Kısmi aralıklarla tamamlanmış kuyuların performansı

Abstract

Petrol ve doğal gaz sahalarında, üretim formasyonunun genellikle belirli bir kısmı akışa açılır. Bu işlem; kuyuya su ve/veya gaz girişini geciktirmek ya da özellikle formasyonun hidrokarbon bakımından yüksek kapasiteli kısımlarını üretmek gibi belli başlı nedenlerden dolayı yapılmaktadır. Akışa açık olan kısım, formasyonun herhangi bir yerinde tek üretim aralığında olabileceği gibi (kısmen tamamlanmış kuyular), birden fazla üretim aralığında da olabilir (kısmi aralıklarla tamamlanmış kuyular). Bir rezervuardan bu şekilde akışa açılmış kuyulara olan akışkan akışının matematiksel olarak çözümlenmesi, petrol mühendisliği literatüründe kuyuların üretim performansının incelenmesi ve kuyu testlerinin analizleri açısından önemlidir. Bu çalışmada, homojen ve anizotropik (farklı geçirgenlikli) bir rezervuardan, kısmi aralıklarla tamamlanmış bir kuyuya olan akışkan akışını tanımlayan matematiksel model geliştirilmiştir. Akışkan olarak tek fazlı, sabit ve küçük sıkıştınlabilirliğe sahip bir sıvı gözönüne alınmıştır. Matematiksel model, iki boyutlu diffiizyon denkleminin Laplace dönüşümü ve değişkenlerin ayrımı yöntemleri kullanılarak çözülmesi ile elde edilmiştir. Bulunan çözüm Fourier-Bessel serileri formunda olup, kişisel bilgisayarlar kullanılarak kuyu basıncı ve kısmi debiler hesaplanabilir. Kuyu hasan ve kuyuiçi depolanması da çözüme literatürdeki yöntemlerle eklenmiştir. Matematiksel model, literatürde varolan kısmen tamamlanmış kuyu modelleri ile karşılaştırılmış ve uyum içinde olduğu belirlenmiştir. Ayrıca, Larsen [30] tarafından geliştirilen kısmi aralıklarla tamamlanmış kuyu modeli ile de karşılaştırılmış ve her iki modelin de basınç davranışlarının aynı olduğu görülmüştür. Ancak Larsen 'in modelinde, erken zamanlardaki kısmi debilerin dağılımında beklenmeyen sonuçlar dikkat çekmiştir. Matematiksel model geliştirilirken, eş akı çözümünden kuyuda sabit bir basınç elde edebilmek için (sonsuz iletken kuyu yaklaşımı), basınç ortalaması yaklaşımı kullanılmıştır. Basınç ortalaması, literatürde varolan diğer yöntemlerle karşılaştırılmış ve diğer yöntemlerden, tümleme zamanı bakımından avantajlı olduğu saptanmıştır. Kısmi aralıklarla tamamlanmış kuyuların üretim performansı da incelenmiştir. Üretim aralığı sayısının ve formasyona dağılımının, penetrasyon oranının ve kuyu hasarının ya da canlandırmanın üretim performansı üzerindeki etkileri grafiklerle verilmiştir. Sonsuz bir rezervuar için elde edilen kararsız akış çözümünden gidilerek erken ve geç zamanlardaki yaklaşımlar bulunmuştur. Son bölümde, yapay bir kuyu testi verisi oluşturularak, literatürde varolan teknikler yardımıyla kuyu testi analizleri yapılmıştır.

In many oil and gas wells, a portion of productive zone is öpen to flow for a number of reasons. The common reasons are to delay gas and/or water coning and to produce hydrocarbon bearing sections of the formation. If only a single portion of producing formation is perforated, such wells are refferred to as restricted-entry, limited-entry ör partially-penetrated wells. in field applications, it is more common to perfbrate the producing formation in several intervals. Wells completed in this way are refferred to as selectively-completed wells. The mathematical modeling of the flow to selectively-completed wells is essential in analyzing the well test data and in predicting well productivity. Transient pressure response of a limited-entry well has been widely studied in the literatüre [1-29]. Almost in ali the studies, only öne perforated interval is considered. Bronş and Marting [3] did treat wells with multiple flow entries for cases where single-interval results can be applied because of symmetric segment distribution. Recently, Larsen [30] and Kamal et. al. [31] have studied the selectively-completed vertical and horizontal wells, respectively. in this study, a new mathematical model is developed to evaluate steady-state, semi steady-state and transient pressure and fractional rate behavior of selectively- completed wells. The model is derived by solving the two-dimensional (2-D) diSlısivity equation. The use of this equation assumes Darcy 's flow, negligible gravitational forces, only öne flowing phase, fluid with constant and small compressibility, and homogeneous but anisotropic formation. The difrusivity equation is given as: 32p l dp k, d2p u,c. do , ^ ," s *+7f+£*=^İ' P = p(r'Z'l) (S'1) where o^ is l for Darcy units and 2.63679x10^ for field units. Öne initial condition and four boundary conditions are required to solve this differential equation. The initial and boundary conditions for a selectively-completed well in an infinite reservoir are: initial condition: t = 0, p(r,z,0) = Pi (S.2) Boundary conditions: l.r = rw, (rf) = ZM-KKMz-(htD,+hpJ]}^i- (S.3) \ ^1 /r=rw i=l a!Z7IKrnpi Xİ where ü(z-a) is the Heaviside Unit Step Function. 2. r ->oo, Iimp(r,z,t) = pi (S.4) r-»co 3.2 = 0, f£(r,0,t) = 0 (S.5) öz 4.z = h, ^(r,h,t) = 0 (S.6) dz where a^ given in first boundary condition is l for Darcy units and 1.127xlO-3 for field units. The initial condition describes constant and initial pressure throughout the reservoir before fluid flow starts. The first boundary condition represents uniform but unequal and unknown flux along each perforated interval and no-flux at unperforated sections. The second boundary condition is the constant and initial pressure in the undisturbed portion of the reservoir. The third and fourth boundary conditions describe the no-flow boundaries at the top and bottom of the reservoir. The sum of the unknown flow rates in each perforated interval is equal to total flow rate in the well. qt = Zqı(t) (s.7) i=l The boundary value problem given above is expressed in terms of the dimensionless variables defined below, a,27ck,h/ \ ._ ". p°=^sr(p'-p) (SJ) l -Tn -T12 -T1N - ^ D2 l -T2ı (S.22) DN l -TNI -TNN qDN "* where, ''.âdrSs^b"^' "J (S'23) T - 2 * V K°^°) p2 j. SdJ rS24^ ^ ~ ^" hr~^"^jr~K~7fi^mj h~ ( } 7t npD. n=1 n A,nR,^n; npDj Ko(V^) Pj= //Av (S-25) J u%(A) The matrix in Eq. S-22 is solved for the unknovvn welbore pressure and rate distribution using the standard numerical techniques [32]. The solution of Eq. S-22 must be converted to real space. in this study, the inversion of the solution in Laplace space was numerically canied out using the Stehfest algoritm [33]. The wellbore storage efFect is incorporated using the Duhamel 's convolution integral expressed as below, P^D(Cp.u)= !;SDfU)^ (S-26) l + uC^^oCu) The early-time and late-time approximations of Eq. S-20 were obtained as below: Early-time solution for line-source approximation is: PwsD(tD) = rL|-^/-Tfl+Sdt (S.27) Vî|_ 2 l 4tDJ XİV The late-time solution is: PwsD(tD) = |(lntD+0.80908) + St (S.28) where St is the total skin factor given as below, S^V-î-S* (S.29) ÛptD Sdt=Zqpj(tD)Sdj (S.30) j=ı c _ 2 l yıqpi(tD)-^. l K0Çk) ^-,R /cin Sp--^-2^-l 2-2. v(ı xR=«LR»j (S'31) it "ptD ı=ı nPDi n=ı n An Jm.%; j=, in Eqs. (S-29) thru (S-31), S^ is the total mechanical skin factor due to damage ör stimulation and Sp is the pseudoskin factor due to selective completion. The semi steady-state solution for selectively-completed wells can be obtained changing the outer boundary condition of transient solution as it follovvs: 2.r = re, ^(r,z,t) = 0 (S.32) ÖT The steady-state solution for selectively-completed wells can be derived to examine the productivity of such wells. in case of steady-state flow, the outer boundary condition is: 2.r = r6, p(r,2) = Pi (S.33) The steady-state solution is also obtained by using the separation of variables method. The final solution is given as, 011 l PWSD Dı 1 -Tıı -Ti2 -TIN %ı D2 l -T2ı (S.34) DN 1 ~TNI "^NN qDN l where TJJ and Ty are defined as in Eqs. S-23 and S-24 except for, XV i.-^- ("S) nwD D-lnr^ (S.36) The solution is verified by comparing the solutions presented in this study with partially-penetrating well solutions in the literatüre (Ref [3,8,14,15,20,23]). Good agreement is observed between the solutions. Also, our solution was compared with the selectively-completed well solution given by Ref [30]. Exceüent agreement was established between the results of. transient-pressure behavior. However, an unexpected behavior in the early time rate distribution was observed in results of Ref [30]. At late times, both solutions give the same rate distribution. in this study, various approaches to compute the response of an infmite- conductivity well are examined. First, a partially penetrating well is considered and infinite-conductivity response is computed by (i) equivalent pressure point, (ii) dividing the öpen interval into N small segments, assuming uniform-flux in each segment, and equating the pressures at the center of each segment, (üi) dividing the öpen interval into N small segments, using pressure averaging for each individual segment and equating the average pressures in the segments, and (iv) using the pressure averaging method for the entire length of the öpen interval. Results obtained from ali four approaches agree well. Although the second and third approaches are similar, the CPU time required to compute the infinite-conductivity responses by the third approach is 30-40 times less than that required by the second approach. Also, the coefficient matrix constructed for the third method is a symmetric matrix having additional computational advantages. Then, the last three approaches are tested for a selectively-completed well. Good agreement is established among the results from former three approaches. it is observed that the fourth approach requires the least computational time. Based on these findings, the last approach is prefferred to approximate the infinite-conductivity response of a selectively-completed well. Another application is made in this study to identify the parameters controlling the productivity of selectively-completed wells. The productivity index of a well is defined by, PI = T~ (S.37) AP where q is the production rate and Ap is the pressure drop. If the productivity ratio is defined as, PI Ratio =|PI|SCW (S.38) *> 'FFW the productivity of selectively-completed wells can be compared with the productivity of fully penetrating wells. Using Eq. S-37 and S-38, productivity ratios were XVİ calculated for several scenarios considering different number and location of perforated intervals, penetration ratio and mechanical skin. Results showed that for a given penetration ratio increasing the number of intervals lessens the loss in productivity due to partial completion. Perforated the formation in several shorter intervals gives better productivity than perforating only one interval. The impact of non-uniform mechanical skin distribution on well productivity was investigated. It was observed that the productivity of selectively-completed wells can be increased to that of fully penetrating wells by stimulating one or more of open intervals. The important reservoir and wellbore parameters such as the permeabilities in horizontal and vertical directions, skin factor, and wellbore storage coefficient are deduced from transient pressure response of oil and gas wells. Transient pressure behavior of selectively-completed wells is examined using the solultion developed in this study. It is observed that three flow regimes characterize the pressure response; an apparent early-time radial flow, a transitional period and the late-time pseudo- radial flow. Hence, a semilog plot of pressure vs time data from a selectively- completed well may exhibit two semilog straight lines. The first straight line is not legitimate but an apparent straight line whose slope can be used to estimate horizontal permeability and total mechanical skin. The second semilog straight line is instrumental in determining the horizontal permeability and total skin factor composed of total mechanical skin and pseudoskin. Pressure derivative behavior of selectively- completed wells are also examined. Type-curve matching technique is tested for its applicability in selectively-completed wells. A method which combines the conventional semilog analysis and type-curve matching methods is presented to determine horizontal and vertical permeabilities, total skin factor, and to decompose total skin factor to its components; mechanical skin and pseudoskin. A simulated well test data is used to test the validity of the proposed technique. The estimated values of horizontal and vertical permeabilites, total skin, mechanical skin and pseudoskin compares well with the input values.