Simulation of Land Subsidence Caused by Coal Mining at the Lupeni Mining Exploitation Using COMSOL Multiphysics

Abstract

Because of its specific nature, mining activity causes numerous negative impacts on the environment, both during the exploitation phase and after it has ended. An important source of income in the Jiu Valley is represented by the Lupeni Mining Exploitation. Like any mining activity, coal exploitation causes various negative effects on the environment. The subsidence phenomenon represents a significant issue associated with coal mining in the Jiu Valley. Underground extraction of mineral deposits induces displacement of the overburden strata. Such displacements result in ground subsidence and modifications of the surface topography. The larger the voids created following the exploitation of useful mineral deposits, the more they affect the surface of the land above the exploitation through sinking, displacement, deformation, and even cracks. Secondary deformations refer to post-mining surface movements induced by delayed rock mass adjustment, manifesting as ground collapse, localized subsoil failure, or uplift driven by groundwater rebound after drainage cessation. In this paper, we aim to study the subsidence phenomenon produced by coal mining at the Lupeni Mining Exploitation using the COMSOL simulation software and applying the Barcelona Basic Model (BBM) and Modified Cam-Clay (MCC) models. Following the simulation, the behavior of the rocks could be observed in order to improve prediction accuracy to support sustainable land management in post-mining areas.

1. Introduction

Because of its inherent characteristics, mining causes numerous environmental harms during operation and after closure [1,2,3,4]. The adverse impacts generated by underground coal mining operations may include the following [5,6,7,8,9]: modifications of landforms, extensive land occupation [10], soil and land degradation, disturbances in groundwater hydrodynamics [11], detrimental effects on atmospheric quality as well as local flora and fauna [12], and the emission of noise, vibrations, and radiation [13].

Subsidence is the phenomenon of slow or sudden sinking of land, which can have many causes, both natural and anthropogenic [14]. The main causes of subsidence are as follows: soil compaction [15,16,17,18], dissolution of underground rocks [19], tectonic activity [20], melting ice [21], Karst phenomena [22], extraction of underground resources [14], extraction of groundwater [23,24,25], mining [14], construction and urban development [15], desiccation and drainage [23], excessive irrigation [15,25], erosion and waterlogging [26], or climate change [21,27].

In recent times, the phenomenon of soil settlement and subsidence has caused serious problems worldwide. These two phenomena are found worldwide in underground coal mining [28,29,30] and are directly influenced by the soil type, hydrological conditions and anthropogenic activities specific to the area [31,32].

The subsidence phenomenon is also influenced by the amount of soil in the roof of the galleries.

The subsidence phenomenon is determined both by the amount and type of soil and by the level of exploitation by modifying the hydrostatic regime of the area [33,34,35,36].

Subsidence can have serious effects on infrastructure, causing cracks in buildings, roads, bridges and pipelines, but can also lead to ecological problems, such as changing river courses or increasing the risk of flooding. It can also affect the geological stability of the region, increasing the risk of landslides or other geomorphological phenomena [25,27,31,37,38,39,40].

Underground extraction of mineral resources induces displacement and deformation within the roof strata, which subsequently propagates upward, compromising the stability and integrity of the surface terrain. The larger the voids created from mining useful mineral deposits, the more they affect the surface of the land above the exploitation through subsidence, displacement, deformation, and even cracks [38,41,42].

Surface displacement is the result of the redistribution of stresses in the rock mass, under the influence of underground excavations created by mining activities, or as an effect of the drying up of aquifer formations [33,41].

When underground excavations, particularly longwall cuttings, exceed critical dimensions relative to the geomechanical stability of the surrounding rock mass, and adequate support or void-filling measures are not implemented, the overburden strata undergo progressive collapse. This process generates a complex suite of geomechanical phenomena, collectively referred to as subsidence effects, which may propagate through the entire overburden sequence and manifest at the ground surface [43,44].

The extent of surface degradation depends on the dimensions of the mined void and the depth of extraction. Deposit thickness, dip, and mining method significantly influence rock mass displacement. Stress redistribution, rock geomechanical properties, and deposit tectonics also play a key role. Mining duration and post-extraction void management further affect subsidence and surface impacts [45,46,47,48,49,50,51].

The formation of sinkholes (cavities or depressions in the ground surface) is a common problem in underground mining areas, such as the area of Lupeni Municipality. The creation of sinkholes leads to the cracking and collapse of buildings, as well as the cracking of the soil on the surface, modifying the hydraulic networks of the soil, which can lead to the formation of water accumulations. In extreme cases, infiltrations through the gaps produced can flood mining galleries, endangering miners. Sinkholes can cause alteration of local ecosystems by modifying the soil and vegetation.

The Lupeni Mining Branch is strategically located in the western part of the Jiu Valley, falling under the administrative jurisdiction of Lupeni Municipality. This mining operation stands out as one of the key coal production sites in the region, playing a significant role in both the local economy and the broader energy sector. With a long-standing history of mining activity, the Lupeni mine contributes substantially to regional employment and supports numerous ancillary industries, while also being closely linked to the infrastructure and development of the surrounding communities.

The coal deposits in the Jiu Valley are part of the geological structure of the Jiu Valley Carboniferous Basin, which belongs to the Petrosani Depression. The formation of these deposits dates back to the Upper Carboniferous period (approximately 300–320 million years ago) and is associated with the Vilcan Coal Group, characterized by sequences of coal seams interbedded with sedimentary rocks.

The basin exhibits a complex anticlinal structure, with multiple folds and faults that influence the dip and continuity of the coal seams. Average seam inclinations range between 25° and 35°, with some sectors exhibiting dips up to 50–60°.

The exploited coal is predominantly bituminous, with a fixed carbon content ranging from 60 to 75%. The thickness of the coal seams varies between 0.5 m and 3–4 m, occasionally exceeding 5 m in the main exploitation areas. Coal layers are interbedded with clays, shales, sandstones, and siltstones, sometimes containing mineral inclusions such as sulfides or ferruginous concretions, which affect mining methods and gallery stability.

From a hydrogeological perspective, the basin contains groundwater associated with permeable layers and fractures, requiring efficient drainage systems in mining operations.

The structural complexity and steep dip of the seams necessitate underground mining methods with extensive and branched galleries, where seam pressure and hydrogeological conditions require the implementation of advanced support and drainage techniques [52,53,54,55,56,57,58].

In the Lupeni mining area, as a result of mining, a continuous subsidence can be observed [59,60] (Figure 1).

The main causes of sinkholes are the deformation of the overlying rocks (shale and clay) that gradually give way, the lack of infrastructure necessary for the maintenance of mining galleries that leads to their collapse after coal mining due to lack of adequate support.

Sinkholes can appear several months or even years after the actual mining, depending on the geological stability. In cases of unstable soil and shallow mining, sinkholes can appear within a few weeks to several months. If the ground is more stable or support techniques are applied (filling with slag, mud, sand), the formation of sinkholes can be delayed and can last several years.

Once the collapse begins, the process can last from several days to several weeks, until the depressions reach their final shape. Sometimes, sinkholes evolve gradually, with successive partial collapses, over a period of months or even years.

2. Materials and Methods

Surface movements resulting from underground coal mining, occurring post-extraction, are termed secondary deformations [61,62]. These deformations generally arise from delayed rock mass readjustment, causing collapse or localized subsoil failure. Superficial structures may experience discontinuous surface manifestations, such as cracks, fissures, and sinkholes. Soil uplift can also occur due to rock mass response to rising groundwater levels following the cessation of mine water drainage [63].

Surface deformations on mined land result from the destabilization of the natural rock mass caused by underground coal extraction. Displacement of rock strata to fill the voids generated by exploitation induces stress redistribution within the rock mass. Depending on the mining depth, surface subsidence develops, initially forming gentle depressions that may evolve into cracks and sinkholes. Post-extraction, these processes gradually diminish over time. Secondary deformations, occurring after mining cessation, are typically linked to groundwater level recovery, causing swelling and reduction in normal stresses in shallow rock layers, or to the collapse of abandoned underground workings [64]. Post-mining studies [65,66,67] indicate that surface anomalies can persist for many years after mine closure, posing potential long-term hazards [68,69].

Romania’s largest coal reserve lies beneath the Jiu Valley Carboniferous Basin. The basin sits at the extreme south of Hunedoara County, within the Petrosani intermontane depression. It stretches for roughly 42 km, with a width that varies from 3 to 7 km [52,70,71].

The coal deposit originated from sedimentary processes, with predominant lithologies in the basin including limestone, marl, clay, clayey sandstone, and conglomerate.

These rocks exhibit compressive strengths ranging from 15 to 16 MPa, occasionally exceeding 50–60 MPa, yet generally display low geomechanical stability and a high susceptibility to collapse [53,54].

Induced subsidence causes damage to industrial and civil buildings as well as communication networks, leading to the drying up of wells, the emergence of new springs and the accumulation of rainwater in the affected areas [23]. In the mining areas of the Lupeni Mining Exploitation, considerable areas of land have been removed from the economic circuit, no longer being suitable for the location of buildings or for agricultural crops, because underground mining also causes the loosening of rocks from the roof of the mining works, which mainly causes the infiltration of surface water and the lowering of the hydrostatic level of the groundwater level. Also, the process of draining mine water leads to a decrease in the hydrostatic level of the soil and the appearance of unsaturated soil surfaces [23,72,73,74,75].

Unsaturated soils show hydromechanical responses, including phenomena such as subsidence and heave, represents a critical consideration in numerous geotechnical and engineering applications [76,77,78].

Soil settlement and heave are strongly controlled by the degree of saturation, matric suction, and the applied loading conditions.

Because long-duration experiments are seldom feasible, numerical models are widely used to study multiphysics coupling [76,79].

In order to study the subsidence phenomenon caused by the Lupeni Mine Exploitation, we will perform a simulation using the Barcelona Basic Model (BBM) and the Modified Cam-Clay model (MCC) implementation with COMSOL. This software was selected due to the research team’s prior experience employing it to address similar engineering problems [80,81] and its versatility as a multiphysics application with built-in robust geoengineering solvers.

By using the BBM and MCC methods, a characterization and simulation of the mechanical behavior of the rock mass and the overburden affected by coal mining was performed. The simulation is performed by integrating the effects of saturation and non-saturation on strength and deformability. Thus, BBM allows the analysis of deformation and settlement processes under non-saturation conditions, influenced by suction variations and wetting–drying cycles, while MCC provides a framework for the study of behavior under conditions of complete saturation, with emphasis on the evolution of plasticity and reaching the critical state. By using the two models in a complementary manner, the study aims to explain the mechanisms of land subsidence in the Lupeni area, to evaluate the impact of mining on the surface and infrastructure and to contribute to the development of safer and more sustainable mining strategies.

2.1. Barcelona Basic Model (BBM)

In geotechnical practice, the Barcelona Basic Model (BBM) is widely used to describe the elastoplastic behavior of unsaturated soils. Developed by Alonso and co-authors, it extends the Modified Cam-Clay (MCC) framework [81].

Its foundations lie in plasticity theory, critical-state soil mechanics, and the Modified Cam-Clay formulation [76].

The Modified Cam-Clay model is used to describe the behavior of saturated clays and allows us to understand the fundamental mechanisms that govern their response to different stress states.

The Modified Cam-Clay (MCC) model was initially formulated for triaxial loading conditions. Experimental observations on soft clays provided the basis for the constitutive relationship describing the evolution of the void ratio e (volumetric strain) as a function of the logarithm of the effective mean stress ${\mathsf{\sigma }}_m^{eff}$, as illustrated in Figure 2. The relationship between the two plots is expressed as follows:

$${\lambda }^{*}={\displaystyle \frac{\lambda }{1+e}}$$

(1)

$$k^{*}={\displaystyle \frac{k}{1+e}}$$

(2)

where k—slope of swelling line;

λ—Slope of NCL (normal consolidation line);

e—Current void ratio.

Figure 2. Constitutive model of material behavior in isotropic consolidation.

Figure 2. Constitutive model of material behavior in isotropic consolidation.

The graph depicts a Normal Consolidation Line (NCL) and a series of swelling lines. During first loading, virgin soil travels along of the NCL. If the soil is consolidated up to a specific stress level, denoted as the preconsolidation pressure $p_c$, and then unloaded along the current swelling line, subsequent reloading initially follows the swelling line until the stress state corresponding to the pre-unloading condition is reached $p_c$. At this point, the soil resumes compression along the NCL, representing primary loading.

Parameters k and λ can be calculated from the expressions given hereunder:

$$\lambda ={\displaystyle \frac{C_c}{2.3}}$$

(3)

$$k=1.3{\displaystyle \frac{1-v_c}{1+v}}{C}_s$$

(4)

where $C_c$ is the one-dimensional compression index.

These parameters are obtained from a standard oedometric test.

The yield surface is smooth, and tensile states are not allowed. Unlike early formulations, the MCC model explicitly represents strain hardening and softening in both normally consolidated and over consolidated soils, with a nonlinear link between volumetric strain and effective mean stress under ideal-plastic bounds. In shear, the material can deform plastically without immediate failure (Figure 3): points 1 and 2 illustrate hardening and softening paths up to the critical state (points 3 and 2, respectively). Beyond that state, shear straining proceeds with constant void ratio e and constant effective mean stress ${\mathsf{\sigma }}_m^{eff}$ under ideal plasticity. Upon unloading, linear response is taken as a linear elastic.

Evolution of the yield surface (hardening/softening) is driven by the current preconsolidation pressure $p_c$:

$$p_c^{i+l}=p_c^i\exp \left[{\displaystyle \frac{-∆{\epsilon }_v^{pl}}{{\lambda }^{*}-k^{*}}}\right]$$

(5)

where $p_c^{i+l}$—current preconsolidation pressure;

-

$∆{\epsilon }_v^{pl}$ increment of volumetric plastic strain.

In addition to parameters κ and λ, the self-weight and Poisson’s ratio, the MCC model requires specifying the following three parameters:

-

$M_{cs}$ slope of the critical state line;

-

OCR—overconsolidation ratio;

-

$e_0$ initial void ratio.

The slope of the critical state line can be determined from the expression:

$$M_{cs}^{+{30}^{\circ }}({\phi }_{cv})={\displaystyle \frac{2\sqrt{3}\sin {\phi }_{cv}}{3-\sin {\phi }_{cv}}}, \mathrm{for} \mathrm{triaxial} \mathrm{compression}$$

(6)

$$M_{cs}^{-{30}^{\circ }}({\phi }_{cv})={\displaystyle \frac{2\sqrt{3}\sin {\phi }_{cv}}{3-\sin {\phi }_{cv}}}, \mathrm{for} \mathrm{triaxial} \mathrm{extension}$$

(7)

where ${\phi }_{cv}$ denotes the constant-volume friction angle—the friction angle at the critical state [82,83].

The BBM is well suited to represent the mechanical response of unsaturated geomaterials, including expansive clays, low-plasticity sands, among others [76,84,85].

With BBM in COMSOL, the effects of unsaturation, suction and moisture variations on the strength, deformability and stability of soils can be studied, extending the classical concepts in MCC for real situations in which the soil is not completely saturated.

Treating suction as a state variable, BBM has been used to examine its influence on elastoplastic behavior; it also characterizes changes in shear strength and consolidation associated with hygroscopic expansion, collapsibility, and suction variations in unsatu-rated soils and rocks [76,86].

Like other elastoplastic constitutive laws, the BBM splits deformation into elastic and plastic components. It uses three stress variable (p,q,s)(p,q,s) and the yield surfaces (LC, SI, CLS)(LC, SI, CLS) to partition the stress space into elastic and plastic domains [77].

Figure 4 presents the three-dimensional representation of the yield surface in the thermo-elasto-plastic BBM within both the p−q−sp−q−s and p′−q−Tp′−q−T stress spaces, where p′p′ denotes net mean stress (total stress minus gas-phase pressure), q represents deviatoric (shear) stress, s is matric suction, and T corresponds to temperature [76,87].

The BBM is already implemented in some simulation software.

2.2. Equations of the BBM

In the BBM, the stress state is written in terms of net stress [76]:

$${\mathsf{\sigma }}^{,}=\mathsf{\sigma }-u_{a}I$$

(8)

where: σ = total tension,

ua = pore air pressure.

In this framework, the invariants are the mean (volumetric) stress p and the deviatoric stress q are defined as [76]:

$$p={\displaystyle \frac{1}{3}}tr({\mathsf{\sigma }}^{\prime }), q=\sqrt{{\displaystyle \frac{3}{2}}}\Vert \xi \Vert$$

(9)

where $\xi ={\mathsf{\sigma }}^{\prime }-1/3tr({\mathsf{\sigma }}^{\prime })I$.

The elastic strain components—volumetric and shear—are expressed as [75]:

$$d{\epsilon }_v^e={\displaystyle \frac{k}{v}}{\displaystyle \frac{dp}{p}}+{\displaystyle \frac{k_s}{v}}{\displaystyle \frac{ds}{(s-p_{a}t)}}$$

(10)

$$d{\epsilon }_s^e={\displaystyle \frac{dq}{3G}}$$

(11)

Notably, the BBM employs two yield surfaces of the form [77]:

$$f_1(p,q,s,p_0^{*})\equiv q^2-M^2(p+p_s)(p_0-p)=0$$

(12)

and

$$f_2(s,s_0)=s-s_0=0$$

(13)

where

$$p_s=k_{s}s$$

(14)

$${\displaystyle \frac{p_0}{p^c}}={\left({\displaystyle \frac{p_0^{*}}{p^c}}\right)}^{\left[\lambda (0)-k\right]/\left[\lambda (s)-k\right]}$$

(15)

$$\lambda (s)=\lambda (0)\left[(1-r)\exp (-{\beta }_s)+r\right]$$

(16)

In the BBM, the yield surface is controlled by the hardening parameters $p_0^{*}$ and $s_0$, which vary with the total plastic volumetric strain increment $d{\epsilon }_v^p$, in the form of [77]:

$${\displaystyle \frac{d{p}_0^{*}}{p_0^{*}}}={\displaystyle \frac{v}{\lambda (0)-k}}d{\epsilon }_v^p$$

(17)

$${\displaystyle \frac{d{s}_0}{s_0+p_{at}}}={\displaystyle \frac{v}{{\lambda }_s-k_s}}d{\epsilon }_v^p$$

(18)

2.3. Equations of Associative Flow Rule

The evolution of plastic strain follows the associative flow rule and can be written as [76]:

$${\dot{\epsilon }}_{vm}^p=\varphi {\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }{\mathsf{\sigma }}^{\prime }}}$$

(19)

where $\varphi$ is a scalar factor. By substituting Equation (9) into Equation (19), we obtain [76]:

$$∆{\epsilon }_v^p=∆\varphi tr\left({\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }{\mathsf{\sigma }}^{\prime }}}\right)=∆\varphi {\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }p}}$$

(20)

The term of $\mathsf{\partial }{f}_1/\mathsf{\partial }{\mathsf{\sigma }}^{\prime }$ can be obtained using the chain rule, following the formulation of Borja and Kavazanjian [63]

$${\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }{\mathsf{\sigma }}^{\prime }}}={\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }p}}{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }{\mathsf{\sigma }}^{\prime }}}+{\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }q}}{\displaystyle \frac{\mathsf{\partial }q}{\mathsf{\partial }{\mathsf{\sigma }}^{\prime }}}={\displaystyle \frac{1}{3}}\left({\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }p}}\right)I+\sqrt{{\displaystyle \frac{3}{2}}}\left({\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }q}}\right)\widehat{n}$$

(21)

where $\widehat{n}=\xi /‖\xi ‖$. The derivates of $f_1$ with respect to $p,q,p_0$, and $p_s$ can be obtained from Equation (12) as

$${\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }p}}=M^2(2p+p_s-p_0)$$

(22)

$${\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }q}}=2q$$

(23)

$${\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }{p}_s}}=M^2(p-p_0)$$

(24)

$${\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }{p}_0}}=M^2(p+p_s)$$

(25)

2.4. Closest Point Projection of the Constitutive Relation

The stress return mapping tensor can be formally defined as [76]:

$${\mathsf{\sigma }}_{n+1}^k={\mathsf{\sigma }}_{n+1}^{tr}-{ℂ}^e:\delta {\epsilon }^p$$

(26)

where ${\mathsf{\sigma }}_{n+1}^{tr}$ is the trial stress for the volumetric and deviatoric components of ${\mathsf{\sigma }}_{n+1}^k$, which are expressed as follows [77]:

$$p\equiv p_{n-1}^k={\displaystyle \frac{1}{3}}tr({\mathsf{\sigma }}_{n+1}^k)=p_{n+1}^{tr}-K∆{\epsilon }_v^p$$

(27)

$$q\equiv q_{n+1}^k=\sqrt{{\displaystyle \frac{3}{2}}}‖{\xi }_{n+1}^k‖=\sqrt{{\displaystyle \frac{3}{2}}}‖{\xi }_{n+1}^{tr}-2\mu ∆{\gamma }^p‖$$

(28)

In the above, $K$ and $\mu$ denote the elastic moduli, which are defined as follows: [77]

$$K={\displaystyle \frac{1+e}{k}}p$$

(29)

$$\mu ={\displaystyle \frac{3K(1-2v)}{2(1+v)}}$$

(30)

Following to Borja and Lee [86], p and q can be computed as: [76]

$$p=p_{n+1}^{tr}-K∆\varphi {\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }p}}=p_{n+1}^{tr}-K∆\varphi M^2(2p+p_s-p_0)$$

(31)

$$q=q_{n+1}^{tr}-3\mu ∆\varphi {\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }q}}=q_{n+1}^{tr}-6\mu ∆\varphi q$$

(32)

The update of hardening parameters $p_0^{*}$ and $s_0$ follows the hardening laws in Equations (17) and (18). Integrating Equations (17) and (18) over a finite time step yields [76]:

$$\begin{array}{l}(p_0^{*}{)}_{n+1}={(p_0^{*})}_n\exp ({\vartheta }^{*}∆{\epsilon }_v^p)\\ ={(p_0^{*})}_n\exp ({\vartheta }^{*}∆\varphi {\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }p}})\\ ={(p_0^{*})}_n\exp \left\{{\vartheta }^{*}∆\varphi M^2\right.\left.(2p+p_s-p_0)\right\}\end{array}$$

(33)

$$\begin{array}{l}{(s_0)}_{n+1}+p_{at}={(s_0)}_n+p_{at})\exp ({\vartheta }^s∆{\epsilon }_v^p)\\ =({(s_0)}_n+p_{at})\exp ({\vartheta }^s∆\varphi {\displaystyle \frac{\mathsf{\partial }{f}_1}{\mathsf{\partial }p}})\\ =({(s_0)}_n+p_{at})\exp \left\{{\vartheta }^s∆\varphi M^2\right.\left.(2p+p_s-p_0)\right\}\end{array}$$

(34)

in which [75]

$${\vartheta }^{*}={\displaystyle \frac{v}{\lambda (0)-k}}$$

(35)

$${\vartheta }^s={\displaystyle \frac{v}{{\lambda }_s-k_s}}$$

(36)

$${\displaystyle \int (}\nabla q)wd\Omega =n{\displaystyle \int (\nabla qw)d\Omega -n{\displaystyle \int \nabla q\nabla wd\Omega }}$$

(37)

3. Results and Discussion

To analyze the subsidence caused by the exploitation of shallow coal deposits in the Lupeni area, we used COMSOL Multiphysics, a software known for its versatile multi-physics models and numerical simulations. It is based on the finite element method, solving individual partial differential equations or complex systems in multi-field scenarios [88]. To evaluate soil settlement and heave, both the MCC and extended BBM solvers were applied, with BBM being particularly suitable for unsaturated soils as it accounts for suction as a parameter.

3.1. Model Definition

For this study, a finite element model of a soil stratum measuring 8 m in width and 6 m in depth was constructed. A 0.8 m wide footing was applied atop the layer, imposing an incrementally increasing normal load. This block will be automatically mirrored by the defined symmetry axis (as defined in the solid mechanics module) to form the actual soil sample that render the results of the simulation.

The soil stratum is underlain by a rigid, perfectly rough base. A fixed boundary condition was imposed on the lower horizontal surface, while a roller boundary condition was applied to the left vertical surface, and a symmetry-type constraint was enforced on the right vertical boundary.

Gravitational loading was implemented via a Gravity node. Pore pressure within the saturated portion of the soil domain was applied using an External Stress node. An additional boundary load was imposed on the upper surface of the soil layer to model the weight of the footing.

Within the Richards Equation interface, a pressure head of 3 m was initially assigned to represent the groundwater level, corresponding to a phreatic surface located 3 m below the soil surface. This pressure head was subsequently increased to 8 m to simulate progressive wetting. The initial configuration establishes the saturated zone below the phreatic line and the unsaturated zone above. To investigate the hydro-mechanical effects of wetting, the groundwater level was raised to fully saturate the soil after the complete application of the footing load.

All the above are visible in Figure 5.

To carry out the simulation, we employed both the BBM and MCC solvers, incorporating the soil properties used in the analysis, as summarized in

Table 1. Properties for the unsaturated soil.

Name	Expression	Value	Description
para	0	0	Parameter
rhos	1800 [kg/m3]	1800 kg/m3	Soil density
rhow	1000 [kg/ m3]	1000 kg/m3	Water density
gammaw	rhow*g_const	9806.7 N/m3	Unit weight of water
muw	1 × 10−3 [Pa*s]	0.001 Pa·s	Water dynamic viscosity
Nu	0.2	0.2	Poisson’s ratio
lambda	0.21	0.21	Compression index
lambda_s	0.125	0.125	Compression index for changes in suction
kappa	0.006	0.006	Swelling index
kappa_s	0.008	0.008	Swelling index for changes in suction
Mb	1.26	1.26	Slope of critical state line
wb	0.4	0.4	Weight parameter
mb	55 [kPa]	55,000 kPa	Soil stiffness parameter
bb	11	11	Plastic potential parameter
sy	110 [kPa]	110,000 Pa	Initial yield value for suction
kb	0.65	0.65	Tension to suction ratio
pref	19 [kPa]	19,000 Pa	Reference pressure
pc0	85 [kPa]	85,000 Pa	Initial consolidation pressure
phi0	0.631	0.631	Initial porosity
e0	phi0/(1−phi0)	1.71	Initial void ratio
K_sat	1 [m/day]	1.1574 × 10−5 m/s	Saturated hydraulic conductivity
alpha	2 [1/m]	21/m	Fitting parameter
S_res	0.24	0.24	Residual degree of saturation
S_sat	1	1	Degree of saturation at full saturation

. The used values are specific to the Jiu Valley Mining Basin.

Also at this stage, the simulation variables such as suction, pore pressure, permeabilities, saturation have to be defined. For the current study, the chosen values are presented in

Table 2. Variables used in the simulation.

Name	Expression	Unit	Description
Suction	−p*(dl.Hp < 0)	Pa	Current suction
PorePressure	p*(dl.Hp ≥ 0)	Pa	Pore Pressure
k_rel	Se		Relative permeability
k	K_sat*muw/gammaw	m3	Soil permeability
Cm	phi0*(S_sat-S_res)*Se*alpha	1/m	Specific moisture capacity
Se	exp (alpha*dl.Hp)*(dl.Hp < 0) + 1*(dl.Hp ≥ 0)		Effective saturation

.

After the definition of the parameters in the “Global Definitions” module of the software, several other data must be defined. The Floating pressure is defined as function of time by points and is presented in Figure 6. The next is the Global water level, also defined as a function of time points, as presented in Figure 7. Finally, the initial suction profile is defined by specific points over time, as shown in Figure 8.

After defining the basic parameters in the “Global Definitions” module of the software, additional data must be introduced to describe the hydro-geotechnical conditions of the model. The first dataset is the Floating pressure, which is defined as a time-dependent function through a series of discrete points, as illustrated in Figure 6; this parameter reflects the evolution of subsurface pressure in relation to drainage or aquifer recharge processes. The next element is the Global water level, also defined as a point-based time function, as shown in Figure 7; this parameter governs the degree of saturation and directly influences the distribution of pore water pressures. Finally, the Initial suction profile must be defined by specific points along the vertical, associated with its variation over time, as illustrated in Figure 8; this profile represents the initial distribution of suction stresses in the unsaturated soil and serves as an essential initial condition for accurately simulating the hydro-mechanical behavior of the ground.

The final step of the model definition is the discretization of the model. We used the calibrate for general physics option with the predefined “finer” size chosen for the definition of the mesh. This generated the following element properties: maximum element size 0.296 m, minimum element size 0.001 m, maximum element growth rate of 1.25, and the curvature factor of 0.25. The resulting grid had 1424 triangles, 761 mesh vertices, 96 edge elements, and 6 vertex elements, as shown in Figure 9.

3.2. Results of the Simulation and Discussion

After running the simulation, a series of results were obtained; they are graphically presented and discussed in this section. The distribution of pore pressure at various groundwater levels within the soil domain is presented in Figure 10.

The values of pore pressure are a key indicator of the hydraulic state of the soil: positive values correspond to saturated zones, while negative values highlight unsaturated regions, where the pore water tension reflects suction effects. Identifying these unsaturated zones is critical, as they strongly influence the mechanical response of the soil mass.

For saturated soils, both the Modified Cam-Clay (MCC) and Barcelona Basic Model (BBM) solvers rely on the effective stress principle, which incorporates pore pressure into the calculation of the stress state to determine yielding and plastic deformation. This ensures that the influence of pore pressure on strength and deformability is consistently accounted for in fully saturated conditions. In contrast, when dealing with unsaturated soils, the BBM introduces an additional complexity by explicitly including matric suction as a state variable in both the yield surface and the plastic potential functions. This extension enables the model to realistically capture the stabilizing effect of suction, which enhances apparent cohesion and alters the stress–strain response under negative pore pressures.

The MCC model, however, does not incorporate suction as a variable, which inherently limits its capacity to simulate unsaturated soil behavior. While MCC remains robust and reliable for saturated soil conditions and critical state analyses, it cannot reproduce the effects of partial saturation, such as suction-induced strength or collapse upon wetting. Therefore, in scenarios where groundwater fluctuations and soil moisture variations are significant, the BBM provides a more realistic and physically consistent representation of soil behavior, whereas MCC may only offer an approximate solution.

Figure 11 compares the spatial distribution of volumetric plastic strain predicted by the two numerical models (solvers) at a groundwater level of 3 m. It can be observed that, when using the MCC (Modified Cam-Clay) model, the plastic zone beneath the footing is noticeably wider than with the BBM (Barcelona Basic Model). This difference reflects the typical behavior reported in previous studies, where the MCC model, calibrated for both saturated and unsaturated soils, tends to extend the plastic zone under similar loads, highlighting its sensitivity to effective stress and soil compressibility.

When the groundwater level rises, the BBM solution shows an interesting phenomenon: additional plasticization occurs, which can be explained by the loss of suction (capillary tension) in the unsaturated soil. This indicates that BBM is highly sensitive to changes in moisture state, capturing the effect of suction on soil strength and generating extra plastic deformation as the water table increases.

In contrast, Figure 12 illustrates the behavior of the MCC model under the same conditions. Unlike BBM, MCC does not show a significant increase in plastic strain. Only minor variations are observed, which are primarily related to changes in effective stress rather than suction loss. This suggests that MCC, being focused on the effective stress–strain relationship, is less sensitive to the effects of suction, with plasticization controlled mainly by changes in soil loading and effective stress.

In essence, the comparative analysis highlights two key points:

Extent of the plastic zone: MCC predicts a wider zone beneath the footing than BBM at the same water level.

Sensitivity to suction and water table changes: BBM responds significantly to rising groundwater with additional plastic strains, whereas MCC does not, indicating a fundamental difference in how the two models capture the influence of unsaturated conditions on soil strength.

The distributions of von Mises stress under fully saturated conditions, as shown in Figure 13 and Figure 14, reveal notable differences between the MCC and BBM constitutive models. Von Mises stress, which represents an equivalent measure of the combined deviatoric stresses in the soil, is commonly used to assess the onset of yielding and plastic flow. By comparing these distributions, it becomes apparent that the two models predict not only different magnitudes of stress but also distinct spatial patterns beneath the loaded foundation.

In the MCC model, the von Mises stress is largely governed by the applied loads and the soil’s effective stress–strain relationship. Because MCC does not explicitly account for suction, the stress distribution is relatively smooth and primarily reflects the redistribution of effective stress in the saturated soil. Peaks in von Mises stress correspond to regions of concentrated load transfer, and the overall pattern emphasizes the role of the stress path in controlling yielding.

By contrast, the BBM incorporates suction as a key constitutive parameter, even under conditions approaching full saturation. As a result, the von Mises stress distribution predicted by BBM differs, especially near the soil–foundation interface and in regions where local saturation changes occur. Suction contributes to the apparent strength of the soil and affects the stress state by providing additional resistance to deformation. Consequently, areas that might plastically yield in MCC may still exhibit higher von Mises stress in BBM due to the stabilizing effect of residual suction.

Taken together, these results highlight the critical role of suction in accurately modeling the mechanical behavior of partially saturated soils. Suction not only influences yielding and the size of plastic zones, as discussed earlier, but also affects the stress distribution within the soil mass. Neglecting suction, as in MCC, can underestimate the soil’s ability to resist deformation in partially saturated conditions, leading to potentially non-conservative predictions of settlements, bearing capacity, or lateral displacements.

In summary, Figure 13 and Figure 14 reinforce the idea that suction is not merely a secondary factor but a fundamental constitutive parameter. Accurately capturing the mechanical response of partially saturated soils requires models like BBM that integrate both effective stress and suction effects, ensuring reliable predictions under variable saturation conditions.

The relationship between footing pressure and settlement, presented in Figure 15, provides important insights into how the two constitutive models—MCC and BBM—predict soil response under loading. In this analysis, the groundwater level was initially set at 3 m, and the footing pressure was incrementally increased up to 130 kPa.

Within the elastic range, both MCC and BBM predict almost identical settlement behavior. This indicates that, for small stresses where the soil response remains elastic, both models have comparable stiffness characteristics. The elastic response is governed primarily by the soil’s Young’s modulus and Poisson’s ratio, and at this stage, the influence of suction (captured by BBM but not by MCC) is minimal because the strains are not large enough to trigger significant yielding.

Once the plastic range is reached, the two models begin to diverge. Plastic strains develop differently in each model due to the way they account for soil strength and the effects of suction. The MCC model, which does not consider suction, predicts larger settlements than BBM. This difference occurs because MCC underestimates the apparent strength contributed by suction in partially saturated soils. Essentially, the soil in MCC “yields” more readily under the same footing pressure, leading to greater vertical deformation.

On the other hand, the BBM incorporates suction as an additional parameter contributing to the soil’s apparent strength. Suction increases the soil’s resistance to deformation, effectively reducing settlements under the same load. This results in smaller predicted settlements compared to MCC, particularly in the early stages of plastic behavior. The difference between the two models highlights the critical role of suction in controlling settlement in unsaturated or partially saturated soils.

In conclusion, Figure 15 demonstrates that while both models capture elastic stiffness similarly, the inclusion of suction in BBM provides a more realistic prediction of settlements in the plastic range. This underscores the importance of selecting a constitutive model that accurately represents both effective stress and suction effects when assessing foundation performance on partially saturated soils.

As the groundwater level progressively rises to the soil surface, the two constitutive models exhibit markedly different responses, as illustrated in Figure 15. The MCC model predicts a decrease in footing displacement, whereas the BBM indicates a significant increase in settlement, reflecting the influence of suction-dependent hydro-mechanical behavior. In the MCC model, the rise in groundwater level reduces the effective stress, producing an upward movement of the soil surface—commonly referred to as heave—illustrated in Figure 16.

Under the conditions considered, both the footing and the surrounding soil experience heaving, which is an upward movement caused primarily by changes in moisture content or pore pressure redistribution. This phenomenon reflects the initial elastic or partially elastic response of the soil, as water infiltrates and alters the stress state. Heaving indicates that the soil expands in response to wetting, and the footing may slightly lift or move upward as the surrounding soil swells.

In contrast, the BBM predicts additional settlement of the footing when wetting occurs, which arises from the simultaneous reduction in suction and effective stress. As the soil becomes wetter, capillary suction decreases, reducing the apparent cohesion and strength of the unsaturated soil. At the same time, the redistribution of pore pressures reduces the effective stress in the soil skeleton. This combination leads to downward deformation of the footing, which is superimposed on the existing stress-induced settlements.

Figure 16 is particularly useful because it isolates the vertical displacement caused solely by wetting, separating it from settlement due to applied loads. By focusing on wetting-induced movements, the figure clearly shows the influence of suction loss and effective stress reduction on foundation performance.

The trends observed in Figure 15 and Figure 16 are consistent with findings reported in the relevant literature on unsaturated soils. In particular, for the BBM, the additional settlement under wetting conditions is often described as soil collapse. This phenomenon occurs because the loss of capillary cohesion reduces the soil’s ability to support applied loads, causing a sudden or accelerated downward movement. Soil collapse under wetting is a well-documented behavior in unsaturated clays and silts, where a decrease in suction—due to rainfall, rising groundwater, or irrigation—can trigger significant settlements even in previously stable soil.

In summary, the BBM effectively demonstrates the dual influence of wetting: initial heaving in elastic regions and additional settlement due to suction loss, consistent with soil collapse phenomena documented in experimental and field studies. This underscores the importance of incorporating unsaturated soil behavior in foundation design to predict both upward and downward movements accurately.

4. Conclusions

This study investigated the phenomenon of land subsidence induced by coal mining activities at the Lupeni Mining Exploitation, focusing on the mechanical response of unsaturated soils through numerical simulations in COMSOL, using its Modified Cam-Clay (MCC) and Barcelona Basic Model (BBM) model solvers. Based on the analyses and results, the following conclusions can be drawn:

• Subsidence Mechanisms—Underground coal extraction at Lupeni significantly alters the geomechanical equilibrium of the rock mass, leading to deformation, settlement, cracking, and in some cases heaving of the ground surface. The extent of these effects depends on the depth of mining, hydrogeological conditions, and geomechanical characteristics of the soil and overlying strata.
• Model Performance—The MCC model provides reasonable predictions for saturated soils but fails to capture suction effects that are essential in unsaturated conditions. In contrast, the BBM, which explicitly incorporates suction as a constitutive parameter, offers a more realistic representation of the mechanical behavior of unsaturated soils and better explains the collapse settlement observed when wetting occurs.
• Simulation Outcomes—The numerical results confirm that groundwater level variation plays a critical role in soil deformation. While the MCC model suggests reduced settlement or even heave under rising water levels, the BBM reveals an increased settlement due to suction loss, aligning more closely with field observations and the literature.
• Practical Implications—The study highlights the importance of using advanced constitutive models, such as BBM, for predicting the behavior of partially saturated soils in mining areas. These models are essential for assessing risks to infrastructure, land stability, and post-mining land use in the Jiu Valley region.
• Future Work—Further research should integrate long-term monitoring data with numerical simulations to refine model calibration, extend the analysis to deeper and more complex geological conditions, and evaluate mitigation strategies such as controlled flooding, backfilling, or reinforcement of surface infrastructure.

In conclusion, the results demonstrate that the subsidence phenomenon at the Lupeni Mining Exploitation is a complex process governed by both mining activities and hydro-mechanical soil interactions. Accurate modeling of unsaturated soil behavior is therefore crucial for sustainable mining practices, effective risk management, and the protection of local communities and ecosystems. The findings underline the importance of using advanced constitutive models to improve the accuracy of subsidence predictions and to support sustainable land management in post-mining areas. Also, a good prediction leads to the discovery of optimal solutions to ameliorate possible material damage and environmental degradation.