AvailableAvailable online online at www.sciencedirect.com at www.sciencedirect.com Procedia Engineering ProcediaProcedia Engineering Engineering 00 (2011) 29 (2012) 000–000 3600 – 3607 www.elsevier.com/locate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE) Numerical Simulation in Whole Space for Resistivity Logging through Casing under Approximate Conditions

Min Geng*, Huaqing Liang, Hongdong Yin, Dejun Liu, Yang Gao

Coll. of Geophysics and Information Engineering, China Univ. of Petroleum-Beijing, Beijing, China

Abstract

Resistivity logging through casing is a kind of logging technique obtaining formation resistivity by measuring casing potential and current. Measurement results are affected by formation parameters, logging tool precision, etc. Based on casing potential and current formula, as well as boundary conditions, longitudinal step-formation model of limited thickness in whole space is established to realize numerical simulation of Russian resistivity logging through casing. Simulation results have lamination ability for formation of different thickness and the approximate degree between apparent and real formation resistivity decreases as the formation thickness decreases. According to actual measuring limitation, numerical simulation is realized again under approximate conditions. The latter simulation results have the similar lamination ability with the former. For thin layer, the approximate degree of the latter is a little worse than the former. Measured data process verifies the feasibility of approximate calculation method.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology

Keywords: Numerical simulation in whole space; Resistivity logging through casing; Approximate condition; Simulation result

1. Introduction

Resistivity logging through casing is a kind of logging technique measuring formation resistivity in metal cased holes. It has lots of advantages, such as large detecting depth, adaptability to formation with low porosity and low mineralization, as well as resistibility to heterogeneity of well and formation. So it becomes main method to analyze remaining oil saturation, determine remaining oil evaluation parameter and evaluate reservoir producing conditions now. From 1990s Kaufman put forward the electrical field

* * Corresponding author.. E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.538 2 Min Geng, Huaqing Liang,Min HongdongGeng et al. Yin, / Procedia Dejun Liu Engineering and Yang 29Gao (2012) / Pro 3600cedia –Engineering 3607 00 (2011) 000–000 3601 model in a borehole with a casing and set up the transmission-line measuring theory [1-2], resistivity logging technique through casing has kept developing [3-5]. Research on numerical simulation of resistivity logging through casing is mainly based on the formation model of unlimited thickness in half space [6-8]. While logging tool works in the formation model of limited thickness in whole space. So it is necessary to carry out numerical simulation of resistivity logging through casing in more reasonable formation model in order to promote the application of resistivity logging technique through casing.

2. Measuring principle of resistivity logging through casing

2.1. Transmission-line model

Electrical field in a cased-hole can be analyzed with the transmission-line model [9]. There is only vertical electric field in cased-hole and radical component in formation, shown in Fig. 1 [10].

casing I(z+dz) z+dz formation

z I(z) main current leakage current injected current

Fig. 1 Current distribution in cased hole

Resistivity of casing is far smaller than that of formation, so most current flows along the casing and only a small part leaks into the formation, called leakage current, which is just the measuring object of resistivity logging through casing. Taking a dz long section of casing as research object, variation rate along casing of potential U and current I is represented by Eqs. (1) and (2). Rz is longitudinal resistance of unit length casing and Rt is lateral resistance of unit length formation. dU = −IR (1) dz z dI U = − dz R t (2) After doing the derivation over Eq. (1), Eq. (3) is obtained with the substitution of Eqs. (1) and (2). d 2U 1 dR dU R − z − z U = 0 dz 2 R dz dz R z t (3) In actual measurement, derivative is replaced by potential difference and Eq. (4) is regarded as the theoretical basis of Russian resistivity logging through casing. Δ2U 1 dR ΔU R − z − z U = 0 Δz 2 R dz Δz R z t (4)

2.2. Calculation formula of formation resistivity

Fig. 2 shows the measurement principle of Russian resistivity logging tool through casing [11].

3602 Min Geng, Huaqing Liang, HongdongMin Geng Yin, et al. Dejun / Procedia Liu and Engineering Yang Gao /29 Pro (2012)cedia 3600Engineering – 3607 00 (2011) 000–000 3

power ’ N B A1 I U U adjacent casing M1 ΔU1 N M2 ΔU2 A2

Fig. 2 Schematic diagram of Russian resistivity logging tool through casing

The tool is composed of two power supply electrodes of A1 and A2, one potential electrode of U and three measuring electrodes of M1, N and M2. Measuring process is divided into two stages. First, casing potential U(IA1), casing current IA1, as well as first-order potential difference ΔUM2N(IA1) andΔUM1N(IA1) are measured by powering to A1 (upper power supply mode). Secondly, the data of U(IA2), IA2, ΔUM2N(IA2) andΔUM1N(IA2) are measured by powering to A2 (lower power supply mode). Then first-order and second- order potential difference between M2 and M1 are calculated from these two groups of data. Eqs. (5) and (6) are obtained with the substitution of all these data to Eq. (4). Eq. (7) is derived through the simultaneous solution of Eqs. (5) and (6) [7-8]. Δ2U (I ) 1 dR ΔU (I ) R M 2M1 A1 − z M 2M1 A1 − z U (I ) = 0 (M M / 2) 2 R dz M M R A1 1 2 z 1 2 t (5) Δ2U (I ) 1 dR ΔU (I ) R M 2M1 A2 − z M 2M1 A2 − z U (I ) = 0 (M M / 2) 2 R dz M M R A2 1 2 z 1 2 t (6) 2 2 ⎡ ⎤ ⎡ΔUM2M1(IA1)−ΔUM2M1(IA1) ΔUM2M1(IA2)+ΔUM2M1(IA2)⎤ U(I A1)ΔUM2M1(I A2 )−U(I A2 )ΔUM2M1(I A1) ρa =k⎢ + ⎥ ⋅⎢ ⎥ I I −ΔU (I )Δ2U (I )+ΔU (I )Δ2U (I ) ⎣ A1 A2 ⎦ ⎣ M2M1 A1 M2M1 A2 M2M1 A2 M2M1 A1 ⎦ (7) where ρa is apparent formation resistivity, k is electrode array coefficient.

3. Numerical simulation in whole space

Numerical simulation of resistivity logging through casing can be implemented according to the following process, setting up formation model first, deriving analytic formula of potential and current in every formation secondly, calculating potential and current of measurement points thirdly and solving formation resistivity finally [12]. Numerical simulation helps to verify the correctness of measurement principle, as well as analyze the influence factors in measurement process.

3.1. Formation model

According to the measuring principle of resistivity logging through casing and Fig. 2, current injected into casing flows in both upward and downward directions and leaks into the formation gradually. Effective formation thickness corresponds to the casing length. Considering the longitudinal step feature of formation simultaneously, numerical simulation of resistivity logging through casing should be done with longitudinal step-formation model of limited thickness in whole space, shown in Fig. 3.

4 Min Geng, Huaqing Liang,Min HongdongGeng et al. Yin, / Procedia Dejun Liu Engineering and Yang 29Gao (2012) / Pro 3600cedia –Engineering 3607 00 (2011) 000–000 3603

z1 ρ1,n1

ρ1,i1 d H1 ρ 1,i1 o 1,1 H ρ2,1 d2,i2 ρ2,i2 H2

ρ2,n2 z2

Fig. 3 Formation model for numerical simulation

where o is the current injected point, namely A1 or A2 in Fig. 2; z1 and z2 are upward and downward directions respectively; H1 and H2 are formation thickness of upper and lower half space respectively; H is the sum of H1 and H2, namely casing length; ρ1,i1 and ρ2,i2 are formation resistivity of i1th and i2th formation in upper and lower half space respectively; d1,i1 and d2,i2 are longitudinal boundary coordinates of i1th and i2th formation in upper and lower half space respectively; n1 and n2 are layer number of formation in upper and lower half space respectively.

3.2. Numerical simulation equations

After doing derivation and simplification over Eq. (2), Eq. (8) is obtained. d 2 I R = z I dz 2 R t (8) In accordance with Eqs. (2) and (8), the analytic formula of current and potential of i1th and i2th layer in upper and lower space can be represented as following [6]. α −α I ( z ) = A e 1,i 1 z1 + B e 1,i 1 z1 1,i1 1 1,i1 1,i1 (9) α −α U (z ) = −ξ A e 1,i1 z1 + ξ B e 1,i1 z1 1,i1 1 1,i1 1,i1 1,i1 1,i1 (10) α −α I ( z ) = A e 2 ,i 2 z 2 + B e 2 ,i 2 z 2 2 ,i 2 2 2 ,i 2 2 ,i 2 (11) α −α U (z ) = −ξ A e 2,i 2z2 + ξ B e 2,i 2z2 2,i2 2 2,i2 2,i2 2,i2 2,i2 (12) where A1,i1, B1,i1, A2,i2 and B2,i2 are coefficients to be solved; ξ1,i1, ξ2,i2, α1,i1 and α2,i2 are feature parameters of formation, which are calculated by Eqs. (13) and (14). R α = z R t (13) ξ = R α t (14) Potential and current value of every measurement point can be calculated in accordance with the boundary conditions shown as Eqs. (15) to (21). Replacing IA1 and IA2 with IM1 and IM2, formation resistivity is obtained by substituting calculated values into Eq. (7) in upper and lower power supply mode. I (0) + I (0) = I 1,1 2 ,1 (15) U (0) = U (0) 1,1 2 ,1 (16) I ( H ) = I ( H ) = 0 1,n1 1 2 ,n 2 2 (17) I (d ) = I (d ) 1,i1 1,i1 1,i1+1 1,i1 (18) 3604 Min Geng, Huaqing Liang, HongdongMin Geng Yin, et al. Dejun / Procedia Liu and Engineering Yang Gao /29 Pro (2012)cedia 3600Engineering – 3607 00 (2011) 000–000 5

U (d ) = U (d ) 1,i1 1,i1 1,i1+1 1,i1 (19) I (d ) = I (d ) 2 ,i 2 2,i 2 2,i 2 +1 2 ,i 2 (20) U (d ) = U (d ) 2 ,i 2 2 ,i 2 2 ,i 2 +1 2 ,i 2 (21) where I is the total current injected into casing in upper or lower power supply mode.

3.3. Numerical simulation results

Fig. 4 shows the geometric size of resistivity logging tool through casing.

A1 M1 N M2 A2 1.3m 0.5m 0.5m 1.3m

Fig. 4 Diagram of electrode arrangement

-7 Setting the following parameters: total injected current I = 7A, casing resistivity rc = 2×10 Ωm, casing radius a = 0.1m and casing thickness Δa = 0.005m. Numerical simulation results, shown in Fig. 5, are obtained in accordance with the size of logging tool.

120 real resistivity 120 real resistivity .m)

apparent resistivity .m)

Ω apparent resistivity 100 100Ω 80 80 60 60 40 40 20

formation resistivity ( resistivity formation 20 formation resistivity ( resistivity formation 0 0 50 100 150 200 250 5 10 15 20 25 formation depth (m) formation depth (m)

(a) (b)

Fig. 5 Numerical simulation results of apparent resistivity: (a) thick layer; (b) thin layer

From Fig. 5a, for thick layer numerical simulation result of apparent resistivity approximates real formation resistivity perfectly and has strong lamination ability. From Fig. 5b, for thin layer some difference between numerical simulation result of apparent resistivity and real formation resistivity exists and the difference becomes bigger as formation becomes thinner. However the result still has apparent lamination ability and the extreme of apparent resistivity also approximates real formation resistivity.

3.4. Numerical simulation under approximate conditions

Upward current IM2 and downward current IM1 are used in numerical simulation, which are unable to be measured in actual underground measurement. In order to get the approximate values of IM2 and IM1 three approximate conditions are adopted. First, in one measuring process distribution ratio of upward to downward current is unchanged in upper and lower supply mode. Second, ratio of current approximates to 6 Min Geng, Huaqing Liang,Min HongdongGeng et al. Yin, / Procedia Dejun Liu Engineering and Yang 29Gao (2012) / Pro 3600cedia –Engineering 3607 00 (2011) 000–000 3605 that of voltage. Third, sum of IM2 and IM1 approximates to that of IA2 and IA1. Figs. 6 and 7 show the numerical simulation results under three approximate conditions of the same formation layers in Fig. 5.

0.02 0.02 0.01 0.01 0 0 50 100 150 200 250 5 10 15 20 25 formation depth (m) formation depth (m)

0.02 0.02 0.01 0.01 0 0 50 100 150 200 250 5 10 15 20 25 formation depth (m) formation depth (m) 0.04 0.04 0.02 0.02 0 50 100 150 200 250 0

total(A) downward(A) upward(A) 5 10 15 20 25 total(A) downward(A) upward(A) formation depth (m) formation depth (m) (a) (b)

Fig. 6 Numerical simulation results of casing current error under approximate conditions: (a) thick layer; (b) thin layer

120 real resistivity 120 real resistivity apparent resistivity .m) 100 .m) apparent resistivity Ω 100Ω 80 80 60 60 40 40 20 20 formation resistivity ( resistivity formation 0 ( resistivity formation 0 50 100 150 200 250 5 10 15 20 25 formation depth (m) formation depth (m) (a) (b)

Fig. 7 Numerical simulation results of apparent formation resistivity under approximate conditions: (a) thick layer; (b) thin layer

Fig. 6 shows the error of IM2 and IM1 under approximate conditions. Maximum current error appears at the position where resistivity ratio of surrounding rock to formation reaches maximum value. In given formation model, resistivity ratio up to 10:1, the maximum error of upward and downward current is about 0.02199A (only 0.3141% of total injected current) and the maximum error of sum of upward and downward current is no more than 0.04020A (only 0.5743% of total injected current). So it is feasible to substitute approximate current values into calculation formula of apparent resistivity. Fig. 7a shows the apparent resistivity response of thick layer under approximate conditions. Numerical simulation result still approximates real formation resistivity perfectly. Fig. 7b shows the apparent resistivity response of thin layer under approximate conditions. Numerical simulation result under approximate conditions is just a little worse than that under real conditions. It indicates that current approximation only has a little influence on apparent resistivity result. The above simulation results provide the powerful basis for calculating apparent resistivity with approximate current values in actual underground measurement.

4. Measured data process 3606 Min Geng, Huaqing Liang, HongdongMin Geng Yin, et al. Dejun / Procedia Liu and Engineering Yang Gao /29 Pro (2012)cedia 3600Engineering – 3607 00 (2011) 000–000 7

Apparent resistivity curve of cased hole, shown as dashed line in Fig. 8, is calculated from data provided by resistivity logging tool and approximate current values of IM1 and IM2 based on the approximate method used in numerical simulation. Its variation trend is similar to that of open hole, shown as solid line in Fig. 8. Therefore the current approximate method is basically feasible.

80 open hole resistivity .m)

Ω 70 cased hole resistivity 60 50 40 30

apparent resistivity ( resistivity apparent 20 10 0 1075 1080 1085 1090 1095 1100 1105 formation depth (m)

Fig. 8 Apparent resistivity curve comparison

5. Conclusion

Longitudinal step-formation model of limited thickness in whole space is established to realize numerical simulation. The following conclusions are obtained. • Apparent resistivity has lamination ability for formation of different thickness. The approximate degree between that and real formation resistivity decreases as the formation thickness decreases. However the extreme of apparent resistivity still approximates real formation resistivity for thin layer. • Apparent resistivity, calculated with approximate current values, has the similar lamination ability with that obtained with the real current values. For thin layer, the approximate degree between apparent and real formation resistivity is a little worse than that under real conditions. • Apparent resistivity curve of cased hole, calculated with approximate current values, has the similar variation trend with that of open hole. The above conclusions establish the foundation for further research of numerical simulation of resistivity logging through casing, correction char acquisition and measured data correction. These are certain to promote the practical application of Russian resistivity logging through casing.

Acknowledgements

The research reported herein is supported by Science Foundation of China University of Petroleum- Beijing with Grant No. FZG201003.

References

[1] Kaufman A A. The electrical field in a borehole with a casing. Geophysics; 1990, 55(1), 29-38. [2] Kaufman A A, Wightman W E. A transmission-line model for electrical logging through casing. Geophysics; 1992, 58(12), 1739-1747. 8 Min Geng, Huaqing Liang,Min Hongdong Geng et al. Yin, / Procedia Dejun Liu Engineering and Yang Gao29 (2012) / Pro cedia3600 Engineering– 3607 00 (2011) 000–000 3607

[3] Vail W B, Momii S T, Woodhouse R, et al. Formation resistivity measurements through metal casing. SPWLA 34th Ann Logging Symposium, Calgary, Alberta, Canada; 1993, 13-16. [4] Schenkel C J, Morrision H F. Electrical resistivity measurement through metal casing. Geophysics; 1994, 59(4), 1072-1082. [5] Benimeli D, Levesque C, Rouault G, et al. A new technique for faster resistivity measurements in cased holes. Paper Y, SPWLA 43rd Annual Logging Symposium; 2002, 2-5. [6] Liu Fuping, Gao Jie, Sun Baodian, et al. Forward calculation of the resistivity logging response through casing by transmission line equation for multi-layer formations. Chinese J. Geophys; 2007, 50(6), 1905-1913. [7] Gao Jie, Liu Fuping, Bao Dezhou, et al. Responses simulation of through-casing resistivity logging in heterogeneous- casing wells. Chinese J. Geophys; 2008, 51(4), 1255-1261. [8] Xie Jinzhuang, Yang Jinghai, Liu Fuping, et al. Study on the environment effect and its correction method for the formation resistivity logging data through casing. Progress in Geophys; 2010, 25(1), 258-265. [9] Gao Jie, Ke Shizhen, Wei Baojun, et al. Introduction to numerical simulation of electrical logging and its development trend. Well logging technology; 2010, 34(1), 1-5. [10] Zheng Lu, Zhang Jiatian, Yan Zhengguo. An overview of the cased hole formation resistivity logging calibration. Petroleum instruments; 2010, 24(1), 7-8. [11] Wang Zhengguo. Study on the Russian through casing resistivity logging and its application. Well logging technology; 2009, 33(4), 374-378. [12] Cheng Xi, He Sanyu, Dang Hailong, et al. Analysis of cased-hole resistivity logging responses. Petroleum instruments; 2007, 21(5), 60-62.