus the wall will not experience shear strength failure.

4.3. Stability Analyses

From Equation (2), it can be inferred that the variation of the moisture field alters soils’ shear strength. The soil slope stability is directly related to the shear strength according to Equation (3). The stability of the Lanxi ancient city wall and the variation law of its safety factors are calculated by invoking the calculation results of the moisture and stress fields in the flood process. The relational curve of the wall’s safety factors varies with the flood level (flood propagation time) from 25 June to 2 July 2017, as shown in Figure 19.

In Figure 19, the safety factors of the wall gradually increase with the rising flood level and decrease with the falling flood level when the small flood peak appears (23 June 2017). Their increase and decrease rates are almost consistent with the changing trend of the flood level. The minimum (1325) of the safety factors appears in the flood process in this section when the flood level drops. From the small flood peak (28.23 m) to the flood level (26.39 m) corresponding to the value (1325), the variation range is 1.91 m, and the safety factor has decreased by 13.62%. The safety factor variation is more intense in the major flood process (around 25 June 2017), and their changing rates are much larger than that of the flood level, especially during the stage of flood level decline. The minimum (1185) of the safety factors appears at this stage, the decline rate is very large, and the decline range of the flood level reaches 4.38 m from 32.04 m (flood peak level) to 27.65 m (flood level corresponding to the safety factor 1185). Meanwhile, the safety factor has decreased by 41.22%. Compared with the previous small flood, the flood level variation range increases by 2.29 times, and the reduction range of the safety factor is 3.03 times.

There are differences in the flood level variation amplitude and rate in the two different floods, which may affect the wall safety factors. To study the effect of flood level variation amplitude and rate on the safety factors, the flood process curves were generalized to form four kinds of generalized curves, as shown in Figure 20. The changing process of the generalization curve 1 is almost the same as that of the natural flood process curve, except that small fluctuations are replaced by straight-line segments. The rise of the Lan River flood level in 2017 has little adverse influence on the wall according to Figure 19, however, its decline process affects the wall stability dramatically. Thus, the generalization curves 2 and 3 were set to compare the effect of the flood level variation range on the wall’s safety factors during the flood retreat period, and the generalization curves 3 and 4 were set to compare the influence of the flood level reduction rate on the safety factors.

According to the above four generalized curves, the changing laws of the safety factors of the ancient city wall were calculated, as shown in Figure 21.

In Figure 21, it is clear that the evolution law of safety factors obtained by the generalization curve 1 and the natural flood level process curve is almost the same. It is worth noting that the generalization curves stayed at the highest water level for 12.5 h, and the safety factors decreased rapidly after increasing to the maximum, indicating that the seepage caused the moisture field variation of the upper fillings, resulting in a significant decrease in soil strength. Because the seepage rate is less than the increasing rate of hydrostatic pressures, in the rising stage, the safety factors increase continuously. Therefore, the long-term high flood level hydrostatic pressure hurts the wall stability, especially on the expansive soil which is more sensitive to water [60].

From the calculation results of generalization curves 2 and 3, it can be inferred that the peak flood level (32.04 m) decreases to 27.65 m and 23.35 m, respectively, at the same decline rate, and the safety factors decrease to 1165 and 1457, respectively. The flood level of the former decreased by 4.39 m, while that of the latter decreased by 8.69 m. The flood level of the latter decreased almost twice as much as that of the former. At this time, the safety factor of the latter increased by 25.06%. The above two curves decrease near the wall foot at the same rate, the safety factors reach the minimum and its changing law with time is the same. This indicates that the safety factors decrease with the decreasing flood level in a certain range, and beyond this range, its values will increase. According to Figure 19, Figure 20 and Figure 21, the safety factors decrease to the minimum when the flood recedes to the wall foot, and they will increase when the flood level drops further. The above phenomena show that the moisture content of the fillings behind decreases in the process of retreating from flood levels, the soils’ matrix suction gradually rises, the shear strength is raised, furthermore, the stability of the soils increases. This is consistent with the above analyses of the variation laws of the maximal shear stresses.

The flood level falling rate of the generalization curve 4 is 1.5 times that of the generalization curve 3. It can be seen from Figure 21 that when both curves fall to the lowest water level, their safety factors are the same. When the flood levels drop near the wall foot at different rates, the minimal safety factor corresponding to the generalization curve 3 is 1165, and that of the generalization curve 4 is 1107, which reduces the safety factor by about 5%. The minimal safety factor of the generalization curve 4 appears 5.5 h earlier than that of the generalization curve 3. This indicates that lowering the maximal flood level near the wall foot at different rates significantly affects the minimal safety factor of the wall in terms of the occurrence time and value. This is consistent with existing research findings [61,62,63].

The process of flood propagation, especially a major flood, has a measurable adverse effect on the stability of the ancient city wall, and this effect mainly occurs during the rapid retreat of major floods. The fundamental reason for the damage to the waterfront ancient city wall is the dramatic seepage effect of the fillings during the process, which causes deformation of the backfill behind the wall. When the filling deformation is too large, the deformed soils compress the wall, leading to displacement or even instability [64]. In this case, to improve the wall stability, efforts must be made to reduce the seepage effect in the fillings. Therefore, it is recommended to install anti-seepage walls or curtains in the fillings behind the wall to enhance its anti-seepage effect. Simultaneously, drainage facilities should be reasonably set up to drain the accumulated water in the fillings behind the wall caused by rainfall or seepage, to prevent excessive PWPs in the fillings from causing damage to the wall.

5. Conclusions

The stability of Lanxi’s ancient city wall has been affected by flooding process for a long time. The paper investigates the wall’s stability-changing process during floods and quantitatively reveals the evolution laws of the seepage, deformation, stress and safety factors varying with the flood level (flood propagation time). The main conclusions obtained are as follows.

(1)

The changing patterns of the fillings’ PWPs are similar to flood level variations, and the rate and range of changes in their values are slightly reduced compared to those of flood levels. The PWP variations in the lower part of the fillings change more dramatically, and their changing amplitudes gradually decrease when the distance between the soil and the wall increases. Meanwhile, the displacement changes of the wall are significantly affected by the deformation of the backfill soil during flood propagation period. The maximal displacement in the horizontal direction is mainly caused by soil extrusion, and that in the vertical direction is primarily caused by soil seepage deformation. When reinforcing the waterfront ancient city walls, the influence of seepage should be fully considered, and anti-seepage and drainage facilities should be reasonably set up.

(2)

The deluge has a significant influence on the stress changes of the walls and fillings below the flood level when the flood moves forward. The maximal shear stresses at Points W3 and W4 on the outer wall are 138.25 kPa and 145.99 kPa, respectively, and the maximal shear stresses of Points WI3 and WI4 on the inner wall are more than 110 kPa. The maximal shear stresses are less than the red sandstone’s shear strength, thus the wall will not experience shear strength failure. However, with the weathering of wall masonry materials, their strength will decrease, which may lead to instability and damage of ancient city walls under the influence of various factors such as flood levels, rainfall and pollution emissions.

(3)

The safety factor variation is more intense in a major flood process, and their changing rates measurably exceed that of the flood level, especially during the stage of flood level decline, and the minimum is 1185. The changing range of the flood level significantly affects the safety factors when it is near and above the wall foot. The minimum of the safety factors decreases with the increasing flood level falling rate when it drops near the wall foot at different rates. Therefore, in the process of protecting waterfront ancient city walls, it is necessary to strengthen the regulation of river water resources to avoid sudden drops in water level during floods. A slower water level change is beneficial for their protection.

(4)

The numerical model established only calculates and analyzes the Lanxi’s ancient city wall, but it has a certain universality and can be applied to the calculation of other waterfront ancient city walls. In the calculation process of the ancient city wall stability, this paper only considers the impact of changes in flood levels. In fact, this is just one of the important factors affecting the stability of ancient city walls. Due to their long construction period, the surrounding living environment and hydrogeological conditions are extremely complex, which cannot be fully considered in the calculation. In the future, more in-depth research will be conducted on the damage mechanism and protection measures of the waterfront ancient city walls, providing a theoretical basis for revealing their instability mechanism and selecting protective measures.