石油礦區中生產井之產率，若超過臨界產率（Critical Production Rate）時，將會發生水錐（Water Coning）現象導致生產井出水，而縮短生產年限。目前文獻中臨界產率解析模式的計算式均假設水錐影響半徑（r1）為已知，而且是直接以生產層邊界（re）作為影響半徑，這種假設並沒有理論根據。因此，本研究的目的是利用數值模擬建立水錐模擬模式，研究水錐所造成之臨界產率及水錐貫穿時間（Breakthrough Time），依此結果建立水錐影響半徑與臨界產率之間的關係。

　　首先，模擬不同臨界產率以及不同滲透率下水錐之變化，繪製出水錐高度（h）對時間（t）變化圖。再以臨界產率解析模式（包含Meyer and Garder（1954）、Schols（1972）），假設水錐影響半徑直接以生產層邊界作為影響半徑，分別計算出臨界產率值後，與模擬所得之臨界產率相互比較。由結果可得知Meyer and Garder（1954）模式所計算之臨界產率較保守，低於模擬之臨界產率，不會產生水錐貫穿出水；Schols（1972）模式所計算之臨界產率在各種地層滲透率下，較模擬出之臨界產率稍微偏高。

　　在水錐貫穿時間的研究方面：也利用數值模式模擬各種產率（含臨界產率）生產時，水錐上升至井底所需之時間，同時，也以Hagoort（1988）解析模式之計算結果與模擬結果相互比較，兩者所計算出來之結果非常接近。

　　接著以臨界產率及水錐貫穿時間模式為基礎，模擬計算水錐影響半徑，推導出達到臨界產率時之無因次化模擬水錐影響半徑對無因次化時間之關係式；同時並研究生產時壓力影響半徑與水錐影響半徑兩者之關連，建立無因次化壓力影響半徑與無因次化模擬水錐影響半徑兩者之關係式，如此更可實際且正確運用在計算水錐臨界產率上。

　Water coning or water cut sometimes appears and shorts the production rate of the well in the oil field when the withdrawal rate exceeds the critical production rate. In literature, the analytical solution of critical rate assumed that the coning effective radius (r1) is given and directly used the reservoir boundary (re). However, the coning effective radius is unknown.

　The purpose of this study is using numerical simulation to build up water coning simulation model, to study the influences of critical rate and breakthrough time by water coning, and to establish the relationship between the coning effective radius and the critical rate according the result.

　First, we simulate the change of water coning on different critical rates and permeabilities to plot the coning height (h) versus time (t). Then we use the critical rate analytical model, including Meyer and Garder (1954), and Schols (1972). The coning effective radius is assumed, and the reservoir boundary is directly used. After calculating the critical values respectively, we compare these critical values with the simulation values. From the results, the calculation of Meyer and Garder (Meyer and Garder, 1954) is relatively conservative, that is, is lower than the simulation and the water breakthrough of the well will not appear. The calculation of Schols (Schols, 1972) under various permeabilities outstrips the simulation result.

　In study of breakthrough time, the time of water coning rising to the bottom of well is compared with the Hagoort’s analytical solution. The results of two calculations are very close.

　About the critical rate and breakthrough time models, we simulate the coning effective radius (rc) and calculate the dimensionless coning effective radius (rcD) versus dimensionless time (tD). Also, we study the relationship between pressure effective radius (ri) and coning effective radius, and obtain the relationship equation of . This equation can be used correctly in calculating the coning critical rate.

1.Abass, H.H., and Bass, D.M.: “The critical production rate in water-coning
systems,” paper SPE 17311, p.351-360, 1988.

2.Addington, D. V.:“An Approach to Gas Coning Correlation for a Large Grid
Cell Reservoir Simulation,”J. Pet. Tech. (November 1981) 2267-2274.

3.Agarwal, R. G., Al-Hussainy, R., and Ramey, H. J., Jr.:“The Influence of
Water Influx in Gas Reservoirs,”J. Pet. Tech. (Nov 1965) 1336-1342.

4.Arthur, M. G.:“Fingering and Coning of Water and Gas in a Homogeneous Oil
Sand,”Trans., AIME(1944) 155-184.

5.Aziz, K. and Flores, J,:“Influence of Production Rate and Oil Viscosity on
Water Coning,”paper 374032 presented at the 25th Annual Technical Meeting
of the Petroleum Society of CIM, Calgary, Alberta, Canada, May 7-10, 1974.

6.Bournazel, C. and Jeanson, B.:“Fast Water Coning Evaluation,”paper SPE
3628 presented at the SPE 46th Annual Fall Meeting, New Orleans, Oct. 3-6,
1971.

7.Bournazel, C., Jeanson, B.: “Fast water-coning evelaution method,” paper
SPE 3628, p.3-6, October 1971.

8.Bruns, J.R., Fetkovich, M. J., and Meitzen, V. C.:“The Effect of Water
Influx on P/Z-Cumulative Gas Production Curves,”J. Pet. Tech. (March 1965)
287-291.

9.Byrne, W. B. and Morse, R. A.:“The Effects of Various Reservoir and Well
Parameters on Water Coning Performance.”paper SPE 4287 presented at the SPE
3rd Numerical Simulation of Reservoir Derformance Symposium, Houston, Jan,
10-12, 1973.

10.CMG,2000.User’s Guide IMEX Advanced Oil/Gas Reservoir Simulator. Computer
Modelling Group Ltd., Calgary,Aalberta Canada,746pp.

11.Chambers, B. Karra, S., and Mortimer, L.:“Use of Type Curve Analysis in
Predicting the Behavior of a Water Drive Reservoir,”J. Can. Pet. Tech.
(January-March 1980) 54-60.

12.Chierici, C. L., Ciucci, G. M., and Long, G.:“Experimental Research on Gas
Saturation Behind the Water Front in Gas Reservoirs Subjected to Water
Drive,”paper 17, proceedings of the World Petroleum Congress, Frankfurt am
Main, Germany, June 25, 1963, 483-498.

13.Dake, L. P.: “Fundamentals of Reservoir Engineering, Elsevier Scientific
Publishing CO., New York City (1978)

14.Fetkovich, M. J.:“Decline Curve Analysis Using Type Curves,”J. Pet. Tech.
(June 1980) 1065-1077.

15.Hoyland, L.A., Papatzacos, P. Skjaeveland, S.M.: “Critical rate for
water coning: Correlation and analytical solution,” paper SPE 15855,
p.59-64, 1986.

16.Lee, J.W.: “Well Testing,” Society of Petroleum Engineering of AIME,
Dallas, Texas (1982).

17.Meyer, H. J. and Searcy, D. F.:“Analog Study of Water Coning,”Trans., AIME
(1956) 302.

18.Meyer, H.L. and Gardner, A.O.: “Mechanics of two Immiscible Fluid in
Porous Media, ” Journal of Applied Physics, Vol. 25, No. 11, p.1400, 1954.

19.Muskat, M.,”Physical Principles of Oil Production”（1949）,Mc Graw-Hill,
New York

20.Muskat, M, and Wyckoff, R. D.:“An Approximate Theory of Water-Coning in
Oil Production,” Trans., AIME(1935)144-163

21.Schols, R.S.: “An Empirical Formula for the Critical Oil Production
Rate,” Erdoel Erdgas, Z., Vol. 88, No. 1, p.6-11, January 1972.

22.Settari, A. and Aziz, K.:“A Computer Model for Two Pirase Coning
Simulation,”Soc. Pet. Eng. J. (June 197) 221-236.

23.Sobocinski, D. P. and Cornelius, A. J.:“A Correlation for Predicting Water
Coning Time,”J. Pet. Tech. (Mat 1965) 594-600.

24.Sobocinski, D.P., Cornelius, A.J.: “A correlation for predicting water
coning time, ” paper SPE 894, p.594-600, 1964.

25.Yang, W., and Wattenbarger, R.A.: “Water coning calculations for vertical
and horizontal wells,” paper SPE 22931, p.459-470, 1991.

26.秦同洛、李璽、陳元千，“實用油藏工程方法”，石油工業出版社，北京，1989年7
月。

27.郎兆新，“油藏工程基礎”，石油大學出版社，山東省東營市，1991年3月。

28.黃瑞鴻、吳柏裕、楊池清，“八掌溪地區氣井水錐現象之研究”， 探採研究彙報，第
十五期，民國八十一年，p.77-84

29.曾繼忠、吳健一、王勝雄，“氣井水錐預測模式及堵水技術探討”，石油季刊，第41卷
第一期，民國九十四年，p.27-42

30.張智星，“MATLAB程式設計與應用”，清蔚科技，2000年