【正文】 致謝 第一作者謝謝兩位學者對公式（ 9）正確性的寶貴評論。 半無限含水層中圓形排水隧道穩(wěn)定地下水流涌水量簡單封閉的分析解決方法在隧道掌子面兩種不同邊界條件 (無水頭以及恒定水頭 )共同的理論框架內(nèi)重新推求。近似解 QA2 由于 r / h≤ ,但結(jié)果穩(wěn)定。圖 4顯示涌水量和r / h 和不同的 b(=H/h)相關(guān)的結(jié)果。如 H ≠ 0，當r/h?1時 )1//ln ( 22 ?? rhrh 可能導致 Q1 值不穩(wěn)定。有趣的是此過量估計是由于當 r/h?1時近似解 QA1迅速增加。 由于 h﹥﹥ r，可得 h﹥﹥ h+ hrhh 222 ??? ，則公式（ 18）可進一步簡化為 )1ln (2Q22A1 ???rhrhhk? (19) 公式（ 19）是由 El Tani (2021)表一中的 Muskat， Goodman 等人提出來的。 El Tani, 2021). (1) 通過假定 ha = h的近似解。如果對地下水位低于地面，則將潛水面作為參考基準面。 （ 2）例 2:總水頭恒定， ha。 表 2 保角映射平面 ??? ????? 11)( iAwz （ 6） 其中 A = h(1α 2)/(1+α 2 ),h為隧道的深度 , α為一個參數(shù)，被定義如下： 212????hr 或 )(1 22 rhhr ???? (7) 則，公式（ 1）可以依據(jù)ξ η坐標系改寫為 02222 ?????? ???? (8) 考慮到邊界條件 ,ζ軸上半徑為ρ的圈上總水頭的解可表達為： ??? ????? 1 4321 c os)(lnCC n nn nCC ????? (9) 其中， C1,C2,C3 與 C4 為常數(shù) ,取值取決于地表和隧道掌子面的邊界條件 。 (1) 例 1：零水頭壓力 ,因此總水頭 =位置水頭 (El Tani,2021) yr ?)(? (4) (2) 例 2：總水頭恒定， ha(Lei, 1999。此處，地表被用作是參考基準面以考慮地下水位在地表上的情況。假定周圍土層滲透系數(shù) k各向同性，地下水流為穩(wěn)定流。 在本研究中 , 我們應當通過集中研究隧道掌子面兩個不同邊界條件 (一個無水頭壓力，另外一個為恒定水頭 )重新梳理半無限含水層中圓形排水隧道穩(wěn)定地下水流涌水量計算的現(xiàn)有解決方法。最近 , El Tani (2021) 在莫比烏斯（德國數(shù)學家）變換公式和傅立葉級數(shù)的基礎(chǔ)上提出了地下水涌水的解析解。涌水量的預測就變成研究邊界條件的差異。 r, y = h), . ha =h (Lei, 1999。 Tunnels。C Co.,Ltd., 3F. 799, AnyangMegavalley, GwanyangDong, DonganGu, Anyang, GyeonggiDo, Republic of Korea Received 19 November 2021。received in revised form 13 February 2021。 Groundwater ?ow。 El Tani, 2021). (1) Approximate solution by assuming ha = h. By simply assuming ha = h and H =0,Eq. (17) can be simpli?ed as )1ln (2Q22A1 ???rhrhhk? (18) Where subscript A means approximate solution. Eq. (18) was indicated as the solution by Rat, Schleiss, Leiin Table 1 of El Tani (2021). (2) Approximate solution in the case of h﹥﹥ r (deep tunnel) For h﹥﹥ r, we have h+ hrhh 222 ??? , and hence Eq. (18) can be further simpli?ed as )2ln(22 rhhkQA ?? (19) Eq. (19) was indicated as the solution by Muskat, Goodman et Table 1 of El Tani (2021). in water in?ow predictions In order to investigate the di?erence in water in?ow predictions among the exact and approximate solutions and the range of applicability of approximate solutions, the relative error, previously shown in Fig. 3 of El Tani (2021), are obtained again from ）（ %100Q 1 1A11 ??? Q Q? or (%)100Q 1 1A22 ??? Q Q? δ1 and δ2 show the di?erences between Q1 (Case 1) andQA1, QA2 (approximate solutions of Case 2) respectively. Here, H = 0 is used, and so this case is that the groundwater level is at/below the ground surface. . Diffierence among solutions (E1 T ani,2021) From Fig. 3, δ1 and δ2 indicate that the approximate solutions, QA1 and QA2, overestimate the in?ow rate by about 10–15% when r/h =. Interestingly the overestimation by the approximate solution QA1 increases drastically as r/h ? 1. This may because the term 0)1//ln ( 22 ??? rhrh and ??1AQ as r/h ? 1. Thus, the approximate solution QA2 seems to give better prediction of groundwater in?ow than , the term 0)1/(1 22 ??? ?? ）（ as r/h? 1, Q1 gives stable results. If H ≠0, however the term )1//ln ( 22 ?? rhrh could cause instability of Q1 as r/h? e?ect is investigated in the next. of H in the underwater tunnel The e?ect of H on the water in?ow prediction in the underwater tunnel is investigated by using the approximate and exact solutions. Fig. 4 shows the results of water in?ow with respect to r/h with di?erent b (=H/h). The in?ow is obtained from Eq. (16) for Q1 or Eq. (19) for QA2 considering ha = h and h ﹥﹥ r. )1()11(kh2Q22221??????rhrhb??? or )2ln( )1(2 2rhbkhQ A ??? The solid line represents the results for Q1,whereas dotted