Introduction

Grotto relics in China are widely distributed with the unique artistic features and historical value. As a global tangible cultural heritage, Yungang Grottoes and Dazu Rock Carvings have been inscribed into the World Heritage List by the United Nations Educational, Scientific and Cultural Organization (UNESCO) [1,2,3]. However, in the past few decades, the Grotto cultural relics have experienced different degrees of damage due to the long-term adverse effects of natural forces, human influences and environmental erosion, among which water seepage and weathering were responsible for the terrible deterioration and greatly threaten to the conservation of grotto relics in China [4,5,6]. Affected by complex geological structures and hydrological conditions, they have been a serious challenge. Previous studies indicated that water is one of the main causes of weathering to the grottoes and the phenomena such as swelling, disintegration and softening occurred after the water penetrated and interacted with the rock, resulting in the reduction of stability. The various forms of occurrence and special action of slight seepage produced a cyclical effect to the Grotto rock mass, and turned to be a dominant factor, accelerating the deterioration and weathering destruction of rock mass [7,8,9,10,11]. The existence of fracture network usually functions as an underground water flow path, thus posing a serious threat to safety of grotto relics. Therefore, the formation and mechanism of water seepage of fractured rock in grotto relics is a key issue to accurately describe the erosion and weathering process.

Fluid flow through macro-scale channels, i.e., faults, joints, fracture and so on, is crucial for evaluating the stability of geotechnical engineering and the mechanical behaviors in rock mass because they are recognized as the main channel for water migration. The finite element (FE) method and finite difference (FD) method were commonly used methods in solving complex problems in seepage mechanics and engineering, evolving from the seepage path, seepage process to hydraulic fracturing and coupling implementation [12,13,14,15,16]. The FE and FD method proved successful in evaluating water seepage from macro-scale, but there is little relevant research associated with the water seepage of fractured rock in grotto relics. Furthermore, the flow process among micro-channels associated with H2O molecules migration was ignored.

Fluid flow through micro-scale channels (small pores), driven by external fields, is critical to many fields of application, i.e., membrane separation, drug delivery, sensors, fuel cells, gas separation, desalination and so on [17,18,19,20,21,22]. Molecular dynamics (MD) simulation has been proved to be an efficient technique to simulate microscopic motions of molecules based on the known physics and the interactions between the molecules [23,24,25]. Using molecular dynamics approach, it was able to tackle the question of the interstitial fluid flow at micro-scale and describe properly the confined transport of H2O molecules through the channels. Pressure difference from either an atomic pressure or hydrostatic pressure gradient, is generated in MD simulation by applying an external constant force on each atom of the liquid flow and has been used to derive liquid transport in many materials, such as carbon nanotubes, graphenes, zeolites, meta-organic frames, polymers and others [26,27,28]. These reports reveal that the pressure difference, pore shape, pore size, temperature, hydrogen-bond (H-bond) and orientation play a vital role in diffusions and ordering of H2O molecules. However, the structure and dynamics of confined water from bulk phase going through the seepage channel of fractured rock in grotto relics obtained from MD simulations has never been reported. The distribution, orientation and motion trajectories of H2O molecules passing through the seepage channel are still unclear.

Because of the particularity of cultural relics, many non-destructive and in-situ methods are restricted, except for field observations [3]. Numerical modeling is the first choice for quantifying water seepage in grotto relics. In this study, we will be confining ourselves to perform FE and MD simulation targeting at visualizing the water seepage through macro-channel and micro-channel related to the direct infiltration of natural rainfall, in which an atomistic configuration of parallel feldspar-quartz platelets mimicking skeleton micro-channel was derived and then the Lennard–Jones(LJ) potentials and Einstein relations was employ, thus to provide a better understanding of the driving processes of internal water transport in grotto relics and a theoretical basis for the conservation of the World Cultural Heritage.

Geological background and simulated conditions

Geological background

Yungang Grottoes (113°20‵E, 40°04‵N) is located at Datong city of Shanxi Province, China, as displayed in Fig. 1. The grottoes are known as one of the largest grotto groups in China and the world-famous stone carving art, including 45 major caves, 252 shrines and approximately 51,000 sculptures. There is an average of 423 mm rain per year with rainfall distribution mostly in July, August and September. The annual average evaporation is 1748 mm, with the maximum of 801 mm in June.

Fig. 1
figure 1

The geological location of Yungang Grottoes in China

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The whole grottoes are divided into three parts: the east (1st–4th Cave), the middle (5th–20th Cave) and the west (21th–53th Cave). The 4th Cave with typical features of seepage hazards (Fig. 2a–f) was chosen as example of seepage channel survey where the blue box shows the actual observed water seepage and the bottom of the grottoes is 10 m higher than the groundwater level, completely in the aeration zone. Precipitation is the only source of replenishment for groundwater, which continues seeping via fractures until arriving in groundwater. Tensile joints were the most developed joints in the east of Yungang Grottoes, appearing in groups, with dip direction nearly erect and extension direction nearly east–west, roughly parallel to the faults. A total of 97 joints inside the 4th Cave were measured and the stereographic diagram for the joint patterns analysis was displayed in Fig. 2g, using upper hemisphere projection, as evidence from the rose diagram for the joint patterns analysis in Fig. 2h. The pole of each facet was shown by a triangle and the average plane was drawn with a circle line. Two groups of the joints in the 4th Cave were obvious: one group was 208°∠68° with direction of S62°E and the other is 265°∠75°with direction of N5°W。

Fig. 2
figure 2

a–f Representation of fractured rock mass and the water seepage phenomenon inside the 4th Cave; c the overview of the 4th Cave; g stereographic diagram for the joint trend analysis using upper hemisphere projection; h Rose diagram for joint pattern analysis

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Measurement of chemical composition

The grotto rock powders were prepared and measured using a X-ray diffraction detector (XRD, D8 Advance) with a scanning range from 3 to 70°. The XRD patterns of the grotto rock powders were analyzed in Fig. 3a for identifying the chemical composition existed in the grotto rocks. It is characteristic of obvious peaks at 20°, 26°, 36°, 39°, 50° and 60°, which could indexed to the standard lattice parameters of quartz. The typical diffraction peaks of feldspar were displayed at 28°, 35° and 42°. The peaks at 8°, 12°, 19° and 24° indicated the existence of biotite and kaolinite. It is mainly composed of 54.39% quartz and 31.20% feldspar according to semi-quantitative calculation as illustrated in Fig. 3b and as a result, feldspar-quartz slab was generated for the current computational studies.

Fig. 3
figure 3

a The XRD patterns b chemical composition of grotto rock

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Computational methods and models

Finite element computational details

Various studies have shown that the critical Reynolds number (Re) for a flow between a Darcy and a non-Darcy flow is 10 and the flow satisfies Darcy`s law when Re ≤ 10 [29]. Re is defined as (Eq. 1) [30]

$$ {\text{Re}} = \frac{\rho Q}{{\mu b}} $$
(1)

where ρ is the density of water, Q is the flow rate, μ is the viscosity of water, b is the aperture of the fracture. In this paper, ρ is 1.0 × 103 kg m−3, μ is 1.005 mPa s [30], Q is 1.0 × 10–5–2.0 × 10−6 m3/s [31] and b is 0.001 m. So, the Re equals to 1.99–9.95 and it was governed by Darcy`s law.

The macro-scale FE model was conducted using commercial available ANSYS software as depicted in Fig. 4. The mesh was generated for all the volume entities with 10-node tetrahedral element to better mesh quality and the converging rate. As a consequence, the FE model was discretized using approximately 8078 tetrahedral elements and 17,139 nodes. The rainfall infiltration was considered in the boundary conditions and the rainfall intensity on the top surface was set as 100 mm d−1. The bottom surface was impermeable boundary. The rainfall infiltrated from the top surface to the underground and it was governed by Darcy’s law (Eq. 2) [32,33,34]:

$$ v = - k\frac{\partial h}{{\partial l}} $$
(2)
Fig. 4
figure 4

The FE model from various perspectives: a overview; b, c section view; d top view

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where v is the seepage rate, k is the permeability h is the total head,l is the length of seepage path.

The flow rate Q can be calculated according to Eq. 3 [32,33,34]:

$$ Q = kw\frac{{h_{1} - h_{2} }}{l} $$
(3)

where w is the cross-section area.

The general governing differential equation for three-dimensional seepage can be expressed mathematically as following (Eq. 4) [35]:

$$ \frac{\partial }{\partial x}\left( {k_{x} \frac{\partial h}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {k_{x} \frac{\partial h}{{\partial y}}} \right) + \frac{\partial }{\partial z}\left( {k_{x} \frac{\partial h}{{\partial z}}} \right) = \gamma_{w} m_{w} \frac{\partial h}{{\partial t}} $$
(4)

where kx, ky and kz are the permeability in the X-, Y- and Z- direction, γw is the specific weight of water and mw is storage curve slope.

MD computational details

The external pressure was applied to drive the fluid flow. All MD simulations are performed by Accerlery Materials Studio software. Figure 5 depicts the initial configuration of parallel feldspar-quartz slabs mimicking skeleton nanochannel with the inter-space (d) of 2.0 nm. A simple point charge-extended (SPC/E) model was applied for H2O molecules because of its excellent description for bulky water. 737 H2O molecules with density equal to = 1.0 g/cm−3 was randomly filled into the left reservoir with dimension of 2.00 × 2.51 × 4.40 nm3. The model consists of a 3 × 2 supercell of quartz slab and a 8 × 5 supercell of feldspar slab, for which position constraints were employed. One-layer Graphene sheet with size of 2.46 × 5.04 nm2 is assigned to motion group and acts as moveable wall (piston) for creating the driven force toward H2O molecules in the left reservoir and the right reservoir (2.00 × 2.51 × 4.40 nm3) was kept empty. Geometry optimization of 5,000 iterations was achieved for energy minimization. Following this, a MD simulation via the isothermal-isometric(NVT) ensemble with the Nose thermostat is performed using Drieding force field at 298 K with a timestep of 1 fs. The intermolecular potential energy includes the LJ 6–12 type potential (EvdW) and the Coulomb potential (Eele) to describe van der Waals and electrostatic interactions, respectively. Their mathematical expressions are displayed in Eq. 5 [36].

$$ U(r) = E_{vdW} + E_{ele} = 4\varepsilon \left[ {\left( {\frac{\sigma }{{r_{ij} }}} \right)^{12} - \left( {\frac{\sigma }{{r_{ij} }}} \right)^{6} } \right] + \frac{{e^{2} }}{{4\pi \varepsilon_{0} }}\sum\limits_{i \ne j} {\frac{{q_{i} q_{j} }}{{r_{ij} }}} $$
(5)
Fig. 5
figure 5

a The schematic diagram of pressure-driven fluid flow in seepage channel and b the specification of the system employed in our study to analyze the transport properties of H2O molecules through seepage channel of fractured rock in grotto relics

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The LJ parameters for the O atoms are σ = 0.354 nm and ε = 0.152 kcal/mol[36].

Results and discussions

The main source of the seepage water in the grottoes was the rainfall infiltration and seepage recharge through joint fissures or joint fractures, mainly depending on the rainfall intensity and permeability [37]. The seepage field calculated from FE method was displayed in the form of color-coded contours of the water head and flow rate distribution in response to the rainfall intensity of 100 mm d−1 (Fig. 6). Joints with a direction of S62°E were recognized to be an effective channel for water seepage. In the fracture zone, water infiltrates quickly from the top surface into the joint fracture with a high value of water heads, in accordance with the actual seepage point in Fig. 2a–f, where the flow rate was ranging from 1.67 × 10–5 to 4.67 × 10–8 m3 s−1. The direct infiltration of natural rainfall acted as the driving force of water migration, providing a framework to construct an atomic model of the pore flow in the subsequent section.

Fig. 6
figure 6

The seepage field calculated from FE method under rainfall intensity of 100 mm d−1: a water head (m) contours of the 4th cave; b the flow rate (m3·s−1) contours of the 4th cave; c water head (m) contours of joint fissures

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The flow direction of water seepage is generally from the low potential area to the high potential area. H2O molecules suffered from a large energy barrier and they should overcome the large occupancy fluctuations to occupy the empty vacancies, which comes from larger energy barrier with strong interactions between water and channel. Figure 7a displayed the time course of energy variation evolving potential energy and its components during NVT simulation. It is found that they showed stable energy levels, a relatively strong electrostatic interaction (attractive) and a relatively weak van der Waal (vdW) interaction (repulsive), leading to a high potential energy barrier. Figure 7b illustrated the variation in the number of H2O molecules entering into seepage channel versus time. The water occupancy increased monotonously with simulation time. During the whole simulation time, there were 270 H2O molecules passing through channel from the initial position along the flow direction, from which the water flux and mass flow rate can be calculated [26, 38], equal to 270 ns−1 and 8.10 × 10–12 g s−1, respectively. The magnitude of mass flow rate was close to the mass flow rate of graphene-based nanochannel with the pore width of 0.7–1 nm [28].

Fig. 7
figure 7

a The time course of energy variation during NVT simulation and b the number of H2O molecules transported through the seepage channel depicted as a function of simulation time

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The flow behavior of H2O molecules in seepage channel involved three steps: (1) permeate into the seepage channel; (2) moving along the channel; (3) flow out of the channel. In order to probe the transport phenomena in detail, three stages (stage I, stage II and stage III) were defined as clarification of H2O molecules in the entrance region, the center region and the exit region, respectively. The typical feature at various stages was demonstrated in Fig. 8a, giving a dynamic process of water transport from the entrance region to the exit region. We divided the simulation box into many slabs along X-axis direction for statistical analysis. Hence, the relative concentration is given by the ratio of concentration in the slab to its average concentration across the entire system, as demonstrated in Fig. 8b. Significant discrepancies emerge at the number and magnitude of the peak values existing in the relative concentration distribution curve. Specifically, at stage I, H2O molecules are uniformly dispersed in the left reservoir with uniform distribution as verified by the multi-peaks. Part of H2O molecules entered into the channel because the relative concentration in the range of 2–4 nm changed from 0 to 2.0, as can be seen at stage II. When the whole channel was filled by H2O molecules, the relative concentration among the channel fluctuated around 1.0, indicating a uniform distribution at stage III.

Fig. 8
figure 8

a The cross-sectional snapshots of H2O molecules configurations through the seepage channel from the left reservoir at various stages and b the related concentration profiles along X -axis direction

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These findings are consistent with the morphologies shown in Fig. 8a.

The orientation distribution of O–H bonds in H2O molecules in a specified stage was also obtained to gain insight into ordering of the confined water, as plotted in Fig. 9. The orientation was defined as the angle between the flow direction (X-axis) and O–H bonds in H2O molecules. We have also included the corresponding orientation distribution at each stage for comparison. As can be seen, for each stage, there was a broader angular distribution and no ordered configuration, suggesting that the orientation behavior of confined H2O molecules responded to pressure-driven flow field without any disruption. The various distributions of may be derived from the fluctuation of H2O molecules and random thermal motion [39]. It was unfavorable for forming H-bonds among confined H2O molecules. A detailed discussion on the H-bond inspection is presented in the following section.

Fig. 9
figure 9

The orientation distribution of H2O molecules in XZ plane at the various stages: a comparison of stage I and stage II; comparison of stage I and stage III

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The aforementioned orientation behavior may be relevant to the H-bonds formation. When the H2O molecules orient themselves parallel to the flow direction, it is in favor of the formation of H-bonds among the confined H2O molecule. For a more detailed discussion, the number and structure of H-bonds for confined H2O molecules were characterized, as shown in Fig. 10. When the distance between the O atom of one H2O molecule, and the H atom of another H2O molecule was less than 3.0 Å, one can assume that a H-bond was formed [18, 40]. Having a precise look at Fig. 10, it has experienced a nearly decreasing trend in the number of H-bonds with ignorance of some fluctuations. The H2O molecules should break some of H-bonds to enter the narrow channel due to steric crowding and large energy barrier, leaving a reduction in the number of H-bonds. This finding agrees well with the results of orientation distribution O–H bonds in H2O molecules. In addition, it is further interpreted as the strong repulse interaction between H2O molecules and feldspar because some reduction has been occurred may be attributed to the disappearance of the H-bonds between H2O molecules and feldspar surface (see Fig. 10c).

Fig. 10
figure 10

a The number of H-bonds depicted as a function of simulation time and b, c the cross-sectional snapshots at stage II and stage III

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Besides analyzing the energy variation, water flux, relative concentration profiles, orientation distribution and H-bonds, further analysis is here focused on describing motion trajectories for elucidating the flow mechanism, by depicting the Cartier coordinates (x, y, z) of O atoms in H2O molecules located at different position along Z -axis direction. Figure 11a gives their initial positions of On (n = 1, 2, 3). The moving trajectories of O1, O2 and O3 in X-, Y- and Z-axis directions were extracted and exhibited in Fig. 11b–d. What is worth mentioning is that the tagged O atoms experienced a zigzag movement instead of linear motion as expected, roughly exhibited the same target direction, moving towards the exit.

Fig. 11
figure 11

a The On (n = 1, 2, 3) diagram in H2O molecules located at different position along Z-axis direction and b–d the moving trajectories of O1, O2 and O3

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The seepage characteristics in grotto relics experienced a complex evolution process and three types can be summarized as water seepage propagation behavior in grotto relics (see Fig. 12): for type 1, the pores interconnected with other pores to form micro-channels and water infiltrates through micro-channels of fracture zone or the small original cracks in the upper area, where the flow rate is low. As a result of washout of atmospheric rainfall, water flows through fracture with a relatively high flow rate (type 2). With the expanding and connecting of fracture or cracks, water seepage maybe turned into a kind of analogous pipe flow, resulting in water seepage hazards (type 3).

Fig. 12
figure 12

Three types of water seepage in grotto relics

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Conclusions

In this study, a new attempt was made to employ FE and MD simulations for elucidating the water infiltration and seepage characteristics through the macro-channel and micro-channel of fractured rock in grotto relics in consideration of the rainfall infiltration. The feldspar-quartz slab was generated for MD simulation because it is mainly composed of 54.39% quartz and 31.20% feldspar in grotto rock, as verified by XRD patterns. The water occupancy increased monotonously with simulation time and there were 270 H2O molecules passing through channel from the initial position along the flow direction, from which the water flux and mass flow rate were calculated. It is found that the broader angular distribution and disordered configuration existed in confined H2O molecules was unfavorable for forming H-bonds among confined H2O molecules, which was verified by the H-bond analysis. The motion trajectories of H2O molecules along Z-direction were investigated and they showed a zigzag movement instead of linear motion as expected. Three types can be summarized as water seepage propagation behavior in grotto relics: for type 1, the pores interconnected with other pores to form micro-channels and water infiltrates through micro-channels in the upper area with a low flow rate; for type 2, water flows through fracture with a relatively high flow rate; for type 3, it turned into a kind of analogous pipe flow as the expanding and connecting of fracture or cracks, resulting in water seepage hazard.