A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs

Dian-Fa Du; Yan-Yan Wang; Yan-Wu Zhao; Pu-Sen Sui; Xi Xia

doi:10.1007/s12182-017-0159-0

A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs

Du, Dian-Fa; Wang, Yan-Yan; Zhao, Yan-Wu; Sui, Pu-Sen; Xia, Xi 2017-04-12 00:00:00 Pet. Sci. (2017) 14:383–394 DOI 10.1007/s12182-017-0159-0 ORIGINAL PAPER A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs 1 2 1 3 1 • • • • Dian-Fa Du Yan-Yan Wang Yan-Wu Zhao Pu-Sen Sui Xi Xia Received: 11 July 2016 / Published online: 12 April 2017 The Author(s) 2017. This article is an open access publication Abstract Horizontal wells are commonly used in bottom the perforation distribution. Wells with denser perforation water reservoirs, which can increase contact area between density at the toe end and thinner density at the heel end wellbores and reservoirs. There are many completion may obtain low production, but the water breakthrough methods used to control cresting, among which variable time is delayed. Taking cumulative free-water production density perforation is an effective one. It is difﬁcult to as a parameter to evaluate perforation strategies is advis- evaluate well productivity and to analyze inﬂow proﬁles of able in bottom water reservoirs. horizontal wells with quantities of unevenly distributed perforations, which are characterized by different param- Keywords Bottom water reservoirs Variable density eters. In this paper, ﬂuid ﬂow in each wellbore perforation, perforation completion Inﬂow proﬁle Cresting model as well as the reservoir, was analyzed. A comprehensive Cumulative free-water production model, coupling the ﬂuid ﬂow in the reservoir and the wellbore pressure drawdown, was developed based on potential functions and solved using the numerical discrete 1 Introduction method. Then, a bottom water cresting model was estab- lished on the basis of the piston-like displacement princi- Bottom water reservoirs are widely distributed on earth and ple. Finally, bottom water cresting parameters and factors hold a large proportion of oil reserves (Islam 1993). Taking inﬂuencing inﬂow proﬁle were analyzed. A more system- China for example, there exist a large number of bottom atic optimization method was proposed by introducing the water reservoirs, most of which are developed using hori- concept of cumulative free-water production, which could zontal wells. Compared with vertical wells, the producing maintain a balance (or then a balance is achieved) between sections of horizontal wells have direct contact with oil stabilizing oil production and controlling bottom water reservoirs, which not only reduces the producing pressure cresting. Results show that the inﬂow proﬁle is affected by drawdown, but also ensures bottom water ﬂowing into the wellbore more smoothly in a form of ‘‘pushing upward’’ (Besson and Aquitaine 1990; Dou et al. 1999; Permadi & Dian-Fa Du et al. 1996; Zhao et al. 2006). Owing to these advantages, it dudf@upc.edu.cn can effectively control bottom water cresting. The need of & Yan-Yan Wang economic and effective development of bottom water 514375721@qq.com reservoirs leads to the appearance of many types of com- School of Petroleum Engineering, China University of pletion methods, such as barefoot well completion, slotted Petroleum, Qingdao 266580, Shandong, China screen well completion and perforation completion Sinopec Petroleum Exploration and Production Research (Ouyang and Huang 2005). Recently, partial completion, Institute, Sinopec, Beijing 100083, China variable density perforation completion and other new Shengli Production Plant, Shengli Oilﬁeld Branch Company, completion methods have been put forward to further Sinopec, Dongying 257051, Shandong, China control bottom water cresting (Goode and Wilkinson 1991; Sognesand et al. 1994). By accurately ﬁnding out the water Edited by Yan-Hua Sun 123 384 Pet. Sci. (2017) 14:383–394 production interval of horizontal wells, adopting plugging were analyzed using the developed model, and a more strategies or properly adjusting the bottom water inﬂow systematic optimization method was proposed by intro- proﬁle, these techniques can effectively prolong the life of ducing the concept of the cumulative free-water produc- production wells. And among all these techniques, perfo- tion, which could realize a balance between stabilizing oil ration completion, including variable density perforation production and controlling bottom water coning. and selectively perforated completion, plays a critical role in alleviating water cresting (Pang et al. 2012). Previously, scholars put more emphasis on the produc- 2 Productivity analysis for horizontal wells tivity evaluation of horizontal wells (Dikken 1990; Novy with variable density perforation completion 1995; Penmatcha et al. 1998) and bottom water cresting (Permadi et al. 1995; Wibowo et al. 2004; Chaperon 1986). For horizontal wells with perforation completed in bottom The published papers mostly focused on horizontal wells water reservoirs, formation ﬂuids ﬁrstly ﬂow into perfora- with open-hole completion. There is little research into tion holes before converging in the horizontal wellbore. horizontal wells with variable density perforation comple- Under this circumstance, the effect of perforation holes is tion, and the ones that exist turned out to be very prob- similar to that of short producing branches in inclined lematic: (1) The method for open-hole completion horizontal wells (Holmes et al. 1998). The real producing horizontal wells was used ignoring the ﬂuid ﬂow in per- part for horizontal wells should be quantities of perforation foration tunnels in these studies (Landman and Goldthorpe holes. Therefore, the perforation holes can be regarded as 1991; Yuan et al. 1996; Zhou et al. 2002). Then, a model ‘‘source-sink’’ term, and this kind of problem can be solved describing the damage zone must be introduced to char- with a source function. acterize the inﬂuence of perforation holes (Umnuaypon- wiwat and Ozkan 2000; Muskat and Wycokoff 2013). 2.1 Analysis of ﬂuid ﬂow near wellbore regions Some scholars utilized the numerical simulation method to discuss the impact of selective perforation on the produc- A model was built to describe the ﬂow of formation ﬂuid tivity of horizontal wells and built a single-phase ﬂow near horizontal wellbores, and the assumptions for this variable density perforation model for horizontal wells by model are as follows: two ﬁltration zones (Li et al. 2010). Since the seepage (1) The formation is homogeneous with a uniform resistance needs to be considered more precisely, espe- thickness; cially in the middle and later periods of the oilﬁeld (2) The horizontal permeability meets the following development, the existing results are somewhat inaccurate. basic relationship: K = K = K , and the vertical x y h (2) Conventional simpliﬁed models cannot analyze for- permeability is K = K . The target reservoir is z v mation pressure thoroughly and predict bottom water inﬁnite in the horizontal plane; cresting. Furthermore, a non-uniformly distributed bottom (3) The single-phase ﬂuid ﬂowing in the reservoir is water inﬂow proﬁle along the wellbore was obtained incompressible and the ﬂuid ﬂow obeys Darcy’s law; without considering the wellbore pressure drop (Guo et al. (4) The wellbore is horizontal, in which the perforations 1992). (3) In order to optimize completion parameters for are unevenly distributed; horizontal wells, oil production is usually viewed as the (5) The perforating direction is perpendicular to the only objective function. It is reasonable for horizontal wells wellbore, and the lengths as well as radius of all the in conventional reservoirs. However, it is not accurate for perforation tunnels are the same. horizontal wells located in bottom water reservoirs because of ignoring bottom water cresting, which decreases the Acoordinate system is established as shown in Fig. 1,in effective production period of wells (Luo et al. 2015). which the heel end of the wellbore is M (x , y , z ). The 0 0 0 0 In this paper, based on the precise consideration of the horizontal part of the wellbore (total length L, m) is divided ﬂuid ﬂow in each perforation, the ﬂow behavior in perfo- into N segments. Therefore, the length of each segment is rations, wellbores, as well as reservoirs, was analyzed. L/N. Coupling the ﬂuid ﬂow in reservoirs and wellbore pressure Different segments have different characteristic drop, a comprehensive model, which can be used to eval- parameters: the perforation density, n (i); the perforation uate productivity of horizontal wells, was developed based depth, l ; bore diameter, D ; phase angle x; and the initial p p on potential functions and solved using the numerical perforation angle x . The heel end is selected as the discrete method. Then, a model describing bottom water origin of this coordinate, and the x direction is parallel with the wellbore. The coordinates of any point (x, y, z)in cresting was established on the basis of the piston-like displacement principle. Finally, both the bottom water the jth perforation tunnel in the ith segment are as fol- cresting behavior and the factors inﬂuencing inﬂow proﬁle lows: 123 Pet. Sci. (2017) 14:383–394 385 i1 > L j x ði; j; tÞ¼ x þ sin h cos a þ sin h cos a þ t l sin c cos v > p 0 k k i i p ij ij > N n ðiÞ k¼1 < i1 L j ð1Þ y ði; j; tÞ¼ y þ sin h sin a þ sin h sin a þ t l sin c sin v ð0 t 1Þ p 0 k k i i p ij ij > N n ðiÞ k¼1 i1 > L j z ði; j; tÞ¼ z þ cos h þ cos h þ t l cos c : p 0 k i p ij N n ðiÞ k¼1 2 2 2 where h is the angle between the kth segment and the o U o U o U þ þ ¼ 0 ð3Þ vertical direction, k ¼ 1; 2; ...; i 1; a is the angle 2 2 02 ox oy oz between the x-axis and the projection of the kth segment pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where U ¼ K K l p,m /s; l is the oil viscosity, in the horizontal plane, k ¼ 1; 2; ...; i 1. h is the angle h v o o mPa s; and also the corresponding parameters of the per- between the ith segment and the vertical direction; a is the angle between the x-axis and the projection of the ith foration become: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ segment in the horizontal plane; c is the angle between ij 0 2 2 2 the jth perforation in the ith producing part of the hori- l ¼ l b cos c þ sin c ; p ij ij zontal well and the z-axis; v is the angle between the x- ij b sin c axis and the projection in the xy plane of the jth perfo- ij sin c ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ij ration in the ith segment of the horizontal well; in this 2 2 1 2 b cos c þ sin c ij ij paper, when all the perforation is perpendicular to the cos c ij horizontal wellbore, then v = p/2. i,j cos c ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ij 2 1 2 According to the classical theory of ﬂow in porous b b cos c þ sin c ij ij media (Cheng 2011), the ﬂuid ﬂow at any point in an In order to simplify the solution procedure, one inﬁnite reservoir obeys the Laplace equation. So we have: assumption that the ﬂow rates of all perforations in each 2 2 2 o p o p K o p segment are equal, is proposed (Wang et al. 2006): þ þ ¼ 0 ð2Þ 2 2 2 ox oy K oz q ðiÞ ra q ¼ ð4Þ where p is the pressure, MPa; K is the vertical perme- Dxn ðiÞ 2 2 ability, lm ; and K is the horizontal permeability, lm . where q is the ﬂow rate of each perforation at the ith Substituting linear transformation z ¼ bz, where p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 3 segment, m /s; q ðiÞ is the ﬂow rate of the ith segment, m / ra b ¼ K =K , into Eq. (2) gives: h v s; Dx is the length of the ith segment, which is equal to L/N, m; and n ðiÞ is the perforation density of the ith segment, holes/cm. Compared with the length of the horizontal wellbore, the (a) (b) L perforation is rather short. Therefore, it can be regarded as an inﬁnitesimal line source. After integrating the point sink solution over the perforation direction, the pressure response of any perforation hole in the formation is obtained. For example, the potential at point M(x, y, z ) caused by the jth perforation in the ith segment can be described as: q ði; jÞ (c) Visible perforation U ðx; y; z Þ¼ ds þ C ij ij Invisible perforation 4pl r Nq ðiÞ ra ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ds þ C ij 2 2 2 0 0 4pl Ln ðiÞ ðx xÞ þðy yÞ þðz z Þ p p p p Fig. 1 Schematic diagram near the horizontal wellbore in a bottom water reservoir ð5Þ 123 386 Pet. Sci. (2017) 14:383–394 with The integration of Eq. (5) is as follows: þ1 0 X r þ r þ l Nq ðiÞ 1ij 2ij 0 0 0 0 0 ra p / ¼ n 2h þ 4nh z ði; j; 0Þ; 2h þ 4nh ij ij pij U ðx; y; z Þ¼ ln þ C ð6Þ ij ij 4pl Ln ðiÞ r þ r l n¼1 p p 1ij 2ij 0 0 0 0 0 z ði; j; 1Þ; x; y; z þ n 4nh þ z ði; j; 0Þ; 4nh ij pij pij where q (i, j) is the ﬂow rate of the jth perforation tunnel in 0 0 the ith segment, m /s; r is the distance between the source þz ði; j; 1Þ; x; y; z pij 0 0 point M ðx; y; z Þ and the target point Mðx; y; z Þ; C is an p ij 0 0 0 0 n 2h þ 4nh þ z ði; j; 0Þ; 2h ij pij integration constant; r is the distance between the heel 1ij end and the target point; r is the distance between the toe 0 0 0 2ij þ4nh þ z ði; j; 1Þ; x; y; z pij end and the target point, and they observe the following 2l expressions, respectively: p 0 0 0 0 0 0 n 4nh z ði; j; 0Þ; 4nh z ði; j; 1Þ; x; y; z þ C ij pij pij i qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ nh 2 2 2 0 0 r ¼ ½x ði; j; 0Þ x þ½y ði; j; 0Þ y þ½z ði; j; 0Þ z 1ij p p ð14Þ ð7Þ The boundary pressure at the oil–water interface is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 0 0 assumed to be constant. Therefore, the potential at any r ¼ ½x ði; j; 1Þ x þ½y ði; j; 1Þ y þ½z ði; j; 1Þ z 2ij p p point of the formation may be expressed as: ð8Þ n ðiÞ N p X X N q ðiÞ ra where x (i, j, 0), y (i, j, 0), z (i, j, 0) and x (i, j, 1), y (i, j, 1), p p p p p U ðx; y; z Þ¼ U ln / ð15Þ ij e ij 4pl L n ðiÞ p p z (i, j, 1) are the coordinates of the left and right ends of the i¼1 j¼1 jth perforation in the ith producing part. Combining the deﬁnition of potential, the pressure at Based on the mirror image reﬂection and superposition any point of the formation can be given as follows: principle, the potential of the jth perforation tunnel for n ðiÞ the ith production segment at point M ðx; y; z Þ is N N p X X Nl q ðiÞ ra 0 o p ðx; y; z Þ¼ p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ln / obtained: wf;ij e ij 4pl L K K n ðiÞ p h v i¼1 j¼1 þ1 Nq ðiÞ ra 0 0 0 0 ð16Þ Uðx;y;z Þ¼ n ð2h þ 4nh z ði;j;0Þ; ij pij 4pl Ln ðiÞ p p n¼1 where p is the boundary pressure, MPa. 0 0 0 0 2h þ 4nh z ði;j;1Þ; x; y; z Þ pij For perforated completion, the perforation tunnels 0 0 0 0 0 þ n ð4nh þ z ði;j;0Þ; 4nh þ z ði;j;1Þ; x ;y;z Þ directly contact the formation. Therefore, the ﬂow potential ij pij pij 0 0 0 0 of some point, which is just located in the perforation, can be n ð2h þ 4nh þ z ði;j;0Þ ;2h ij pij obtained using Eq. (15). In this case, there are two points 0 0 0 þ 4nh þ z ði;j;1Þ; x ;y ;z Þ pij involved: the target point and the source point (perforation 0 0 0 n ð4nh z ði;j;0Þ; 4nh point). To get rid of singularity phenomenon, the central hole ij pij 0 0 0 in the wall is chosen as the jth perforation’s target point when z ði;j;1Þ; x ;y ;z Þþ C ð9Þ pij i calculating the distance between two perforation points. with Only calculating the pressure of one central point for the segment and regarding it as the pressure of the whole r þ r þ l 1ij 2ij segment will result in deviation when analyzing the seg- n ðe ; e ; x; y; z Þ¼ ln ð10Þ ij 0 1 r þ r l 1i 2i ment’s pressure of a horizontal wellbore. In this paper, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ taking the average over all the perforations’ pressure in the 2 2 2 r ¼ ½xði; j; 0Þ x þ½yði; j; 0Þ y þ½e z ð11Þ 1i 0 same segment and using the average value as the repre- qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sentative pressure of that segment, the pressure of the ith 2 2 2 r ¼ ½xði; j; 1Þ x þ½yði; j; 1Þ y þ½e z ð12Þ 2i 1 segment is as follows: Dxn ðiÞ According to the superposition principle, the potential at p ðiÞ¼ p ði; jÞð17Þ any point of the inﬁnite formation created by all the per- wf wf Dxn ðiÞ j¼1 foration tunnels of the horizontal well is: n ðiÞ where p (i) is the ﬂow pressure for all the perorations in N wf X X N q ðiÞ ra the ith segment, MPa. U ðx; y; z Þ¼ ln / þ C ð13Þ ij ij 4pl L n ðiÞ p p i¼1 j¼1 123 Pet. Sci. (2017) 14:383–394 387 2 2 Dp ði; jÞ¼ q v ðÞ i; j v ði; j þ 1Þ There are 2N variables required to be calculated by acc s s 13 2 2 analyzing pressure distribution along the wellbore: p (i) wf ¼ 3:5215 10 q ði; jÞ q ði; j þ 1Þ L L and q (i), (i = 1, 2, …, N). However, it only contains ra ð20Þ N equations in the ﬂow model [Eq. (15)]. Therefore, one more model is needed to describe pressure drop along the where q is the liquid density, kg/m ; D is the wellbore wellbore (Li et al. 1996, 2006). diameter, m; q (i,j) is the ﬂow rate along the wellbore, m / d; v ði; jÞ is the average velocity of the jth perforation in the ith segment, m/s; and f ði; jÞ is the friction coefﬁcient of fric 2.2 Wellbore pressure drop model the jth perforation in the ith segment. In Eq. (19), one parameter, called the friction coefﬁ- For perforated horizontal wells, the main idea to develop a cient, is introduced. The calculation of the friction coefﬁ- wellbore pressure drop model is to divide the horizontal cient is dependent on the Reynolds number. If the Reynolds wellbore into several segments, and each segment is sub- number is less than or equal to 2000, the ﬂow is laminar; divided into several smaller parts that only include one otherwise, it is turbulent ﬂow. And the expression for the perforation (Su and Gudmundsson 1994). Reynolds number is as follows: According to the analysis of the pressure drop in a wellbore, the total pressure loss can be written as: Qq Re ¼ 7:3682844 10 rl dp ðiÞ¼ dp ðiÞþ dp ðiÞþ dp ðiÞþ dp ðiÞð18Þ wf fric acc mix G where Re is the Reynolds number; Q is the axial ﬂow rate where dp (i) is the friction loss of the ith segment, MPa; fric along the wellbore, m /d; l is the viscosity of the ﬂowing dp (i) is the acceleration loss of the ith segment, MPa; acc ﬂuid, mPa s; and r is the wellbore radius, m. dp (i) is the mixing loss of the ith segment, MPa; and mix In addition, we also introduced a criterion when calcu- dp (i) is the gravity loss of the ith segment, MPa. lating the mixing loss between two perforation holes. The It should be noted that, in the previous section, a speciﬁc calculation method for frictional loss and mixing mechanical ﬁeld described by the ﬂow model is estab- losses is listed in Table 1. In Table 1, q is the critical rate, lished based on the potential function. Therefore, when c m /d; e is the surface roughness, mm. the potential function is used to deal with ﬂow problems, When calculating the mixing loss, one parameter, Dp , per the pressure loss caused by viscous force has been con- is introduced. It is the frictional loss showing up after sidered. As we all know, the mathematical expression of perforating and can be ﬁgured out using Eq. (19). ﬂuid potential is Kp/l, where K is the formation perme- The friction loss, acceleration loss and the mixing loss ability, p represents pressure, while l means the ﬂuid of the ith segment are listed below: viscosity, and it is used to describe the viscous force, which will lead to pressure loss along the perforation. So Dxn ðiÞ q q ði; jÞ 13 L it means that the viscous force has been taken into con- dp ðiÞ¼ 1:0862 10 f ði; jÞ ð21Þ fric fric D n ðiÞ j¼1 sideration in the ﬁrst model. In other words, when building wellbore pressure here, only four kinds of pres- Dxn ðiÞ sure loss should be calculated. dp ðiÞ¼ Dp ði; jÞ acc acc The calculation method of frictional loss and accelera- j¼1 Dxn ðiÞ tion loss between two perforation tunnels is expressed, p 13 2 2 ¼ 3:5215 10 q ði; jÞ q ði; j þ 1Þ respectively. L L j¼1 Dl qv ði; jÞ 13 s Dp ¼ 1:34 10 f ði; jÞ ð22Þ fric fric D 2 qq ði; jÞ n ðiÞ 13 L ¼ 1:0862 10 f ði; jÞ ð19Þ fric D n ðiÞ dp ðiÞ¼ Dp ði; jÞð23Þ mix mix j¼1 Table 1 Calculation method for friction coefﬁcient and mixing loss between two perforation holes Flow pattern Discriminant Calculation of friction coefﬁcient f (i, j) (Cheng Calculation of mixing losses Dp ði; jÞ (Cheng fric mix standard 2011) 2011) q ðiÞ 7 ra Laminar ﬂow q ði; jÞ\q L c Dp ði; jÞ 0:31 10 Reði; jÞ per Re(i;j) Dxn ðiÞq ði;jÞ p L hi hi 2 q ðiÞ 3 ra Turbulent q ði; jÞ [ q 1:11 L c 6:9 e 0:76 10 Reði; jÞ 1:8lg þ Dxn ðiÞq ði;jÞ p L Reði;jÞ 3:7D ﬂow 123 388 Pet. Sci. (2017) 14:383–394 In this paper, an iterative method is used to solve the If the horizontal well is inclined, the gravity loss is above-mentioned model, and the iterative process is shown non-ignorable, and the wellbore pressure drop model in Fig. 2. becomes: Dxn ðiÞ Dxn ðiÞ p p > X X q q ði; jÞ q > L 13 2 2 1:0862 10 f ði; jÞ þ 3:5215 10 ½q ði; jÞ q ði; j þ 1Þ > fric L L 5 5 D n ðiÞ D > j¼1 j¼1 n ðiÞ Dp ði; jÞ¼ þ qgDx cos h þ Dp ði; jÞ q ðiÞ 6¼ 0 ð24Þ wf i mix ra > j¼1 Dxn ðiÞ > 2 q q ði; jÞ > L 1:0862 10 f ði; jÞ þ qgDx cos h q ðiÞ¼ 0 > fric i ra : 5 D n ðiÞ j¼1 2.3 Coupling model 3 Analysis of inﬂow proﬁles in bottom water reservoirs When a horizontal well begins to produce reservoir ﬂuids, the ﬂuids in the perforations connect the wellbore and oil The bottom water rises fastest in the vertical plane of the reservoir together. Therefore, perforation is considered as horizontal wellbore (Cheng et al. 1994). In other words, the an inﬁnitesimal linear sink, which directly contacts reser- bottom water will ﬁrstly break through into the wellbore in voir and wellbore. Meanwhile, pressure responses are this plane due to the highest pressure gradients in this proﬁle. generated in the whole reservoir. The generated pressure Therefore, a complicated 3-D problem can be turned into a responses near the perforations are associated with oil 2-D problem in the xz proﬁle where the wellbore lies. The inﬂow in the radial direction of the horizontal wellbore, formation between the wellbore and the oil–water interface which can be calculated by utilizing the reservoir ﬂow is discretized according to the division of the horizontal model. Since the pressure in perforation holes is relevant to wellbore in the reservoir ﬂow model. And the total number of the wellbore pressure, a coupling relationship exists grids in the vertical direction is n , which is shown in Fig. 3. between the reservoir ﬂow model and the wellbore pressure The rise of bottom water is treated as a piston-like drop model. ﬂooding process. That is to say, there is an obvious inter- According to the reservoir ﬂow model, a model is face between the oil zone and the water zone. The oil– developed to calculate steady-state productivity of the water contact moves upward to the horizontal wellbore in horizontal well with variable density perforation, in which the vertical direction. Once it reaches any point of the the pressure drop along the horizontal wellbore is consid- wellbore, water breakthrough occurs there. According to ered: the material balance theory (Xiong et al. 2013), we have: n ðkÞ N p > N X X 1 l q ðkÞ < ra p ðiÞ¼ p / k wf e kj l 4pkDx n ðkÞ k ¼ 1; 2;...;N; i ¼ 1; 2;...; ð25Þ p p k¼1 j¼1 Dx p ðiÞ¼ p ði 1Þþ 0:5½Dp ði 1ÞþDp ðiÞ wf wf wf wf where p (i) is the wellbore pressure in the ith segment, wf ½S ði; k þ 1Þ S udxdydz ¼ v dxdydt ð26Þ w wc w MPa; N is the number of divided segments along the Based on Darcy’s law, the water rise velocity is as wellbore; and / is a function corresponding to the hori- kj follows: zontal wellbore as well as the oil–water interface. 123 Pet. Sci. (2017) 14:383–394 389 Start Input formation and well completion parameters Divide segments and establish the coefficient matrix A Set the initial pressure p and margin of error ε wf0 Solve the matrix equation and get the flow rate q(i) for each segment p = p wf0 wf1 Solve the wellbore pressure drop model and get the new pressure p wf1 for each segment NO |p p | <ε wf1 wf0 YES Output production and pressure End Fig. 2 Flowchart for solving the coupling model z(1, 1) z(2, 1) z(3, 1) z(4, 1) z(5, 1) z(6, 1) z(7, 1) z(N, 1) z(1, 2) z(2, 2) z(3, 2) z(4, 2) z(5, 2) z(6, 2) z(7, 2) z(N, 2) z(1, n ) z(2, n ) z(3, n ) z(4, n ) z(5, n ) z(6, n ) z(7, n ) … z z z z z z z z(N, nz) Fig. 3 Schematic diagram of the physical model for bottom water cresting 123 390 Pet. Sci. (2017) 14:383–394 density of each case is 2 shots/m. In Case 1, the perforation K K opði; kÞ rw v v ¼ ð27Þ is uniformly distributed. In Cases 2 and 3, the perforation l oz density at the heel end of the horizontal well is larger than where S (i, k ? 1) is the water saturation of the (i, k ? 1) that at the toe end of the horizontal well, while the perfo- grid at time t in the longitudinal proﬁle; S is the connate wc ration density is denser at the toe end than that at the heel water saturation; K is the relative permeability to water; rw end for Cases 4–6. and l is the water viscosity, mPa s. The simulation results for all the six cases are shown in According to the results of grid discretization, the ver- Figs. 5, 6, 7, 8 and 9. Figure 5 shows the pressure distri- tical pressure gradients between any two contiguous grids bution along the horizontal wellbore. In fact, there exists a are as follows: pressure drop along the perforation hole. However, it has little relationship with our research object; thus, the rele- Dpði; kÞ¼ pði; k þ 1Þ pði; kÞ k ¼ 1; 2; ...; n 1 Dpði; kÞ¼ p ðiÞ pði; kÞ k ¼ n vant calculation was not carried out in this work. In order wf z to better identify their characteristics, the pressure distri- ð28Þ bution curves of only three cases (Cases 1, 2 and 5) are Substituting Eqs. (28) into (27) gives the rise velocity of plotted together in Fig. 5. It can be seen that the pressure bottom water: distribution curves are steep near the heel end of the hor- izontal wellbore, while relatively ﬂat at the toe end. In K K Dpði; kÞ rw v v ði; kÞ¼ ð29Þ addition, the greater the pressure drop is, the denser the l Dz perforations will be. Near the toe end, the pressure of Case According to the established mathematical model, the 5 is higher than those of Case 1 and Case 2, indicating a pressure at any grid between the oil–water contact surface denser perforation and a greater pressure drop in this and the horizontal wellbore can be written as: location, while near the heel end, the difference of these three curves is smaller. Figure 6 shows the friction and Nl p ði; kÞ¼ p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qði; kÞ/ ð30Þ wf e ik acceleration losses (Case 1) along the wellbore, respec- 4pL K K h v i¼1 tively. At any point of the horizontal wellbore, the friction loss is greater than the acceleration loss, and the former is Combining with Eq. (26), the time required for the nearly six times as much as the latter, which means the water rising from the (k ? 1)th grid to the kth grid is friction loss plays a leading role. obtained: Figure 7 gives the ﬂow rate distribution along the hor- ul ½S ði; kÞ S ðDzÞ w wc izontal wellbore. For the horizontal well with uniformly tði; kÞ¼ ð31Þ K K Dpði; kÞ rw v distributed perforation, the ﬂow rate at the toe end is lower than that at the heel end, due to the inﬂuence of the well- The breakthrough time at the ith segment is: bore pressure drop. These six cases have the same number of perforations, and thus, the production is also similar, t ¼ tði; kÞð32Þ k¼1 especially for Cases 1–3. For Case 3, although the differ- ence of the ﬂow rate between the heel end and the toe end where u is the porosity; n is the number of meshes in the is larger compared with the other ﬁve cases (Fig. 7), the longitudinal direction between the wellbore and the oil– production rate of the horizontal well is slightly larger water interface. (Fig. 8). Therefore, in order to maximize the production of the horizontal well, a perforation scheme with a larger perforation density at the toe end should be adopted. It 4 Case study should be noted that a larger perforation density at the toe end does not necessarily result in a higher production rate. Using the developed model, the well productivity and The reason is that this kind of perforation scheme will lead water breakthrough for a horizontal well in a bottom water to much lower ﬂow rate at the toe end. In addition, Fig. 7 drive reservoir were evaluated. Table 2 lists the bottom also shows that the ﬂow rate distribution of Case 6 is more water drive reservoir properties and its drilling and com- uniform compared with the other ﬁve cases. Due to the pletion parameters. inﬂuence of the pressure drop along the wellbore and the The steady-state productivity of the horizontal well with reduced end effect, the ﬂow rate distribution of Case 6 has variable density perforation completion was evaluated, and a more uniform distribution of cresting height, which will also the bottom water inﬂow proﬁle was calculated. A set delay the occurrence of bottom water breakthrough. Sim- of basic variable density perforation cases are designed and ulation results indicate that reducing the perforation density are shown in Fig. 4. For simplicity, the average perforation at the heel end is helpful for obtaining an evenly advancing 123 Pet. Sci. (2017) 14:383–394 391 Table 2 Parameters for the Parameter Value Parameter Value bottom water drive reservoir and its drilling and completion Reservoir thickness, m 28 Bottom hole pressure, MPa 20 parameters Pressure at the oil–water interface, MPa 25 Horizontal well length, m 600 Horizontal permeability, lm 0.2 Wellbore diameter, cm 17.45 Vertical permeability, lm 0.05 Water avoidance height, m 21 Oil density, kg/m 845 Relative wellbore roughness 0.0001 Oil viscosity, mPa s 15.4 Initial perforation phase angle p/2 Perforation density, shots/m 2 Angle between the wellbore and the x-axis 0 3.5 0.03 0.008 Friction loss 3.0 Acceleration loss 0.006 2.5 0.02 2.0 0.004 1.5 0.01 1.0 0.002 Case 1 Case 2 Case 3 0.5 Case 4 Case 5 Case 6 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Distance to the heel end, m Distance to the heel end, m Fig. 6 Friction loss and acceleration loss along the horizontal Fig. 4 Cases for variable density perforation wellbore for Case 1 21.8 1.10 Case 1 Case 1 Case 2 Case 3 21.5 Case 2 Case 5 Case 4 Case 5 Case 6 0.90 21.2 20.9 0.70 20.6 0.50 20.3 0.30 20.0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Distance to the heel end, m Distance to the heel end, m Fig. 7 Flow rate distribution along the horizontal wellbore for Fig. 5 Pressure distribution along the horizontal wellbore different cases water proﬁle on the vertical plane of the horizontal breakthrough location and time of the whole wellbore can wellbore. be calculated by the developed model, as shown in Table 3. The detailed distribution of bottom water breakthrough Figure 10 illustrates the distribution of the bottom water time along the wellbore is shown in Fig. 9. Compared with rising height along the wellbore for three different cases, Case 1 (evenly distributed perforation), the redistribution which is in good agreement with the results in Fig. 9. The of perforating density changes both the breakthrough bottom water cresting height of Case 2 is shown in Fig. 11. location and breakthrough time simultaneously. It is It shows that the effect of the perforation density on bottom obvious that the larger perforation density means a shorter water cresting height becomes more serious with the breakthrough time. Meanwhile, the bottom water increase of time. The deformation of the water ridge is Pressure, MPa Perforation density, shots/m Flow rate, m /d/m Friction loss, MPa Acceleration loss, MPa 392 Pet. Sci. (2017) 14:383–394 Case 1 Case 2 Case 5 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 0 100 200 300 400 500 600 Distance to the heel end, m Fig. 8 Production rate of the horizontal well for different cases Fig. 10 Bottom water rising height for different cases (production 700 time, 150 days) Case 1 Case 2 Case 3 horizontal well with a higher production rate will have a shorter water-free production period. So there exists a Case 4 Case 5 Case 6 contradiction between increasing the well production and delaying the breakthrough time of bottom water. In order to 400 comprehensively evaluate the effect of both controlling bottom water cresting and stabilizing oil production, a new parameter, called the cumulative free-water production, is deﬁned as follows: Q ¼ q T ð33Þ o o where Q is the cumulative oil production without water, 0 100 200 300 400 500 600 m ; q is the rate when the horizontal well produces stea- Distance to the heel end, m dily, m /d; and T is the production time without water, d. Fig. 9 Distribution of breakthrough time for different cases Obviously, the cumulative oil production without water of the horizontal well could consider both the well pro- duction and effective production time. obvious, which reﬂects on the degree of water crest According to the calculation results (Table 4), the Q of asymmetry. Cases 4 and 5 is higher than those of other Cases owing to According to the above results, the perforation the more uniform inﬂow proﬁle. Therefore, for the hori- scheme with higher production (Cases 2 and 3) will zontal well studied in this paper, the perforation advance the time of the bottom water breakthrough and scheme with denser perforation hole at the toe end and thus have a negative effect on the development of bottom sparser perforation hole at the heel end is a more appro- water reservoirs using horizontal wells. In other words, the priate choice. Table 3 Bottom water breakthrough location and time Case Distance between the Breakthrough time at the ﬁrst Breakthrough time at the point Breakthrough time at the point breakthrough point and the heel breakthrough point, d 100 m away from the heel, d 500 m away from the heel, d end, m 1 65 205.3 208.5 271.9 2 55 180.8 187.1 328.5 3 60 190.9 196.4 294.2 4 105 226.0 226.2 256.5 5 265 239.1 253.4 243.6 6 335 242.1 256.0 284.9 Breakthrough time, d Production rate, m /d Rising height, m Pet. Sci. (2017) 14:383–394 393 point of the distribution curve will move upward, which 30 d 70 d 110 d means the water breakthrough time is delayed. Under these 150 d 190 d circumstances, the horizontal well obtains a longer effec- tive production period. It is meaningful to take cumulative oil production without water as a parameter to evaluate the perforation strategies for horizontal wells. With this parameter, both the effects of productivity of the horizontal well and bot- tom water breakthrough time can be considered compre- hensively. However, in order to simplify the calculation process, some assumptions have been introduced in this model, some of which are ideal ones and different from the 0 100 200 300 400 500 600 actual reservoirs. Further work needs to be done in order to Distance to the heel end, m extend the application of this model. Fig. 11 Rising height of bottom water at different production times (Case 2) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give Table 4 Parameters required for calculating Q of different cases appropriate credit to the original author(s) and the source, provide a 3 4 3 link to the Creative Commons license, and indicate if changes were Case Breakthrough location, m T ,d q,m /d Q ,10 m o o made. 1 65 205.3 367.1 7.5 2 55 180.8 367.6 6.6 3 65 190.9 369.0 7.0 References 4 105 226.0 361.0 8.2 5 265 239.1 351.4 8.4 Besson J, Aquitaine E. Performance of slanted and horizontal wells on an anisotropic medium. In: SPE European petroleum conference, 21–24 October. The Hague, Netherlands; 1990. doi:10.2118/20965-MS. Chaperon I. A theoretical study of coning toward horizontal and 5 Conclusions vertical wells in anisotropic formations: subcritical and critical rates. In: SPE annual technical conference and exhibition, 5–8 In this paper, a model coupling ﬂuid ﬂow in reservoirs and October. New Orleans, Louisiana; 1986. doi:10.2118/15377-MS. pressure drop along the wellbore was developed to obtain Cheng LS. Higher seepage mechanics. Beijing: Petroleum Industry Press; 2011 (in Chinese). the distribution of ﬂow rate along the horizontal wellbore Cheng LS, Lang ZX, Zhang LH. Reservoir engineering problem of and to determine the horizontal well productivity. In this horizontal wells coning in bottom-water drive reservoir. J Univ model, the ﬂuid ﬂow at each perforation hole was taken Pet China. 1994;18(4):43–7 (in Chinese). into account, and the perforation tunnels were treated as Dikken BJ. Pressure drop in horizontal wells and its effect on production performance. In: SPE India oil and gas conference many inﬁnitesimal sections. After that, a bottom water and exhibition, 17–19 February. New Delhi, India; 1990. doi:10. cresting model was developed based on the piston-like 2118/39521-MS. displacement principle, which could be used to calculate Dou H, Lian S, Guan C. The experimental studies of physical the bottom water breakthrough location, breakthrough time simulation of bottom water reservoirs with barrier and permeable interbed on horizontal well. In: SPE western regional meeting, and the shape of water cresting at different times. 26–27 May. Anchorage, Alaska; 1999. doi:10.2118/55995-MS. For a horizontal well with uniform density perforation Goode PA, Wilkinson DJ. Inﬂow performance of partially open completed in a bottom water reservoir, the pressure drop horizontal wells. J Pet Technol. 1991;43(8):983–7. doi:10.2118/ along the wellbore may signiﬁcantly affect the water 19341-PA. Guo BY, Molinard JE, Lee RL. A general solution of gas/water breakthrough mode. Speciﬁcally, the bottom water will coning problem for horizontal wells. In: SPE European ﬁrstly break through at the heel end of the wellbore. petroleum conference, 16–18 November. Cannes, France; Neglecting the end effects, water breakthrough occurs in 1992. doi:10.2118/25050-MS. proper sequence from heel end to toe end. The bottom Holmes JA, Barkve T, Lund O. Application of a multi-segment well model to simulate ﬂow in advanced wells. In: European water breakthrough at a speciﬁc position will happen ear- petroleum conference, 20–22 October. The Hague, Netherlands; lier if the perforation density is increased. When a hori- 1998. doi:10.2118/50646-MS. zontal well with denser perforation density at the toe end Islam MR. Oil recovery from bottom-water reservoirs. J Pet Technol. and sparser perforation density at the heel end, the lowest 1993;45(6):514–6. doi:10.2118/25394-PA. Rising height, m 394 Pet. Sci. (2017) 14:383–394 Landman MJ, Goldthorpe WH. Optimization of perforation distribu- streaks. In: SPE/DOE improved oil recovery symposium, 21–24 tion for horizontal wells. In: SPE Asia-Paciﬁc conference, 4–7 April. Tulsa, Oklahoma; 1996. doi:10.2118/35440-MS. November. Perth, Australia; 1991. doi:10.2118/23005-MS. Sognesand S, Skotner P, Hauge J. Use of partial perforations in Li XG, Wang QH, Li Y. Numerical simulation model of perforated Oseberg horizontal wells. In: SPE annual technical conference well completions. J Univ Pet China. 1996;20(2):48–53 (in and exhibition, 25–28 September. New Orleans, Louisiana; Chinese). 1994. doi:10.2118/28569-MS. Li GS, Song J, Xiong W, et al. Simulation model and calculation of Su Z, Gudmundsson JS. Pressure drop in perforated pipes: experi- seepage ﬂow ﬁeld for high pressure waterjet perforated wells. ments and analysis. In: SPE Asia Paciﬁc oil and gas conference, Pet Explor Dev. 2006;32(6):97–100 (in Chinese). 7–10 November. Melbourne, Australia; 1994. doi:10.2118/ Li H, Chen DC, Meng HX. Optimized models of variable density 28800-MS. perforation in the horizontal well. Pet Explor Dev. 2010;37(3): Umnuayponwiwat S, Ozkan E. Water and gas coning toward ﬁnite- 363–8 (in Chinese). conductivity horizontal wells: cone buildup and breakthrough. Luo X, Jiang L, Su Y, Huang K. The productivity calculation model In: SPE rocky mountain regional/low-permeability reservoirs of perforated horizontal well and optimization of inﬂow proﬁle. symposium and exhibition, 12–15 March, Denver, Colorado; Petroleum. 2015;1(2):154–7. doi:10.1016/j.petlm.2015.04.002. 2000. doi:10.2118/60308-MS. Muskat M, Wycokoff RD. An approximate theory of water-coning in Wang RH, Zhang YZ, Bu YH, et al. A segmentally numerical oil production. Trans AIME. 2013;114(1):144–63. doi:10.2118/ calculation method for estimating the productivity of perforated 935144-G. horizontal wells. Pet Explor Dev. 2006;33(5):630–3 (in Novy RA. Pressure drops in horizontal wells: when can they be Chinese). ignored. SPE Reserv Eng. 1995;10(1):29–35. doi:10.2118/ Wibowo W, Permadi P, Mardisewojo P, et al. Behavior of water 24941-PA. cresting and production performance of horizontal well in Ouyang LB, Huang B. A comprehensive evaluation of well-comple- bottom water drive reservoir: a scaled model study. In: SPE Asia tion impacts on the performance of horizontal and multilateral Paciﬁc conference on integrated modelling for asset manage- wells. In: SPE annual technical conference and exhibition, 9––12 ment, 29–30 March, Kuala Lumpur, Malaysia; 2004. doi:10. October. Dallas, Texas; 2005. doi:10.2118/96530-MS. 2118/87046-MS. Pang W, Chen DC, Zhang ZP, et al. Segmentally variable density Xiong J, He HP, Xiong YM, et al. The effect of partial completion perforation optimization model for horizontal wells in heteroge- parameters in horizontal well on water coning. Nat Gas Geosci. neous reservoirs. Pet Explor Dev. 2012;39(2):230–8. doi:10. 2013;24(6):1232–7 (in Chinese). 1016/S1876-3804(12)60036-6. Yuan H, Sarica C, Brill JP. Effect of perforation density on single Penmatcha VR, Arbabi S, Aziz K. A comprehensive reservoir/ phase liquid ﬂow behavior in horizontal wells. In: SPE wellbore model for horizontal well. In: SPE India oil and gas international conference on horizontal well technology, 18–20 conference and exhibition, 17–19 February. New Delhi, India; November, Calgary, Alberta, Canada; 1996. doi:10.2118/37109- 1998. doi:10.2118/39521-MS. MS. Permadi P, Lee RL, Kartoatmodjo RST. Behavior of water cresting Zhao G, Zhou J, Liu X. An insight into the development of bottom under horizontal wells. In: SPE annual technical conference and water reservoirs. J Can Pet Technol. 2006;45(4):22–30. doi:10. exhibition, 22–25 October. Dallas, Texas; 1995. doi:10.2118/ 2118/06-04-CS. 30743-MS. Zhou ST, Ma DQ, Liu M. Optimization of perforation tunnel Permadi P, Gustiawan E, Abdassah D. Water cresting and oil distribution in perforated horizontal wells. J China Univ Pet (Ed recovery by horizontal wells in the presence of impermeable Nat Sci). 2002;26(3):52–4 (in Chinese). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Petroleum Science Springer Journals http://www.deepdyve.com/lp/springer-journals/a-new-mathematical-model-for-horizontal-wells-with-variable-density-AoMERbBL0Q

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References (33)

R. Novy (1995)
Pressure Drops in Horizontal Wells: When Can They Be Ignored?
Spe Reservoir Engineering, 10
LS Cheng (2011)
Higher seepage mechanics
I. Chaperon (1986)
Theoretical Study of Coning Toward Horizontal and Vertical Wells in Anisotropic Formations: Subcritical and Critical Rates
Software - Practice and Experience
H. Yuan, C. Sarica, J. Brill (1996)
Effect of Perforation Density on Single Phase Liquid Flow Behavior in Horizontal Wells
W. Wibowo, P. Permadi, P. Mardisewojo, P. Sukarno (2004)
Behavior of Water Cresting and Production Performance of Horizontal Well in Bottom Water Drive Reservoir: A Scaled Model Study
E. Ozkan (2000)
Water and Gas Coning toward Finite-Conductivity Horizontal Wells: Cone Buildup and
B. Guo, J. Molinard, R. Lee (1992)
A General Solution of Gas/Water Coning Problem for Horizontal Wells
Europace
Z. Su, J. Gudmundsson (1994)
Pressure drop in perforated pipes: Experiments and analysis
M. Muskat, R. Wycokoff (1935)
An Approximate Theory of Water-coning in Oil Production
Transactions of the AIME, 114
L. Ouyang, W. Huang (2005)
A Comprehensive Evaluation of Well-Completion Impacts on the Performance of Horizontal and Multilateral Wells
(1994)
Reservoir engineering problem of horizontal wells coning in bottom-water drive reservoir
G. Zhao, J. Zhou, X. Liu (2004)
An Insight Into the Development of Bottom Water Reservoirs
X. Ju (2013)
The Effect of Partial Completion Parameters in Horizontal Well on Water Coning
Natural Gas Geoscience
M. Landman, W. Goldthorpe (1991)
Optimization of Perforation Distribution for Horizontal Wells
J. Holmes, T. Barkve, O. Lund (1998)
Application of a Multisegment Well Model to Simulate Flow in Advanced Wells
V. Penmatcha, K. Aziz (1998)
A Comprehensive Reservoir/Wellbore Model for Horizontal Wells
Wang Rui-he, Zhang Yu-zhe, Bu Yu-huan, Liu Jilin, Zhang Rui (2006)
A segmentally numerical calculation method for estimating the productivity of perforated horizontal wells
Petroleum Exploration and Development
P. Permadi, E. Gustiawan, D. Abdassah (1996)
Water Cresting and Oil Recovery by Horizontal Wells in the Presence of Impermeable Streaks
ST Zhou, DQ Ma, M Liu (2002)
Optimization of perforation tunnel distribution in perforated horizontal wells
J China Univ Pet (Ed Nat Sci), 26
P. Goode, D. Wilkinson (1991)
Inflow Performance of Partially Open Horizontal Wells
Journal of Petroleum Technology, 43
Ben Dikken (1990)
Pressure drop in horizontal wells and its effect on production performance
Journal of Petroleum Technology, 42
S. Sognesand, P. Skotner, J. Hauge (1994)
Use of Partial Perforations in Oseberg Horizontal Wells
Software - Practice and Experience
M. Islam (1993)
Oil Recovery From Bottomwater Reservoirs
Journal of Petroleum Technology, 45
Z. Sheng (2002)
OPTIMIZATION OF PERFORATION TUNNELS' DISTRIBUTION IN PERFORATED HORIZONTAL WELLS
Journal of the University of Petroleum,China
W. Pang, Dechun Chen, Zhang Zhongping, Lifu Jiang, Changheng Li, Xu Zhao, B. Wang (2012)
Segmentally variable density perforation optimization model for horizontal wells in heterogeneous reservoirs
Petroleum Exploration and Development, 39
P. Permadi, R. Lee, R. Kartoatmodjo (1995)
Behavior of Water Cresting Under Horizontal Wells
Software - Practice and Experience
J. Besson (1990)
Performance of Slanted and Horizontal Wells on an Anisotropic Medium
Europace
C. Lt (2010)
Optimized models of variable density perforation in the horizontal well
Petroleum Exploration and Development
Hongen Dou, S. Lian, Changzhi Guan (1999)
The Experimental Studies of Physical Simulation of Bottom Water Reservoirs With Barrier and Permeable Interbed on Horizontal Well
Xianbo Luo, Lifu Jiang, Yanchun Su, Kai Huang (2015)
The productivity calculation model of perforated horizontal well and optimization of inflow profile
Petroleum, 1
L. Gensheng, S. Jian, Xiong Wei, Huang Zhongwei, Niu Jilei (2005)
Simulation model and calculation of seepage flow field for high pressure waterjet perforated wells
Petroleum Exploration and Development
(2011)
Higher seepage mechanics. Beijing: Petroleum Industry
(1996)
Numerical simulation model of perforated well completions

Publisher: Springer Journals
Copyright: Copyright © 2017 by The Author(s)
Subject: Earth Sciences; Mineral Resources; Industrial Chemistry/Chemical Engineering; Industrial and Production Engineering; Energy Economics
ISSN: 1672-5107
eISSN: 1995-8226
DOI: 10.1007/s12182-017-0159-0
Publisher site: See Article on Publisher Site

Abstract

Pet. Sci. (2017) 14:383–394 DOI 10.1007/s12182-017-0159-0 ORIGINAL PAPER A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs 1 2 1 3 1 • • • • Dian-Fa Du Yan-Yan Wang Yan-Wu Zhao Pu-Sen Sui Xi Xia Received: 11 July 2016 / Published online: 12 April 2017 The Author(s) 2017. This article is an open access publication Abstract Horizontal wells are commonly used in bottom the perforation distribution. Wells with denser perforation water reservoirs, which can increase contact area between density at the toe end and thinner density at the heel end wellbores and reservoirs. There are many completion may obtain low production, but the water breakthrough methods used to control cresting, among which variable time is delayed. Taking cumulative free-water production density perforation is an effective one. It is difﬁcult to as a parameter to evaluate perforation strategies is advis- evaluate well productivity and to analyze inﬂow proﬁles of able in bottom water reservoirs. horizontal wells with quantities of unevenly distributed perforations, which are characterized by different param- Keywords Bottom water reservoirs Variable density eters. In this paper, ﬂuid ﬂow in each wellbore perforation, perforation completion Inﬂow proﬁle Cresting model as well as the reservoir, was analyzed. A comprehensive Cumulative free-water production model, coupling the ﬂuid ﬂow in the reservoir and the wellbore pressure drawdown, was developed based on potential functions and solved using the numerical discrete 1 Introduction method. Then, a bottom water cresting model was estab- lished on the basis of the piston-like displacement princi- Bottom water reservoirs are widely distributed on earth and ple. Finally, bottom water cresting parameters and factors hold a large proportion of oil reserves (Islam 1993). Taking inﬂuencing inﬂow proﬁle were analyzed. A more system- China for example, there exist a large number of bottom atic optimization method was proposed by introducing the water reservoirs, most of which are developed using hori- concept of cumulative free-water production, which could zontal wells. Compared with vertical wells, the producing maintain a balance (or then a balance is achieved) between sections of horizontal wells have direct contact with oil stabilizing oil production and controlling bottom water reservoirs, which not only reduces the producing pressure cresting. Results show that the inﬂow proﬁle is affected by drawdown, but also ensures bottom water ﬂowing into the wellbore more smoothly in a form of ‘‘pushing upward’’ (Besson and Aquitaine 1990; Dou et al. 1999; Permadi & Dian-Fa Du et al. 1996; Zhao et al. 2006). Owing to these advantages, it dudf@upc.edu.cn can effectively control bottom water cresting. The need of & Yan-Yan Wang economic and effective development of bottom water 514375721@qq.com reservoirs leads to the appearance of many types of com- School of Petroleum Engineering, China University of pletion methods, such as barefoot well completion, slotted Petroleum, Qingdao 266580, Shandong, China screen well completion and perforation completion Sinopec Petroleum Exploration and Production Research (Ouyang and Huang 2005). Recently, partial completion, Institute, Sinopec, Beijing 100083, China variable density perforation completion and other new Shengli Production Plant, Shengli Oilﬁeld Branch Company, completion methods have been put forward to further Sinopec, Dongying 257051, Shandong, China control bottom water cresting (Goode and Wilkinson 1991; Sognesand et al. 1994). By accurately ﬁnding out the water Edited by Yan-Hua Sun 123 384 Pet. Sci. (2017) 14:383–394 production interval of horizontal wells, adopting plugging were analyzed using the developed model, and a more strategies or properly adjusting the bottom water inﬂow systematic optimization method was proposed by intro- proﬁle, these techniques can effectively prolong the life of ducing the concept of the cumulative free-water produc- production wells. And among all these techniques, perfo- tion, which could realize a balance between stabilizing oil ration completion, including variable density perforation production and controlling bottom water coning. and selectively perforated completion, plays a critical role in alleviating water cresting (Pang et al. 2012). Previously, scholars put more emphasis on the produc- 2 Productivity analysis for horizontal wells tivity evaluation of horizontal wells (Dikken 1990; Novy with variable density perforation completion 1995; Penmatcha et al. 1998) and bottom water cresting (Permadi et al. 1995; Wibowo et al. 2004; Chaperon 1986). For horizontal wells with perforation completed in bottom The published papers mostly focused on horizontal wells water reservoirs, formation ﬂuids ﬁrstly ﬂow into perfora- with open-hole completion. There is little research into tion holes before converging in the horizontal wellbore. horizontal wells with variable density perforation comple- Under this circumstance, the effect of perforation holes is tion, and the ones that exist turned out to be very prob- similar to that of short producing branches in inclined lematic: (1) The method for open-hole completion horizontal wells (Holmes et al. 1998). The real producing horizontal wells was used ignoring the ﬂuid ﬂow in per- part for horizontal wells should be quantities of perforation foration tunnels in these studies (Landman and Goldthorpe holes. Therefore, the perforation holes can be regarded as 1991; Yuan et al. 1996; Zhou et al. 2002). Then, a model ‘‘source-sink’’ term, and this kind of problem can be solved describing the damage zone must be introduced to char- with a source function. acterize the inﬂuence of perforation holes (Umnuaypon- wiwat and Ozkan 2000; Muskat and Wycokoff 2013). 2.1 Analysis of ﬂuid ﬂow near wellbore regions Some scholars utilized the numerical simulation method to discuss the impact of selective perforation on the produc- A model was built to describe the ﬂow of formation ﬂuid tivity of horizontal wells and built a single-phase ﬂow near horizontal wellbores, and the assumptions for this variable density perforation model for horizontal wells by model are as follows: two ﬁltration zones (Li et al. 2010). Since the seepage (1) The formation is homogeneous with a uniform resistance needs to be considered more precisely, espe- thickness; cially in the middle and later periods of the oilﬁeld (2) The horizontal permeability meets the following development, the existing results are somewhat inaccurate. basic relationship: K = K = K , and the vertical x y h (2) Conventional simpliﬁed models cannot analyze for- permeability is K = K . The target reservoir is z v mation pressure thoroughly and predict bottom water inﬁnite in the horizontal plane; cresting. Furthermore, a non-uniformly distributed bottom (3) The single-phase ﬂuid ﬂowing in the reservoir is water inﬂow proﬁle along the wellbore was obtained incompressible and the ﬂuid ﬂow obeys Darcy’s law; without considering the wellbore pressure drop (Guo et al. (4) The wellbore is horizontal, in which the perforations 1992). (3) In order to optimize completion parameters for are unevenly distributed; horizontal wells, oil production is usually viewed as the (5) The perforating direction is perpendicular to the only objective function. It is reasonable for horizontal wells wellbore, and the lengths as well as radius of all the in conventional reservoirs. However, it is not accurate for perforation tunnels are the same. horizontal wells located in bottom water reservoirs because of ignoring bottom water cresting, which decreases the Acoordinate system is established as shown in Fig. 1,in effective production period of wells (Luo et al. 2015). which the heel end of the wellbore is M (x , y , z ). The 0 0 0 0 In this paper, based on the precise consideration of the horizontal part of the wellbore (total length L, m) is divided ﬂuid ﬂow in each perforation, the ﬂow behavior in perfo- into N segments. Therefore, the length of each segment is rations, wellbores, as well as reservoirs, was analyzed. L/N. Coupling the ﬂuid ﬂow in reservoirs and wellbore pressure Different segments have different characteristic drop, a comprehensive model, which can be used to eval- parameters: the perforation density, n (i); the perforation uate productivity of horizontal wells, was developed based depth, l ; bore diameter, D ; phase angle x; and the initial p p on potential functions and solved using the numerical perforation angle x . The heel end is selected as the discrete method. Then, a model describing bottom water origin of this coordinate, and the x direction is parallel with the wellbore. The coordinates of any point (x, y, z)in cresting was established on the basis of the piston-like displacement principle. Finally, both the bottom water the jth perforation tunnel in the ith segment are as fol- cresting behavior and the factors inﬂuencing inﬂow proﬁle lows: 123 Pet. Sci. (2017) 14:383–394 385 i1 > L j x ði; j; tÞ¼ x þ sin h cos a þ sin h cos a þ t l sin c cos v > p 0 k k i i p ij ij > N n ðiÞ k¼1 < i1 L j ð1Þ y ði; j; tÞ¼ y þ sin h sin a þ sin h sin a þ t l sin c sin v ð0 t 1Þ p 0 k k i i p ij ij > N n ðiÞ k¼1 i1 > L j z ði; j; tÞ¼ z þ cos h þ cos h þ t l cos c : p 0 k i p ij N n ðiÞ k¼1 2 2 2 where h is the angle between the kth segment and the o U o U o U þ þ ¼ 0 ð3Þ vertical direction, k ¼ 1; 2; ...; i 1; a is the angle 2 2 02 ox oy oz between the x-axis and the projection of the kth segment pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where U ¼ K K l p,m /s; l is the oil viscosity, in the horizontal plane, k ¼ 1; 2; ...; i 1. h is the angle h v o o mPa s; and also the corresponding parameters of the per- between the ith segment and the vertical direction; a is the angle between the x-axis and the projection of the ith foration become: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ segment in the horizontal plane; c is the angle between ij 0 2 2 2 the jth perforation in the ith producing part of the hori- l ¼ l b cos c þ sin c ; p ij ij zontal well and the z-axis; v is the angle between the x- ij b sin c axis and the projection in the xy plane of the jth perfo- ij sin c ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ij ration in the ith segment of the horizontal well; in this 2 2 1 2 b cos c þ sin c ij ij paper, when all the perforation is perpendicular to the cos c ij horizontal wellbore, then v = p/2. i,j cos c ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ij 2 1 2 According to the classical theory of ﬂow in porous b b cos c þ sin c ij ij media (Cheng 2011), the ﬂuid ﬂow at any point in an In order to simplify the solution procedure, one inﬁnite reservoir obeys the Laplace equation. So we have: assumption that the ﬂow rates of all perforations in each 2 2 2 o p o p K o p segment are equal, is proposed (Wang et al. 2006): þ þ ¼ 0 ð2Þ 2 2 2 ox oy K oz q ðiÞ ra q ¼ ð4Þ where p is the pressure, MPa; K is the vertical perme- Dxn ðiÞ 2 2 ability, lm ; and K is the horizontal permeability, lm . where q is the ﬂow rate of each perforation at the ith Substituting linear transformation z ¼ bz, where p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 3 segment, m /s; q ðiÞ is the ﬂow rate of the ith segment, m / ra b ¼ K =K , into Eq. (2) gives: h v s; Dx is the length of the ith segment, which is equal to L/N, m; and n ðiÞ is the perforation density of the ith segment, holes/cm. Compared with the length of the horizontal wellbore, the (a) (b) L perforation is rather short. Therefore, it can be regarded as an inﬁnitesimal line source. After integrating the point sink solution over the perforation direction, the pressure response of any perforation hole in the formation is obtained. For example, the potential at point M(x, y, z ) caused by the jth perforation in the ith segment can be described as: q ði; jÞ (c) Visible perforation U ðx; y; z Þ¼ ds þ C ij ij Invisible perforation 4pl r Nq ðiÞ ra ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ds þ C ij 2 2 2 0 0 4pl Ln ðiÞ ðx xÞ þðy yÞ þðz z Þ p p p p Fig. 1 Schematic diagram near the horizontal wellbore in a bottom water reservoir ð5Þ 123 386 Pet. Sci. (2017) 14:383–394 with The integration of Eq. (5) is as follows: þ1 0 X r þ r þ l Nq ðiÞ 1ij 2ij 0 0 0 0 0 ra p / ¼ n 2h þ 4nh z ði; j; 0Þ; 2h þ 4nh ij ij pij U ðx; y; z Þ¼ ln þ C ð6Þ ij ij 4pl Ln ðiÞ r þ r l n¼1 p p 1ij 2ij 0 0 0 0 0 z ði; j; 1Þ; x; y; z þ n 4nh þ z ði; j; 0Þ; 4nh ij pij pij where q (i, j) is the ﬂow rate of the jth perforation tunnel in 0 0 the ith segment, m /s; r is the distance between the source þz ði; j; 1Þ; x; y; z pij 0 0 point M ðx; y; z Þ and the target point Mðx; y; z Þ; C is an p ij 0 0 0 0 n 2h þ 4nh þ z ði; j; 0Þ; 2h ij pij integration constant; r is the distance between the heel 1ij end and the target point; r is the distance between the toe 0 0 0 2ij þ4nh þ z ði; j; 1Þ; x; y; z pij end and the target point, and they observe the following 2l expressions, respectively: p 0 0 0 0 0 0 n 4nh z ði; j; 0Þ; 4nh z ði; j; 1Þ; x; y; z þ C ij pij pij i qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ nh 2 2 2 0 0 r ¼ ½x ði; j; 0Þ x þ½y ði; j; 0Þ y þ½z ði; j; 0Þ z 1ij p p ð14Þ ð7Þ The boundary pressure at the oil–water interface is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 0 0 assumed to be constant. Therefore, the potential at any r ¼ ½x ði; j; 1Þ x þ½y ði; j; 1Þ y þ½z ði; j; 1Þ z 2ij p p point of the formation may be expressed as: ð8Þ n ðiÞ N p X X N q ðiÞ ra where x (i, j, 0), y (i, j, 0), z (i, j, 0) and x (i, j, 1), y (i, j, 1), p p p p p U ðx; y; z Þ¼ U ln / ð15Þ ij e ij 4pl L n ðiÞ p p z (i, j, 1) are the coordinates of the left and right ends of the i¼1 j¼1 jth perforation in the ith producing part. Combining the deﬁnition of potential, the pressure at Based on the mirror image reﬂection and superposition any point of the formation can be given as follows: principle, the potential of the jth perforation tunnel for n ðiÞ the ith production segment at point M ðx; y; z Þ is N N p X X Nl q ðiÞ ra 0 o p ðx; y; z Þ¼ p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ln / obtained: wf;ij e ij 4pl L K K n ðiÞ p h v i¼1 j¼1 þ1 Nq ðiÞ ra 0 0 0 0 ð16Þ Uðx;y;z Þ¼ n ð2h þ 4nh z ði;j;0Þ; ij pij 4pl Ln ðiÞ p p n¼1 where p is the boundary pressure, MPa. 0 0 0 0 2h þ 4nh z ði;j;1Þ; x; y; z Þ pij For perforated completion, the perforation tunnels 0 0 0 0 0 þ n ð4nh þ z ði;j;0Þ; 4nh þ z ði;j;1Þ; x ;y;z Þ directly contact the formation. Therefore, the ﬂow potential ij pij pij 0 0 0 0 of some point, which is just located in the perforation, can be n ð2h þ 4nh þ z ði;j;0Þ ;2h ij pij obtained using Eq. (15). In this case, there are two points 0 0 0 þ 4nh þ z ði;j;1Þ; x ;y ;z Þ pij involved: the target point and the source point (perforation 0 0 0 n ð4nh z ði;j;0Þ; 4nh point). To get rid of singularity phenomenon, the central hole ij pij 0 0 0 in the wall is chosen as the jth perforation’s target point when z ði;j;1Þ; x ;y ;z Þþ C ð9Þ pij i calculating the distance between two perforation points. with Only calculating the pressure of one central point for the segment and regarding it as the pressure of the whole r þ r þ l 1ij 2ij segment will result in deviation when analyzing the seg- n ðe ; e ; x; y; z Þ¼ ln ð10Þ ij 0 1 r þ r l 1i 2i ment’s pressure of a horizontal wellbore. In this paper, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ taking the average over all the perforations’ pressure in the 2 2 2 r ¼ ½xði; j; 0Þ x þ½yði; j; 0Þ y þ½e z ð11Þ 1i 0 same segment and using the average value as the repre- qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sentative pressure of that segment, the pressure of the ith 2 2 2 r ¼ ½xði; j; 1Þ x þ½yði; j; 1Þ y þ½e z ð12Þ 2i 1 segment is as follows: Dxn ðiÞ According to the superposition principle, the potential at p ðiÞ¼ p ði; jÞð17Þ any point of the inﬁnite formation created by all the per- wf wf Dxn ðiÞ j¼1 foration tunnels of the horizontal well is: n ðiÞ where p (i) is the ﬂow pressure for all the perorations in N wf X X N q ðiÞ ra the ith segment, MPa. U ðx; y; z Þ¼ ln / þ C ð13Þ ij ij 4pl L n ðiÞ p p i¼1 j¼1 123 Pet. Sci. (2017) 14:383–394 387 2 2 Dp ði; jÞ¼ q v ðÞ i; j v ði; j þ 1Þ There are 2N variables required to be calculated by acc s s 13 2 2 analyzing pressure distribution along the wellbore: p (i) wf ¼ 3:5215 10 q ði; jÞ q ði; j þ 1Þ L L and q (i), (i = 1, 2, …, N). However, it only contains ra ð20Þ N equations in the ﬂow model [Eq. (15)]. Therefore, one more model is needed to describe pressure drop along the where q is the liquid density, kg/m ; D is the wellbore wellbore (Li et al. 1996, 2006). diameter, m; q (i,j) is the ﬂow rate along the wellbore, m / d; v ði; jÞ is the average velocity of the jth perforation in the ith segment, m/s; and f ði; jÞ is the friction coefﬁcient of fric 2.2 Wellbore pressure drop model the jth perforation in the ith segment. In Eq. (19), one parameter, called the friction coefﬁ- For perforated horizontal wells, the main idea to develop a cient, is introduced. The calculation of the friction coefﬁ- wellbore pressure drop model is to divide the horizontal cient is dependent on the Reynolds number. If the Reynolds wellbore into several segments, and each segment is sub- number is less than or equal to 2000, the ﬂow is laminar; divided into several smaller parts that only include one otherwise, it is turbulent ﬂow. And the expression for the perforation (Su and Gudmundsson 1994). Reynolds number is as follows: According to the analysis of the pressure drop in a wellbore, the total pressure loss can be written as: Qq Re ¼ 7:3682844 10 rl dp ðiÞ¼ dp ðiÞþ dp ðiÞþ dp ðiÞþ dp ðiÞð18Þ wf fric acc mix G where Re is the Reynolds number; Q is the axial ﬂow rate where dp (i) is the friction loss of the ith segment, MPa; fric along the wellbore, m /d; l is the viscosity of the ﬂowing dp (i) is the acceleration loss of the ith segment, MPa; acc ﬂuid, mPa s; and r is the wellbore radius, m. dp (i) is the mixing loss of the ith segment, MPa; and mix In addition, we also introduced a criterion when calcu- dp (i) is the gravity loss of the ith segment, MPa. lating the mixing loss between two perforation holes. The It should be noted that, in the previous section, a speciﬁc calculation method for frictional loss and mixing mechanical ﬁeld described by the ﬂow model is estab- losses is listed in Table 1. In Table 1, q is the critical rate, lished based on the potential function. Therefore, when c m /d; e is the surface roughness, mm. the potential function is used to deal with ﬂow problems, When calculating the mixing loss, one parameter, Dp , per the pressure loss caused by viscous force has been con- is introduced. It is the frictional loss showing up after sidered. As we all know, the mathematical expression of perforating and can be ﬁgured out using Eq. (19). ﬂuid potential is Kp/l, where K is the formation perme- The friction loss, acceleration loss and the mixing loss ability, p represents pressure, while l means the ﬂuid of the ith segment are listed below: viscosity, and it is used to describe the viscous force, which will lead to pressure loss along the perforation. So Dxn ðiÞ q q ði; jÞ 13 L it means that the viscous force has been taken into con- dp ðiÞ¼ 1:0862 10 f ði; jÞ ð21Þ fric fric D n ðiÞ j¼1 sideration in the ﬁrst model. In other words, when building wellbore pressure here, only four kinds of pres- Dxn ðiÞ sure loss should be calculated. dp ðiÞ¼ Dp ði; jÞ acc acc The calculation method of frictional loss and accelera- j¼1 Dxn ðiÞ tion loss between two perforation tunnels is expressed, p 13 2 2 ¼ 3:5215 10 q ði; jÞ q ði; j þ 1Þ respectively. L L j¼1 Dl qv ði; jÞ 13 s Dp ¼ 1:34 10 f ði; jÞ ð22Þ fric fric D 2 qq ði; jÞ n ðiÞ 13 L ¼ 1:0862 10 f ði; jÞ ð19Þ fric D n ðiÞ dp ðiÞ¼ Dp ði; jÞð23Þ mix mix j¼1 Table 1 Calculation method for friction coefﬁcient and mixing loss between two perforation holes Flow pattern Discriminant Calculation of friction coefﬁcient f (i, j) (Cheng Calculation of mixing losses Dp ði; jÞ (Cheng fric mix standard 2011) 2011) q ðiÞ 7 ra Laminar ﬂow q ði; jÞ\q L c Dp ði; jÞ 0:31 10 Reði; jÞ per Re(i;j) Dxn ðiÞq ði;jÞ p L hi hi 2 q ðiÞ 3 ra Turbulent q ði; jÞ [ q 1:11 L c 6:9 e 0:76 10 Reði; jÞ 1:8lg þ Dxn ðiÞq ði;jÞ p L Reði;jÞ 3:7D ﬂow 123 388 Pet. Sci. (2017) 14:383–394 In this paper, an iterative method is used to solve the If the horizontal well is inclined, the gravity loss is above-mentioned model, and the iterative process is shown non-ignorable, and the wellbore pressure drop model in Fig. 2. becomes: Dxn ðiÞ Dxn ðiÞ p p > X X q q ði; jÞ q > L 13 2 2 1:0862 10 f ði; jÞ þ 3:5215 10 ½q ði; jÞ q ði; j þ 1Þ > fric L L 5 5 D n ðiÞ D > j¼1 j¼1 n ðiÞ Dp ði; jÞ¼ þ qgDx cos h þ Dp ði; jÞ q ðiÞ 6¼ 0 ð24Þ wf i mix ra > j¼1 Dxn ðiÞ > 2 q q ði; jÞ > L 1:0862 10 f ði; jÞ þ qgDx cos h q ðiÞ¼ 0 > fric i ra : 5 D n ðiÞ j¼1 2.3 Coupling model 3 Analysis of inﬂow proﬁles in bottom water reservoirs When a horizontal well begins to produce reservoir ﬂuids, the ﬂuids in the perforations connect the wellbore and oil The bottom water rises fastest in the vertical plane of the reservoir together. Therefore, perforation is considered as horizontal wellbore (Cheng et al. 1994). In other words, the an inﬁnitesimal linear sink, which directly contacts reser- bottom water will ﬁrstly break through into the wellbore in voir and wellbore. Meanwhile, pressure responses are this plane due to the highest pressure gradients in this proﬁle. generated in the whole reservoir. The generated pressure Therefore, a complicated 3-D problem can be turned into a responses near the perforations are associated with oil 2-D problem in the xz proﬁle where the wellbore lies. The inﬂow in the radial direction of the horizontal wellbore, formation between the wellbore and the oil–water interface which can be calculated by utilizing the reservoir ﬂow is discretized according to the division of the horizontal model. Since the pressure in perforation holes is relevant to wellbore in the reservoir ﬂow model. And the total number of the wellbore pressure, a coupling relationship exists grids in the vertical direction is n , which is shown in Fig. 3. between the reservoir ﬂow model and the wellbore pressure The rise of bottom water is treated as a piston-like drop model. ﬂooding process. That is to say, there is an obvious inter- According to the reservoir ﬂow model, a model is face between the oil zone and the water zone. The oil– developed to calculate steady-state productivity of the water contact moves upward to the horizontal wellbore in horizontal well with variable density perforation, in which the vertical direction. Once it reaches any point of the the pressure drop along the horizontal wellbore is consid- wellbore, water breakthrough occurs there. According to ered: the material balance theory (Xiong et al. 2013), we have: n ðkÞ N p > N X X 1 l q ðkÞ < ra p ðiÞ¼ p / k wf e kj l 4pkDx n ðkÞ k ¼ 1; 2;...;N; i ¼ 1; 2;...; ð25Þ p p k¼1 j¼1 Dx p ðiÞ¼ p ði 1Þþ 0:5½Dp ði 1ÞþDp ðiÞ wf wf wf wf where p (i) is the wellbore pressure in the ith segment, wf ½S ði; k þ 1Þ S udxdydz ¼ v dxdydt ð26Þ w wc w MPa; N is the number of divided segments along the Based on Darcy’s law, the water rise velocity is as wellbore; and / is a function corresponding to the hori- kj follows: zontal wellbore as well as the oil–water interface. 123 Pet. Sci. (2017) 14:383–394 389 Start Input formation and well completion parameters Divide segments and establish the coefficient matrix A Set the initial pressure p and margin of error ε wf0 Solve the matrix equation and get the flow rate q(i) for each segment p = p wf0 wf1 Solve the wellbore pressure drop model and get the new pressure p wf1 for each segment NO |p p | <ε wf1 wf0 YES Output production and pressure End Fig. 2 Flowchart for solving the coupling model z(1, 1) z(2, 1) z(3, 1) z(4, 1) z(5, 1) z(6, 1) z(7, 1) z(N, 1) z(1, 2) z(2, 2) z(3, 2) z(4, 2) z(5, 2) z(6, 2) z(7, 2) z(N, 2) z(1, n ) z(2, n ) z(3, n ) z(4, n ) z(5, n ) z(6, n ) z(7, n ) … z z z z z z z z(N, nz) Fig. 3 Schematic diagram of the physical model for bottom water cresting 123 390 Pet. Sci. (2017) 14:383–394 density of each case is 2 shots/m. In Case 1, the perforation K K opði; kÞ rw v v ¼ ð27Þ is uniformly distributed. In Cases 2 and 3, the perforation l oz density at the heel end of the horizontal well is larger than where S (i, k ? 1) is the water saturation of the (i, k ? 1) that at the toe end of the horizontal well, while the perfo- grid at time t in the longitudinal proﬁle; S is the connate wc ration density is denser at the toe end than that at the heel water saturation; K is the relative permeability to water; rw end for Cases 4–6. and l is the water viscosity, mPa s. The simulation results for all the six cases are shown in According to the results of grid discretization, the ver- Figs. 5, 6, 7, 8 and 9. Figure 5 shows the pressure distri- tical pressure gradients between any two contiguous grids bution along the horizontal wellbore. In fact, there exists a are as follows: pressure drop along the perforation hole. However, it has little relationship with our research object; thus, the rele- Dpði; kÞ¼ pði; k þ 1Þ pði; kÞ k ¼ 1; 2; ...; n 1 Dpði; kÞ¼ p ðiÞ pði; kÞ k ¼ n vant calculation was not carried out in this work. In order wf z to better identify their characteristics, the pressure distri- ð28Þ bution curves of only three cases (Cases 1, 2 and 5) are Substituting Eqs. (28) into (27) gives the rise velocity of plotted together in Fig. 5. It can be seen that the pressure bottom water: distribution curves are steep near the heel end of the hor- izontal wellbore, while relatively ﬂat at the toe end. In K K Dpði; kÞ rw v v ði; kÞ¼ ð29Þ addition, the greater the pressure drop is, the denser the l Dz perforations will be. Near the toe end, the pressure of Case According to the established mathematical model, the 5 is higher than those of Case 1 and Case 2, indicating a pressure at any grid between the oil–water contact surface denser perforation and a greater pressure drop in this and the horizontal wellbore can be written as: location, while near the heel end, the difference of these three curves is smaller. Figure 6 shows the friction and Nl p ði; kÞ¼ p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qði; kÞ/ ð30Þ wf e ik acceleration losses (Case 1) along the wellbore, respec- 4pL K K h v i¼1 tively. At any point of the horizontal wellbore, the friction loss is greater than the acceleration loss, and the former is Combining with Eq. (26), the time required for the nearly six times as much as the latter, which means the water rising from the (k ? 1)th grid to the kth grid is friction loss plays a leading role. obtained: Figure 7 gives the ﬂow rate distribution along the hor- ul ½S ði; kÞ S ðDzÞ w wc izontal wellbore. For the horizontal well with uniformly tði; kÞ¼ ð31Þ K K Dpði; kÞ rw v distributed perforation, the ﬂow rate at the toe end is lower than that at the heel end, due to the inﬂuence of the well- The breakthrough time at the ith segment is: bore pressure drop. These six cases have the same number of perforations, and thus, the production is also similar, t ¼ tði; kÞð32Þ k¼1 especially for Cases 1–3. For Case 3, although the differ- ence of the ﬂow rate between the heel end and the toe end where u is the porosity; n is the number of meshes in the is larger compared with the other ﬁve cases (Fig. 7), the longitudinal direction between the wellbore and the oil– production rate of the horizontal well is slightly larger water interface. (Fig. 8). Therefore, in order to maximize the production of the horizontal well, a perforation scheme with a larger perforation density at the toe end should be adopted. It 4 Case study should be noted that a larger perforation density at the toe end does not necessarily result in a higher production rate. Using the developed model, the well productivity and The reason is that this kind of perforation scheme will lead water breakthrough for a horizontal well in a bottom water to much lower ﬂow rate at the toe end. In addition, Fig. 7 drive reservoir were evaluated. Table 2 lists the bottom also shows that the ﬂow rate distribution of Case 6 is more water drive reservoir properties and its drilling and com- uniform compared with the other ﬁve cases. Due to the pletion parameters. inﬂuence of the pressure drop along the wellbore and the The steady-state productivity of the horizontal well with reduced end effect, the ﬂow rate distribution of Case 6 has variable density perforation completion was evaluated, and a more uniform distribution of cresting height, which will also the bottom water inﬂow proﬁle was calculated. A set delay the occurrence of bottom water breakthrough. Sim- of basic variable density perforation cases are designed and ulation results indicate that reducing the perforation density are shown in Fig. 4. For simplicity, the average perforation at the heel end is helpful for obtaining an evenly advancing 123 Pet. Sci. (2017) 14:383–394 391 Table 2 Parameters for the Parameter Value Parameter Value bottom water drive reservoir and its drilling and completion Reservoir thickness, m 28 Bottom hole pressure, MPa 20 parameters Pressure at the oil–water interface, MPa 25 Horizontal well length, m 600 Horizontal permeability, lm 0.2 Wellbore diameter, cm 17.45 Vertical permeability, lm 0.05 Water avoidance height, m 21 Oil density, kg/m 845 Relative wellbore roughness 0.0001 Oil viscosity, mPa s 15.4 Initial perforation phase angle p/2 Perforation density, shots/m 2 Angle between the wellbore and the x-axis 0 3.5 0.03 0.008 Friction loss 3.0 Acceleration loss 0.006 2.5 0.02 2.0 0.004 1.5 0.01 1.0 0.002 Case 1 Case 2 Case 3 0.5 Case 4 Case 5 Case 6 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Distance to the heel end, m Distance to the heel end, m Fig. 6 Friction loss and acceleration loss along the horizontal Fig. 4 Cases for variable density perforation wellbore for Case 1 21.8 1.10 Case 1 Case 1 Case 2 Case 3 21.5 Case 2 Case 5 Case 4 Case 5 Case 6 0.90 21.2 20.9 0.70 20.6 0.50 20.3 0.30 20.0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Distance to the heel end, m Distance to the heel end, m Fig. 7 Flow rate distribution along the horizontal wellbore for Fig. 5 Pressure distribution along the horizontal wellbore different cases water proﬁle on the vertical plane of the horizontal breakthrough location and time of the whole wellbore can wellbore. be calculated by the developed model, as shown in Table 3. The detailed distribution of bottom water breakthrough Figure 10 illustrates the distribution of the bottom water time along the wellbore is shown in Fig. 9. Compared with rising height along the wellbore for three different cases, Case 1 (evenly distributed perforation), the redistribution which is in good agreement with the results in Fig. 9. The of perforating density changes both the breakthrough bottom water cresting height of Case 2 is shown in Fig. 11. location and breakthrough time simultaneously. It is It shows that the effect of the perforation density on bottom obvious that the larger perforation density means a shorter water cresting height becomes more serious with the breakthrough time. Meanwhile, the bottom water increase of time. The deformation of the water ridge is Pressure, MPa Perforation density, shots/m Flow rate, m /d/m Friction loss, MPa Acceleration loss, MPa 392 Pet. Sci. (2017) 14:383–394 Case 1 Case 2 Case 5 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 0 100 200 300 400 500 600 Distance to the heel end, m Fig. 8 Production rate of the horizontal well for different cases Fig. 10 Bottom water rising height for different cases (production 700 time, 150 days) Case 1 Case 2 Case 3 horizontal well with a higher production rate will have a shorter water-free production period. So there exists a Case 4 Case 5 Case 6 contradiction between increasing the well production and delaying the breakthrough time of bottom water. In order to 400 comprehensively evaluate the effect of both controlling bottom water cresting and stabilizing oil production, a new parameter, called the cumulative free-water production, is deﬁned as follows: Q ¼ q T ð33Þ o o where Q is the cumulative oil production without water, 0 100 200 300 400 500 600 m ; q is the rate when the horizontal well produces stea- Distance to the heel end, m dily, m /d; and T is the production time without water, d. Fig. 9 Distribution of breakthrough time for different cases Obviously, the cumulative oil production without water of the horizontal well could consider both the well pro- duction and effective production time. obvious, which reﬂects on the degree of water crest According to the calculation results (Table 4), the Q of asymmetry. Cases 4 and 5 is higher than those of other Cases owing to According to the above results, the perforation the more uniform inﬂow proﬁle. Therefore, for the hori- scheme with higher production (Cases 2 and 3) will zontal well studied in this paper, the perforation advance the time of the bottom water breakthrough and scheme with denser perforation hole at the toe end and thus have a negative effect on the development of bottom sparser perforation hole at the heel end is a more appro- water reservoirs using horizontal wells. In other words, the priate choice. Table 3 Bottom water breakthrough location and time Case Distance between the Breakthrough time at the ﬁrst Breakthrough time at the point Breakthrough time at the point breakthrough point and the heel breakthrough point, d 100 m away from the heel, d 500 m away from the heel, d end, m 1 65 205.3 208.5 271.9 2 55 180.8 187.1 328.5 3 60 190.9 196.4 294.2 4 105 226.0 226.2 256.5 5 265 239.1 253.4 243.6 6 335 242.1 256.0 284.9 Breakthrough time, d Production rate, m /d Rising height, m Pet. Sci. (2017) 14:383–394 393 point of the distribution curve will move upward, which 30 d 70 d 110 d means the water breakthrough time is delayed. Under these 150 d 190 d circumstances, the horizontal well obtains a longer effec- tive production period. It is meaningful to take cumulative oil production without water as a parameter to evaluate the perforation strategies for horizontal wells. With this parameter, both the effects of productivity of the horizontal well and bot- tom water breakthrough time can be considered compre- hensively. However, in order to simplify the calculation process, some assumptions have been introduced in this model, some of which are ideal ones and different from the 0 100 200 300 400 500 600 actual reservoirs. Further work needs to be done in order to Distance to the heel end, m extend the application of this model. Fig. 11 Rising height of bottom water at different production times (Case 2) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give Table 4 Parameters required for calculating Q of different cases appropriate credit to the original author(s) and the source, provide a 3 4 3 link to the Creative Commons license, and indicate if changes were Case Breakthrough location, m T ,d q,m /d Q ,10 m o o made. 1 65 205.3 367.1 7.5 2 55 180.8 367.6 6.6 3 65 190.9 369.0 7.0 References 4 105 226.0 361.0 8.2 5 265 239.1 351.4 8.4 Besson J, Aquitaine E. Performance of slanted and horizontal wells on an anisotropic medium. In: SPE European petroleum conference, 21–24 October. The Hague, Netherlands; 1990. doi:10.2118/20965-MS. Chaperon I. A theoretical study of coning toward horizontal and 5 Conclusions vertical wells in anisotropic formations: subcritical and critical rates. In: SPE annual technical conference and exhibition, 5–8 In this paper, a model coupling ﬂuid ﬂow in reservoirs and October. New Orleans, Louisiana; 1986. doi:10.2118/15377-MS. pressure drop along the wellbore was developed to obtain Cheng LS. Higher seepage mechanics. Beijing: Petroleum Industry Press; 2011 (in Chinese). the distribution of ﬂow rate along the horizontal wellbore Cheng LS, Lang ZX, Zhang LH. Reservoir engineering problem of and to determine the horizontal well productivity. In this horizontal wells coning in bottom-water drive reservoir. J Univ model, the ﬂuid ﬂow at each perforation hole was taken Pet China. 1994;18(4):43–7 (in Chinese). into account, and the perforation tunnels were treated as Dikken BJ. Pressure drop in horizontal wells and its effect on production performance. In: SPE India oil and gas conference many inﬁnitesimal sections. After that, a bottom water and exhibition, 17–19 February. New Delhi, India; 1990. doi:10. cresting model was developed based on the piston-like 2118/39521-MS. displacement principle, which could be used to calculate Dou H, Lian S, Guan C. The experimental studies of physical the bottom water breakthrough location, breakthrough time simulation of bottom water reservoirs with barrier and permeable interbed on horizontal well. In: SPE western regional meeting, and the shape of water cresting at different times. 26–27 May. Anchorage, Alaska; 1999. doi:10.2118/55995-MS. For a horizontal well with uniform density perforation Goode PA, Wilkinson DJ. Inﬂow performance of partially open completed in a bottom water reservoir, the pressure drop horizontal wells. J Pet Technol. 1991;43(8):983–7. doi:10.2118/ along the wellbore may signiﬁcantly affect the water 19341-PA. Guo BY, Molinard JE, Lee RL. A general solution of gas/water breakthrough mode. Speciﬁcally, the bottom water will coning problem for horizontal wells. In: SPE European ﬁrstly break through at the heel end of the wellbore. petroleum conference, 16–18 November. Cannes, France; Neglecting the end effects, water breakthrough occurs in 1992. doi:10.2118/25050-MS. proper sequence from heel end to toe end. The bottom Holmes JA, Barkve T, Lund O. Application of a multi-segment well model to simulate ﬂow in advanced wells. In: European water breakthrough at a speciﬁc position will happen ear- petroleum conference, 20–22 October. The Hague, Netherlands; lier if the perforation density is increased. When a hori- 1998. doi:10.2118/50646-MS. zontal well with denser perforation density at the toe end Islam MR. Oil recovery from bottom-water reservoirs. J Pet Technol. and sparser perforation density at the heel end, the lowest 1993;45(6):514–6. doi:10.2118/25394-PA. Rising height, m 394 Pet. Sci. (2017) 14:383–394 Landman MJ, Goldthorpe WH. Optimization of perforation distribu- streaks. In: SPE/DOE improved oil recovery symposium, 21–24 tion for horizontal wells. In: SPE Asia-Paciﬁc conference, 4–7 April. Tulsa, Oklahoma; 1996. doi:10.2118/35440-MS. November. Perth, Australia; 1991. doi:10.2118/23005-MS. Sognesand S, Skotner P, Hauge J. Use of partial perforations in Li XG, Wang QH, Li Y. Numerical simulation model of perforated Oseberg horizontal wells. In: SPE annual technical conference well completions. J Univ Pet China. 1996;20(2):48–53 (in and exhibition, 25–28 September. New Orleans, Louisiana; Chinese). 1994. doi:10.2118/28569-MS. Li GS, Song J, Xiong W, et al. Simulation model and calculation of Su Z, Gudmundsson JS. Pressure drop in perforated pipes: experi- seepage ﬂow ﬁeld for high pressure waterjet perforated wells. ments and analysis. In: SPE Asia Paciﬁc oil and gas conference, Pet Explor Dev. 2006;32(6):97–100 (in Chinese). 7–10 November. Melbourne, Australia; 1994. doi:10.2118/ Li H, Chen DC, Meng HX. Optimized models of variable density 28800-MS. perforation in the horizontal well. Pet Explor Dev. 2010;37(3): Umnuayponwiwat S, Ozkan E. Water and gas coning toward ﬁnite- 363–8 (in Chinese). conductivity horizontal wells: cone buildup and breakthrough. Luo X, Jiang L, Su Y, Huang K. The productivity calculation model In: SPE rocky mountain regional/low-permeability reservoirs of perforated horizontal well and optimization of inﬂow proﬁle. symposium and exhibition, 12–15 March, Denver, Colorado; Petroleum. 2015;1(2):154–7. doi:10.1016/j.petlm.2015.04.002. 2000. doi:10.2118/60308-MS. Muskat M, Wycokoff RD. An approximate theory of water-coning in Wang RH, Zhang YZ, Bu YH, et al. A segmentally numerical oil production. Trans AIME. 2013;114(1):144–63. doi:10.2118/ calculation method for estimating the productivity of perforated 935144-G. horizontal wells. Pet Explor Dev. 2006;33(5):630–3 (in Novy RA. Pressure drops in horizontal wells: when can they be Chinese). ignored. SPE Reserv Eng. 1995;10(1):29–35. doi:10.2118/ Wibowo W, Permadi P, Mardisewojo P, et al. Behavior of water 24941-PA. cresting and production performance of horizontal well in Ouyang LB, Huang B. A comprehensive evaluation of well-comple- bottom water drive reservoir: a scaled model study. In: SPE Asia tion impacts on the performance of horizontal and multilateral Paciﬁc conference on integrated modelling for asset manage- wells. In: SPE annual technical conference and exhibition, 9––12 ment, 29–30 March, Kuala Lumpur, Malaysia; 2004. doi:10. October. Dallas, Texas; 2005. doi:10.2118/96530-MS. 2118/87046-MS. Pang W, Chen DC, Zhang ZP, et al. Segmentally variable density Xiong J, He HP, Xiong YM, et al. The effect of partial completion perforation optimization model for horizontal wells in heteroge- parameters in horizontal well on water coning. Nat Gas Geosci. neous reservoirs. Pet Explor Dev. 2012;39(2):230–8. doi:10. 2013;24(6):1232–7 (in Chinese). 1016/S1876-3804(12)60036-6. Yuan H, Sarica C, Brill JP. Effect of perforation density on single Penmatcha VR, Arbabi S, Aziz K. A comprehensive reservoir/ phase liquid ﬂow behavior in horizontal wells. In: SPE wellbore model for horizontal well. In: SPE India oil and gas international conference on horizontal well technology, 18–20 conference and exhibition, 17–19 February. New Delhi, India; November, Calgary, Alberta, Canada; 1996. doi:10.2118/37109- 1998. doi:10.2118/39521-MS. MS. Permadi P, Lee RL, Kartoatmodjo RST. Behavior of water cresting Zhao G, Zhou J, Liu X. An insight into the development of bottom under horizontal wells. In: SPE annual technical conference and water reservoirs. J Can Pet Technol. 2006;45(4):22–30. doi:10. exhibition, 22–25 October. Dallas, Texas; 1995. doi:10.2118/ 2118/06-04-CS. 30743-MS. Zhou ST, Ma DQ, Liu M. Optimization of perforation tunnel Permadi P, Gustiawan E, Abdassah D. Water cresting and oil distribution in perforated horizontal wells. J China Univ Pet (Ed recovery by horizontal wells in the presence of impermeable Nat Sci). 2002;26(3):52–4 (in Chinese).

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A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs

A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs

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A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs

A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs

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