## Abstract

Drainage to lower water pressure is an effective measure for preventing and controlling water ingress when mining above a confined aquifer. The deep limestone aquifer in the Huaibei mining area, China, generally has high pressure, low permeability and variable water abundance, so it is difficult to meet single-borehole drainage requirements. In order to achieve good drainage, and take into account engineering and environmental protection requirements, a multi-objective optimization model of group borehole drainage was established. The model takes the minimization of single-hole flow and borehole numbers as the objective functions, and the drawdown in drainage boreholes and the water level control point as the constraint conditions. The particle swarm optimization algorithm was used to solve the model. The results indicate that, for a low permeability aquifer, measures such as using partially penetrating wells, increasing the number of drainage boreholes appropriately and reducing individual borehole yield have good drainage effects. The extent of drilling and amount of drainage are also relatively small. This is all to the good for the drainage. When the optimization results were applied to coal-face drainage in Huaibei the outcome was good.

## HIGHLIGHTS

The multi-objective optimization model of group borehole drainage for low permeable aquifer is established. Under the premise of ensuring the safety of coal mining, the research results protect the groundwater resources to the greatest extent and reduce the amount of drilling work.

The optimization model has certain guiding significance for the determination of single-hole flow, the degree of drainage borehole penetrating the aquifer and the number of boreholes when it is applied to the drainage of low permeability aquifer.

The research results are of guiding significance to avoid floor water inrush during mining above a confined aquifer.

## INTRODUCTION

Coal is a reliable and affordable energy source in many countries; for example, China, and supports about 40% of world electricity generation. Coal production, however, is constrained by various problems, including water inrushes to the working face during mining (Yuan 2019). Most of China's coal comes from North China type coal fields at present and recently, with increasing mining depth, the threat from the karst fissure water from Carboniferous Taiyuan formation and Ordovician limestone has increased to mining of the lower coal group (Wu *et al.* 2014, 2017). According to incomplete statistics, most major water inrush incidents in China in recent years have been caused by karst fissure water in the coal seam floor penetrating the working face. Many scholars have studied the prevention and control of karst fissure inrushes with fruitful results. They have produced forward theories including ‘water inrush coefficient’ (Guo & Liu 1989; Li *et al.* 2018), ‘key stratum’ (Qian *et al.* 1995, 2003; Lu *et al.* 2018), ‘progressive intrusion’ (Wang *et al.* 1996; Lu *et al.* 2015), ‘plate model’ (Zhang 2005; Kostecki & Spearing 2015) and ‘down three zone’ (Li 1999; Sun *et al.* 2012; Yin *et al.* 2016), etc., which all reveal water inrush mechanisms from different aspects. All of these theories have played a positive guiding role in mine safety improvements. The rock mass is so complex, however, that major water inrush incidents still occur from the floor from time to time.

Huaibei coalfield is of the North China type and mining of No. 10 coal seam is generally threatened by the confined waters of the underlying Taiyuan Formation limestone. As mining goes deeper, the aquifer water pressure and risk of water inrush increase. At present, the aquifer is commonly drained at a certain flow by constructing drainage boreholes on the ground or in the roadway. This can reduce the confined water level to below the ‘safe water head’, resulting in safe mining. The method is economical and easy, so it has been used widely in mining the Lower Coal Group in the Huaibei mining area, with good results.

Previous studies have shown that ‘high water pressure and low permeability’ are characteristics of the deep Taiyuan Formation limestone aquifer in the Huaibei mining area (Chen *et al.* 2012; Zheng 2018). Due to the aquifer's low permeability, the aquifer's water pressure decreases rapidly and the water level falls quickly during pumping. The cone of depression is V-shaped and the radius of influence is small. In order to improve drainage, measures like infill drainage boreholes are generally taken, but the increased numbers of drainage boreholes has increased coal mine production costs and, at the same time, drainage has also damaged the groundwater resources, with a serious impact on sustainable development of the mining area. Therefore, in order to achieve safe production, the total volume of water drained and the amount of drilling should be reduced as far as possible, to optimize the balance between safety, economy and environmental benefits. This was the fundamental starting point for the drainage optimization project.

There has been considerable research into water resource management to date (Sulis *et al.* 2011; Wang *et al.* 2012a, 2012b; Dolatshahi-Zand & Khalili-Damghani 2015; Zhao & He 2015; Song *et al.* 2016). However, there are few drainage optimization designs for mining above confined limestone aquifers. Meng *et al.* (2015) established the optimal design model for single-hole drainage and the optimal drainage borehole layout. The limestone aquifer generally has the characteristics of high water pressure, low permeability and uneven water abundance in deep mining, so it is hard to meet the single-hole drainage requirements. There are no reports, however, on the optimization of group borehole drainage in such aquifers. The purpose of this study was to fill this gap, therefore, by optimizing borehole numbers and flow in the group borehole drainage, so that drainage is achieved while the cost of coal mining and degree of environmental damage are reduced effectively. The optimization model proposed can be used to study the design and calculation of group borehole drainage on the basis of the aquifer's hydrogeological characteristics. Because the analytical solution of the drawdown of inclined and/or horizontal wells is relatively complex, the optimization model reported here was established on the basis of the unsteady flow theory of vertical wells.

## RESEARCH METHOD

### Unsteady groundwater flow into a well

Depending on the degree of aquifer exposure and the water inflow conditions, drainage boreholes can be either fully or partially penetrating. When a drainage borehole is open through the entire thickness of an aquifer, it is fully penetrating, otherwise it is a partially penetrating well (Figure 1).

*r*, between the observation and drainage wells is , the drawdown is given by Equation (1):in which:where

*Q*is the borehole yield (m

^{3}/d),

*r*the line distance between the calculation point and the drainage borehole (m),

*S*the drawdown (m);

*K*the aquifer permeability coefficient (m/d);

*M*the full aquifer thickness (m); the elastic release coefficient; the Theis well function;

*u*the independent well function variable;

*ζ*the additional resistance coefficient of non-integrity;

*d*and

*l*respectively are the distance from the confining layer to the top and bottom of the borehole filter (m);

*z*the distance between the confining layer and the bottom of the observation borehole (m); and

*t*the drainage time (d).

*Q*

_{1},

*Q*

_{2}, … ,

*Q*

_{n}, respectively, the drawdown at any point can be calculated using the seepage superposition principle – that is, it is the sum of the drawdowns caused at this point by each individual drainage borehole – Equation (6):

### Optimization problem

#### Water level control point and safe water level

*M*is the distance from the coal seam floor to aquifer roof, that is, the aquifer cap thickness (m); and

**′***T*the critical water inrush coefficient (MPa/m). In the ‘detailed rules for the prevention and control of water in coal mines’ (China State Administration of Coal Mine Safety 2018),

_{s}*T*is taken as 0.06 MPa/m in sections with structural damage and 0.1 MPa/m in complete sections;

_{s}*p*is the head that the aquifer cap can bear safely (MPa). After calculating

*p*, the safe head can be determined according to the conversion relationship between pressure and head.

In a real drainage project, the working face boundary is taken as the drainage boundary, and the end of the working face is selected as the water level control point within the drainage boundary to calculate the safe water level. The aquifer cap floor's elevation – that is, that of the aquifer roof – *H*_{d} — is determined first at each water level control point together with initial water level elevation, *H*_{0.} Then the safe pressure is converted to a safe water level using *H*_{s} = *p* × 100 + *H*_{d}. Finally, using *S*_{s} = *H*_{0}-*H*_{s}, the safe drawdown level at each water level control point is determined.

#### Optimization model

- (1)
Objective function

The drainage roadway is generalized as a line segment with dip angle *θ*. When the coordinates of points *A* and *B* at its ends are (*x _{p}*

_{1},

*y*

_{p}_{1}) and (

*x*

_{p}_{2},

*y*

_{p}_{2}), the roadway's linear equation

*y*=

*ax*+

*b*can be obtained, where

*a*is the line's slope,

*b*its intercept, and [

*x*

_{p}_{1},

*x*

_{p}_{2}] is the abscissa range of the straight line segment. The roadway's total length is

*L*.

The drainage roadway has *n* equally spaced drainage boreholes; *F*_{1}, *F*_{2}.. *F _{i}*. .

*F*. that are

_{n}*D*apart. The coordinates of any drainage borehole are (

*x*

_{ci},

*y*

_{ci}). The working face has

*m*water level control points –

*C*

_{1},

*C*

_{2}…

*C*

_{j},

*C*

_{m}– the coordinates of any control point are (

*x*

_{j},

*y*

_{j}). The distance from any drainage borehole to any water level control point is recorded as

*r*

_{ij}. It is assumed that each drainage borehole's yield is the same and =

*Q*. The minimum optimization models of

*Q*and

*n*are min(

*Q*) and min(

*n*). The drainage borehole and water level control points are shown in Figure 2.

On the premise of safe mining, the single-hole flow and the number of boreholes are minimized, and the best drainage borehole position in the roadway will affect the realization of optimization directly.

- (2)
Constraint conditions

- (1)
Borehole number

- (1)

*x*

_{1},

*y*

_{1}). Because drainage boreholes must be arranged along the drainage roadway, they must be within the drainage roadway's bounds. Using the roadway end point coordinates and its straight line equation, the constraint conditions for the number

*n*of drainage boreholes in this group can be written as Equation (9):

- (2)
Yield

When the drainage borehole water level drops to the aquifer roof (*S*_{max}), the drainage borehole yield is recorded as *Q*_{max} and is the maximum value constraint condition of single-hole flow in optimization – that is, the upper limit of flow. If the drainage borehole radius is *r*_{w}, and taking account of mutual interference between drainage boreholes, the flow constraints are as follows:

*r*

_{(i−1)D}is the distance from the first to the

*i*-th borehole (m), and the other distances are the same as in Figure 2.

- •
- (3)
Drawdown at control points

- (3)

The drawdown at all control points shall reach the safe drawdown level in order to meet the requirements of safe mining. For the *j*-th water level control point, the safe drawdown is , and its constraint conditions are as follows:

#### Model solving

Because of the problem of draining the high pressure, low permeability aquifer, a dense pattern of drainage boreholes is generally used to ensure the safety of the whole working face. However, if too many boreholes are drilled, mining costs will rise while, if the number is reduced, the yield from each must be increased. Equally, due to the low permeability, the yield from any one borehole should not be too large, or the effect of drainage is poor. The optimization objective, therefore, is to use as few drainage boreholes as possible while minimizing the single-hole yield under the constraint conditions – that is, there are multiple objectives. For this study, the multi-objective particle swarm optimization (MOPSO) algorithm proposed by Coello & Lechuga (2002) was used. It extends particle swarm optimization to multi-objective issues. MOPSO is recognized as a classical approach. At its core is choosing the best combination among the multiple mutual restricting objectives. The establishment and calculation procedures are shown in Figure 3.

## EXAMPLE APPLICATION AND DISCUSSION

### Engineering background

Suntuan coal mine is in the southwest of the Huaibei coalfield, in northern Anhui Province, China. The No. 10 coal seam is 35 to 55 m above the top of the confined Taiyuan Formation limestone aquifer (Figure 4). The formation comprises 12 limestone horizons (*L*_{1}-*L*_{12}) in the coalfield, and its total thickness is 85 to 130 m. Of this, the limestone horizons account for 45 to 70 m, or about 60% of the total. The limestone's water-bearing horizons *L*_{1} to *L*_{4} are the main factor affecting the No. 10 seam's mining safety. The aquifer below horizon *L*_{4} is far enough from the seam for its impact on mining to be small.

The trend of the mine stratum is approximately north-south, and it dips to the east at about 10° in a monoclinal structure. There are many faults, most of which are tensional and torsional. Little flushing fluid leaks into the fault fracture zone during drilling. Pumping test data from the fault zone show that typical borehole inflow, *q*, is 0.016 to 0.00,048 *L*/(s.m), and *K* is 0.023 to 0.0017 m/d. As the permeability of the faults is low, the water-bearing upper member of Taiyuan Formation is a closed and weak aquifer, making it feasible to use drainage to lower the head.

During exploration, seven boreholes were drilled to carry out pumping tests on *L*_{1}–*L*_{4} aquifer. The hydrogeological results are shown in Table 1.

Borehole . | Aquifer roof elevation H_{d}(m)
. | Initial water level elevation H_{0} (m)
. | Inflow q (L/s.m)
. | Permeability coefficient K (m/d)
. |
---|---|---|---|---|

22-23-1 | −261.2 | +19.91 | 0.058 | 0.21 |

27 − 1 | −320.59 | +21.12 | 0.055 | 0.17 |

07 observation 3 | −210.32 | +18.39 | 0.77 | 2.99 |

14 observation 1 | −205.35 | +1.22 | 1.55 | 3.93 |

25-26-S1 | −878.13 | +14.22 | 0.0082 | 0.026 |

13-14-S4 | −565.36 | +14.20 | 0.0052 | 0.027 |

11 observation 2 | −220.35 | +21.1 | 0.093 | 0.28 |

Borehole . | Aquifer roof elevation H_{d}(m)
. | Initial water level elevation H_{0} (m)
. | Inflow q (L/s.m)
. | Permeability coefficient K (m/d)
. |
---|---|---|---|---|

22-23-1 | −261.2 | +19.91 | 0.058 | 0.21 |

27 − 1 | −320.59 | +21.12 | 0.055 | 0.17 |

07 observation 3 | −210.32 | +18.39 | 0.77 | 2.99 |

14 observation 1 | −205.35 | +1.22 | 1.55 | 3.93 |

25-26-S1 | −878.13 | +14.22 | 0.0082 | 0.026 |

13-14-S4 | −565.36 | +14.20 | 0.0052 | 0.027 |

11 observation 2 | −220.35 | +21.1 | 0.093 | 0.28 |

As can be seen, *q**=* 0.0052 to 1.55 *L*/(s.m), and *K**=* 0.026 to 3.93 m/d, with plenty of water and good connectivity in the shallow sub-crop, but becoming weaker with increasing depth. The yield, *q*, and permeability, *K*, from different boreholes also differ, further illustrating the upper limestone aquifer's heterogeneity.

Working faces 1013 and 1011 are in 101 mining area, which is divided into several blocks by large faults – F9, F10 and F11. The two faces are between F9 and F10 (Figure 4), and their hydrogeological conditions are relatively independent. Beneath the working faces is the Taiyuan Formation aquifer, where the total thickness of *L*_{1}–*L*_{4} is 50 m. Of 126 water exploration boreholes in the upper part of the aquifer (*L*_{1}–*L*_{4}) in the mining area, only four yielded more than 5 m^{3}/h, while most were dry. Since mining began in 2007, the Taiyuan Formation limestone has been drained. Prior to mining of the two working faces, the water level in observation borehole 14-observation 1 was −50 m, and the minimum combined coal seam and aquifer thickness was 45 m. The lowest working face elevation was calculated as – 515 m, so the maximum head on the base of the aquifer cap was 4.65 MPa, and the water inrush coefficient about 0.103, exceeding the critical value in the ‘detailed rules for the prevention and control of water in coal mines’ (China State Administration of Coal Mine Safety 2018), so there is danger of water ingress. The plan was to allow 1 year for drainage to lower the head, and the goal was to reduce the head to the critical water inrush coefficient value to ensure mining safety.

Because of the aquifer's drainage characteristics and to ensure the safety of the two working faces, a special drainage roadway was constructed (Figure 5), which directly exposes the limestone. The roadway floor elevation varies from −541.9 to −540.7 m, and it is 390 m long. According to the hydrogeological exploration report, the aquifer thickness (*L*_{1}–*L*_{4}) in the mining area is 50 m, *K* is 0.03 m/d, and *μ** (the storage coefficient) is 1 × 10^{−6}. The drainage borehole radius, *r*_{w}, is 0.05 m, and the maximum single-borehole drawdown, *S*_{max}, 490 m.

### Optimization model establishment and solution

Water level control points should be at the upper and lower ends of the roadway far from the drainage borehole and the deepest working face – see Figure 2. Control points measure the head at the base of the coal seam's aquifer cap; that is, *z**=* 0 and *d**=* 0 for the drainage borehole. According to the Huaibei mining regulations, the working face critical water inrush coefficient is 0.06 MPa/m, as noted, and the safe water pressure at the control point is 2.7 MPa (from Equation (7)). The safe water levels and drawdowns at the control points can be calculated from this and are shown in Table 2.

Control point . | x (m)
. | y (m)
. | Base of aquifer cap H_{d}(m)
. | Safe head at control point H_{s}(m)
. | Safe drawdown at control point S_{s}(m)
. |
---|---|---|---|---|---|

C_{1} | 360.26 | 971.06 | −360 | −90 | 40 |

C_{2} | 381.85 | 790.39 | −390 | −120 | 70 |

C_{3} | 1,263.51 | 774.97 | −425 | −155 | 105 |

C_{4} | 1,144.15 | 643.54 | −450 | −180 | 130 |

C_{5} | 1,427.15 | 338.74 | −540 | −270 | 220 |

C_{6} | 1,787.30 | 269.39 | −565 | −295 | 245 |

Control point . | x (m)
. | y (m)
. | Base of aquifer cap H_{d}(m)
. | Safe head at control point H_{s}(m)
. | Safe drawdown at control point S_{s}(m)
. |
---|---|---|---|---|---|

C_{1} | 360.26 | 971.06 | −360 | −90 | 40 |

C_{2} | 381.85 | 790.39 | −390 | −120 | 70 |

C_{3} | 1,263.51 | 774.97 | −425 | −155 | 105 |

C_{4} | 1,144.15 | 643.54 | −450 | −180 | 130 |

C_{5} | 1,427.15 | 338.74 | −540 | −270 | 220 |

C_{6} | 1,787.30 | 269.39 | −565 | −295 | 245 |

The drainage boreholes were drilled along roadway section *AB*. The coordinates of *A* and *B* are (2,008.42,582.83) and (2,146.8,408.48), respectively, and the roadway can be generalized into a linear equation as *y**=* −1.23*x* + 3,057.4(2,008.42 ≤ *x* ≤ 2,146.8). The optimization model is solved using the MOPSO algorithm based on Matlab software – the results are shown in Table 3. Taking the incomplete well as an example (*l*/M = 0.8), all calculation results meeting the constraints can be obtained, as shown in Figure 6. That is, *n* and *Q* (the total flow *n* * *Q* is known, of course). According to the optimization principle (*n*, *Q* and *n* * *Q* are minimum), the final optimization result is *n* = 5, *Q* = 104.9 m^{3} /d, and the specific results are shown in the third row of Table 3.

l/M
. | n
. | First borehole coordinates x (m)
. | Spacing D (m)
. | Yield Q (m^{3}/d)
. | Drilling quantities l × n (m)
. | Total yield Q × n (m^{3}/d)
. |
---|---|---|---|---|---|---|

1 (fully penetrating well) | 4 | 2,012.2 | 62.17 | 131.05 | 200 | 524.2 |

0.8 (partially penetrating well) | 5 | 2,008.4 | 51.23 | 104.9 | 200 | 524.5 |

0.6 (partially penetrating well) | 6 | 2,008.4 | 39.02 | 87.33 | 180 | 523.98 |

0.4 (partially penetrating well) | 8 | 2,036.1 | 24.76 | 65.91 | 160 | 527.31 |

0.2 (partially penetrating well) | 16 | 2,034.1 | 11.69 | 32.9 | 160 | 526.4 |

l/M
. | n
. | First borehole coordinates x (m)
. | Spacing D (m)
. | Yield Q (m^{3}/d)
. | Drilling quantities l × n (m)
. | Total yield Q × n (m^{3}/d)
. |
---|---|---|---|---|---|---|

1 (fully penetrating well) | 4 | 2,012.2 | 62.17 | 131.05 | 200 | 524.2 |

0.8 (partially penetrating well) | 5 | 2,008.4 | 51.23 | 104.9 | 200 | 524.5 |

0.6 (partially penetrating well) | 6 | 2,008.4 | 39.02 | 87.33 | 180 | 523.98 |

0.4 (partially penetrating well) | 8 | 2,036.1 | 24.76 | 65.91 | 160 | 527.31 |

0.2 (partially penetrating well) | 16 | 2,034.1 | 11.69 | 32.9 | 160 | 526.4 |

Table 3 shows that when the drainage borehole is fully penetrating, the optimal number of drainage boreholes, *n*, is 4, the spacing between them 62.17 m, the yield from a single borehole 131.05 m^{3}/d, and the total yield 524.2 m^{3}/d. For partially penetrating boreholes, the amount of drilling decreases, but more boreholes are needed and the single-borehole yield decreases, and there is little difference in the total yield. However, the total amount of drilling decreases step by step.

It can be seen from the above that, for a low permeability, heterogeneous aquifer, a single, low yield borehole can achieve the drainage required. However, when the yield from a single borehole is low, many are needed and construction costs are high. In this case, the aquifer is a composite of horizons *L*_{1}–*L*_{4}. Generally, the yield from *L*_{3}–*L*_{4} is the highest. This is because *L*_{3}–*L*_{4} accounts for a large proportion of the thickness of *L*_{1}–*L*_{4}.If the drilling depth is insufficient, it becomes difficult to penetrate to *L*_{3}–*L*_{4}. The optimization calculation results show, however, that when *l*/*M**=* 0.6, the number of boreholes and extent of drilling are moderate, and the total yield is also lowest, which is an ideal design scheme.

### Parameters

- (1)
Aquifer permeability

To determine the influence of aquifer permeability on drainage, when *μ** and *K* are known, the optimal *n* and *Q* are needed – see Figure 7.

As *K* increases gradually, *n* first decreases then rises. It is at its lowest when *K**=* 0.03 m/d. This shows that the effect of drainage in low permeability aquifers is better when measures like installing intermediate drainage boreholes and reducing individual borehole yields are taken. In addition, as *K* increases, borehole yields, *Q*, generally increase but, above some high *K* value, increasing *n* has little effect on *Q*.

- (2)
Aquifer storage coefficient

*μ**on the optimization results, the optimal

*n*and

*Q*are needed – Figure 6.

*μ**reflects the aquifer's available water content. So, as

*μ**increases, the optimal number of drainage boreholes under the constraint conditions also increases, while the single-borehole yield decreases and the total yield increases.

### Drainage effect

In reality, in the mine, the optimization results were noted and the partially penetrating borehole scheme adopted. Individual borehole yields were reduced, the number of drainage boreholes was increased appropriately and other measures were taken to deal with the aquifer drainage problems. In the two roadway drilling fields, six cross limestone layer drainage boreholes were installed (Figure 8). One borehole is in the third limestone horizon, but the other five are all in the horizon *L*_{4}. Individual boreholes yield between 52 and 100 m^{3}/d.

In the drainage roadway, the sub-horizontal drainage boreholes along the limestone layer, which are generally long – it is difficult and expensive to drill in the roadway – combined with the drainage holes penetrating the layer, can achieve better drainage. For this reason, four sub-horizontal drainage boreholes were constructed along the 170°, 196°, 206° and 222° azimuth in the drainage line of No. 3 drilling field. The drainage boreholes are essentially parallel to the trend of horizon *L*_{4}, and their average length is 140 m.

The total discharge in 2015 from the limestone in mining area 101 was about 490,000 m^{3} at an average yield of about 56 m^{3}/h. The water level in borehole 14 observation 1, which is on the ground outside the mining area, fell from −50 to −170 m – that is, 120 m – Figure 8(b). The maximum daily fall was 2.4 m. The head measured at the working face is 0.2 MPa (corresponding to a water level of −320 m), and the elevation of the lowest point of the working face mining section is −401.1 m, so that the maximum working face water inrush coefficient is 0.026 MPa/m, basically reaching the goal of drainage mining.

For the inclined shafts in the actual project, the conversion method is often used to calculate drawdown, which introduces calculation errors. It is also noted that this model is not suitable for calculating drawdown in horizontal drainage boreholes or along the limestone horizons. However, the model is of some help in determining individual borehole yield, the degree of borehole penetration into the aquifer and the number of boreholes needed, when applied to the drainage of a low permeability aquifer.

## CONCLUDING REMARKS

Based on the aquifer's low permeability, and the requirements of economy and environmental protection, a multi-objective optimization model for multi-borehole drainage was established. The model takes the minimum single-borehole yield and number of boreholes as the objective function, and the drainage borehole and water level control point drawdown as the constraint conditions. The model was applied to a coal mining face in the Huaibei mining area. While ensuring safety in coal mining, the research results protect the groundwater resources to the greatest extent and reduce the amount of drilling required. The main conclusions are as follows:

- (1)
When drainage borehole penetration of the aquifer (

*l*/*M*) is reduced, the number of optimized drainage boreholes,*n*, increases gradually, but the yield from individual boreholes decreases while the total yield remains about the same. At the same time, the total amount of drilling required decreases step by step. Because this is a low permeability, heterogeneous aquifer, it is better to drain it using larger numbers of partially penetrating boreholes, and reduce the yield of individual boreholes appropriately. - (2)
The hydrogeological parameters affect the optimization results significantly. With increasing

*K*values in the aquifer, the optimum number of drainage boreholes first decreases then increases, while the individual borehole yield,*Q*, increases before leveling out. As*μ**increases, the optimal number of drainage boreholes increases and the individual borehole yield decreases. - (3)
The upper Taiyuan Formation is a compound aquifer comprising horizons

*L*_{1}–*L*_{4}, of which*L*_{3}and*L*_{4}have good yields. This suggests that drainage boreholes should penetrate the third limestone horizon (*L*_{3}).

## ACKNOWLEDGEMENTS

This study was partially supported with research grants from the major projects of natural science research in Higher Education Institutions (KJ2019ZD11) and the National Natural Science Foundation of China (41977253). The data used in this paper can be accessed by contacting the corresponding author directly.