1 Introduction

In recent decades, the growth of population, accompanied by overdrawing of groundwater resources for agricultural, industrial and drinking demands, has affected the environment severely. Subsidence is one of the undesirable phenomena, particularly occurring in the alluvial plains. As a precursor, this morphologic behavior (Stephen 2011; Glopper 1989) is a result of natural and human factors (Galloway 2001) caused a large number of impacts such as morphologic chaos, damage of the facilities, building, surficial infrastructures (Simeoni et al. 2017), streets, highways and airport line (Clanton and Amsbury 1975), of the irrigation and drainage networks, stream networks (Barends et al. 1995) and water transmission pipelines, reduction in Aquifer storage coefficient (Galloway et al. 1999; Clanton and Amsbury 1975), combination of fresh and saline water (Chen et al. 2010), destroying underground structures such as tunnels, aqueducts, underground mines (Fathi et al. 2017), etc. Evaluating the subsidence phenomenon is an important step in managing the mentioned hazards (Hu et al. 2004). In recent decades, there is a growing need for the study and analysis of solid earth hazards and also studies of the sedimentary basin of cities focused to mainly on the mechanisms controlling basin formation and subsequent subsidence evolution (Cloetingh and Podlachikov 2000).

According to the UNESCO Working Group of subsidence, the oldest known subsidence occurred in the United States in 1900 (Galloway et al. 2016). Since 1965, UNESCO began The International Hydrological Decade, as man’s first concerted attempt to take stock of his diminishing available resources of freshwater, which in the following years the study of subsidence became one of its main issues (Raymond 1969).

Since then, further studies have been carried out in advanced countries, particularly in the United States and Japan, which resulted in optimizing water usage and changing consumption pattern to stop subsidence occurrence in many cases. The subsidence was reported due to excessive groundwater extraction in many areas, including the City of Mexico (Calderhead et al. 2011; Carreón-Freyre et al. 2005), Stockholm (Sundell et al. 2016), Shanghai (Hu et al. 2004), Tianjin (Lixin et al. 2012),California (Galloway et al. 1999), Bangkok (Phien-wej et al. 2006), Venice (Brambati et al. 2003) and Jakarta (Hasanuddin et al. 2009).

In Iran and for the first time, the regional land subsidence due to the groundwater withdrawal was reported in 1967 in Rafsanjan plain. Decreasing every meter of groundwater level caused the subsidence of between 5 and 15 cm for this plain (Iranian Ministry of Energy 2009). Also in recent years, there are close to 600 plains which more than half of them are likely to be exposed to subsidence. The land subsidence is reported in many different plains of Iran such as Kerman and Rafsanjan plains (Motagh et al. 2008, 2017), Damaneh and Kashan Plains (Ghazifard et al. 2016, 2017), Damghan plain (Afzali et al. 2013).

The geotechnical and the hydrogeological factors of the aquifer control the rate and extent of land subsidence. Typically the occurrence of land subsidence of the earth in a wide zone is due to compacting of the fine-grained sediments as a result of the reduction of groundwater level and local subsidence because of soil compaction, loading or presence of unstable soils. Consolidation caused by the increment of the effective stress is studied in two ways. When the increase in effective stress due to the lowering of the groundwater table is greater than the amount of stress experienced by the particles, the arrangement of the particles will be changed and a relatively irreversible settlement will occur, known as a non-elastic settlement. If the maximum pressure which soil strata have been subjected to be less than the previously experienced pressure, the changes will be reversible, known as an elastic settlement. However, the amount of 0elastic settlement compared with inelastic one is negligible..

There are many techniques to study the subsidence. Some of this techniques including GPS (Sneed and Galloway 2000; Bell et al. 2002), continuous GPS (CGPS) (Simeoni et al. 2017), leveling (Bell et al. 2002), optical remote sensing, borehole extensometer (Pope and Burbey 2004) and synthetic aperture radar interferometry (Motagh et al. 2017; Ghazifard et al. 2016, 2017; Hoffmann et al. 2001; Bell et al. 2002; Galloway and Burbey 2011).

In recent decades, an extensive increase in groundwater harvesting has been occurred in Iran, due to reduced rainfall and increased water demand for the expansion of the city, agricultural and industrial activities. Therefore, many parts of Iran are facing the danger of subsidence, which at some locations has reached even 30 cm per year (Motagh et al. 2017). Many studies are presented about land subsidence in Iran such as Motagh et al. 2017; Ghazifard et al. 2016, 2017; Davoodijam et al. 2015; Akbari and Motagh 2012; Dehghani et al. 2009, 2013; Motagh et al. 2008; Anderssohn et al. 2008. However, this paper focuses on Isfahan Metropolis, which is known as a historic city in the world. Although Isfahan is a historic and populous Iranian city, no land subsidence studies have been conducted. This may be due to the urban area as well as the high speed of building construction and lack of GPS or Radar data in the city of Isfahan. In this research, the progressive subsidence due to groundwater withdrawal from 2000 to 2014 was evaluated by available data in Isfahan Metropolis.

2 Study area

Isfahan is the second largest city of Iran that is located in in the middle part of Iran in a semi-arid region. The metropolis of Isfahan is well-known for its ancient buildings such as Masjed-e Jame of Isfahan (Jame Mosque), Sio-Se-Pol Bridge, Naghshe Jahan square, and Chahar-Bagh which has a global reputation (Seyedashrafi et al. 2017; Hemmatiha et al. 2018).

In recent years, there has been evidence of subsidence in the city and even near the ancient buildings and in the future, the ancient buildings could be at risk of subsidence. As a result, studies are required to determine the potential of subsidence and high-risk areas. In this study, the potential for occurrence and pattern of subsidence in Isfahan Metropolis, with an area of 180 km2, has been studied (Fig. 1). However, in recent years, a series of environmental, geological, archeological and social hazards have appeared in the city due to reduced rainfall and drying of the river in the past years. These risks are progressing over time due to the lack of remedial measures. Cracking on the historic bridge of Sio-Se-Pol is one that could be due to land subsidence (Fig. 2).

Fig. 1
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Location of the study area

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Fig. 2
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Cracking into the structure of the Sio-Se-Pol Bridge affected by the local subsidence

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The Isfahan plain is surrounded by desert in the north and east, and its western and southern parts are surrounded by Sayyed Mohammad and Soffeh Mountains. The reason for the existence of this city should be due to the waters that originate from the high Zagros Mountains called Zardkouh Bakhtiari which created the Zayandehrud River, and as a result, the beautiful city of Isfahan on both sides of the Zayandehrud River has been developed. The city of Isfahan is formed on a relatively flat plain with a 2% dip toward north-east.

2.1 Geology

The study area is geologically located on the margin of the structural zone of central Iran and within the subzone of Sanandaj Sirjan. Its base is covered with Jurassic shale (Dadkhah et al. 2016). The young Formations in this area belong to the Quaternary period and the outcrops of the study area are sedimentary rocks. Deposits belong to the occupy the highest elevation in the region including black shales with intercalations of Triassic limestone (in the western and southern parts of the region), black shale containing ammonites with intercalations of the Jurassic conglomerate, limestone belong to Cretaceous and Quaternary sediments as shown in the southern part of Fig. 3. The main parts of the study area are covered with alluvial and flood plain deposits (Qm) and also south of Qm is mainly covered by old river plain deposits (QM,rg).

Fig. 3
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Geological map of Isfahan (Geological survey and mineral exploration of Iran 2013)

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2.2 Hydrogeology of the aquifer

The Isfahan Plain hydrogeology is affected by Zayandehrud River, which flows in the southern part of the city. The issue of subsidence is directly influenced by the change in the hydrological regime of the aquifer. Considering the location of the Zayandehrud River, the depth of the beddings and the thickness of the sediments from the logs of exploratory boreholes, the study area is divided into two parts in the north and south of the Zayandehrud River.

2.3 South of Zayandehrud River hydrogeology

Based on an investigation of exploratory boreholes near the Azadi Square to Islamabad (southern part of Isfahan), an unconfined aquifer is formed by alluvium deposits and weathered shales.

The aquifer around Azadi Square has the lowest thickness and its thickness gradually increases toward the west. The main recharging sources of this aquifer in southern parts of Isfahan are annual precipitation and flow leakage from irrigation of green spaces in the south of Isfahan. The groundwater flows from the south toward the north in this part.

2.4 North of Zayandehrud River hydrogeology

In the north of Isfahan, the aquifer consists of two fine-grain layers separated by a coarse one. Generally, the groundwater table in many places is situated on the boundary between the upper fine-grained layer and the middle coarse layer, and the main direction of groundwater flow is toward the north and north-east. The fluctuations of groundwater table in the north aquifer are mainly dependent on recharge resources such as seasonal rainfall, amount of infiltration and withdrawing rate of the operating wells. In the last decade, the irregularity of water flowage into the Zayandehrud River was the main source of the aquifer’s recharge and discharge in the north part of the city.

The exploitation of aquifer resources in Isfahan is carried out by several wells. There is no Qanat (ancient tunnels to transport water from highlands lo lowlands) and springs, and the number of exploited wells in Isfahan is 2587; their locations are shown in Fig. 4. The depth of the wells is between 7 m in the south and to more than 270 m in the northern area. According to the water level readings from the wells, the average water level in the Isfahan aquifer has been steadily decreased from 1999 to 2015. By using Theissen method the oscillation long-term hydrograph has been drawn (Fig. 5).

Fig. 4
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Location of wells in the study area

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Fig. 5
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Long-term hydrograph of the aquifer in the study area

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Based on water well inventory and assessment in the study area, the main discharge of the Isfahan aquifer is mostly attributed to agricultural and green space consumptions, which is about 70.4 MCM, with an average decline in groundwater head of about 0.88 m annually (Behab Sad Asia 2015).

3 Materials and methods

By identifying the physical, mechanical and chemical characteristics of soil strata, the required data was obtained from geophysical and geotechnical investigations.

The depth of the exploratory boreholes varies from 7 to 42 m, often drilled with the continuous sampling method. The soil samples were collected from the boreholes depending on the required parameters, as well as recording the water table. The undisturbed and disturbed samples were obtained from various depths of soil strata according to ASTM D (1586-98), Cline (1944), and Tan (2005), and then tests are conducted. The Locations of exploratory boreholes in the study area are shown in Fig. 6. The geotechnical tests including sieve analysis (ASTM D2487), soil Description (ASTM D2488-09a), soil thickness, Atterberg limits, plasticity index, void ratio, and consolidation parameters were conducted. Also, the geological map of the study area has been modified based on field studies and existence geotechnical data (Fig. 7). Further, other parameters are obtained based on data obtained from geophysics and geotechnical tests. Finally, the subsidence was calculated at the site of the boreholes from 2000 to 2014 and was expanded using the best interpolation method for the entire study area in ArcGIS software. Different types of interpolation models including global polynomial interpolation (GPI), inverse distance weighting (IDW), local polynomial interpolation (LPI), radial basis functions (RBF), and geostatistical methods (kriging) are conducted and the best interpolation models with the lowest root mean square error (RMSE) and mean absolute error (MAE) are selected.

Fig. 6
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Distribution map of exploratory boreholes drilled in the study area

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Fig. 7
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The modified geological map of the study area

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To calculate the RMSE and MAE of the interpolation models, one of the data point is deleted and its value is estimated by other data points. This is done for all data points and at the end of the RMSE and MAE is calculated for the interpolation model. The RMSE and MAE were calculated from Eqs. (1), and (2), respectively. These equations are used by some researcher for evaluation of prediction models (e.g. Khanlari et al. 2016; Ajalloeian and Kamani 2017).

$${\text{RMSE}} = \sqrt {\frac{1}{\text{n}}\mathop \sum \limits_{{{\text{i}} = 1}}^{\text{n}} \left( {{\text{Y}}_{\text{i}} - {\text{X}}_{\text{i}} } \right)^{2} }$$
(1)
$${\text{MAE}} = \frac{{\mathop \sum \nolimits_{{{\text{i}} = 1}}^{\text{n}} \left| {{\text{Y}}_{\text{i}} - {\text{X}}_{\text{i}} } \right|}}{\text{n}}$$
(2)

where Xi is the predicted value of subsidence by interpolation models, Yi is the measured value of subsidence by geotechnical data, and n is data numbers.

3.1 The depth of bedrock based on geophysical studies

Based on geophysical results, the thickness of sediments in the southeastern areas is less than 5 m; in the vicinity of the Sio-Se-Pol Bridge it is about 40 m, and in the southwestern part of the study area is more than 120 m. From the northwest towards the central part, the thickness of sediments increases to about 280 m, but towards the east part, it decreases to about 220 m (Fig. 8).

Fig. 8
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The plan of geo-electrical profiles in Isfahan

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3.2 Fluctuations in groundwater head

According to the piezometers data obtained from the observation wells and boreholes, the groundwater fluctuations in Isfahan Aquifer over a 14-year period (2000–2014) indicate that the groundwater head in the aquifer is dependent on the seasonal rainfall, withdrawing volume and recharging from Zayanderud River in water flow periods. By using the geotechnical and hydrogeological data obtained from exploration and observation wells, the information layers in ArcGIS software (Shape-file) were derived. The amount of subsidence was calculated considering the groundwater oscillations, thickness and characteristics of the sedimentary deposits in the study area, which has experienced a decline in groundwater head from 2000 to 2014.

3.3 Consolidation test

In this study the one-dimensional consolidation tests were conducted according to ASTM D 2535-04, BS 1998: 5, ASTMD 4546, and AASHTO T216 Standard methods for determining soil consolidation properties.

Subsidence as a result of soil consolidation is a time-dependent process that relies on the hydraulic conductivity of soil, thickness and drainage conditions. As a result of low conductivity clay, this process sometimes takes which contribute to consolidation settlement in the soil. In fact, when the vertical stress increases in a fine-grained soil, initial pore water pressure is equal to the increment in normal stress. Over time, the effective stress increases due to decrement in pore-pressure, and soil subsidence occurs.

The subsidence amount due to the initial consolidation in the soil can be calculated based on the assumption of a one-dimensional consolidation in fine-grained layers. Due to the rise in the thickness of the aquifer in Isfahan Plain during sedimentation along the time and the decreasing trend in groundwater level in recent years, the fine-grained deposits are normally consolidated. Therefore, the summation of the subsidence at any location on the ground surface can be calculated by Eq. (3).

$${\text{S}}_{1} = \frac{{{\text{C}}_{\text{c}} {\text{H}}}}{{1 + {\text{e}}_{^\circ } }}{ \log }\left( {\frac{{{\text{P}}_{^\circ } +\Delta {\text{P}}}}{{{\text{P}}_{^\circ } }}} \right)$$
(3)

where; S1 is the amount of settlement in a normal consolidated fine-grained layer, Cc is the compression index, H is the thickness of clay layer, \({\text{P}}_{^\circ }\) is the initial pore water pressure, and \({\text{e}}_{^\circ }\) is the initial void ratio.

Cc is an important factor for calculating the soil settlement which is determined by the consolidation test directly from the gradient of the void ratio (e) regarding the effective stress logarithm (log σ) (Eq. 4).

$${\text{c}}_{\text{c}} = \frac{{{\text{e}}_{1} - {\text{e}}_{2} }}{{{{\log\upsigma }}_{2}^{\prime } - {{\log\upsigma }}_{1}^{\prime } }} = \frac{{\Delta {\text{e}}}}{{{ \log }\left( {\frac{{{{\upsigma }}_{2}^{\prime } }}{{{{\upsigma }}_{1}^{\prime } }}} \right)}}$$
(4)

Regarding the borehole logs, the thickness of fine-grained soils has been calculated. Also, Cc was directly obtained from the laboratory consolidation tests in any sedimentary layer. In layers without consolidation parameters, Cc values have been estimated by backward elimination. Various experimental relationships have been presented for estimating Cc values from simple laboratory parameters (Nagavaj and Murty 1985; Slamet and Abdelazim 2012).

Based on the statistical analysis, among the parameters such as liquid limit (LL), water content (W  %), Plasticity Index (PI), specific gravity (Gs) and void ratio (e), LL and Gs have a significant effect on Cc values in study area (P value is less than 0.05). As is shown in Table 1, there is a direct relationship between Cc and the variables LL and Gs.

Table 1 Results of the statistical analysis of backward elimination
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Moreover, the values of R Square and Adjusted R Square were obtained for the fitted model is 0.55 and 0.54, respectively, which show fairly suitable results in accordance with the data. Therefore, the final equation used in this study to estimate the Cc values from available laboratory results is presented by Eq. 5.

$${\text{C}}_{\text{c}} = 0.00198 {\text{LL}} + 0.02413 {\text{G}}_{\text{s}} - 0.5562$$
(5)

3.4 Aquifer physical load factor

Permanent subsidence depends on factors such as the thickness of compressible fine-grained sediments, inelastic skeletal specific-storage (Sskv), and water level decrement and therefore the increase of the effective stress (Hung et al. 2010). The Sskv is inversely related to the effective stress (Leake and Galloway 2007). Different methods have been developed to assign and estimate Sskv such as using the results of pumping test, stress–strain analyses based on extensometer data, the results of the recursive analysis of simulated models and consolidation test (Mishra 2010; Hoffmann 2003; Riley 1969; Jorgensen 1980). In this study, distribution of the thickness of fine-grained compressible sediments in an inelastic state for Isfahan aquifer has been investigated.

Sskv is a portion of the special storage of an unconfined aquifer, caused by the irreversible deformation of the aquifer system skeleton (Eq. 6) (Joseph et al. 1984).

$${\text{S}}_{\text{skv}} = {{\frac{\Delta b}{\Delta h}} \mathord{\left/ {\vphantom {{\frac{\Delta b}{\Delta h}} b}} \right. \kern-\nulldelimiterspace} b}$$
(6)

In which b is the thickness of the aquifer layer, Δb is the change in the thickness of the layer (in compression value) and Δh is the change in the applied stress.

The non-elastic skeletal specific storage (Sskv), can be calculated using Eq. 7.

$${\text{S}}_{\text{skv}} = \frac{{0.434{\text{C}}_{\text{c}} {{\upgamma }}_{\text{w}} }}{{\left( {1 + {\text{e}}0} \right){{\upsigma }}^{\prime } }}$$
(7)

\({{\upsigma }}^{\prime }\), Cc, γw and \({\text{e}}_{^\circ }\) are the effective stress value, the compression index, the unit weight of water and the initial void ratio of soil, respectively. The skeletal specific storage and tension have an inverse relationship. In deep sediments, \({{\upsigma }}^{\prime }\) is very large, and the decrement in the pore water pressure (u) after pumping of groundwater caused negligible changes in \({{\upsigma }}^{\prime }\). However, \({{\upsigma }}^{\prime }\) is relatively low for shallow sediments and even a meager change in u caused significant \({{\upsigma }}^{\prime }\) changes.

The irreversible deformations for coarse-grain materials are negligible compared to the amount of fine-grains ones. For unconsolidated clays or silty sediments, the Sskv value may be about 10–100 times larger than elastic skeletal specific-storage (Sske) (Rily 1998), and the consequent elastic subsidence can be ignored compared with the plastic one. Therefore, the effect of time on reaching the ultimate limit of elastic deformation usually lasts several days, but it may take several years for non-elastic deformations regarding the thickness of the fine-grained layers (Hoffmann 2003). Therefore, the specific storage capacity of the compacted sediments is the key factor which determines the aquifer’s compressibility.

4 Results and discussion

In order to find out the geotechnical characteristics, the thickness of the sediments layers and their lateral changes were determined from the results of exploration in different parts of the city. In this research, the required geotechnical data have been collected from 266 boreholes drilled to study various construction projects including buildings, bridges, subways, etc. Moreover, groundwater level records were collected from more than 1000 observation wells.

Available geotechnical data including borehole locations, thickness and soil classification, atterberg limits, void ratio, unit weight of soil, groundwater heads, etc., were introduced as information layers in the Arc-GIS software. Using Eq. (3), subsidence values resulting from the consolidation of fine-grained deposits proportional to the level of groundwater fluctuations during 2000-2014 for each region were determined and presented in the Arc-GIS software.

4.1 The thickness of the aquifer layers

Based on field observations and geotechnical data, the bedrock of Isfahan city is Jurassic Shale; outcrops of this shale unit are visible in the east, southeast, south and west parts of the city. Based on geophysical studies carried out by a geo-electric method in 2015, the depth of the bedrock in Isfahan varies from at least 5 m in the south to more than 250 m in the central and northern parts of the city (Behab Sad Asia 2015).

According to the Bedrock topography (Fig. 9), in the southern and western parts of the study area, the depths of the bedrock gradually decreases, and in some cases, even the outcrops of rock are visible on the ground surface.

Fig. 9
figure 9

Bedrock topography map of Isfahan plain

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Geotechnical Investigations indicate that the thickness of the topsoil layer varies from 0.4 to 2.5 m in the study area. This layer, especially in the north and center of the city, has covered a thick fine-grained layer consisted of clay and marl with intercalation of sand or gravel. Under the fine-grained layer, there is a layer of relatively coarse deposits including gravel, sand, and silt. These upper layers have covered a fine-grained layer. The characteristics of aquifer materials due to the limited depth of the boreholes have not been correctly identified yet.

In the south of Isfahan plain, north of Soffeh Mountain, and southern parts of Zayandehrud River, the main portion of the aquifer consists of coarse-grained layers. Obviously, due to the nature and genesis of the sediments in the rivers, the sediment deposits of Isfahan floodplain has an inter-fingering structure, and in order to draw a complete 3D model of such a heterogeneous deposition, a systematic exploration with a proper pattern of close exploratory boreholes is required.

Figure 10 shows the distribution and thickness of fine-grained sediments (CL, CH, ML, MH, CL-ML) based on available data up to a maximum depth of 42 m in Isfahan aquifer.

Fig. 10
figure 10

The diversity and thickness of fine-grained sediments

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4.2 The decline in groundwater head

According to the available hydrological data and the statistical records of groundwater head from 2000 to 2014, the highest decline occurred in the western parts of Isfahan (Fig. 11). Moreover, the general trend of lowering the groundwater table was observed from the southeast toward the northwest, decreasing from 4 to 36 m, respectively.

Fig. 11
figure 11

The decline in groundwater head in Isfahan aquifer from 2000 to 2014

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4.3 Subsidence measurement

The subsidence was calculated at the site of the boreholes and it was presented in Table 2. Also, this data was used for interpolation for the other locations of the study area which have not any geotechnical data.

Table 2 The statistical parameters of data
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In Fig. 12 the value of RMSE for each interpolation models is shown. Three of the best interpolation models are IDW with power 2 (IDW2), RBF with 4 sectors (RBF-4SEC) and universal kriging (KR-UNI) which it’s RMSE equal to 8.84, 8.88 and 8.96, respectively. For three of the best interpolation models, more studies are conducted using MAE and cross-validation curve. The MAE of IDW2, RBF-4SEC, and KR-UNI is equaled to 5.53, 5.82 and 5.96, respectively. Then, according to Fig. 13, RMSE and MAE, the best interpolation models are IDW2 which have the lowest value of RMSE and MAE. Therefore, the map of subsidence pattern was created using the IDW2. Finally, the pattern of subsidence in Isfahan was created by the best interpolation model and presented in Fig. 13.

Fig. 12
figure 12

The value of RMSE for each interpolation models

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Fig. 13
figure 13figure 13

The cross-validation curve for a IDW2, b RBF-4SEC and c KR-UNI

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The maximum subsidence due to the groundwater withdrawal was observed in the fine-grained sediments in the west and northwest parts of Isfahan are about 75 cm over the period of 15 years (2000–2014) (Fig. 14a). The dominant sediments in these two areas are CH and CL. In recent years, there are some reports about the cracking of residential buildings in Khaneh-Isfahan and Ashegh-Abad districts, located in the northwest of Isfahan metropolitan area (Fig. 14b).

Fig. 14: a
figure 14

Interpolation map of subsidence occurrence in Isfahan; b cracking in roof and wall of a house in Ashegh-Abad; c cracking in Golestan street area

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The subsidence map (Fig. 14a) was evaluated using field studies. The results showed that this map correlates with the reality of the region. For example, on the Ashegh-Abad and Golestan Street area, there is evidence of subsidence in buildings as shown in Fig. 14b, c, respectively. Also, no evidence of subsidence has been seen on the Jei and Atashgah areas. In the area of Ashegh-Abad, land subsidence currently estimated to be about 20 cm, while in the Jei area, land subsidence is not recognizable. In the west of Jei, the abnormal anomaly is seen that it is due to the type of soil (OSS unit) caused more expected settlement. In Fig. 14a, the land subsidence in Ashegh-Abad area is about 70 cm while the land subsidence in the region of Jei is less than 5 cm over the period of 15 years. In Atashgah, the subsidence is about 30 cm by the model although no evidence of the subsidence has been seen in the field study. It is maybe due to the time-consuming process that the visible subsidence will occur in the future or lack of accuracy of the model in this location.

Due to the discordant dropping of groundwater head, the heterogeneity of the texture, the thickness, and characteristics of the aquifer materials, asymmetric subsidence has occurred. For investigating the effects of existing operational wells on the subsidence propagation, the location of wells has been plotted on subsidence map (Fig. 15).

Fig. 15
figure 15

Adaptation of subsided areas regarding the density of wells

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Although there are so many operational wells at the south-central part of the city, the subsidence rate is very low. This condition could be due to the presence of high permeable coarse-grained deposits.

However, despite the relatively low density of operational wells, the maximum occurrence of settlement in the west and north of the study area is due to the low hydraulic conductivity of materials, the high thickness of fine-grained sediments and high withdrawal of groundwater.

4.4 Determining the subsidence rate

The Sskv parameter was calculated based on subsidence values and the effective stress variations at the aquifer using the relationship provided by Leake and Prudic 1991 (Eq. 8).

$$\Delta {\text{b}}_{\text{inelastic}} = \frac{{{\text{S}}_{\text{skv}} \cdot {\text{b}}_{0} }}{{ \Delta _{\text{w}} }} \cdot\Delta {{\upsigma }}^{\prime }$$
(8)

which, bo is the thickness of the fine-grained layer.

In this study, the Sskv value was calculated in the fine-grained layers of each borehole in 2000 and 2014, separately. The results are depicted in the years 2000 and 2014 in the study area (Figs. 16 and 17).

Fig. 16
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Interpolation map of the Sskv values of Isfahan city in 2000

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Fig. 17
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Interpolation map of the Sskv values of Isfahan city in 2014

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Considering the Sskv influence on effective stress, the irregular distribution of compacted sediments at various points leads to different Sskv values. Therefore, the rate of settlement varies in different locations. Over time, the decreasing level of groundwater and the consequent increment in the effective stress of compactable sediments diminish Sskv coefficient. Therefore, the value of Sskv is time-dependent, and, as shown in Figs. 16 and 17, the calculated Sskv values in 2014 compared to those in 2000 have decreased.

5 Conclusion

The purpose of this study was to investigate the land subsidence phenomenon in the Isfahan metropolitan which is an important historic and tourist city of Iran. This study was conducted based on in situ and laboratory studies, and the data from 266 exploratory boreholes with different depths from 7 to 42 m, and oscillations records of the water level from more than 1000 observation wells were used.

In layers with no data of consolidation parameters, Cc values have been estimated by backward elimination. Based on the regression results, among LL, PI, W %, Gs and e parameters, LL and Gs have a significant effect on Cc values in the study area. Subsidence values resulting from the consolidation of fine-grained deposits according to the level of groundwater withdrawal during 2000–2014 for each region were determined. Also, the best interpolation model is selected for evaluation of land subsidence pattern in Isfahan.

Maximum subsidence in the fine-grained sediments due to the groundwater withdrawal in the west and northwest parts of Isfahan is about 75 cm over the 15 years period. Based on the unified classification system, CH and CL are two main soil categories in this region. However, despite the relatively low density of operational wells, the maximum occurrence of settlement in the west and north of the study area was observed due to the low hydraulic-conductivity of materials, the high thickness of fine-grained sediment and enormous water extraction.

According to the results, the subsidence is different in each region of the study area and is estimated to be from 0 to 70 cm and the average subsidence in Isfahan is about 9 cm. The results indicate that the hazard of subsidence in Isfahan is very serious and also evidence of subsidence has been observed in recent years. In the vicinity of some ancient monuments and old buildings, the level of subsidence hazard is remarkable.

The Sskv value was calculated in the fine-grained layers. Considering the Sskv influence on effective stress, the irregular distribution of compacted sediments at various points leads to different Sskv values. Therefore, the rate of settlement varies in different locations. Decreasing the level of groundwater and the consequent increment in the effective stress of compactable sediments diminish Sskv coefficient over time.

Regarding the calculation of Sskv in 2000 and 2014, progressive subsidence due to groundwater withdrawal in northwest, center, and southeast of Isfahan Metropolis is expected.