İki paralel levha arasında non-newtonian akışın modellenmesi

Abstract

Sondajın güvenle ilerlemesi için sondaj mühendisinin düşey dizi hareketleri sonucu oluşan surge ve swab basınçlarını doğru hesaplayan bir modele ihtiyacı vardır. Bu çalışmada, sondaj sırasında düşey dizi hareketleri sonucu non-Newtonian Power-law akışkan akışının neden olduğu sürtünme basınç kaybını veren model analitik çözüm kullanılarak geliştirilmiştir. Newtonian akışkanlar için iki paralel levha arasında akış (slot) geometrisi ve anüler geometri arasındaki kriter, bu çalışmada kullanılan non- Newtonian Power-law akışkanlar içinde kabul edilerek analitik çözüm elde edilmiştir. Literatür örneklerinde ve saha uygulamalarında, Dodge ve Metzner' in çalışmaları sonucu Power-law akışkanlar için tanımladıkları akmazlık terimi, Newtonian akışkanlar için sürtünme basınç kaybını veren ifadedeki akmaz lık terimi yerine kullanılarak, Power- law akışkanlar için sürtünme basınç kaybı hesaplanmaktadır. Ancak, analitik çözüm bütünüyle Power-law akışkanlar için yapılmadığından sonuç tam değil yaklaşıktır. Bu nedenle, yapılan bu çalışmada Power-law akışkanlar için düşey dizi hareketleri sonucu oluşan sürtünme basınç kayıplarını hesaplayan analitik çözüm geliştirilmiştir. Geliştirilen bağıntı ve uygulamada kullanılan bağıntının sonuçları karşılaştırılmıştır. Karşılaştırma sonucunda, literatürde verilen ve uygulamada kullanılan yöntemle hesaplanan sonuçların, geliştirilen bağıntı sonuçlarından daha büyük olduğu belirlenmiştir. Oysa daha düşük basınç kaybı değeri, daha büyük manevra hızlarına olanak sağlar. Bu ise, sondajın daha kısa sürede ve daha az maliyetle tamamlanmasıdır. Ancak, akış davranış indeksinin bire eşit olması halinde sonuçların birbirine eşit olduğu, birden daha küçük akış davranış değerlerinde farklı sonuçların hesaplandığı belirlenmiştir. Ortalama hız V, değerinin düşük değerleri için yapılan hatanın özellikle swab basınçlarını hesaplarken daha büyük olduğu görülmüştür. Dizi hareket hızının Vp, yüksek değerleri için basınç kayıplarınında yüksek olduğu gözlenmiştir. Geliştirilen denklem ile sadece pozitif ortalama hız değerleri için değil, aynı zamanda sadece dizi hareketi sonucunda oluşan negatif ortalama hız değerleri içinde surge basınçlarının hesaplanabildiği gösterilmiştir.

The science of fluid mechanics is very important to the drilling engineers. Extremely large fluid pressures are created in the long slender wellbore and tubular pipe strings by the presence of drilling mud or cement. The precence of these pressures must be considered in almost every well problem encountered. The subsurface fluid pressure can be developed for three common well conditions. These well conditions include, 1. A static condition in which both the well fluid and the central pipe string are at rest, 2. A circulating operation in which the fluids are being pumped down the central pipe string and up the annulus, 3. A tripping operation in which a central pipe string is being moved up or down through the fluid. The second and third conditions are complicated by the non-Newtonian behavior of drilling muds and cements than the Newtonian's. The displacement of drilling fluid by the movement of the drill strings or casing produces pressure variations in the borehole. The velocity of this displacement governs the magnitude of the pressure change which can add or subtract to the hydrostatic pressure of the mud, to produce surge or swab pressure, respectively. Thus, surge pressure is a pressure increase caused by running drill strings into a liquid filled borehole; while swab pressure is a pressure decrease that occurs when drill strings are pulled out of a liquid filled borehole. Excessive surge and swab pressure can lead to serious well problems. Pressure reduction due to swabbing is a major source of blowouts [3], [5], Swabbing can lead to formation fluid invading the well annulus (a kick). High surge pressure can lead to loss of mud either suddenly by fracturing the formation or slowly by continuous efflux to permeable formation. The drilling fluid enters the fractures and this produces a drop in the fluid level, causing a reduction in the well bore hydrostatic pressure [8], [9]. This reduction in mud hydrostatic pressure allows formation fluids to enter the well bore which may lead to a blowout. VI The need for calculating surge pressures has long been recognized in the industry and several steady-state [8] / [9] / [1Q] / as well as unsteady-state (also called dynamic models) [11], [12], [14], models of this phenomenon exist in the literature. tn a comperative study of the steady state and the dynamic models, point out that the steady-state models are conservative in their prediction of surge pressure, because, unlike the dynamic models, several factors are omitted, such as fluid inertia, fluid compressibility, pipe longitudinal elasticity, and pipe distance from bottom. [13] Most drilling fluids are too complex to be characterized by a single value for viscosity. Fluids that do not exhibit a direct proportianality between shear stress and shear rate are classified as non-Newtonian. Since the Power-law fluids used in this study, Bingham plastic fluids definition and properties will not be given. The non-Newtonian Power- law model is defined by The Power-law model requires two parameter for fluid characterization. However, the Power-law model can be used to represent a pseudoplastic fluid (nl). Above equation is valid only laminar flow. The parameter K usually is called the consistency index of the fluid, and the parameter n usually is called ether the Power-law exponent or the flow-behavior index. The deviation of the dimensionless flow-behavior index from unity characterizes the degree to which the fluid behavior is non-Newtonian. The units of consistency index K depend on the value of n. [20] A theoretical analysis of coutte flow of Power-law fluids for surge and swab pressure is conducted for closed- end casing or drill strings. Using the mathematical relationships developed, a familly of curves is obtained which is a function of n, and K. Knowledge of the mud rheological properties, pipe velocity, pipe end hole geometry is needed to determine the surge and swab pressure. Annular flow also can be approximated using equations developed for flow through rectangular slots. The slot flow equations are much simpler to use and are reasonably accurate as long as the ratio between drilstring and annulus d.,/d2 > 0.3. This minimum ratio almost always is exceeded in rotary drilling applications. In this study, a laminar coutte flow model for Power-law fluids through a slot (two parallel plates, with one plate moving and the VII other stationary) is considered. Based on the developed model, equations and graphs are employed to determine the surge or swab pressure. The following assumptions are made in the derivation of the model equations: - the drillstring is placed concentrically in the casing or open hole - sections of open hole are circular in shape and of known diameter - the drilling fluid is incompressible - steady-state one dimensional flow takes place - the flow regime is laminar - the flow is isothermal and the fluid properties are constant. Consider the flow of a Power-law fluid between two flat parallel plates representing a slot. The top plate is moving with a constant velocity, Vp, and the lower plate is stationary. Two dinstinct flow areas occur. 1. A region within the boundary limits: 0 < y < Xh. The rate of shear in this region is negative, and the velocity profile is i (1+-) n 2. a region within the boundary limits: Xh < y < h. The shear rate in this region is positive. The velocity profile is v r HT (i+-) (i+-i) V2= K a\ [ (h-Xh) a - (y-Xh) n ] -Vp (1 + -) n The volumetric flow rate of fluid through a slot can be obtained from following equation, q = W f V dy VIII substituting for V, and V2 in above equation, the flow rate is K rlT {2*1) (2*1) (2*1) q= K aL Wh n [(1-X) a +X n]-WVph(l-X) (2 + -İ) n Using the relationship between mean velocity and flow rate on above equation, and solving for the frictional pressure loss gradient is d (V+Vp(l-X))(2 + ±) dL h{a*x) 12*1) (2*1) X n +{1-X) At y=A.h, V1 = V2 hence dPt_ K EVi-iW dL h{n+1) a*l) (i*l) n [(1-X) * -X n ] Dimensionless pressure loss gradient are equal to each other in above the last two equation and using this relationship X is obtained as following form. (2 + -İ)... (2 + i) X=[(l-X) (1+2, Vp(l + ±)[X » +(1~X) «] (JL) 1-X) a - ] n+1 [V+VA1-X)] (2 + 1) * II X can be determined for given pipe velocity and mean velocity by iteration, or by any other convenient numerical technique. IX The algorithm to determine surge and swab pressure is outlined below. 1. For a given set of pipe velocity, mean velocity, and flow behavior index n, the value A. is determined. 2. Substituting the X value in frictional pressure loss gradient equation, surge or swab pressure is obtained. If there is no fluid flow is due only to pipe movement. In this case A. is equal to one, and mean velocity is considered as the minimum mean velocity. Using velocity profile equation which is developed for slot flow, the minimum mean velocity for swab cases and surge cases can be obtained. The minimum mean velocity for swab case is: V< = in+1) V min (212+1) p Similarly, minimum mean velocity for surge case is: V. = - {n+1) V mi1 (2i2+l) p To calculate the frictional pressure loss gradient for Power-law fluids, Dodge and Metzner defined an apparent viscosity concept and used the frictional pressure loss gradient equation for Newtonian fluids. The apparent viscosity for Power-law fluids, in field units, is defined as, Kid -d ) (1~'n) ^+~^ m - K 2 x' [ n p 144F1-n) 0.0208 The frictional pressure loss gradient equation in field units is: 2*1 v - T7~ I lit dL literature 1440OO (d ~d ) {n+1) X The model proposed in this study has been compared with the current model in the literature. For small values of mean velocity, the frictional pressure loss gradient calculated from current model in the literature is much greater than that predicted from our model. Two models are agree well at intermediate and high mean velocity range. Therefore, the equations currently used to determine safe pipe movement speed is conservative. The pipe can be moved faster without causing serious problem in the wellbore. A faster pipe movement will reduce the drilling time and costs. Although, the new equations developed in this thesis are more complex, they can be easily implemented using programmable calculators or personnal computers. Thus, the proposed model can be utilized for field applications.