# Rate Decline Analysis for Horizontal Wells with Multiple Sections.

1. IntroductionIn the past decades, horizontal wells have been widely used in exploration and development of low permeability reservoirs and unconventional reservoirs [1-7]. Rate decline analysis for oil and gas reservoirs is needed for well performance evaluation and reservoir characterization [8]. Arps [9] proposed a series of decline charts to make it convenient for decline curves extrapolation. Then, Fetkovich [10] developed theoretical model and log-log curves to demonstrate both infinite and finite reservoir solutions. To analyze rate decline data where pressure drop and flow rate vary significantly, Blasingame [11] established method for transforming variable-rate system into equivalent constant system using material balance time function.

Many researchers investigated transient rate behavior of horizontal wells. Pratikno et al. [12] provided new type curves for rate decline analysis of fractured well. For analyzing tight oil reservoir, Clarkson and Pedersen [13] provided a method which combines traditional techniques (e.g., flow regime analysis, analytical model history matching, and type curve matching). Duong [14, 15] developed a novel approach to predict future rate for fracture-dominated well in unconventional reservoirs. With considering slippage and desorption in shale gas reservoir, Nobakht et al. [16] established an analytical methodology to analyze rate transient behavior. Belyadi et al. [17] evaluated the productivity of Marcellus Shale wells based on rate transient analysis. On the basis of the dynamic drainage area concept, Qanbari and Clarkson [18] developed a new rate-transient-analysis technique for shale reservoirs. Kuchuk et al. [19] presented the rate decline response of fractured horizontal wells for both conventional and unconventional formations.

In the last decade, numerous research works have been presented on RDA of horizontal wells [20-27]. Although most models in previous study focus on rate decline analysis with one whole section, they did not specifically address the problem that production rate distributes nonuniformly along horizontal wellbore. This may be caused by several reasons. On one hand, due to selective completion, total production rate is contributed from less than the entire well interval. On the other hand, even if the entire well interval is open, on account of reservoir heterogeneity and nonuniform formation damage, production rate of each section along horizontal wellbore is not the same. Some of horizontal sections have little or no contribution. Since this phenomenon exists in horizontal wells with selective completion and open-hole completion, when analyzing transient rate behavior, nonuniform distribution of parameters subject to each section (e.g., length, production rate, skin factor, and location) should be taken into consideration [28-30].

Pressure transient analysis (PTA) is a reliable method for evaluating well performance and reservoir characteristics of wells with multiple sections. Kamal et al. [31] presented an analytical model which allows for a horizontal well to consist of multiple sections. A new flow regime was found, which behaves as a horizontal line with the value of 0.5 divided by number of horizontal sections on pressure-derivative curve. Other PTA models are proposed in different works [32-36]. He et al. [37] proposed a method for estimating malfunctioning horizontal wellbore through bottom hole pressure data. Qin et al. [38] developed novel approach to evaluate nonuniform production distribution of multifractured horizontal wellbore based on PTA. During past years, although plenty of work has been developed for PTA of horizontal wells and multifractured horizontal wells with multiple sections [39-43], rate decline analysis of HWMS is still rarely documented.

The major objective of this paper is to establish an analytical model for analyzing transient rate behavior of HWMS. The effects of some sensitive parameters on decline curves are further investigated (i.e., production rate distribution along wellbore, number of horizontal sections, length of horizontal sections, and spacing between adjacent horizontal sections).

2. Physical Model of Horizontal Well with Multiple Sections

Figure 1 illustrates the physical model of HWMS (top view and side view). Major assumptions are as follows:

(i) Formation is circular-bounded with a constant thickness h, porosity [phi], horizontal permeability [k.sub.h], vertical permeability [k.sub.v], initial pressure [p.sub.i], and total compressibility [C.sub.t].

(ii) The HWMS consists of n sections. Due to formation damage, each section owns specific properties (e.g., length [L.sub.k], production rate [q.sub.k], and skin factor [S.sub.k]).

(iii) The fluid is assumed to be single-phase oil or gas in this model. Total production rate equals to q.

(iv) The gravity effect and capillarity effect are negligible.

3. Solution of HWMS Model

In the circular-bounded reservoir, the analytical solution for pressure drop of HWMS at any point can be derived on the basis of the instantaneous source solutions raised by Gringarten and Ramey [44] and Newman product method [45].

Assuming that an HWMS is located in a circular-bounded formation, horizontal wellbore can be regarded as continuous-line-sources with length of L. Through the integration of point-source functions, the line-source solutions can be derived.

For a single point source in the infinite-slab reservoir, it can be considered as a result of plane sources in three different directions (x-, y-, and z-).

By using Newman product method, the point-source function can be obtained:

[mathematical expression not reproducible] (1)

[[eta].sub.h] = [k.sub.h]/[phi][mu][C.sub.t] (2)

[[eta].sub.v] = [k.sub.v]/[phi][mu][C.sub.t] (3)

[beta] = [square root of ([[eta].sub.v]/[[eta].sub.h])] (4)

where dV represents flow rate per volume. [[eta].sub.h] and [[eta].sub.v] are diffusivity in horizontal and vertical directions, respectively.

Through the integration of point-source function, the line-source solution is given by

[mathematical expression not reproducible] (5)

where ds represents rate per unit length when t equals to [tau].

When there is a continuous flow with the rate of [q.sub.k] from the k-th section located at point ([x.sub.k], [y.sub.k], [z.sub.k]), the pressure drop at point (x, y, z) can be expressed as

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[h.sup.*] = h [square root of ([k.sub.h]/[k.sub.v])] = h/[beta] (10)

where [h.sup.*] means the formation thickness considering permeability anisotropy and [beta] is anisotropy coefficient.

Through the principle of superposition, pressure drop caused by multiple sections can be obtained:

[mathematical expression not reproducible] (11)

denoting

q = [n.summation over (k=1)] [q.sub.k] (12)

The dimensionless length and production rate of the k-th section, dimensionless length of horizontal well, and dimensionless time are defined as

[mathematical expression not reproducible] (13)

Other dimensionless variables are defined as follows:

[mathematical expression not reproducible] (14)

As a result, dimensionless pressure drop of an HWMS with n sections is given by

[mathematical expression not reproducible] (15)

[mathematical expression not reproducible] (16)

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible] (18)

Since formation damage caused by drilling and completion cannot be ignored, dimensionless pressure drops resulted by skin effect can be written as

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

By converting solution into Laplace domain, dimensionless pressure drop with considering the effect of wellbore storage can be expressed as

[mathematical expression not reproducible] (21)

On the basis of Laplace transformation, analytical solution for unsteady-state pressure drop of HWMS with incorporating skin effect and wellbore-storage effect at any point can be derived:

[[bar.p].sub.CD] ([C.sub.D], S, u) = [[bar.p].sub.D]/1 + [C.sub.D][u.sup.2][[bar.p].sub.D] (22)

where [[bar.P].sub.CD] and [[bar.p].sub.D] represent the dimensionless pressure with and without considering wellbore storage, respectively, and u is the Laplace transform variable. Through numerical inversion algorithm developed by Stehfest [46] and Duhamel principle [47], rate solution of HWMS can be obtained.

The circular-boundary effect can be taken into consideration by using the method from Pratikno et al. [12]:

[mathematical expression not reproducible] (23)

By using Duhamel principle [47], dimensionless rate solution of HWMS under constant pressure located in circular-bounded formation in Laplace space can be expressed as

[[bar.q].sub.D] (s) = 1/[s.sup.2] [[bar.p].sub.CD] (s) (24)

[[bar.q].sub.D] can be converted into the real space using the Stehfest inversion algorithm [46].

Furthermore, three types of rate-material balance time curves can be plotted.

(i) The normalized dimensionless decline production curve ([q.sub.Dd])

(ii) The normalized dimensionless decline production integral curve ([q.sub.Ddi])

(iii) The dimensionless decline production integral derivative curve ([q.sub.Ddid])

4. Model Comparison

In this part, Blasingame type curves are developed by use of HWMS model proposed in this paper and HWSS model in previous study.

For HWSS model, an HWSS with length of L and constant rate of q is located in circular-bounded reservoir. Since HWSS only consists of one section, in this case, n equals one. For HWMS model, length of each section equals 0.1L and production rate of each section is q/n. Two cases are further discussed (i.e., n=1, n=5). When n equals 1, producing section is located at the heel of horizontal wellbore. When n equals 5, sections are distributed uniformly along wellbore.

By comparing type curves from HWMS model with that from HWSS model, clear distinctions can be observed on Blasingame compound type curves except for the boundary-dominated flow period, shown in Figure 2. Three kinds of curves (i.e., [q.sub.Dd], [q.sub.Ddi], and [q.sub.Ddid]) belonging to compound type curves are shown in Figure 3. Compared to Figures 3(a) and 3(b), the use of production integral derivative smooths the derivative curve of measured data and makes it easier to identify different features as shown in Figure 3(c). Therefore, it is necessary and significant to establish the RDA model of HWMS to analyze the rate transient behavior, which enable us to better evaluate production performance.

5. Effect of Sensitive Parameters on Decline Curves

To understand the effects of main parameters on rate transient behaviors of a HWMS, we develop the Blasingame compound type curves to perform the sensitivity analysis through the RDA model proposed in this paper (i.e., production rate distribution along wellbore, number of horizontal sections, length of horizontal sections, and spacing between adjacent horizontal sections).

5.1. Production Rate Distribution along Wellbore. Dimensionless production rate is defined as the production rate of each producing section divided by total production rate of all sections ([q.sub.kD] = [q.sub.k]/[q.sub.total]). A horizontal well with length of L consists of three producing sections, and length of each section equals 0.1L. Three sections located at the heel, middle, and toe of wellbore, respectively, and production rate are unequal for these sections. Figure 4 shows the compound type curve. The red curves represent the normalized dimensionless decline production curves ([q.sub.Dd]), the green curves means the normalized dimensionless decline production integral curve ([q.sub.Ddi]), and the blue curves denote the dimensionless decline production integral derivative curve ([q.sub.Ddid]). The effects of production rate distribution along wellbore on production characteristic are obvious especially for the production integral derivative curve during the transient flow regime. The results tell us that we can estimate the production distribution of different locations along horizontal wellbore through the history matching of long-term production data. If the field production data can match with these three curves at the same time, the results can be reliable.

5.2. Number of Horizontal Sections. For number of horizontal sections, five cases are further discussed (i.e., n=1, 2, 3, 4, 5). In all cases, each section has the same length ([L.sub.k] = 0.1L), production rate ([q.sub.k] = [q.sub.total]/n), and skin factor, respectively. All producing sections distribute uniformly along wellbore. The compound type curves of different numbers of horizontal sections are shown in Figure 5. As the number of horizontal sections decreases, all curves move downward. There exist big differences between the results for n=1 and n=5 during the transient flow period. Although the distinctions between the normalized production curves ([q.sub.Dd]) and the normalized production integral curve ([q.sub.Ddi]) are not obvious enough, the production integral derivative curves ([q.sub.Ddid]) show clear differences among them. Therefore, the number of horizontal sections is sensitive and can be determined through matching the production data with the compound type curves in this paper.

5.3. Length of Horizontal Sections. Dimensionless length of horizontal sections equals length of each section divided by length of horizontal wellbore ([L.sub.kD] = [L.sub.k]/L). Two sections with the same dimensionless length are located at the heel and toe of horizontal well. Both sections have the same production rate, skin factor, etc. Figure 6 shows the effect of length of horizontal sections on type curves. The normalized production curves and the normalized production integral curve are quite different especially when dimensionless length of horizontal sections become smaller. The production integral derivative curves differ from other cases for different [L.sub.kD], which is benefit of estimating effective producing length of the horizontal well.

5.4. Spacing between Adjacent Horizontal Sections. Dimensionless spacing between adjacent horizontal sections is defined as the spacing between adjacent horizontal sections divided by length of horizontal wellbore ([DELTA][x.sub.D] = [DELTA]x/L). Five cases are designed to discuss the influences of dimensionless spacing on decline curves. As the distance from the second producing section to the heel becomes farther, decline types move up, shown in Figure 7. Though the influences of spacing between horizontal sections on type curves are not obvious compared to other parameters, it is possible for us to distinguish them based on the production integral derivative curves during the transient flow regime.

6. Conclusions

This paper establishes the RDA model of a HWMS and the compound rate decline type curves to analyze the effect of production distribution, number and length of producing horizontal sections, and spacing between sections on rate transient behaviors.

(1) Since different horizontal segments make unequal contribution to total production rates in fact, the effect of these phenomena on rate decline behaviors should be considered.

(2) The effects of production rate distribution along wellbore on rate characteristic are obvious especially for the production integral derivative curve during the transient flow regime, which makes it possible to estimate the production distribution through the history matching of production data.

(3) The normalized production curves and the normalized production integral curve are quite different especially when dimensionless length of horizontal sections become smaller. The production integral derivative curves differ from other cases for different [L.sub.kD], which is benefit of estimating effective producing length of the horizontal well.

(4) Although the distinctions between the normalized production curves and the normalized production integral curve are not obvious, the production integral derivative curves show clear differences among them. Therefore, the number of horizontal sections can be determined through matching the production data.

In conclusion, the proposed RDA model enables engineers to evaluate the rate performance of horizontal wells and interpret reservoir and well parameters more effectively and accurately (e.g., estimating the production distribution of different locations along horizontal wellbore and effective producing length, permeability, well controlled area and reserves, and skin factor) through the history matching of long-term production data. If the field production data can match well with these three curves at the same time, the results can be reliable.

Nomenclature C: Wellbore-storage coefficient, [atm.sup.-1] [C.sub.D]: Dimensionless wellbore-storage coefficient [C.sub.t]: Total compressibility, [atm.sup.-1] h: Formation thickness, cm [h.sup.*]: Formation thickness considering permeability anisotropy, cm [h.sub.D.sup.*]: Dimensionless formation thickness considering permeability anisotropy [k.sub.h]: Horizontal permeability, D [k.sub.v]: Vertical permeability, D L: Length of horizontal well, cm [L.sub.D]: Dimensionless length of horizontal well [L.sub.k]: Length of the k-th section, cm [L.sub.kD]: Dimensionless length of the k-th section n: Number of horizontal sections, dimensionless p: Pressure, atm [p.sub.CD]: Dimensionless pressure drop with considering wellbore-storage effect [p.sub.D]: Dimensionless pressure drop [p.sub.i]: Initial reservoir pressure, atm [p.sub.S]: Dimensionless pressure drop caused by skin effect [p.sub.SD]: Dimensionless total pressure drop with considering skin effect q: Total production rate, [cm.sup.3]/s [q.sub.Dd]: Normalized dimensionless decline production [q.sub.Ddi]: Normalized dimensionless decline production integral [q.sub.Ddid]: Normalized dimensionless decline production integral derivative [q.sub.k]: Production rate of the k-th section, [cm.sup.3]/s [q.sub.kD]: Dimensionless production rate of the k-th section s: Laplace-transformation parameter S: Skin factor [S.sub.k]: Skin factor of the k-th section t: Time, s [t.sub.D]: Dimensionless time u: Laplace transform variable x, y, z: Cartesian coordinates [x.sub.D], [y.sub.D], [z.sub.D]: Dimensionless Cartesian coordinate [x.sub.k], [y.sub.k], [z.sub.k]: Coordinates of the center of the k-th section [x.sub.kD], [y.sub.kD], [z.sub.kD]: Dimensionless coordinates of the center of the k-th section [phi]: Porosity, fraction [[eta].sub.h]: Diffusivity in horizontal direction, [cm.sup.3]/s [[eta].sub.v]: Diffusivity in vertical direction, [cm.sup.3]/s [mu]: Fluid viscosity, cP [beta]: Anisotropy coefficient [tau]: Time variable [DELTA]p: Pressure drop caused by all sections [DELTA][p.sub.k]: Pressure drop caused by the k-th section [[bar.p].sub.CD]: Dimensionless pressure in Laplace space with considering wellbore-storage effect [[bar.p].sub.D]: Dimensionless pressure in Laplace space without considering wellbore-storage effect [[bar.q].sub.D]: Dimensionless rate in Laplace space.

https://doi.org/10.1155/2018/2047365

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the National Science and Technology Major Projects (No. 2017ZX05013001-005, No. 2017ZX05009-004, and No. 2017ZX05069) and Science Foundation of China University of Petroleum, Beijing (No. C201605), for their financial support.

References

[1] E. Ozkan, R. Raghavan, and S. D. Joshi, "Horizontal-well pressure analysis," SPE Formation Evaluation, vol. 4, no. 4, pp. 567-16378, 1989.

[2] Z. Chen, X. Liao, X. Zhao, X. Dou, L. Zhu, and L. Sanbo, "A finite-conductivity horizontal-well model for pressure-transient analysis in multiple-fractured horizontal wells," SPE Journal, vol. 22, no. 4, pp. 1112-1122, 2017.

[3] Z. Rui, C. Li, F. Peng et al., "Development of industry performance metrics for offshore oil and gas project," Journal of Natural Gas Science and Engineering, vol. 39, pp. 44-53, 2017

[4] W. Ren, G. Li, S. Tian, M. Sheng, and L. Geng, "Adsorption and surface diffusion of supercritical methane in shale," Industrial & Engineering Chemistry Research, vol. 56, no. 12, pp. 3446-3455, 2017.

[5] Z. Chai, B. Yan, J. E. Killough, and Y. Wang, "An efficient method for fractured shale reservoir history matching: The embedded discrete fracture multi-continuum approach," in Proceedings of the SPE Annual Technical Conference and Exhibition, ATCE 2016, UAE, September 2016.

[6] T. Guo, Y. Li, Y. Ding, Z. Qu, N. Gai, and Z. Rui, "Evaluation of acid fracturing treatments in shale formation," Energy & Fuels, vol. 31, no. 10, pp. 10479-10489, 2017.

[7] Z. Rui, F. Peng, K. Ling, H. Chang, G. Chen, and X. Zhou, "Investigation into the performance of oil and gas projects," Journal of Petroleum Science and Engineering, vol. 38, pp. 12-20, 2017.

[8] Z. Rui, J. Lu, Z. Zhang et al., "A quantitative oil and gas reservoir evaluation system for development," Journal of Natural Gas Science and Engineering, vol. 42, pp. 31-39, 2017

[9] J. Arps, "Analysis of decline curves," Transactions of the AIME, vol. 160, no. 1, pp. 228-247, 1945.

[10] M. Fetkovich, "Decline curve analysis using type curves," Journal of Petroleum Technology, vol. 32, no. 6, pp. 1065-1077, 1980.

[11] T. A. Blasingame, T. L. McCray, and W. J. Lee, "Decline curve analysis for variable pressure drop/variable flowrate systems," in Proceedings of the SPE Gas Technology Symposium, Paper no. SPE 21513, Houston, Tex, USA, January 1991.

[12] H. Pratikno, J. A. Rushing, and T. A. Blasingame, "Decline curve analysis using type curves--fractured wells," in Proceedings of the SPE Annual Technical Conference and Exhibition, SPE-84287-MS, pp. 2023-2036, Society of Petroleum Engineers, Denver, Colo, USA, October 2003.

[13] C. R. Clarkson and P. K. Pedersen, "Tight oil production analysis: adaptation of existing rate-transient analysis techniques," in Proceedings of the Canadian Unconventional Resources and International Petroleum Conference, Calgary, Alberta, Canada, 2010.

[14] A. N. Duong, "Rate-decline analysis for fracture-dominated shale reservoirs," SPE Reservoir Evaluation and Engineering, vol. 14, no. 3, pp. 377-387, 2011.

[15] A. N. Duong, "Rate-decline analysis for fracture-dominated shale reservoirs: part 2," in Proceedings of the SPE/CSUR Unconventional Resources Conference--Canada, Calgary, Alberta, Canada, 2014.

[16] M. Nobakht, C. R. Clarkson, and D. Kaviani, "New and improved methods for performing rate-transient analysis of shale gas reservoirs," SPE Reservoir Evaluation and Engineering, vol. 15, no. 3, pp. 335-350, 2012.

[17] H. Belyadi, S. Yuyi, and J. Junca-Laplace, "Production analysis using rate transient analysis," in Proceedings of the SPE Eastern Regional Meeting, Morgantown, West Virginia, USA, 2015.

[18] F. Qanbari and C. R. Clarkson, "Rate-transient analysis of liquid-rich tight/shale reservoirs using the dynamic drainage area concept: Examples from North American reservoirs," in Proceedings of the SPE Low Perm Symposium 2016, USA, May 2016.

[19] F. Kuchuk, K. Morton, and D. Biryukov, "Rate-Transient analysis for multistage fractured horizontal wells in conventional and un-conventional homogeneous and naturally fractured reservoirs," in Proceedings of the SPE Annual Technical Conference and Exhibition, Dubai, UAE, 2016.

[20] J. F. Manrique and B. D. Poe, "A unique methodology for evaluation of multifractured wells in stacked-pay reservoirs using commingled production and rate transient analysis," in Proceedings of the SPE Annual Technical Conference and Exhibition, Anaheim, California, U.S.A., 2007

[21] R. O. Bello and R. A. Wattenbarger, "Rate transient analysis in naturally fractured shale gas reservoirs," in Proceedings of the CIPC/SPE Gas Technology Symposium 2008 Joint Conference, Calgary, Alberta, Canada, 2008.

[22] D. Ilk, J. A. Rushing, A. D. Perego, and T. A. Blasingame, "Exponential vs. hyperbolic decline in tight gas sands -understanding the origin and implications for reserve estimates using arps' decline curves," in Proceedings of the SPE Annual Technical Conference and Exhibition, ATCE 2008, pp. 4637-4659, USA, September 2008.

[23] P. P. Valko and W. J. Lee, "A better way to forecast production from unconventional gas wells," in Proceedings of the SPE Annual Technical Conference and Exhibition, Florence, Italy, 2010.

[24] D. M. Anderson, P. Liang, and V. O. Mangha, "Probabilistic forecasting of unconventional resources using rate transient analysis: case studies," in Proceedings of the SPE Americas Unconventional Resources Conference, Pittsburgh, Pa, USA, 2012.

[25] B. D. Poe, "Production performance evaluation of wells completed in unconventional reservoirs using capillary pressure data and relative permeability effects," in Proceedings of the SPE Annual Technical Conference and Exhibition, Amsterdam, The Netherlands, 2014.

[26] M. A. Torcuk, B. Kurtoglu, P. Fakcharoenphol, and H. Kazemi, "Theory and application of pressure and rate transient analysis in unconventional reservoirs," in Proceedings of the SPE Annual Technical Conference and Exhibition, New Orleans, La, USA, 2013.

[27] H. A. Al-Khaldi, O. H. Al-Obathani, and N. M. Anisur Rahman, "A thorough investigation into the transient-pressure data from horizontal oil wells equipped with electrical submersible pumps," in Proceedings of the SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition, Dammam, Saudi Arabia, 2017.

[28] Y. He, S. Cheng, J. Qin et al., "Successful application of well testing and electrical resistance tomography to determine production contribution of individual fracture and water-breakthrough locations of multifractured horizontal well in changqing oil field, China," in Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, Te, USA, 2017.

[29] Y. He, S. Cheng, L. Li et al., "Waterflood direction and front characterization with four-step work flow: a case study in changqing oil field China," SPE Reservoir Evaluation and Engineering, vol. 20, no. 3, pp. 708-725, 2017

[30] Y. He, S. Cheng, L. Li et al., "Waterflood direction and front characterization with multiple methods: A case study in Changqing Oilfield, China," in Proceedings of the SPE Oil and Gas India Conference and Exhibition, India, November 2015.

[31] M. Kamal, I. Buhidma, S. Smith, and W. Jones, "Pressure-transient analysis for a well with multiple horizontal sections," in Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, Texas, 1993.

[32] T. Yildiz and E. Ozkan, "Transient pressure behaviour of selectively completed horizontal wells," in Proceedings of the SPE Annual Technical Conference and Exhibition, New Orleans, La, USA, 1994.

[33] F. J. Kuchuk, C. Lenn, P. Hook, and P. Fjerstad, "Performance evaluation of horizontal wells," in Proceedings of the 1998 Asia Pacific Conference on Integrated Modelling for Asset Management, pp. 231-243, March 1998.

[34] B. Toufik, D. Tiab, and S. Jokhio, "Effect of non-uniform skin on finite conductivity horizontal well," in Proceedings of the 2003 SPE Production and Operations Symposium, pp. 511-522, USA, 2003.

[35] T. Yildiz, "The impact of nonuniform formation damage on horizontal well performance," in Proceedings of the SPE International Symposium and Exhibition on Formation Damage Control, Lafayette, La, USA, 2006.

[36] S. J. Al Rbeawi and D. Tiab, "Effect of the number and length of zonal isolations on pressure behavior of horizontal wells," in Proceedings of the SPE Production and Operations Symposium, Oklahoma City, Oa, USA, 2001.

[37] Y. He, S. Cheng, J. Qin et al., "A semianalytical approach to estimate the locations of malfunctioning horizontal wellbore through bottom-hole pressure and its application in Hudson oilfield," in Proceedings of the in Proceedings of the SPE Middle East Oil Gas Show and Conference, Manama, Bahrain, 2017.

[38] J. Qin, S. Cheng, Y. He et al., "Estimation of non-uniform production rate distribution of multi-fractured horizontal well through pressure transient analysis: model and case study," in Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, Te, USA, 2017

[39] Y. He, S. Cheng, S. Li et al., "A semianalytical methodology to diagnose the locations of underperforming hydraulic fractures through pressure-transient analysis in tight gas reservoir," SPE Journal, vol. 22, no. 3, pp. 924-939, 2017

[40] H. Tang, Z. Chai, B. Yan, and J. Killough, "Application of multi-segment well modeling to simulate well interference," in Proceedings of the Unconventional Resources Technology Conference, Austin, Te, USA, July 2017

[41] H. Tang, A. R. Hasan, and J. Killough, "A fully-coupled wellbore-reservoir model for transient liquid loading in horizontal gas wells," in Proceedings of the SPE Annual Technical Conference and Exhibition 2017, USA, October 2017

[42] J. Qin, Y. Liu, Y. Feng, Y. Ding, L. Liu, and Y. He, "New well pattern optimization methodology in mature low-permeability anisotropic reservoirs," Journal of Geophysics and Engineering, vol. 15, no. 1, pp. 93-105, 2018.

[43] Y. He, S. Cheng, L. Hu et al, "A pressure transient analysis model of multi-fractured horizontal well in consideration of unequal production of each fracture," Journal of China University of Petroleum (Edition of Natural Science), vol. 41, no. 4, pp. 116-123, 2017.

[44] A. C. Gringarten and H. J. Ramey Jr., "The use of source and green's function in solving unsteady-flow problem in reservoir ," SPE Journal, vol. 13, no. 5, pp. 285-296, 1973.

[45] A. B. Newman, "Heating and cooling rectangular and cylindrical solids," Industrial and Engineering Chemistry, vol. 28, no. 5, pp. 545-548, 1936.

[46] H. Stehfest, "Remark on algorithm 368 [D5]: Numerical inversion of Laplace transforms," Communications of the ACM, vol. 13, no. 10, p. 624, 1970.

[47] L. G. Thompson and A. C. Reynolds, "Analysis of variable-rate well-test pressure data using duhamel's principle," SPE Formation Evaluation, vol. 1, no. 5, pp. 453-469, 1986.

Jiazheng Qin (iD), (1,2) Shiqing Cheng (iD), (1,2) Youwei He, (1,2) Dingyi Li, (1,2) Jia Zhang, (1,2) Dong Feng, (1,2) and Haiyang Yu (iD), (1,2)

(1) State Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum, Beijing 102249, China

(2) MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

Correspondence should be addressed to Shiqing Cheng; chengsq@cup.edu.cn

Received 19 October 2017; Accepted 19 November 2017; Published 19 November 2018

Academic Editor: Mohammad Sarmadivaleh

Caption: Figure 1: Physical model of HWMS (top view and side view).

Caption: Figure 2: Blasingame compound type curves of HWMS and HWSS.

Caption: Figure 3: Blasingame type curve according to the solution of constant rate: (a) [q.sub.Dd]; (b) [q.sub.Ddi]; (c) [q.sub.Ddid].

Caption: Figure 4: The effect of rate distribution on type curves.

Caption: Figure 5: The effect of number of horizontal sections on type curves.

Caption: Figure 6: The effect of length of two sections on type curves.

Caption: Figure 7: The effect of spacing between adjacent horizontal sections on type curves.

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Title Annotation: | Research Article |
---|---|

Author: | Qin, Jiazheng; Cheng, Shiqing; He, Youwei; Li, Dingyi; Zhang, Jia; Feng, Dong; Yu, Haiyang |

Publication: | Geofluids |

Geographic Code: | 9CHIN |

Date: | Jan 1, 2018 |

Words: | 4759 |

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