PETROLEUM EXPLORATION AND DEVELOPMENT Volume 37, Issue 1, February 2010 Online English edition of the Chinese language journal Cite this article as: PETROL. EXPLOR. DEVELOP., 2010, 37(1): 99–103.
RESEARCH PAPER
A model of production data analysis for horizontal wells Wang Xiaodong1,*, Hu Yongle2, Ding Yiping1 1. China University of Geosciences, Beijing 100083, China; 2. PetroChina Research Institute of Petroleum Exploration and Development, Beijing 100083, China; 3. Exploration & Production Research Institute, Sinopec, Beijing 100083, China
Abstract: The production data analysis is one of hot spots of reservoir engineering recently, which combines conventional reservoir engineering analysis and modern well test analysis. Based on the theory of transient porous flow, the analysis provides a mathematical model, which not only reasonably predicts the production decline of oil or gas wells but also effectively evaluates the physical properties of reservoirs. Theoretical charts of Agarwal-Gardner production decline curves and derivative curves for common horizontal wells are presented by solving the transient 3D porous flow in the bounded reservoir. The Agarwal-Gardner curves can be divided into two stages: the derivatives are dispersed in the early transient decline stage and drawn to Arps harmonious decline in the late pseudo-steady state stage. The influence of vertical location of horizontal wells on production decline in the mid-long term is negligible in a homogenous reservoir. A field example illustrates that, given production data, the results in this article can be directly used to evaluate the reservoir characteristics and effectively forecast the long-term productivity for horizontal wells. Key words: horizontal well; production data analysis; rate decline; material balance time; transient 3D porous flow
Introduction The research on production performance of oil and gas wells has been one of core tasks of reservoir engineers. The traditional production data analysis methods, such as Arps production decline equation, are mostly empirical methods. However the modern production data analysis generally develops an appropriate production analysis mathematical model based on the oil and gas transient porous flow theory, and it is used to summarize performance characteristics of well production data, including perdurability, bottom flow pressure, production and cumulative production, to evaluate the reservoir macroscopic properties and predict future production decline. In productivity evaluation, the modern production data analysis method not only overcomes oversimplification of the conventional reservoir engineering, but also eliminates over-theorization of the modern well test analysis, and essentially is composed of both advantages of the two. In the 1990s, a number of scholars were interested in researching new production data analysis type curves, aroused by a master’s thesis[1] of Texas A & M University. So relevant researches were started and formed serial charts, such as Fetkovich-McCray type curves[1], Palacio-Blasingame type curves[2], Agarwal-Gardner type curves[3], NPI (Normalized Pressure Integral) type curves[4] and Flowing Material Balance Equation[5], etc. Basically, the results of these studies estab-
lished a theory of production data analysis system. However, the porous flow theory of horizontal wells is so complex that there is no satisfactory research results on production data analysis. Araya and Ozkan[6] worked out dimensionless fluid productivity index curves of horizontal wells, which was only fit for constant rate or constant bottom flow pressure, while Anderson et al[7] presented an Agarwal-Gardner production decline type curve of horizontal wells as qualitative illustration yet without theoretical derivation and typical curves, let alone discussion about the main impact parameters. Domestically, the research of horizontal well productivity has focused on steady-state production analyses[8−11], with less research on production data analysis theory. Wang Xiaodong and Liu Ciqun[12] looked into the general characteristics of horizontal wells production decline curves, but they did not form more efficient type curves. By 3D transient flow, this paper presents production performance of horizontal wells in the bounded reservoir. Through material balance time, the reciprocal value of the dimensionless bottomhole pressure in constant rate condition is converted into the data of dimensionless production function in a constant bottom pressure condition. After complex analytical solution and numerical calculation, the rate decline curves of Agarwal-Gardner type are obtained, and the analysis method of horizontal well production data is developed.
Received date: 18 Mar. 2009; Revised date: 28 Oct. 2009. * Corresponding author. E-mail:
[email protected] Foundation item: Important National Science & Technology Specific Program During the Eleventh Five-Year Plan Period (2008ZX05009-004-03). Copyright © 2010, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.
Wang Xiaodong et al. / Petroleum Exploration and Development, 2010, 37(1): 99–103
1 Material balance time and generalized flow equation for pseudo-steady state In analysis of the production data, current dimensionless material balance time (tmDA) is defined as the ratio of dimensionless cumulative production to dimensionless bottom flow rate as follows:
tmDA =
1 qD (tDA )
tDA
∫ qD (τ )dτ
(1)
0
where, dimensionless time based on drainage area is defined as: 3.6 K × 24 t tDA = φ μ Ct π( re 2 − rw 2 ) Dimensionless rate of oil well is defined as:
qD =
−3
1.842 × 10 q(t ) Bμ Kh( pi − pwf )
where, the integral term in Eq.(1) is dimensionless cumulative production obviously. The purpose of the use of material balance time is to convert the reciprocal of dimensionless bottomhole pressure in the constant bottom rate into dimensionless production function in the constant bottom flow pressure. The conversion accuracy by means of the dimensionless material balance time was discussed by Anderson and Mattar[13]. Also conversion results are asymptotic in the early linear flow and middle radial flow regimes, and become more and more precise along with time elapse, so as to be clearly accurate in the late pseudo-steady state period. According to the Duhamel convolution, the current pseudo-steady state equation expresses an approximate relationship of variable production and variable bottom flow pressure: pwD (tmDA ) = mNtmDA + bpss (2) qD (tmDA )
Fig. 1 Comparison of conversion effects of horizontal well production
center. A model is shown as Fig. 1 in the literature [12]. Use dimensionless time based on the drainage area, the governing equation for the 3D transient flow in constant production is presented as: 2 1 ∂ ⎛ ∂pD ⎞ 1 ∂pD 2 ∂ pD = (3) ⎜ rDL ⎟ + LD 2 rDL ∂rDL ⎝ ∂rDL ⎠ ∂zD πreDL 2 ∂tDA The initial conditions are: pD=0, tDA=0 (4) The constant rate condition for point convergence in the internal boundary is: ⎛ ⎞ 1 zwD +ε 2 ∂p lim ⎜ lim rDL D dzwD ⎟ = ε →0 ⎝ rDL →0 ε ∫zwD −ε 2 ∂rDL ⎠ zD − zwD > ε 2 ⎪⎧ 0 (5) ⎨ − 1 z ⎪⎩ D − z wD ≤ ε 2 Respectively, the vertical boundary condition and radial outer boundary condition are: ∂pD (6) = 0 , zD=0, 1 ∂zD
∂pD = 0 , rDL= reDL ∂rDL
(7)
where, dimensionless bottom flow pressure is defined as: Kh( pi − pwf ) pwfD = 1.842 × 10−3 qBμ
where, dimensionless variables are defined as follows: Kh( pi − p ) r L pD = , rDL = , LD = −3 L h 1.842 × 10 qBμ
As a specific case for the vertical wells in pseudo-steady state, the Eq. (2) has been derived by Palacio and Blasingame[2]. Significantly, the generalized pseudo-steady-state flow equation is not only applied in a single condition of constant bottom pressure or constant rate in the classical porous flow theory but also extended to the variable production and bottom flow pressure situations. It is not difficult to show that the generalized pseudo-steady-state flow equation is recognized for any type wells in the scope of the classical porous flow theory, also applies for horizontal wells.
z r z , zwD = w , reDL = e h L h y r x 2 2 2 rDL =xDL +yDL , xDL = , yDL = , rwDL = w L L L The governing equations of the 3D flow for the point convergence in cylinder media are constituted by the Eqs. (3) to (7). According to the principle of superposition, if a horizontal well is treated as a line convergence with infinite conductivity, we can obtain its solution by the integral to the point convergence. So firstly we acquired the solution under point convergence, and then calculated the integration to derive the last solution. Through the Laplace transformation, Fourier transformation of finite cosine integral and integral superposition theorem, the dimensionless bottomhole pressure, ~ p wfD ( s) , is presented in Laplace domain as:
2 Mathematical model of transient flow for horizontal wells in a cylindrical formation We consider a circular bounded anisotropic reservoir with the thickness h and drainage radius re; the upper and lower boundary is a non-flow plate, and a horizontal well is in the
zD =
Wang Xiaodong et al. / Petroleum Exploration and Development, 2010, 37(1): 99–103 +1
s p% wfD ( xDL , s) =
1 Gu ( xDL ,α , ε 0 )dα + 2 −∫1
∞ +1
∑ ∫ Gu ( xDL ,α , ε n )
n =1 −1
dα cos( β n zrD )cos( β n zwD )
(8)
where,
β n = nπ
ε 0 = s / πre2
ε n = s / πre2 + β n 2 LD 2
(n=1, 2, …, ∞)
zrD = zwD + rwDL LD
Gu ( xDL ,α , ε n ) = K 0 ⎡ ( xDL − α ) 2 ε n ⎤ + ⎢⎣ ⎥⎦ K1 ( reDLε 0 ) ⎡ I0 ( xDL − α )2 ε n ⎤ ⎦⎥ I1 (reDLε 0 ) ⎣⎢
Fig. 2 Agarwal-Gardner type curves of horizontal wells with different reDL
Eq.(8) is represented as dimensionless bottom pressure of the horizontal well in bounded reservoir. After integrated it along horizontal well length, the bottom pressure is given as: +1 +1
s p% wfD ( s) =
1 Gu ( β ,α , ε 0 )dα dβ + 4 −∫1 −∫1
+1 +1
1 ∞ ∑ Gu (β ,α , ε n )dα dβ cos(β n zrD )cos(β n zwD ) 2 n =1 −∫1 −∫1
(9)
Using Duhamel convolution and with consideration of bottomhole constant pressure-variable production and constant production-variable pressure, we can obtain the relationship between dimensionless production qD(tD) at constant bottom pressure and dimensionless pressure pwfD(tD) at constant rate in Laplace domain, and the equation is written as 1 sq%D ( s ) = (10) sp% wfD ( s ) According to Eq. (10), we can apply the numerical inversion to calculate dimensionless pressure pwfD(tD) at constant rate, and then acquire dimensionless production qD(tD) at constant bottom pressure. After the algorithm of the special functions[12], we can use material balance time to calculate qD(tmDA) and 1/pwfD(tDA). They are in a good consistency, and the compared results are shown in Fig. 1. It is shown that 1/pwfD(tDA) can be represented as qD (tmDA) by using material balance time, while pwfD(tDA) is bottom pressure function at the constant rate in the transient well test analysis, which is familiarized by reservoir engineers and obtained easily.
3 Agarwal-Gardner type curves for horizontal wells Using Stehfest numerical inversion[14] to calculate Eq. (9), pwfD(tDA) is directly obtained, and its reciprocal value can be used to acquire the Agarwal-Gardner type curves of the production data analysis for horizontal wells are shown as Figs. 2 and 3. In Fig. 2, zwD is equal to 0.5, and LD is 5.0, while reDL is a variable. From the characteristics of derivative curves, the type curves can be divided into two parts: the derivatives are
Fig. 3 Agarwal-Gardner type curves of horizontal wells with different LD
dispersed in the early transient decline period and drawn to Arps harmonious decline in the late pseudo-steady state period. Given reDL and zwD, we can calculate the impact of different values, LD, and the results are shown in Fig. 3. It is indicated that the impact of dimensionless length on the horizontal well production is obvious in the early production period, while it is weakening in the middle to later period. Especially, the derivative curves will be normalized in this period, which is a good method to choose an appropriate theoretical curve by type curve matching. For an isotropic reservoir, if given reDL and LD, we calculate the influence of different values of zwD. And the results show that the effect of zwD in the mid-late term is weak and can be negligible.
4
Application methods
According to the basic data, the specific application methods are given. The essential parameters for this case are listed as below: the reservoir effective thickness is 9.5 m, the wellbore radius is 0.1 m, the initial pressure is 15.2 MPa, the bottomhole flowing pressure is 12.3 MPa, the reservoir temperature is 355.4 K, the initial water saturation is 0.34, the formation water compressibility factor is 5.12×10−4 MPa−1, the rock compressibility coefficient is 5.80×10−4 MPa−1, the porosity is 0.22, the total compressibility is 3.4×10−4 MPa−1, the volume factor is 1.12 m3/m3 and the oil viscosity is 7.8 mPa⋅s. The detail analysis procedure is as follows: Step 1: Analyze the transient well test data of the target well, obtain LD by the typical curve matching. If no transient
Wang Xiaodong et al. / Petroleum Exploration and Development, 2010, 37(1): 99–103
test data, we could estimate LD according to the reservoir thickness, horizontal wells perforated well length and reservoir permeability anisotropy. Step 2: Process the production data of the target well including bottomhole flowing pressure pwf and well production q, and then calculate the cumulative production Np and the material balance time tm, where, tm = Np/q. Step 3: Calculate qd = q /(pi−pwf), and then draw the curve of qd−tm. Step 4: Calculate 1/[d(1/qd)/dlntm], and then draw the curve of 1/[d(1/qd)/dlntm]−tm. Step 5: Complete type curve matching, obtain the matching parameters, and then calculate results. Fig. 4 shows the matching effect of the actual production data and the theoretical curve of a horizontal well, and the required parameters formula and calculated results are shown as bellows. The reservoir permeability is: 1.842 μ B ⎡ q (t ) ⎤ K= ⎢ ⎥ = h( pi − pwf ) ⎣ qD ⎦ MP 1.842 × 7.8 × 1.12 × (50.2) MP = 29.3 ×10−3 μm2 9.5(15.2 − 12.3) The oil in place under the well control is: ⎛ t ⎞ ⎡ q (t ) ⎤ 1 Gi = ⎜ ⎟ ⎢ ⎥ = 2πCt ( pi − pwf ) ⎝ tmDA ⎠ MP ⎣ qD ⎦ MP 1 2π × 3.4 × 10
−4
× (15.2 − 12.3)
×
( 4 500 )MP × ( 50.2 )MP = 3 646.35 ×104 m3 The drainage area is: 0.01Gi Bo 0.01 × 3 646.35 × 1.12 A= = = 29.61 km 2 φ (1 − S wi )h 0.22 × (1 − 0.34) × 9.5 The well productivity skin factor is: ⎡ 0.1 × (12 000)MP ⎤ ⎡ r (r ) ⎤ ScA = ln ⎢ w eDL MP ⎥ = ln ⎢ ⎥ = −0.94 6 A/ π ⎦ ⎣ ⎣⎢ 29.61 × 10 / π ⎦⎥
Through analysis, we can acquire the formation permeability, oil in place under the well control, drainage area and well productivity skin factor etc. At the same time, according to the matching theoretical curves, future production in the current system can be forecasted.
5
Conclusions
Based on the theory of transient porous flow, the method of type curve matching for analyzing production data combines the advantages of both the conventional reservoir engineering and the modern well test analysis, and can be used to acquire the formation permeability, oil in place under well control, drainage area, and well productivity skin factor. By defined new dimensionless variables, the Agarwal-Gardner production decline curves and their derivative curves for horizontal wells are presented by using the integral transformation and Duhamel convolution to solve the transient 3D porous flow equations in a bounded reservoir. According to the characteristics of the derivative curves, it is shown that the Agarwal-Gardner type curves can be divided into two parts: the derivatives are dispersed in the early transient decline period and drawn to Arps harmonious decline in the late pseudo-steady state period. Consequently, the application method of Agarwal-Gardner type curves for horizontal wells is illustrated step by step through a field case. It should be noted that the research is performed for a well in an oil reservoir. If the dimensionless variables are changed, the result can be also applied to a gas reservoir. If the horizontal well length and the anisotropy of the equivalent wellbore radius are adjusted, the result can be extended to the conditions with anisotropy in horizontal direction.
Nomenclature tDA—dimensionless time based on drainage area qD—dimensionless production of oil well -3
K—reservoir permeability, 10
μm2
t—elapse time, d φ—formation porosity, fraction μ—fluid viscosity, mPa·s Ct—system compressibility, MPa
-1
rw—wellbore radius, m re—drainage radius, m q(t)—horizontal well production rate, m3/d B—fluid volume factor, m3/m3 h— reservoir thickness, m pi—initial formation pressure, MPa pwf—bottom hole following pressure, MPa mN, bpss—constant x, y, z—Cartesian coordinate, m p—pressure function, MPa L—horizontal wellbore half length, m r—radial coordinates, m zw—elevation of the horizontal wellbore in vertical direction, m τ, α, β—intermediate integration variable s—Laplace transformation Im(x), Km(x)—modified Bessel function of first and second kind,
Fig. 4 A matching case of production data analysis for a horizontal well
order m, where m=0,1 A—drainage area, km2
Wang Xiaodong et al. / Petroleum Exploration and Development, 2010, 37(1): 99–103
Gi—well controlling OOIP, 104 m3 ScA—productivity skin factor, dimensionless
77690, 2002. [6]
Blasingame T A, Johnston J L, Lee W J. Type-curve analysis using the pressure integral method. SPE 18799, 1989.
Subscriptions
[7]
Mattar L, McNeil R. The ‘Flowing’ material balance. JCPT, 1998, 37(2): 52–55.
D—dimensionless
[8]
McCray T L. Reservoir analysis using production decline data
L—based on horizontal half length
and adjusted time. Texas: Texas A & M University, College
MP—matching parameters
Station, 1990. [9]
References
Ling Zongfa, Wang Lijuan, Hu Yongle, et al. Flood pattern optimization of horizontal well injection. Petroleum Exploration and Development, 2008. 35(1): 85–91.
[1]
Agarwal R G, Gardner D C, Kleinsteiber S W, et al. Analyzing
[10] Liu Yuetian, Zhang Jichang. Methodology for horizontal well
well production data using combined type curve and decline
pattern design in anisotropic oil reservoirs. Petroleum Explo-
curve concepts. SPE 57916, 1999. [2]
[11] Luo Wanjing, Wang Xiaodong, Li Fanhua. Productivity of
and radial flow. Calgary: Canadian International Petroleum
horizontal wells with segmental perforation. Petroleum Ex-
Conference, 2003. [3] [4]
Anderson D M, Stotts G W J, Mattar L. Production data
ploration and Development, 2009.36(1): 97–102. [12] Palacio J C, Blasingame T A. Decline-curve analysis using
analysis: challenges, pitfalls, diagnostics. SPE 102048, 2006.
type curves-analysis of gas well production data. SPE 25909,
An Yongsheng, Wu Xiaodong, Han Guoqing, et al. A produc-
1993.
tivity prediction model for the complex well with formation damage. Petroleum Exploration and Development, 2008. 35(5): 625–629. [5]
ration and Development, 2008.35(5): 619–624.
Anderson D M, Mattar L. Material-balance-time during linear
[13] Stehfest H. Numerical inversion of Laplace transforms. Communications of ACM, 1970.13(1): 47–49. [14] Wang Xiaodong, Liu Ciqun. The rate decline curves of hori-
Araya A, Ozkan E. An account of decline-type-curve analysis
zontal wells and their applications. Petroleum Exploration and
of vertical, fractured, and horizontal well production data. SPE
Development, 1996.23(4): 50–53.