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Journal of Engineering Mathematics 27: 443-454, 1993. 443( 1993 Kluwer Academic Publishers. Printed in the Netherlands.The finite deformation of a reinforced packer, membrane theoryC. ATKINSON* and B. PELTIERSchlumberger Cambridge Reserach, High Cross, Madingley Road, Cambridge CB3 OEL, UK*Permanent address: Department of Mathematics, Imperial College of Science, Technology and Medicine,London, SW7 2BZ, UKReceived 9 October 1992; accepted in revised form 23 April 1993Abstract. Using a membrane theory, an analytical solution is reached for the deformation of a packer membranereinforced with inextensible cords. This model gives the deformation of the membrane and the levels of strain andstress in the membrane and the cords. The model enables one to study the influence of the various designparameters on the behaviour and integrity of the packer. Based on the outputs of the model, packer designsuggestions are made which are expected to be of practical significance.Notation2h0initial membrane thickness (L); W strain energy (ML -T 2); Al meridian extension; 2 radial extension; A2maximum A2, i.e. packer expansion ratio; 21, initial membrane length (L); P relative inflation pressure (ML T2);r tension in cords (ML T-2); A initial spacing between cords (L)1. IntroductionTo carry out pressure testing or hydraulic fracturing tests in an oil well, composite inflatablepackers are often used to seal off and apply specified pressure to the wall of either uncasedor cased wellbores. To undergo the requisite expansion (3:1 expansion), the packers need tobe designed to be failsafe and reusable at high pressures. This paper is concerned with amembrane analysis of composite packers under actual downhole conditions and serves as anexample of how the large deformation theory of reinforced composites can be applied.To attempt a relatively simple analysis of the deformation of the packer as it is deformeddownhole, we discuss below the consequences of applying a membrane theory of a fibrereinforced rubber packer. The basic theory for reinforced membranes has been outlined byKydoniefs 1. This theory enables one to calculate the expanded shape of the membrane asa function of the internal pressure given the expression for the strain energy function of theelastomer (this is to be obtained from experiment) and the shape and distribution of thecords. The theory also enables the tension in the cords and in the elastometer material to becalculated.2. General equationsAxisymmetric deformations are considered of an initially cylindrical membrane of uniformthickness 2ho composed of an elastic homogeneous isotropic and incompressible material444 C. Atkinson and B. Peltierpossessing a strain energy function W = W(I, ,12) and reinforced by two families of perfectlyflexible and inextensible cords.The cords of the two families are assumed to form constant angles +a with the generatorsof the undeformed membrane. It is also assumed that the lengths of the intercepts on anyone cord of the one family by two adjacent cords of the other family are independent of theposition on the surface and that these lengths are small compared with the radii of curvatureat any point of the deformed or undeformed membrane. Moreover, it is assumed that no twocords of the same family are brought into contact as a result of the deformation and thatintersecting cords of the two families do not move relative to each other at their point ofintersection.As any cylindrical membrane can be deformed into a circular cylinder without extensionon its surface, it can be assumed, without loss of generality, that the undeformed membraneis a circular cylinder.We refer the deformation to cylindrical polar coordinates and denote by (p, 0, 7),p = constant, the coordinates in the undeformed configuration of a point which hascoordinates (r, 0, z) in the deformed state. Because the deformation is axially symmetricr = r(q) , z = z(-7) .(1)The elements of length in the undeformed and deformed configurations will be denoted bydS and ds, respectively. The angle of dS with the generator of the undeformed membranewill be denoted by a and the element of the deformed meridian by d. It can then be shown1, 2 that() = Al(cos(a) + A2(sin(a)2(2)where the notationAl = d/d7, A2= rp (3)has been used.From the symmetry of the system and the incompressibility of the material it follows thatAl, A2and A3= (AlA2)- 1are the principal extension ratios in terms of which the straininvariants I are given byI, = A + A + A (4)2 = A2+ A2+ A3-2(5)Since the cords are inextensible, for a = a we obtainA2(cos(a)2+ A2(sin(a)2= 1 (6)which gives A in terms of A2and a.Let +-/3(7) be the angles of the two families of cords with the meridian of the deformedsection. Then, if dS, ds are on the same cord,r sin(a) dS = p sin(,3) ds(7)The finite deformation of a reinforced packer, membrane theoryandcos(/5) ds = d = A, cos(a) dS (8)From these relations we obtainsin() = A2, sin(a) (9)andcos(/3) = A cos(a) (10)Let A and denote the distance between adjacent cords in the undeformed and deformedmembrane, respectively. FromA = pcos(a) dO, =rcos(,/)dOand the above relations for sin and cos , we get6= A1A2(11)A further assumption is that the meridian C of the deformed membrane does not intersectthe z-axis and it has no finite element parallel or perpendicular to the z-axis. Moreover, andwithout loss of generality, it is assumed that the tangent to C is nowhere parallel orperpendicular to the z-axis, except perhaps at an end point of C; otherwise C may be dividedinto segments for which this condition does hold. This condition will be relaxed somewhatwhen we consider the situation when the expander meets the casing. With this assumptionthe equations of equilibriumd drd(rT) = T2 (12)andK1T1+ K2T2= P (13)which are valid for any axisymmetric deformation of a membrane, can be written in the formddr (rT,) = T2 (14)andK1T1+ K2T2= P (15)where T, T2are the stress resultants, per unit length of the deformed membrane, in thedirections of the meridian and the circle of latitude, respectively, K. the principal curvatures,K1being the curvature of the meridian, and P is the internal pressure. If co a = 19.50 (39)Multi layered cordsIt should be noted that the two layers of cords crossing each other (a 0) is a design featurewhich is not required for the integrity of the packer. The cord reinforcement is required tosupport the high meridian tension which takes place in the membrane sections located in theannulus between the borewall and the packer mandrel. This meridian reinforcementrestrains the membrane from expanding axially; no azimuthal (hoop) reinforcement isneeded since the radial expansion is constrained by the borewall. Therefore, a value of a = 0is acceptable and even desirable in view of the previous considerations on the influence of aon the inflation pressure P, the cords tension r and the elastomer shear strain y.From equation (11) and (6) the spacing between cords is2 2hA N1- A2sin a= AA A2= A 2 (40)cos aThe derivative of by A2d8 A 1 -2A sin2adA2cosa 1 n2a (41)dA2cos a 1- 2si 2shows the presence of a maximumi 10-. *:. : :. . .: :1.,00.,I .I- I-.- . .- . - I .I. -. -.11 ?. iO.1 II - .1 -. -. -?511-1- . 1. .- . . . . . .-? . . . . 1 .1The finite deformation of a reinforced packer, membrane theory 453Table 1. Maximum spacing between cordsa A2 (AlA2)mx (ilA2)A2=300 d/dA20 VA,2A23.00104.07 2.92 2.60123.4 2.46 2.40152.7 2.00 1.961A2 -= sin (42)The influence of a on the maximum spacing between cords max is shown in Table 1. Thesecond column shows the value Ah at which 6max occurs.Two layers of closely spaced cords are often used in order to limit the extrusion of theelastomer between cords. Unfortunately, this method tends to defeat its own purpose:closely spaced cords leave little elastomer material in-between cords resulting in destructivelevel of shear in the surrounding elastomer. This leads to the debonding of the cords whichthen are no more maintained equally spaced and tend to bundle leaving large gaps ofunsupported elastomer subject to extrusion under high inflation pressure. This has beenshown experimentally to be the case by Newburn 5.Design suggestionsIn view of these considerations the following design approach is proposed. The membrane isto be composed of a single layer of cords laid along the meridian of the packer (a = 0). Thespacing between the cords is to be such as to leave space for a hole or a notch between eachtwo cords as shown in Fig. 5 in order to limit the level of strain due to A2in the elastomerin-between the cords. This feature is to avoid debonding of the cords and thereby maintainthe cords equally spaced.Experimental evidence 6 indicates that in-homogeneities in the elastomer can result innon-uniform stretching of the elastomer leading to nonequal spacing of the cords andUnstretched StretchedK C=Unstretched StretchedFig. 5.454 C. Atkinson and B. Peltierextrusion of the elastomer. Although it is expected that the hole or notch arrangement willby itself be sufficient to keep the cords equally spaced, as a further remedy to the problem itmight be useful to constrain the maximum spacing between cords by a series of flexible wiresplaced radially in the hole or notch between cords. As an example, a maximum packer ex-pansion ratio of 3 would lead to a length of wire limiting the local expansion ratio to, say, 3.3.Borewall shearThe two ends of the packer are free to move axially; one end is built to slide along thepacker mandrel; the other end is fixed to the mandrel which can move axially with littlerestraint from the compliant string holding the packer tool in the hole. As a consequence,the mandrel offers no axial support to the membrane and the axial force resulting from thepressure across the membrane over the annulus section is entirely supported by the frictionbetween the membrane and the borewall. When a high differential pressure is imposedacross the packer, the axial force applied on the packer results in high level of shear stress atthe interface between the membrane and the borewall which could lead, especially withopenhole in weak formations, to slippage, high deformation and failure of the packer.4. ConclusionUsing a membrane theory, an analytical solution has been reached for the deformation of apacker membrane reinforced with inextensible cords. This model gives the deformation ofthe membrane and the levels of strain and stress in the membrane and the cords. Although,as it is, the model shows the influence of the various design parameters on the behaviour andintegrity of the packer, the model should now be checked against experimental data toascertain the limit of its validity.The main issue for the integrity of the membrane seems to be the extrusion of theelastomer in-between un-equally spaced cords. The traditional design of two layers of cordscrossing each other does not guarantee equally spaced cords in the inflated state because ofdebonding of the cords. It is suggested that a single layer of cords laid along the meridian ofthe packer and inbedded in the elastomer with some added compliance in-between the cords(hole or notch) is more likely to offer a solution.AcknowledgementThe authors would like to thank Dale Newburn for a series of stimulating discussions.References1. A.D. Kydoniefs, Finite axisymmetric deformations of an initially cylindrical membrane reinforced withinextensible cords. Q.J.M.A.M. 23 (1970) 481-488.2. A.J.M. Spencer, The static theory of finite elasticity. J. Inst. Maths. Applics. 6 (1970) 164-200.3. A.C. Pipkin, Z. Angew. Math. Phys. 19 (1968) 818.4. A.D. Kydoniefs, Finite axisymmetric deformations of an initially cylindrical elastic membrane enclosing a rigidbody. Q.J.M.A.M. 22 (1969) 219-331.5. D. Newburn, Private communication. July 1991.6. L. Muller, Private communication. July 1991.过油管封隔器设计学 生：陈 聪，机械工程学院指导教师：周志宏，机械工程学院摘要本文研究的封隔器是油气井作业工具中使用广泛的一种井下工具，其作用是在油井深度方向实现分层作业。随着油气田开采的深入，地层情况越来越复杂，分层作业越来越普遍，对封隔器的使用要求也越来越高。封隔器系统失效的情况时有发生，给生产造成损失。本文概述了油田常用的封隔器的类型、参数、命名及其工作原理。讨论了目前封隔器所用的胶筒的发展现状；分析了钢片叠层的密封机理；分析了过油管封隔器的工作原理以及结构特征并对各重要零部件进行了强度校核，进而完成封隔器的设计，得出结论。关键词 过油管 封隔器 钢片叠层 Design of the thru-tubing packerAbstract Packer of this study is widely used in oil and gas wells operating a downhole tool, its role is to realize the depth direction in the well stratified operation.With the deepening exploiation of oil and gas field, stratigraphic situation more complex, layered work more and more common, the use of packer demand more and more, packer system failure situations have occurred, causing damage to production. This article outlines the common oilfield packer type, parameters, name and the working principle; discuss the current packer cones used in the development of the status quo; analyse seal mechanism of the stacked steel and the thru-tubing packer works and structural characteristics and strength check of all major components, and then complete the thru-tubing packer Design, draw a conclusion.Keyword thru-tubing packer stacked steelXV江大学毕业设计(论文)任务书学院（系） 机械工程 专业 过程装备与控制工程 班级 装备091 学生姓名 陈 聪 指导教师/职称 周志宏/教授 1毕业设计(论文)题目：过油管封隔器设计2毕业设计（论文）起止时间：2013年4月1日2013年6月8日3毕业设计(论文)所需资料及原始数据（指导教师选定部分）资料：i. 程心平，陈桂和. 高效膨胀式封隔器的设计和试验-国外石油机械1999年03期ii. Thru-Tubing Inflatable Systems, Baker Oil Tools, iii. George Arnold, Enrico Suwandi. Innovative Inflatable System Enables the Underbalanced Deployment of Sand Control Screens, SPE84900原始数据：未膨胀外径：70mm膨胀外径：200mm密封段长度：1.5m4毕业设计(论文)应完成的主要内容(1)过油管膨胀式封隔器发展概述(2)过油管膨胀式封隔器工作原理(3)过油管膨胀式封隔器参数计算(4)过油管膨胀式封隔器强化钢片设计计算(5)过油管膨胀式封隔器强化钢片应力分析(6)密封计算(7)装配图与部分零件图设计5毕业设计(论文)的目标及具体要求总装配图1张；零件图若干张；三维装配图。6、完成毕业设计(论文)所需的条件及上机时数要求ANSYS或Comsol软件，上机100小时任务书批准日期 2013 年 3 月10日 教研室(系)主任(签字) 任务书下达日期 2013 年 4月1日 指导教师(签字) 完成任务日期 年 月 日 学生（签名） II增强封隔器有限变形的薄膜理论 C.阿特金森*和B.珀尔帖 著 陈聪 译摘要利用薄膜理论，来为用不可伸长的帘线加固的封隔器薄膜的形变求解。此模型给出了该膜的形变以及帘线和薄膜应力形变的程度。该模型可使人研究封隔器在各种设计参数及其完整性封隔影响下的行为。基于模型的输出，对封隔器的设计提出了建议，预计将有实际意义。符号2h初始膜厚度（L）；W 应变能（MLT）； 经络延伸； 径向延伸;最大值 ，即封隔的膨胀比；2l 初始膜的长度（L）；P相对充气压力（MLT）；张力线（ML T）； 初始线之间的间距（L）1. 引言开展压力测试或油井的水力压裂试验中，复合膨胀性封隔器也常常用于密封指定压力的任一裸眼井的或套管井。设计成安全和可重复使用的封隔器需要进行必要压力扩展（3:1增大）。本文重点在于研究复合式封隔器在井下实际条件下的薄膜变形，并作为一个例子分析如何应用变形理论来增强复合材料的强度。尝试对井下封隔器变形做一个相对简单的分析，下面我们将应用纤维膜理论来讨论增强橡胶对封隔器带来的影响。在Kydoniefs1中已阐述了增强薄膜的基本理论。这个理论能计算薄膜膨胀后的形状，给出了弹性体内部压力（这是从实验中获得）和帘线形状和分布的应变能函数的表达式。该理论还能在弹性体材料对绷直的帘线进行计算。2. 通用方程轴对称柱的变形被认为是具有应变能函数W = W（I，I）及被具有完全弹性和不可伸长帘线这两个系列特性增强的具有均相各向同性弹性及不可压缩性的材料构成的初始圆柱形薄膜厚度2h。假设未发生薄膜变形的母线上两个系列的帘线形成恒定的角度。还假定在一个系列任一条帘线上的截距的长度与另一系列相邻的帘线在膜上的位置是无关的，并且与变形或未变形的薄膜上的任一点的曲率半径相比是很小的。此外，假设没有两个同一系列的帘线接触作为形变的结果，并且两个系列相交的帘线在交点处不相互移动。任一圆柱形薄膜能在其表面变形为没有扩展的圆柱面，则可不失一般性的假设未变形的薄膜是圆柱面形状。形变涉及圆柱形极坐标并记为（，），值=常量，在未变形结构中的一点在形变状态下具有坐标（r，z）。由于形变是轴对称的，有 （1）未变形和变形组态中要素的长度，将分别表示为dS的和ds。未薄膜变形的母线和dS的角度，将表示为a而且变形的子午线要素将表示为d。然后，它可以表示为1，2 （2）其中被应用的符号 （3）根据系统的对称性和材料的不可压缩性，遵循，和，应变的不变量I的主要延伸率无论在哪可由下式给出 （4） （5）由于帘线不可伸长，因此给出，和，我们得到a=，有 （6）设（）为两个系列帘线与部分变形子午线的角度。然后，如果dS存在，则ds是在相同的帘线上， rsin()dS=sin()ds （7）和 cos()ds=d=cos()dS （8）从这些关系中，我们得到 sin()= sin() （9）和 cos()= cos() （10）在未变形和变形的薄膜上，分别用和表示相邻的帘线之间的距离。根据 =cos()d ，=rcos()d和上述关系中的sin和cos，得到 = （11）进一步假设，变形薄膜的子午线C不与z轴相交，而且没有平行或垂直于z轴的有限元。此外，不失一般性，假定为C的切线是都不平行或垂直于z轴，也许除了上面C的端点，否则C可以在条件不成立下被划分成段。当我们认为膨胀器符合套管的情况下，这种条件将有所放宽。有了这个假设，则平衡方程为 (rT) =T （12）和 T + T= （13）对任何轴对称变形的膜是有效的，可以写成如下形式： (rT) =T （14） 和 T+ T= P （15）在经络和纬度圈的方向，变形的膜每单位长度的应力合成分别用T1，T2表示。是主曲率，是经络的曲率，P是内部压力。如果/ 2表示变形的薄膜的子午线切线和对称轴所形成的角，则 =cos() （16） =cos() /r （17）通过使用（3），（16）和（17），方程（14）和（15）被还原为 （18）应力合成T可以分解成两部分： （19）其中T是由于材料的形变产生，用应变能函数W可以表示为 （20）和 （21）T 是由于帘线中的张力造成的，这些表达式可以写成如下形式： （22）通过使用（19）（22），第一个平衡方程（18）可解为 3 （23）其中，A为积分常数。自定义的不变量（4）和（5）与，根据（6）和给出的表达式（22）和（23）我们得到的T做为的函数。连同表达式（20）和（21），在不管值如何时，给出的力的T和积分常数A。它仍然结合第二个方程（18），在一组给定的边界条件下，无论怎么变化取得。2.1应用如上文所述，我们仍然需要对（18）方程进行第二次整合对给定的问题给出求解方法。对于恒定压力P下加载的薄膜，有 （24）其中，C为任意常数。封隔器自由端中= 1，在轴向方向上的相应的力应为零。故 （25）由= 1。因此，常数 （26）因为给定的压力膜的最大膨胀比，当=时除了=0，则（24）式可写为 （27）从方程（20）和（22），回代 （28）运用（23）和（27），给出了常数A，（24）式给出了cos()和形状以及在必要条件下，运用适当的非线性方程，运用下面列出和解出的积分给出薄膜的其他属性。如果在扩展的膜的情况下，根据自由指定的初始长度求解膜的方程。应变能函数进一步进行求解，我们需要弹性体的能量密度的表达式。这已经从实验测试中推导了出来。在国际单位（SI）中，表示为 (29)使用此表达式，根据方程（20）和（21）我们可以推断在弹性体力的函数。2.2. 扩展的膜的形状一旦角已获得（24）的形状的膜，可确定最大膨胀的膜已经被发现的情况下，在指定的压力下扩张的条件，判断膜的初始长度被指定为2 l。 (30)为了实现这个方程的一个解决方案，它有助于解决问题的值的范围在其中一个解决方案可能会出错。 = 0的情况下，下限的压力P提供最大膨胀比可推导出 (31)变形的膜的形状，通过绘制，膨胀比r / ，对z的轴向变形的坐标系中的坐标确定 (32)和用于根据是否2的增加或减少的函数，=0用于z= 0的加号或减号。长度由下式给出的变形的（半）子午线 （33）正如上文所述的自由膨胀封隔器，一个必须解决的最大膨胀比方程（30）。但是，对于该情况中，当封隔器膨胀对一个刚性壳体的程序是不同的4。对于这种情况下指定和中央部，其是在与墙壁接触的一半长度由下式给出 （34）在上述方程中的积分符号之外因子评价在最大膨胀比。膜的其余部分的形状被确定为以上惟现在由方程（32）给出由z+ z取代。3. 数值结果和工程方面的考虑从薄膜理论预测的结果来看，三种情况下的膨胀被认为是帘线角具有四个值。的影响，从设计角度来看，值的影响具有特别意义。其中0，100，12，15），对于每一个已被封隔器充气膜触摸borewall对应于2=3的膨胀比的地步。计算，然后一直延伸到触壁压力加0.1 MPa和触壁的压力加上封隔器的运行工况仿真每个值10.0兆帕。在下列实施例中使用的数值是10厘米（p值=0.05），直径1米（L= 0.5）与1厘米的（h= 0.005）厚的膜的材料的应变能函数表示的（29）的长封隔器加强线每间隔5毫米（= 0.005）。只出现在最后的计算和。需要注意的是在变形的配置的帘线的角度3可以由关系式（9），即 （35）一旦已被确定为位置的函数的对膜的扩展率。通胀的压力和线张力增加值的一个所需要的封隔器充气压力P的增加值达到相同的膨胀比， =3。图1a示出的形状的自由变形的膜上面的四个值的。各自的通胀压力（0.143，0.190，0.237和0.639兆帕）增加了33，66和347，对于通胀压力=0。这样的效果，结合（是大致成正比1/cos）的直接作用，封隔器的性能是不利的，因为它导致增加帘线的张力水平。图1b显示在帘线的最大张力再次增加了82，154和809，相对于帘线的张力对= 0。需要注意的是达到的最高水平的帘线张力膜满足墙上的地步。 半径（米） 半径（米） 半径（米）图1。 P=接触壁面的压力。由于通胀压力超越了触摸侧壁值增加时，膜被限制由borewall;最大剪应力保持不变（图。1c，2c和3c的），因为它是由膨胀比控制，本身就限制了由borewall张力线的增加，虽然相对的下降的影响，其效果仍然十分可观，即使在压力高达触壁压力加10兆帕（+20，+34和+74;图3B）。另外值得一提的是半长的膜和borewall的由原来的0.359米至0.295米（-18）0.262米（-27）和0.175米（-51之间的接触的效果）四个各自的值，在高企的通胀压力