An Innovative Finite Tube Method for Coupling of Mine Ventilation Network and Gob Flow Field: Methodology and Application in Risk Analysis

Abstract

Explosions and fires originated from longwall gob due to the formation of methane-air mixture have been a severe threat to coal miner’s lives. Many numerical studies on coal mine fire and explosion hazards have focused on the airflow in roadways and mine gobs. However, most of these studies isolate the gob from its surrounding roadways, and the network analysis and the CFD method are applied independently to model the two classes of airflows. This approach greatly limits the ability of simulating mine ventilation flow, especially unable to consider the effects of gob boundary conditions on air exchange between the gob and the surrounding airways. An innovative finite tube method (FTM) is developed to couple the one-dimensional mine ventilation network (MVN) and the 2D/3D gob flow field (GFF). In FTM, GFF is discretized into a finite number of flow tubes each of which is formed by any two adjacent stream lines. These tubes, representing the gob’s field flow, connect the MVN into a new coupling network. To solve the coupling model between MVN and GFF, an iterative solution technique is developed in which the MVN analysis is used to evaluate the boundary pressures for GFF simulation and in turn the FTM feeds the GFF results back to the coupling network. Based on the FTM approach, a model for gas migration in gob has been established for delineating the hazard zones of explosive methane concentration and spontaneous combustion. A computer program is developed to implement the FTM simulation. An illustrative example with five flow tubes representing the GFF is created to verify the stability and convergence of the FTM solution process. A simulation example also indicates that the accuracy of FTM is improved by 12% compared with previous method. Results of an application case show that the program is capable of quantitatively evaluating the gob’s risk zones prone for spontaneous combustion and gas explosion as well as performing risk analysis for various ventilation scenarios.

Introduction

Mine gob, a large zone of broken rock rubble left by longwall coal mining, has long been a safety concern in the coal industry due to the air flowing through the gob area regardless of a bleeder or bleederless ventilation system being used. Firstly, methane emitted into the gob can reach an explosive level (~ 5 to ~ 15%) [1, 2]. Secondly, the air flowing through gob provides oxygen for coal oxidation, possibly leading to spontaneous combustion [3, 4]. Both problems could potentially cause mine fires and explosions—the most feared hazards in underground coal mines. Therefore, the knowledge of the airflow distribution and the location of the hazard zones in longwall gob are very important for preventing and controlling these hazards.

Unlike airflow in roadways that can be accurately measured, it is nearly impossible to observe and measure the airflows in a mine gob due to its inaccessibility. Many numerical studies have been performed to simulate gob air leakage using computational fluid dynamics (CFD) programs. Yuan and Smith [5] used ANSYS Fluent software to simulate and examine the risk of spontaneous coal combustion in a two-panel longwall district using a bleeder ventilation system. Krawczyk and Janus [6] applied a two-dimensional model to analyze the propagation of methane from the longwall gob. Xia et al. [7] developed a coupled 2D numerical model to study the controlling factors of symbiotic risks between coalbed methane and spontaneous combustion in longwall mining gobs using program COMSOL. Basically, most of the studies isolated the gob from mine ventilation network (MVN) with only predefined pressure boundary conditions. In fact, MVN is not static in a dynamic longwall mining process, and it consequently changes the boundary conditions and the gob flow field (GFF). As the working face advances, the changing gob permeability and airflow pattern inside the gob [7, 8] also affect the boundary condition. The predefined static boundary conditions used in the previous studies cannot truly reflect the time-dependent changes. The progression of the mining face and the changing ventilation schemes will result in changes of the risk zones in the gob in both locations and sizes that may require different prevention and control measures. Therefore, the mine ventilation system in terms of MVN should be incorporated in the CFD model to prescribe the transient boundaries. However, MVN is a one-dimensional model while the CFD is either a 2D or 3D model. It is very challenging to couple the 1D network flow model and the 2D/3D field flow model to make the mine ventilation simulation realistic.

Past studies have tried to simulate the gob ventilation by integrating the MVN and GFF into one model. Brunner [9] discretized the gob area into many vertical airways in order to integrate GFF into MVN. However, using such airways to represent the gob flow is much less accurate than the CFD method. Some latest studies have incorporated the network analysis theory with CFD modeling to solve the above problem. Wedding [10] incorporated the network model into commercial CFD program, and Wu [11] added the CFD model into the MVN program, which in some degree did create a simple integration. But the coupling relationship between GFF and MVN has not been well considered in the studies by Wedding and Wu.

Though many mathematical models have been developed to study airflows in gob, very little modeling effort has been made to fully consider the coupled boundary conditions between MVN and GFF. The flow mechanisms of air leaking in and out gob areas are not fully understood. Particularly, the difficulty exists in how to coordinate the two distinctive disciplines of the 1D network flow and the 2D/3D gob field flow in an integrated approach. The objective of this research is to fill these knowledge gaps using a proposed novel finite tube method (FTM). This new method can couple the MVN and GFF for the purpose of numerically simulating the airflows in both gob areas and surrounding roadways simultaneously. The development and validation of FTM are detailed in Sects. 2, 3, and 4. A methane migration model in the MVN and gob areas is carried out in Sect. 5 based on the FTM. A case study is presented in Sect. 6 to demonstrate the capability of FTM in analyzing the risks of spontaneous coal combustion and methane explosion in longwall gobs.

Models of Airflow in Gob Areas

The layout of a typical active longwall panel (Fig. 1) is used in this paper to describe the working scheme of the coupled MVN and GFF method. The U-shaped longwall face ventilation is commonly used in coal mines, often for preventing the potential spontaneous combustion problems. In a U-shaped ventilation system, it is intended that all the air from the intake entry flows only through the longwall face without leaking into the gob area behind the face, and leaves to the return entry. The gob in the U-shaped ventilation scheme is not intended to be ventilated; however, some air will inevitably leak into and out the gob from the boundary L1. The flow distribution of the leaked air in the gob area depends on the tightness of the caved materials, the pressure source at intake T-junction, and the pressure sink at the return T-junction. Spontaneous combustion could only occur in a zone where the heat produced by coal oxidation (limited by available oxygen) is more than that dispersed by ventilation air. Therefore, the zone of potential spontaneous combustion in the gob is bounded by the upper and lower air velocity limits located on and away from the longwall face sides, respectively. The wider is this zone (Ws), the higher is the risk for spontaneous combustion. It also takes a certain time (ts) for a coal oxidation process to develop into a spontaneous combustion event. Therefore, a fast mining rate (us) to ensure Ws/us < ts is the key to effectively prevent the spontaneous combustion in the longwall gob. In this study, longwall face will be represented by MVN branches and the gob be represented using GFF.

Fig. 1
figure1

Longwall face layout and gob element meshing

Distribution of Gob Hydraulic Conductivity

Airflow and gas migration in gob highly depend on the porosity of the caved rocks and coal debris left there. Research has shown that the porosity or hydraulic conductivity in gob generally meets the characteristic of the O-ring distribution [7, 8]. The distribution function of the rock mass bulking factor, permeability, caved zone height, porosity, and hydraulic conductivity at a particular point (x, y) in gob can be obtained using the following equations.

$$ {K}_P\left(x,y\right)={K}_{P,\min }+\left({K}_{P,\max }-{K}_{P,\min}\right)\cdotp \exp \left\{-{a}_1\left(x-{x}_0+ ut\right)\cdotp \left[1-\exp \left(\xi \cdotp {a}_0{d}_0\right)\right]\right\} $$
(1)
$$ k\left(x,y\right)=\frac{d^2{\left[{K}_P\left(x,y\right)-1\right]}^3}{150{K}_P\left(x,y\right)} $$
(2)
$$ H\left(x,y\right)=\frac{hK_p\left(x,y\right)}{K_p\left(x,y\right)-1} $$
(3)
$$ \varepsilon \left(x,y\right)=1-\frac{1}{K_p\left(x,y\right)} $$
(4)
$$ E\left(x,y\right)=k\left(x,y\right)\frac{g}{\upsilon } $$
(5)

In the equations, Kp refers to the ratio of the volume after and before rock broken; Kp, min and Kp, max are the compaction and initial bulking factors, respectively; a0 and a1 are the bulking factor attenuation ratios in the dip and strike directions of the face (m−1), respectively; u is the advance rate of the longwall face (m/day); t is the advance time (day); x0 is the distance from the panel’s setup room to Y coordinate axis (m); ξ is the geometric adjustment factor; the term of (x − x0 + ut) and d0 in Eq. (1) are, respectively, the distances from point (x, y) to the longwall face and the supporting coal pillar (m), where d0 = y − |y − 0.5 W|, W is the width of the gob (m); k is the permeability (m2); d is the average particle diameter of gob (m); H is the caved zone height (m); h is the thickness of the mining coal seam (m); ε is the porosity of gob; υ is the air kinematic coefficient of viscosity (m2/s); E is the hydraulic conductivity (m/s); g is the acceleration of gravity (m/s2).

Finite Element Method–Based Gob Flow Field Model

FEM, one of the important CFD methods [12], is used to solve the gob flow field in this modeling effort. In some area of the gob, the air velocity is very low that can be represented by linear seepage model while in areas with higher velocity, air flow is governed by nonlinear model. Therefore, a common flow model should be chosen to represent both the linear and nonlinear flow characteristics according to air velocity automatically.

Steady Linear Seepage Model

In general, airflow in most of the gob areas is low and can be assumed to obey the linear Darcy’s law:

$$ V=-E\nabla P $$
(6)

where V is the flow velocity (m/s) and P is the pressure (Pa). Considering that the air flow in the gob is steady and incompressible, the continuity equation can be obtained as Eq. (7).

$$ \nabla V=0 $$
(7)

Since the length and width of the longwall gob are significantly larger than its height, the problem can be regarded as a 2D plane model. Therefore, the Hamilton operator in Eqs. (6) and (7) is\( \nabla =i\frac{\partial }{\partial x}+j\frac{\partial }{\partial y} \). Substituting Eq. (6) into Eq. (7) results in Eq. (8)—the GFF model’s differential equation with the boundary conditions imposed to make the solution unique.

$$ \left\{\begin{array}{c}\frac{\partial }{\partial x}\left(E\frac{\partial P}{\partial x}\right)+\frac{\partial }{\partial y}\left(E\frac{\partial P}{\partial y}\right)=0\kern1.25em \left(x,y\right)\in G\\ {}P={p}_0\kern9em \left(\mathrm{on}\ {L}_1\right)\\ {}E\frac{\partial P}{\partial x}{n}_x+E\frac{\partial P}{\partial y}{n}_y=0\kern3em \left(\mathrm{on}\ {L}_2\right)\end{array}\right. $$
(8)

In Eq. (8), G is the domain of the gob area; p0 is the input pressures for this model; L2 is the perimeter of the gob without air leakage; nx and ny represent the x and y components of the outer unit normal vector of L2, respectively. When the gob area is discretized into triangular elements using the FEM approach shown in the insert graph of Fig. 1, the pressure function P(x, y) in the gob is represented by interpolating function P (x, y) for each element and the interpolant is given by

$$ {P}_{\Delta}\left(x,y\right)={N}_i{p}_i+{N}_j{p}_j+{N}_k{p}_k $$
(9)

where pi, pj, and pk are pressures of each triangular element’s nodes which are counterclockwisely numbered around the triangle; Ni, Nj, and Nk are shape functions which only have relations to the coordinates of each element and can be described as Ni = 0.5(ai + bix + ciy) / A, Nj = 0.5(aj + bjx + cjy) / A, Nk = 0.5(ak + bkx + cky)) / A, where A is the area of each triangular element; ai = xjyk− xkyj, bi = yj− yk, ci = xk− xj, aj = xkyi− xiyk, bj = yk− yi, cj = xi− xk, ak = xiyj− xjyi, bk = yi− yj, ck = xj− xi. Based on the discretization, the standard Galerkin finite element method [12] is used to discretize the differential equation of Eq. (8) into the algebraic equations shown in Eq. (10).

$$ {\sum}_{\Delta i}\frac{E_{\Delta}}{4{A}_{\Delta}}\left[\left({b}_i{b}_i+{c}_i{c}_i\right){p}_i+\left({b}_i{b}_j+{c}_i{c}_j\right){p}_j+\left({b}_i{b}_k+{c}_i{c}_k\right){p}_k\right]=0,i=1,2,\dots, S $$
(10)

In Eq. (10), ∑i is a summation operation of all elements which are connected to the node i; S is the counts of gob nodes not on the boundary L1, i.e., the unknown number of the equations. The pressure values of gob nodes on L1 also appear in Eq. (10) but as known values. Equation (10), a linear system of equations about pi (i = 1, 2,…, S), is the governing equations of the steady linear seepage model in gob. When all gob node pressures are solved from Eq. (10), the gob airflow velocities can be determined by substituting P (x, y) into Eq. (6).

Nonlinear Seepage Model

In some gob areas, particularly that adjacent to the working face, the air velocity could be fairly high, the pressure-velocity relationship is nonlinear and can be represented by Bachmat equation [13]:

$$ -E\nabla P=\left|1+\frac{\left|V\right|\beta d}{\varepsilon \upsilon}\right|V $$
(11)

In the equation, β is a dimensionless particle shape coefficient of medium. There are two terms on the right side of Eq. (11), linear (i.e., V) and nonlinear (i.e., the second part) terms related to V. In gob areas far away from the face, V is very low and the nonlinear term is almost zero, making Eq. (11) to be equivalent to Darcy’s law as shown in Eq. (6). Therefore, Eq. (11) is able to describe both linear and nonlinear seepage, which can better describe the flow characteristics of air anywhere in the gob area than Eq. (6). However, after replacing Eq. (6) with Eq. (11), the nodal pressures cannot be determined directly by Eqs. (8) and (10). When Eq. (11) is rewritten to Eqs. (12) and (13), an iterative solution method is used to solve the nonlinear seepage model [14].

$$ V=\hbox{-} E\hbox{'}\nabla P $$
(12)
$$ E\hbox{'}=E{\left|1+\frac{\beta d}{\varepsilon \upsilon}\left|V\right|\right|}^{-1} $$
(13)

In Eqs. (12) and (13), E′ can be treated as an apparent coefficient of hydraulic conductivity corrected by a velocity dependent factor. Factor E′ makes Eq. (12) in the same form as Eq. (6), meaning that Eq. (10) can also be used for Eq. (12) as E is replaced by E′. Therefore, it is easy to derive the iteration equations of Eqs. (14) and (15) from Eqs. (12) and (13).

$$ {V}^i=\hbox{-} {E}^i\nabla P $$
(14)
$$ {E_{\Delta}}^i={\left|1+\frac{\beta d}{\varepsilon \upsilon}\left|{V}_{\Delta}^{i-1}\right|\right|}^{-1}{E}_{\Delta}^0 $$
(15)

In Eqs. (14) and (15), the subscript means that the variable is for element. The term Ei is the ith corrected value from its original E0 calculated by Eq. (5). Equations (14) and (15) indicate that the nonlinear seepage flow in the gob can be treated as a Darcy flow but with a set of modified hydraulic conductivities. The computational flow chart to implement the proposed mathematical model is shown in Fig. 2. In the chart, all the operations about E0, Ei, and Vi are for each element; Pgi is a vector of the gob nodal pressures determined in the ith iteration by solving Eq. (10); D is the maximum absolute value of difference between Pgi and Pgi −1. The iteration continues until D is less than a preset tolerance, ε.

Fig. 2
figure2

Computational flow of solving nonlinear seepage-based linear seepage model

Model of Air Distribution in the MVN

A mine ventilation network, analogous to 1D electrical network, is simplified into a closed and directed graph where the longwall face is generally represented by one branch with high resistance without considering the longwall gob. In order to formulate a mine ventilation network into a set of solvable linear and nonlinear equations, the minimum resistance spinning tree algorithm is applied [15]. Figure 3 shows an example of simplified MVN for demonstrating mathematical formulation process. In a given network with M nodes and N branches, there are N − M + 1 chords and adding each of chords to the tree branches can result in one mesh also one nonlinear equation. Air flowing through a connected network satisfies Kirchhoff’s first and second laws, which are (1) the total quantity of air leaving a node must be equal to that entering this node and (2) the algebraic sum of the pressure changes through a mesh is zero. They can be expressed as Eqs. (16) and (17).

Fig. 3
figure3

A simplified mine ventilation network: arrow denotes the direction of airflow

$$ \sum \limits_{j=1}^N{b}_{ij}{q}_j=0,i=1,2,\dots, M-1 $$
(16)
$$ \sum \limits_{j=1}^N{c}_{ij}{R}_j\left|{q}_j\right|{q}_j-\sum \limits_{j=1}^n{c}_{ij}{h}_f\left({q}_j\right)=0\kern0.50em i=1,2,\dots, N-M+1 $$
(17)

In Eq. (16), qj represents the air quantity in branch j (m3/s); bij = 0 when branch j is not connect to node i, bij = 1 when air in branch j enters node i, bij = − 1 when air in branch j leaves node i. In Eq. (17), Rj is the resistance of the branch j (N s2/m8); cij = 0 if branch j is not in mesh i, cij = 1 when air in branch j flows in the same direction of mesh i or cij = − 1 when airflow is in the opposite direction of mesh i; hf (qj) is the characteristic curve function of the fan placed in branch j (Pa), such as branch 8 in Fig. 3. In an MVN analysis, the air quantities in the N branches are the unknowns to be solved. The total number of linear and nonlinear equations in Eqs. (16) and (17) is also N—ensuring that the MVN has a set of unique solution. Among the equations, Eq. (17) results in N – M + 1 nonlinear equations, Newton method is used to solve the equations iteratively [16, 17]. Once the air quantities in the branches are determined, the pressures at the network nodes can be calculated based on a reference zero pressure node.

Finite Tube Method for the Coupled Model of MVN and GFF

For implementing the finite tube method, the longwall face will be divided into a sufficient number of subbranches. The nodal pressures along the longwall face determined by the MVN solution are the input as boundary condition for simulating the GFF as described in Eq. (8).

Research Basis—Two Existing Integrated Models

Two integrated models of MVN and GFF from previous studies of Brunner [9] and Wu [11] are the important stepstones of this study. In order to understand these two models better, the same longwall face layout is used to interpret the princple of the two models. In both models, the longwall face is divided into subbranches numbered 3 to 6 and the gob flow field is two-dimensional, as shown in Fig. 4. But different discretization methods for gob areas are used in the two models.

Fig. 4
figure4

Two integrated models of MVN and GFF from previous studies. a Integrated network model based on leakage branch [9]. b Integrated but noncoupling model based on shared nodal pressures [11]: branch 1 representing the simplified part of the airway network; branches 2 and 7 representing the intake and return entries; arrows in the boundary representing the leakage direction; the airway branches in solid lines; the gob’s leakage branches and triangular elements in dashed lines

Figure 4 a shows the Brunner’s [9] integrated model where the gob is discretized into enough branches called leakage branch. Once the working face and the gob boundary are divided into the same number of fine branches by the nodes 3 to 7, the airway network and the gob area become an entirety and form a new network. Before solving the new network using ventilation network analysis theory described in Sect. 3, the resistance value of each leakage branch can be determined based on the presumption that the caved gob area can be described as a porous medium of variable grain size, thickness and consolidation. However, due to the lack of rigorous proof for representing the gob by leakage branches, this integrated model in solving GFF is as not strict as the CFD modeling, especially without the ability to model the path or distribution of the leakage.

Figure 4 b shows the other integrated but noncoupling model [11] where GFF is solved by the FEM-based CFD approach as described in Sect. 2. This model makes the simulation of mine ventilation network and gob flow more convenient by sharing pressures of nodes 3 to 7 between the longwall face and gob boundary. As shown in Fig. 4b, the nodes of the longwall face branch use the same number as the nodes on the air leakage boundary of the gob. In Sect. 2, Eq. (8) has shown that the nodal pressures on the air leakage boundary of gob are the boundary conditions for solving the gob flow field. In this model the pressures of nodes 3 to7 are determined by network calculation, and then automatically used in gob flow simulation by FEM. However, the results of the gob flow field are not fed back to the ventilation network calculation so it is a noncoupling approach. Although this noncoupling model implemented both methods of ventilation network analysis for airways and CFD-based simulation for GFF, it does not fully model the interaction between MVN and GFF.

The finite tube method proposed in the next section retains the advantages of CFD in solving the GFF, and uses a more rigorous approach to reconstruct the leakage branches in gob.

Finite Tube Method and Coupling Model

In fluid mechanics, streamline is the path of an imaginary particle as it moves with the flow. Since the streamline is traced out by a moving particle, at every point along the path the velocity is tangent to the path. Since there is no velocity component normal to the path, mass cannot cross a streamline. Therefore, the mass contained between any two streamlines remains the same throughout the flow field. As shown in Fig. 5, if a set of streamlines start at points that form a closed loop, these streamlines will form a streamtube. The tube is impermeable since the walls of the tube are made up of streamlines, and there is no flow normal to a streamline. The flow field in gob can be treated as a large number of streamtubes. In the example longwall section, the streamtubes start at the intake side of boundary L1 and end at the return side of boundary L1.

Fig. 5
figure5

Streamtube formed with streamlines: the red arrows tangential to the streamline show the direction and magnitude of the flow velocity

In a 2D gob space, a streamtube becomes a smooth-edged quadrilateral with two streamlines. According to fluid mechanics, streamline can be plotted using stream function value and the points on a streamline have the same stream function values. The flux in the streamtube can be determined by the stream function values of the two surface streamlines as the difference of any two streamlines’ stream function values represents the flux of the corresponding streamtube per unit thickness. In order to obtain the stream function values in the gob area, the partial differential equation described in Eq. (18) must be solved.

$$ \left\{\begin{array}{cc}\frac{\partial }{\partial x}\left(\frac{1}{E}\frac{\partial \psi }{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{1}{E}\frac{\partial \psi }{\partial y}\right)=0& \left(x,y\right)\in G\\ {}\psi =0& \mathrm{on}\kern0.5em {L}_2\\ {}{\psi}_i={\psi}_j+{V}_{ij}{d}_{ij}& \mathrm{on}\kern0.5em {L}_1\end{array}\right. $$
(18)

In Eq. (18), stream function ψ equals zero on boundary L2 with no air leakage; ψi and ψj represent stream function values of two adjacent nodes on L1; dij is the distance between nodes i and j (m); Vij represents the component of velocity vector in the direction of the outer normal to the edge between nodes i and j (m/s). For instance, in Fig. 4b, ψ3 = 0 and ψ4 = ψ3 + V34d34. It is easy to see that all the input parameters Vij required in Eq. (18) are part of the results of Eq. (8). It should be noticed that Eq. (18) is equivalent to Eq. (8) in describing the GFF and the finite element method for solving Eq. (8) is still applicable for Eq. (18) since their differential equations are identical in form. The difference is that the pressures and fluxes are used as the boundary conditions by Eqs. (8) and (18), respectively. Therefore, once the GFF is gained by solving Eq. (8) whose pressure boundary condition is provided by MVN, the stream function values of the gob area will be gained and then could be used to form flow tubes.

Since a streamline must start from intake boundary and end at the return boundary on L1 of the gob, any two adjacent streamlines will form a streamtube as a leakage branch between the intake and return boundaries. If two nodes are added on the boundary for each streamtube, each streamtube can be treated as a branch and is coupled into the network. The term of qj in Eqs. (16) and (17) can also represent the flow rate of tube j as shown in Eq. (19).

$$ {q}_j=H\hbox{'}\left(|{\psi}_{j,1}-{\psi}_{j,2}|\right) $$
(19)

In Eq. (19), H′ is the average height of caving zone (m); ψj,1 and ψj,2 are stream function values of the two streamlines defining the tube j. Because the air and any other gases in the longwall face and gob are all assumed to be incompressible, the volume flow rate (m3/s) is still used in Eq. (19). When using the finite tube method (FTM), only a finite number of tubes are needed to approximate the gob flows even though there are infinite streamlines connecting the intake and return boundaries. FTM turns the 2D gob flow into 1D tube flow, generating a coupling network between MVN and GFF. Substituting Eq. (19) into Eqs. (16) and (17) results in the equations for the coupling network. Recalculating the coupling network with the same network analysis method will produce a new boundary condition to GFF. An iterative solution process is applied between the network and GFF simulations before the MVN and GFF are fully coupled when the final results of the overall mine ventilation system are achieved as shown in Fig. 6. The iteration convergence is monitored by checking the evolution of gob nodal pressures or total flux of leakage. In Fig. 6, QLi represents the ith calculated total flux of air leaked into the gob. Iterations continue until the difference D between QLi and QLi − 1 is lower than a preset tolerance, ɛ.

Fig. 6
figure6

Iteration flowchart of solving the coupling between MVN and GFF

Program Development

A program, integrated Mine Ventilation Simulator (i-MVS), with a parallel computational method PARDISO [18] and visual interface development approach ObjectARX [11], was developed to encapsulate the FTM model with 2D gob in VC++. Through i-MVS, the finite element meshing, ventilation network modeling, and dynamically forming the flow tubes can be accomplished.

The program is able to generate and refine the meshes for the gob area. The finite element method has the advantage of low requirements for mesh generation, and can adapt to the complex mesh composed of different shapes and sizes of elements. Generally speaking, the finer the elements, the higher is the accuracy, but the more the computation is needed. In order to ensure the computation accuracy without increasing the amount of calculation, a rational mesh generation scheme is to use fine mesh in the gob areas with large hydraulic conductivity gradient and coarse mesh in areas with small hydraulic conductivity gradient. Therefore, i-MVS is able to divide the gob automatically into isosceles right triangular elements by using structured grid generation method first, as that used in Fig. 4b. In this step, the edge of triangular element can be a little longer than that used in current gob simulation experience. The principle of mesh refinement in i-MVS is to make the difference between the hydraulic conductivities of any two adjacent elements not exceed a preset small value. Therefore, these characteristics ensure that i-MVS has strong robustness and fast computing ability. Even for a complicated case in Sect. 6, it only takes about 10 min to complete the simulation.

Example Analysis and Improvement by the FTM

To validate the proposed FTM method, especially its mathematical models, it is applied to the simple example shown in Fig. 4b with a uniform gob hydraulic conductivity E = 0.5 m/s, and five flow tubes to approximate the gob flow. The element size in gob is 60 m, and the average caving height is 15.8 m. The fan’s characteristic curve function is hf(q1) = 614.5 + 3.4206q1 − 0.0161q12 − 0.0061q13. The resistances for branches 2 and 7 are 0.2083 N s2/m8, and the resistance per meter length between nodes 3 and 7 is 2.083 × 10−4 N s2/m9. The use of only five flow tubes and 60-m coarse triangular elements to represent the gob flow here is to make the simulation fast and easy to check its results. In Sect. 6, a complicated case will be analyzed based on finer meshing and larger number of flow tubes in which the FTM will be implemented for the analysis of methane migration in the gob. Some of the other input parameters for the simulations in this section, and Sect. 6 are listed in the Table 1. The simulation results of this example are shown in Figs. 7, 8, 9, and 10.

Table 1 Parameters used in the numerical simulations
Fig. 7
figure7

Simulation results of gob nodal pressures (a) and stream function values (b)

Fig. 8
figure8

Operating point of the fan

Fig. 9
figure9

The coupling network (blue color) of the MVN and GFF: airflow rates (m3/s) beside the branches and tubes and nodal pressures (Pa) inside the circles

Fig. 10
figure10

Convergence process of iterations between simulations of MVN and GFF. a Deviation of the gob nodal pressures and total leakage flux. b Evolution of the total leakage flux and total pressure drop along the working face

Figure 7 shows the nodal pressures and stream function values in the gob. The larger differential pressure observed near the longwall than those far deep in the gob indicates the reasonableness of the solution. Figure 8 shows the fan operating point for the example mine. Figure 9 shows the coupling network of MVN and GFF. In Fig. 9, the area between any two adjacent red streamlines denotes a tube or a branch with the same flux of 0.775 m3/s. The blue streamlines are the centerlines for the streamtubes. Figure 10 shows that the computation converged after 14 iterations to meet the preset convergence tolerance of 10−6. As shown in Fig. 6, each iteration includes a ventilation network simulation and a subsequent GFF simulation. Figure 10 b shows that the calculated final pressure drop (58.5 Pa) after 14 iterations along the working face is 6.8 Pa less than that (65.4 Pa) obtained from the first iteration. The large difference of 6.8 Pa, 12% of the total pressure drop (58.5 Pa), is the result of the fact that the ventilation network analysis has not been coupled to gob flow tubes because gob simulation has not been performed in the first iteration. Starting from the second iteration, the simulated network is coupled with gob flow tubes and the convergence trend in pressure drop can be clearly observed. It is easy to see that the results obtained after the third iteration gradually approach to the final results, that is, the results of the coupling model. And the results obtained after the first iteration are those from the noncoupling model described in Sect. 4.1. Therefore, 12% represents the improvement by the FTM method for the simulation of GFF.

Methane Migration Model in the MVN and Gob

Methane is a dangerous gas commonly found in coal mines, especially in working faces and gob areas. A good knowledge of the methane concentration distribution in the longwall face and the gob is very useful for gas control. A functionality for simulating methane migration in the MVN and gob has been added to the developed FTM program in this section. It should be pointed out that the methane migrations in the ventilation network and longwall gob are highly dependent on the airflow distributions.

Methane Migration in the MVN

The methane in underground coal mines is mainly emitted from the coal and surrounding rock strata due to mining activities. In a MVN, methane will transport with airflow throughout the mine ventilation system. When a branch is divided into a number of sequentially connecting short branches, the quantity of methane emitting into the branch can be evenly divided into its two nodes of each branch. Then, a mass conservation of the methane at each node can be formulated as

$$ \sum \limits_{j=1}^M{r}_{ij}{q}_{ij}^{\hbox{'}}{C}_j^{\hbox{'}}-{C}_i^{\hbox{'}}\sum \limits_{j=1}^M{r}_{ij}{q}_{ij}^{\hbox{'}}+\frac{1}{2}\sum \limits_{j=1}^N{b}_{ij}{I}_j^{\hbox{'}}=0,i=1,2,\dots, M $$
(20)

where Ci′ is the methane concentration at node i (mol/m3); q′ij is the flow rate of the branch connecting nodes i and j (m3/s); rij = 1 when air entering node i from node j, otherwise rij = 0; Ij is the methane source emitting into branch j (mol/s). Taking zero concentration for the intake nodes as boundary condition, for instance C1′ = 0 in Fig. 3, all of the other Ci′ can be determined by solving Eq. (20). It is assumed that the methane concentration in a branch is the same as that at its intake node.

Methane Migration in Gob

In the gob area, a gradual distribution of methane emitted from the residual coal or neighboring coal seams and overburden rock strata is formed. Understanding the migration behavior is very important to suppress a potential explosive hazard from methane. In order to establish the migration model during longwall mining, the gas release rate and transportation model will be established in this section.

Methane Release Intensity

Methane in gob is mainly emitted from the coal debris, the overlying gob, the underlying and overlying coal seams, and the surrounding gas-bearing rock strata. The gas emission rate per unit gob area (mol/(m2 h)) from a longwall mining operation can be expressed as [7]

$$ {W}_{CH_4}\left(x,y\right)=w{\hbox{'}}_0+{w}_0+\sum \limits_{i=1}^2{w}_i\exp \left(-{\lambda}_i\frac{L}{u}\right) $$
(21)
$$ L=x-{x}_0+ ut $$
(22)

where w′0 and w0 are the methane release intensities from the local channel point and the underlying coal seam of gob (mol/(m2 h)), respectively; w1 and w2 are the initial methane release intensities from the residual coal and the overlying gob or liberation coal seam (mol/(m2 h)); λi is the attenuation coefficient of gob methane release (day−1); L represents the distance from a point (x, y) in the gob to the working face (m) and which can be calculated by Eq. (22). The methane emission rate at any point per unit volume and unit time during advancing time can be defined as

$$ {w}_{CH_4}=\frac{W_{CH_4}\left(x,y\right)}{H} $$
(23)

Methane Migration Model

As shown in Fig. 1, in the longwall face of the U-shaped ventilation system, the air leaked into the gob will eventually return to the working area, and some of the methane accumulated in the gob will inevitably be brought back into the working area by the returned leakage air. Therefore, the distribution of methane concentration in the gob areas adjacent to the longwall face is of high importance for preventing the methane explosive zone from penetrating into the working face. Strictly speaking, the methane emission in gob is an unsteady process. However, if the longwall face is assumed to advance continuously at a constant speed u, fixed methane emission intensity at a fixed distance behind the working face is always maintained, as shown in Eq. (21). Therefore, a steady methane concentration field is always behind the longwall face under the steady GFF modeled in Sect. 2.2. Equation (24), with known gas flow velocity V′ determined by FTM and boundary conditions, uniquely defines the methane convection-diffusion problem in gob. It should be pointed out that although Eqs. (24) and (8) are both steady-state models, once the parameters of ventilation network and gob change, new fields of gas flow and methane concentration in gob can be obtained by the FTM method.

$$ \left\{\begin{array}{cc}V\hbox{'}\cdotp \nabla {C}_{CH_4}-{K}_{CH_4}\cdotp {\nabla}^2{C}_{CH_4}=w{}_{CH4}& \left(x,y\right)\in G\\ {}{C}_{CH_4}={\overline{C}}_{CH_4}& \mathrm{on}\kern0.5em {\Gamma}_{\mathrm{C}}\\ {}n\cdotp \left(-{K}_{CH_4}\nabla {C}_{CH_4}\right)=0& \mathrm{on}\kern0.5em {\Gamma}_{\mathrm{q}}\end{array}\right. $$
(24)

In Eq. (24), CCH4 represents the methane concentrations (mol/m3); KCH4 is the methane diffusion coefficient tensor (m2/s); V′ is the gas flow velocity (m/s), V′ = V/ε; ГC represents the inflow part of the boundary L1; Гq represents boundaries of L2 and the air outflow part of boundary L1; n is the unit normal vector of Гq. The gas diffusion coefficients in porous media are different from that in normal flow and the item [kij]CH4 of KCH4 is determined by the following equation [7]:

$$ {\left[{k}_{ij}\right]}_{{\mathrm{CH}}_4}=\left({a}_T\left|V\hbox{'}\right|+\tau \cdotp {D}_{CH_4}\right){\delta}_{ij}+\left({a}_L-{a}_T\right){v}_i{v}_j/\left|V\hbox{'}\right| $$
(25)

where the values of subscripts i and j are 1 or 2, representing coordinate axes X and Y, respectively; DCH4 is molecular diffusion coefficient (m2/s); τ is tortuosity factor of porous media; δij = 0 when i ≠ j, δij = 1 when i = j; aT and aL are the longitudinal and transvers dispersivity degrees (m).

For most convection-diffusion process, using the standard Galerkin finite element method in the convection-dominated problem leads to an oscillatory solution. Methane transportation in gob described by Eq. (24) is a typical case of such convection-diffusion process. Based on the same mesh generation results in Sect. 2, the upwind finite element method detailed in literatures [12, 19] has been used to discretize the differential equation of Eq. (24) into algebraic equations with their unknowns being the nodal methane concentrations in this research. Programming work of the program i-MVS has largely benefited from these literatures.

Program Applications

Case Study and Simulated Conditions

The developed numerical simulation model and the program i-MVS are applied to a typical longwall panel with U-shaped face ventilation. The simplified network model with each branch’s resistance is shown in Fig. 11a. The fan characteristic curve shown in Fig. 8 is used in this case. The panel is 1440 m long, 180 m wide, and 6 m high from the bottom of the coal seam. The branch between nodes 3 and 6 represents the longwall face. An entry connecting the intake entry (between nodes 4 and 3) and return entry (between nodes 5 and 6) is placed at the panel recovery end for various production and safety reasons. A ventilation regulator, in the form of adjustable door, is installed in this connecting entry to control the air quantities that flows to the longwall face. The designed airflow in the longwall face is 25 m3/s. With an advance rate of 3 m/day, the contour map of the 270-m-long gob for the evolution of the hydraulic conductivities when the advance time is 90 days is shown in Fig. 11a. The physical parameters of the model including methane release parameters are listed in Table 1.

Fig. 11
figure11

Basic conditions of the simulation example. a Branch resistances (10−3 N s2/m8) and hydraulic conductivity (m/s) distribution in the gob at the advance time t = 90 days. b Refined grid of the upper corner in the gob area near node 6

In Fig. 11a, the gob area was totally meshed into 12,932 elements. First, the gob was divided into isosceles right triangular elements with edge length of 3 m. The difference of hydraulic conductivities between any two adjacent elements is not more than 0.2 m/s after mesh refinement. Finer meshes were automatically generated mainly in the two gob corners where most of airflow enters and leaves the gob, which would contribute to get an accurate simulating of air flow in those areas since the porosity varies greatly in these places as described in Eqs. (1) and (4). Figure 11 b shows the mesh refinement effect of a part area of 12-m depth and 18-m width at the upper corner of the gob, where the hydraulic conductivities are shown in triangles.

Simulation Results

The simulation results of the coupling of the MVN and GFF are shown in Fig. 12. The gob is divided into 50 flow tubes represented by their center streamlines, and the flux of each tube is 0.0266 m3/s. These tubes in gob are color-coded according to the magnitude of gas velocity. The calculated total air leakage between longwall face and the gob area is 1.33 m3/s (79.8 m3/min). In the boundary, the blue tube lines connecting to the network clearly show the distribution of air leakage along the boundary—the more dense the lines, the greater the leakage; the less sparse the lines, the smaller the leakage. The distribution of the tubes shows that (1) the air velocity in gob is getting smaller with the increasing distance to the longwall face and (2) the air leakage occurs mainly near the two ends of the longwall face.

Fig. 12
figure12

The coupling of MVN and GFF when the advance time is 90 days: pressures (Pa) in nodal circles and flow rates (m3/s) beside the branches and tubes which are color-coded with air velocity magnitudes (m/s)

Gob Flow Behavior and the Prone Zone for Spontaneous Combustion

Spontaneous combustion mostly occurs in gob where the heat generated by the low-temperature oxidation of coal is not adequately dissipated by conduction or convection in the coal incubation period, which can be tested in the laboratory [20]. Spontaneous combustion in gob is highly affected not only by the coal properties but also by ventilation and gob characteristics. Prone zone for spontaneous combustion is a term used to denote an area with the high possibility of spontaneous combustion hazard in gob. The criteria of the air velocity (0.0017–0.004 m/s) has been commonly used to delineate the prone zone for spontaneous combustion because the higher air velocity could carry heat away quickly while a lower air velocity could not provide sufficient oxygen for the self-heating process [8]. For this simulation case in Fig. 12, the yellow zone, representing the gob area with higher spontaneous combustion risk, has a longer width near the panel edges along the panel longitudinal direction. Once the prone zone for spontaneous combustion is determined, a proper longwall advancing rate to prevent spontaneous combustion can be obtained accordingly. The prone zone for spontaneous combustion changes dynamically with the advancement of longwall face. To prevent spontaneous combustion in gob, the lifetime of the prone zone must shorter than the incubation period of spontaneous combustion. Therefore, a minimum longwall advancing rate can be calculated using the length of the prone zone and incubation period of spontaneous combustion. The program carried out in this study can identify the prone zone for spontaneous combustion and further help develop good measures to prevent spontaneous combustion.

Identification of the Zone Prone for Methane Explosion

As the methane concentration in a longwall gob area varies, there is an explosive zone with its methane concentration ranging between 5% and 15% sandwiched between the fuel lean zone on the face side and the fuel rich zone deep in the gob. In this case, the calculated distributions of methane concentrations in the MVN and gob are shown in Fig. 13. The fuel lean, explosive, and fuel rich zones are plotted as the green, yellow, and blue colors in Fig. 13, respectively, where the maximum methane concentration in blue area is 50.5%. Each contour line of the methane concentration in the gob area forms an arch shape behind the longwall face. A series of continuous yellow arches form the methane explosive potential zone range 5–15%. It should be noted that a 16.5-m-long section of the longwall face on the return entry side is in the explosive and fuel-rich zones where the methane concentration ranges from 5 to 18.8%. The methane concentration in the longwall face increases rapidly from 0.5 to 0.81% in the last 16.5-m-long section. An explosive gas zone, an unavoidable and dangerous zone for longwall operations employing bleederless ventilation system, does exist in the longwall gob, and its tip extends to the longwall face near the return entry. Ignition sources including spontaneous combustion in the yellow area may spark a methane explosion. Prior to the 16.5-m-long section, the methane concentration in the longwall face increases gradually and is only 0.2% at the middle point of the face. Using such simulation results, in addition to determining a proper retreating rate of longwall face to prevent spontaneous combustion, proper inert scheme and auxiliary ventilation system can also be determined to ensure the safe operation in a longwall mining face.

Fig. 13
figure13

Methane concentration in the ventilation system: the MVN and boundary are color-coded with Branch CCH4 bar; the gob is color-coded with Gob CCH4 bar; the yellow area with a methane concentration range of 5–15% represents the prone explosive zone

Scenario Analysis

Figure 14 shows the simulation results of an accidental regulator failure in the connection entry between nodes 4 and 5 (see Fig. 11). In the simulation, the resistance of the branch with the regulator is 1.8379 N s2/m8 including 0.0125 N s2/m8 of the roadway and 1.8254 N s2/m8 of the regulator. The failure of the regulator significantly reduces the branch’s resistance to 0.0125 N s2/m8, which make the total resistance drops from 0.3984 to 0.3077 N s2/m8 as shown in Fig. 15. The changes in air quantity and pressure to the operating point of the fan are small, from (34.2 m3/s, 467.3 Pa) to (36.7 m3/s, 415.3 Pa), respectively. However, the regulator failure caused a serious short-circuit of the longwall ventilation system. In comparison with the simulation results before the failure (see Figs. 12 and 13), with a 28.8 m3/s of flow short-circuited to the connection entry, the airflow entering the longwall face after this event is reduced from 26.3 to 7.9 m3/s—insufficient to ventilate the longwall face. Among the air flowing to the longwall face, the air leaked into the longwall gob is reduced from 79.8 to 22.5 m3/min. Figure 14 shows that the methane concentration in the tailgate T-junction will be 2.49% which is 3 times higher than that (0.81%) before the failure of the regulator—resulting in a more dangerous condition as the methane concentration is required to be kept below 1%.

Fig. 14
figure14

Results of the coupling of the MVN and GFF when the regulator damaged: methane concentrations (%) in MVN are in nodal circles; flow rates (m3/s) are beside the branches and tubes; the branches and boundary lines are color-coded with Branch CCH4 bar; the tubes in gob are color-coded with Gob CCH4 bar

Fig. 15
figure15

Operating points of the fan before (a) and after (b) the regulator damaged

Conclusions

A good understanding of the air flow in the longwall gob behind its operating face is important for mine safety management. A proper ventilation system is to prevent the methane explosive zone from penetrating into the working face and to reduce the size of the coal spontaneous combustion prone zone. A novel method of FTM for the coupling model of MVN and GFF has been developed for studying the airflows in the mine ventilation system and the longwall gob as an entirety. Several conclusions can be drawn from this research.

  1. 1.

    An innovative finite tube method for coupling of mine ventilation network and gob flow field has been proposed. In FTM, the gob is discretized into finite flow tubes by the streamlines of the gob flow field, which are coupled with the mine ventilation network to form a new network. This coupled simulation technique makes an improvement on simulation of airflow in mine ventilation.

  2. 2.

    A program, i-MVS, has been developed based on the proposed mathematical models to implement the required computations. The FTM method can be used to generate appropriate ventilation plans for longwall mining operations and bring better user experience to current mine ventilation simulation software, because the i-MVS program has the following three important advantages over the current mine ventilation simulation practices: The gob flow field is simulated with the ventilation network simultaneously to eliminate the separate and tedious manual input for the gob boundary conditions; the prone zones for gas explosion and coal spontaneous combustion can be quantitatively determined; the methane concentration in the ventilation network, especially at the tailgate T-junction, can be predicted.

  3. 3.

    One application case has been conducted using the developed program. It produces converging and realistic results about the air flows in the longwall face and the gob. The program can be used to generate mining operation and ventilation plans to prevent spontaneous combustion event and explosive methane conditions in the longwall face area.

In this research, the FTM is currently used for bleederless ventilation system in China. In fact, the gob air leakage in a bleeder ventilation system, a ventilation method widely used in the USA, is much more than that in a bleederless system. In current network modeling practice for such a bleeder system, gob leakage is often represented inadequately. Such a simulation may give an acceptable approximation to the airflows in the main airways but may be considerably in error on, or close to, each working face. The FTM could be easily modified to correctly simulate such bleeder ventilation system.

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Funding

This work is supported by the National Natural Science Foundation of China [grant numbers 51974232, 51574193] and Fundamental Research Funds of Shaanxi Province, China [grant number 2017JM5039].

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Correspondence to Fengliang Wu.

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Wu, F., Luo, Y. An Innovative Finite Tube Method for Coupling of Mine Ventilation Network and Gob Flow Field: Methodology and Application in Risk Analysis. Mining, Metallurgy & Exploration 37, 1517–1530 (2020). https://doi.org/10.1007/s42461-020-00255-0

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Keywords

  • Network flow
  • Field flow
  • Mine ventilation network
  • Coal mine gob
  • Risk zones