- Email: [email protected]
Reproducing kernel method for numerical simulation of downhole temperature distribution
Applied Mathematics and Computation 297 (2017) 19–30
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Reproducing kernel method for numerical simulation of downhole temperature distribution Ming-Jing Du a, Yu-Lan Wang a,∗, Chao-Lu Temuer a,b a b
Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, PR China Department of Mathematics, Shanghai Maritime University, Shanghai 201306, PR China
a r t i c l e
i n f o
Keywords: Reproducing kernel Singularly partial differential equation Heat transfer Analytical solution
a b s t r a c t This paper research downhole temperature distribution in oil production and water injection using reproducing kernel Hilbert space method (RKHSM) for the first time. The aim of this paper is that using the highly accurate RKHSM can solve downhole temperature problems effectively. According to 2-D mathematical models of downhole temperature distribution, the analytical solution was given in a series expansion form and the approximate solution was expressed by n-term summation of reproducing kernel functions which initial and boundary conditions were selected properly. Numerical results of downhole temperature distribution with multiple pay zones, in which different radial distance and different injection–production conditions (such as injection rate, injection temperature, injection time, oil layer thickness), were carried out by mathematical 7.0, and numerical results correspond to general knowledge and show that use RKHSM to research downhole temperature distribution is feasible and effective. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Being informed of downhole temperature distribution is very important in oil producing and water injecting. The study of dynamic temperature log has been a very useful information in oil production field. Temperature log is used to search for pay zone at the beginning, later people find there is no difference in heat conduction performance between oil and water, soon they realized that it can evaluate the dynamic of produce well zone by measuring and analyzing temperature anomalies. At present, temperature logging is still mainly on qualitative analysis, the quantitative application temperature data is also still in a stage of development. Scholars at home and abroad have made a lot of numerical simulation of downhole research in the wellbore. It is worth mentioning that water injection is an effective and important measure adopted in oil field to keep formation energy, enhance crude oil recovery ratio, insure long-term high and stable production. With the deepening of oil field development and the extension of exploitation time, the viscosity of oil in the well become stickier and the production is reduced, lead to a phenomenon of halting production and a large number of stored oil in underground cannot be mined, but water injection can save this problem. By injecting water, pressure in oil layer is improved, which can guarantee the stability and persistence in crude oil production, and then realize the high oil field production [1–8,15–17]. Since the late 1930s, with the development of temperature measurement technology, numerical simulations of downhole temperature distributions for various kinds of wells have been previously developed by many researchers. Finite difference ∗
Corresponding author. Fax: +86 4716575863. E-mail address: [email protected] (Y.-L. Wang).
http://dx.doi.org/10.1016/j.amc.2016.10.036 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
20
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
0.8 T( r , z )
1.0 0.6 0.4 0.5 z
0.0 0.5 r 1.0
0.0
Fig. 1. Exact solution of the test.
method, finite element method and analytical method (semi-analytical method) have been used in downhole temperature field simulation. Ramay was the first to develop the theoretical solution for estimating well bore fluid temperature [1]. He presented an approximate solution of the wellbore heat transfer problem involving injection of hot or cold fluids. The solution provides an estimate of the fluid, tubing and casing temperatures. Based on Ramey theory, some scholars have calculated the distribution of underground temperature field and numerical simulation of downhole temperature distribution [2,3]. Due to the test of oil well temperature production has many complicated and influence factors, it has been a hot topic. Song and Shi simplify the complex well tube structure, they considered the heat transfers in the wellbore as continuous medium conduction and convection in a vertical tube while the well is producing, the heat transfers in the pay zones were treated as conduction and convection in porous media. The heat transfers surrounding adjacent formation were treated as conduction in porous media, they built the temperature field models [4] and used the alternating direction implicit method (ADI) to discrete downhole temperature field model; Ren used the full implicit form of finite difference method to discrete and solve the model [5]; Li used the finite difference method to solve the mathematical models [6]; Yang used the implicit finite difference method to solve the mathematical models [7]; Xiao used the alternating direction implicit method (ADI) and speedup method (on the radial shaft adopts evenly spaced, in stratum adopts logarithmic interval) to discrete temperature field model in water injection [8]. As we known, RKHSM is simple and effective. In recent years, there has been a growing interest use RKHSM to solve mathematical problems [9–14,18], but there is no one apply this method into oil well production problems. This paper try to study the numerical simulation of downhole temperature distribution in oil production and water injection with RKHSM. the numerical simulations are given in Figs. 4–7 and Figs. 9–12 by the mathematical 7.0, which are the downhole temperature distribution with multiple pay zones in different distance and different injection–production conditions (injection rate, injection temperature, injection time and oil layer thickness). The paper is organized as follows: Section 2 introduces the mathematical models of downhole temperature distribution and reproducing kernel Hilbert space method. Section 3 gives the numerical results and discussion. Section 4 is the conclusions of this paper. 2. Mathematical modeling 2.1. The mathematical model of well bore temperature field In the wellbore (r = 0), considering the heat transfers as continuous medium conduction and convection in a vertical tube, the temperature field model as [4]
∂ 2 (λl T ) ∂ 2 (λl T ) 1 ∂ (λl T ) ∂ (cl ρl T ) ∂ (cl ρl νz T ) + + = + . r ∂r ∂t ∂z ∂ z2 ∂ r2
(1)
At the axis(r = 0), the temperature field model as
∂ 2 (λl T ) ∂ 2 (λl T ) ∂ (cl ρl T ) ∂ (cl ρl νz T ) +2 = + . 2 ∂t ∂z ∂z ∂ r2
(2)
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
21
0.001 Error
1.0
0.000 −0.001 0.5 z
0.0 0.5 r 1.0
0.0
Fig. 2. Error of the test.
Fig. 3. A simple stratigraphic model of three areas.
In the pay zones or injection layers, considering the heat transfers as conduction and convection in porous media, the temperature field model as
∂ ((cρ )∗ T ) ∂ 2 (λc T ) ∂ 2 (λc T ) 1 ∂ (λc T ) ∂ (cl ρl νr T ) = + + − . ∂t r ∂r ∂r ∂ z2 ∂ r2
(3)
22
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
Fig. 4. The fluid temperature distribution in the well axis.
Fig. 5. Effect of different radial distance inside oil wellbore.
In adjacent formation, considering the heat transfers as conduction in porous media, the temperature field model as
∂ ((cρ )∗∗ T ) ∂ 2 (λr T ) ∂ 2 (λr T ) 1 ∂ (λr T ) = + + . ∂t r ∂r ∂ z2 ∂ r2
(4)
In the four partial differential equations, T(r, z) is the temperature, λl = Fw λw + (1 − Fw )λ0 is the thermal conductivity of mixed fluid inside the oil wellbore, λw , λ0 are the thermal conductivity of water and oil, Fw is water saturation. λc is the thermal conductivity of pay zones. λr is the thermal conductivity of adjacent formation. z is the depth of the oil wellbore, r is the radial distance. ρl cl = Fw ρw cw + (1 − Fw )ρ0 c0 , ρw , ρ0 are the density of water and oil, cw , c0 are the specific heat of water and oil. (cρ )∗ is equivalent specific heat of pay zones. (cρ )∗∗ is equivalent specific heat of adjacent formation. ν z is the velocity of fluid inside the oil wellbore, ν r is the velocity of fluid in pay zones. The initial temperature of the formation and wellbore is at the geothermal condition which can be written as T |t=0 = T0 = a + bz, a is the formation temperature when ∂ (λ T ) z = 0, b is the geothermal gradient, the boundary of adiabatic condition is ∂1r |r=0 = 0, the upper boundary of adiabatic condition is
∂ ( λ1 T ) ∂ ( λ1 T ) ∂ z |z=H = 0, the under boundary of adiabatic condition is ∂ z |z=0 = 0.
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
23
Fig. 6. Effect of different radial distance near borehole wall.
Fig. 7. Effect of different radial distance faraway from borehole wall.
In order to solving Eqs. (1)–(4), we introduce the reproducing kernel spaces. 2.2. Reproducing kernel Hilbert spaces method (RKHSM) In this paper, RKHSM is given to solve the four partial differential equations with prescribed initial and boundary conditions. There are several reproducing kernel spaces(RKHS): W21 [0, 1], W23 [0, 1], W23 [0, 1] W23 [0, 1], W21 [0, 1] W21 [0, 1]. (1) Space W21 [0, 1] = { f | f is one-variable absolutely continuous function, f ∈ L2 [0, 1]}. An inner product in W21 [0, 1] is given by
f (x ) , g(x )W21 = f (0 )g(0 ) +
1 0
f (x ) · g (x )dx ,
f (x ), g(x ) ∈ W21 [0.1].
The space W21 [0, 1] is a reproducing kernel space and its reproducing kernel is {1}
Rx ( y ) = (2) Space W23 [0, 1] =
1 + x,
y>x
1 + y,
x > y.
f | f, f , f are one-variable absolutely continuous functions, f (0 ) = f (0 ) = 0, f ∈ L2 [0, 1] .
24
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
Fig. 8. A simple model of mixed water and oil with three areas.
Fig. 9. Effect of different injection rate on the temperature profile.
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
Fig. 10. Effect of different injection temperature on the temperature profile.
Fig. 11. Effect of different injection time on the temperature profile.
An inner product in W23 [0, 1] is given by
f (x ) , g(x )W23 = f (0 )g (0 ) +
0
1
f (x ) · g (x )dx,
f (x ), g(x ) ∈ W23 [0.1].
25
26
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
The space W23 [0, 1] is a reproducing kernel space and its reproducing kernel is
{3}
Rx ( y ) =
y2 (−5xy2 + y3 + 10x2 (3 + y ))/120,
y > x,
x (x − 5x y + 30y + 10xy ))/120,
x > y.
2
3
2
2
2
(3) The reproducing kernel space W = W23 [0, 1] W23 [0, 1]
=
f (x, y )|
∂ n+m f (x, y ) are two-variable complete continuous functions, n = 0, 1, 2, ∂ xn ∂ ym
∂ p+q m = 0, 1, 2, f (x, y ) ∈ L2 (D ), p = 0, 1, 2, q = 0, 1, 2, f (x, 0 ) = f (x, 0 ) = f (0, y ) = f (0, y ) = 0 ∂ x p ∂ yq
=
f (x, y )|
∞ ∞
ci j g(1i ) (x ) · g(2j ) (y ) : ∀
i=1 j=1
∞ ∞
|ci j |2 < ∞
i=1 j=1
(i )
Where D = [0, 1] × [0, 1], and gk (k = 1, 2 ) is a complete orthogonal function system of W23 [0.1]. ∞ ∞ ( j) (l ) (i ) (k ) j=1 ci j g1 (x ) · g2 (y ) ∈ W, g(x, y ) = i=1 j=1 dkl g1 (x ) · g2 (y ) ∈ W. An inner product: f, g = {3} {3} ( j ) (l ) ∞ ∞ ¯ (i ) (k ) ¯ i=1 j=1 i=1 j=1 ci j dkl g1 , g1 1 g2 , g2 2 = i=1 j=1 ci j di j , so the reproducing kernel of W is R (ξ ,η ) (x, y ) = Rξ (x ) × {3} { 1 } {1} Rη (y ); By the same way, we continue get W1 = W21 [0, 1] W21 [0, 1], the reproducing kernel of W1 is R(ξ ,η ) (x, y ) = Rξ (x ) × {1}
For any f (x, y ) = ∞ ∞ ∞ ∞
∞ ∞ i=1
Rη ( y ).
2.3. Analytical solution and approximate solution ∂ 2 (λ T )
In order to solve Eq. (1), we let (LT )(r, z ) = ∂ z2l + (5), the solution of (1) turned into the solution of (5).
∂ 2 ( λl T ) ∂ (λ T ) ∂ (cl ρl T ) ∂ (cl ρl νz T ) + 1r ∂ rl , F (r, z ) = − , then (1) turned into ∂t ∂z ∂ r2
(LT )(r, z ) = F (r, z ),
(5)
Where L: W(D) → W1 (D) is bounded linear operator and L−1 is existent. W(D) is a RKHS with the repro{3} {1} ducing kernel R(ξ ,η ) (r, z ), W1 (D) is a RKHS with the reproducing kernel R(ξ ,η ) (r, z ). Then it can be noted that ϕi (r, z ) = R{(1r },z ) (r, z ), ψi (r, z ) = L∗ ϕi (r, z ), B = b, where b = [ψ1 (r, z ), ψ2 (r, z ), . . .]T , = [ζ1 , ζ2 , . . .]T , B = i
i
(Lψi (r, z )|(r,z)=(r j ,z j ) )i, j=1,2, ..., L∗ is the adjoint operator. If B−1 is existent, the matrix (Lζ j (r, z )|(r,z)=(ri ,zi ) )i, j=1,2,... is an identity matrix. According to Refs. [9–14], If {ri , zi }∞ is distinct densely points in D and L−1 is existent, then T (r, z ) = i=1 ∞ F ( r , z ) ζ ( r, z ) is an analytical solution of (5) , and the approximate solution is Tn (r, z ) = nj=1 F (r j , z j )ζ j (r, z ), j = j j j j=1 1, . . . , n [10,13,14].
·
Theorem 2.1. (Convergence analysis) If Tn (r, z ) −→ T (r, z )(n → ∞ ), then Tn (r, z) → T(r, z)(n → ∞). Proof. For any (r, z) ∈ D,
| Tn (r, z ) − T (r, z ) |=| Tn (ξ , η ) − T (ξ , η ), K(ξ ,η ) (r, z ) |≤ Tn (ξ , η ) − T (ξ , η ) K(ξ ,η ) (r, z ) .
(6)
then there exists C > 0 so that
| Tn (r, z ) − T (r, z ) | ≤ C Tn (ξ , η ) − T (ξ , η ) . ·
Since Tn (ξ , η ) −→ T (ξ , η )(n → ∞ ), hence, it holds Tn (r, z) → T(r, z)(n → ∞).
(7)
Theorem 2.2. (Error estimate) If for every given (r, z) ∈ [a, b] × [a, b], there is always {(ri , zi )}∞ satisfying r − ri = i=1 ∞ ∂ Kr (ξ ,η ) ∂ Kz (ξ ,η ) 1 M , then | T ( r, z ) − T ( r, z ) | < , where M = F ( r , z ) ζ ( r, z ) | + | . n j j j r= ζ z= θ j=n+1 n n ∂r ∂z
1 n,
z − zi =
Proof. Implying LT (ri , zi ) = LTn (ri , zi ), so we obtain
|LT (r, z ) − LTn (r, z )| = |LT (r, z ) − LT (ri , zi ) − [LTn (r, z ) − LTn (ri , zi )]|
(8)
Application reproducing kernel property, we have T (r, z ) = T (ξ , η ), K(r,z ) (ξ , η ), LT (r, z ) = T (ξ , η ), LK(r,z ) (ξ , η ) We also have
LT (r, z ) − LTn (r, z ) = LT (r, z ) − LT (ri , zi ) − [LTn (r, z ) − LTn (ri , zi )] = T (ξ , η ), LK(r,z ) (ξ , η ) − LK(ri ,zi ) (ξ , η ) − Tn (ξ , η ), LK(r,z ) (ξ , η ) − LK(ri ,zi ) (ξ , η ) = T (ξ , η ) − Tn (ξ , η ), LK(r,z ) (ξ , η ) − LK(ri ,zi ) (ξ , η )
(9)
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
27
Table 1 Numerical results. (r, z)
Approximate solution
Exact solution
Absolute error
(0.08, 0.08) (0.16, 0.16) (0.24, 0.24) (0.32, 0.32) (0.40, 0.40) (0.48, 0.48) (0.56, 0.56) (0.64, 0.64) (0.72, 0.72) (0.80, 0.80) (0.88, 0.88) (0.96, 0.96)
0.370239 0.373791 0.389475 0.406848 0.429949 0.45938 0.495906 0.540463 0.594175 0.658376 0.734607 0.824709
0.370239 0.373791 0.389479 0.406854 0.429947 0.459352 0.495822 0.540293 0.593907 0.658038 0.734332 0.824746
1.87642 × 10−8 2.68195 × 10−7 4.03275 × 10−6 5.52503 × 10−6 1.95128 × 10−6 2.87421 × 10−5 8.40834 × 10−5 1.6952 × 10−4 2.68744 × 10−4 3.38042 × 10−4 2.7483 × 10−4 3.68595 × 10−5
Table 2 Physical parameters. Category
Density/(kg/m3 )
Specific heat/(J/kg ·◦ C )
Thermal conductivity/(W/km ·◦ C )
Water Oil Sandstone matrix Mudstone skeleton
10 0 0 800 2650 2450
4220 1920 820 820
0.622 0.148 2.100 2.800
Moreover
|T (r, z ) − Tn (r, z )| = |L−1 [LT (r, z ) − LTn (r, z )]| ≤ |T (ξ , η ) − Tn (ξ , η ), L−1 LK(r,z ) (ξ , η ) − L−1 LK(ri ,zi ) (ξ , η )| ≤ T − Tn K(r,z ) (ξ , η ) − K(ri ,zi ) (ξ , η )
(10)
It is noted that we take norm of K(r,z ) (ξ , η ) − K(ri ,zi ) (ξ , η ) for variable (ξ , η). The function K(r, z) (ξ , η) is derived on (r, z) in the interval of [a, b] × [a, b]. So we have K(r,z ) (ξ , η ) − K(ri ,zi ) (ξ , η ) = ∂ Kr∂(ξr ,η ) |r=ζ (r − ri ) + ∂ Kz∂(ξz ,η ) |z=θ (z − zi ). Hence
∂ Kr (ξ , η ) ∂ Kz (ξ , η ) |T (r, z ) − Tn (r, z )| ≤ T − Tn |r=ζ (r − ri ) + |z=θ (z − zi ) ∂r ∂z ∞ 1 M ∂ K (ξ , η ) ∂ Kr (ξ , η ) = F (r j , z j )ζ j (r, z ) |r=ζ + r |z=θ n ≤ n ∂z j=n+1 ∂ r
(11)
2.4. Numerical test In order to show RKHSM is effective, the exact solution is used as a test to prove the high accuracy of the method. In Eq. (1), select the above conditions and boundary conditions, let λl = 1, D = (0, 1 ) × (0, 1 ), the exact solution of the equation is given T (r, z ) = e−1 · cosh r · coshz [19]. By mathematica 7.0, the number of nodes in this text is 12. Fig. 1 is the exact solution, Fig. 2 is the error, and Table 1 shows us the numerical results, exact solution T(r, z), and absolute errors at t = 1. 3. Numerical simulation results and discussion 3.1. Numerical simulation of downhole radial temperature distribution Fig. 3 is a simple stratigraphic model of three areas: wellbore, pay zones, adjacent formations, the lines in Fig. 3 correspond to the lines in Figs. 5–7. We use the common body structure of vertical oil well and assume that the fluid in the oil wellbore is the Newton fluid, the radius of oil well is 0.1 m, the geothermal gradient is 0.025 ◦ C/m, ignore the influence of casing and cement, the initial formation temperature is 15 ◦ C when the depth z = 0 km, the liquid producing capacity is 0.005 m/s. Table 2 shows us the physical parameters. Table 3 shows us the formation parameters. Through RKHSM, Numerical simulation results of downhole temperature distribution with multiple pay zones with different distance from producing oil wellbore are given in the form of Figs. 4–7 by mathematical 7.0, and the number of nodes is 12. Compared with the different radial distance, we get the following analysis conclusions.
28
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30 Table 3 Formation parameters. Horizon
z/(km )
Oil saturation/(%)
Water saturation/(%)
Porosity/(%)
Adjacent formation1 Pay zone1 Adjacent formation2 Pay zone2 Adjacent formation3
1.0 0 0–1.030 1.030–1.035 1.035–1.065 1.065–1.075 1.075–1.100
0.0 0.2 0.0 0.2 0.0
1.0 0.8 1.0 0.8 1.0
0.3 0.2 0.3 0.2 0.3
Fig. 12. Effect of different oil layer thickness on the temperature profile.
3.1.1. In the wellbore Fig. 4 is the fluid temperature distribution in the well axis which is the green line in Fig. 5, Fig. 5 is the temperature distribution inside oil wellbore. Obviously, there are two pay zones when r = 0.07 m, and the temperature in the well axis is higher than r = 0.07 m, this is because they are exchanged each other by the continuously heat fluid which come from underground when r = 0.07 m. 3.1.2. Near borehole wall Fig. 6 shows the temperature of near borehole wall is deviated from original temperature as the surrounding temperature has changed. Compared r = 0.16 m with r = 0.19 m, the closer distance from the borehole wall, the more temperature is changed. Because geothermal fluid constantly flowing come from far place in the pay zones, temperature near the pay zone is close to the geothermal temperature. 3.1.3. Faraway from borehole wall Fig. 7 presents the temperature of faraway from borehole wall, there is no difference when r = 0.76 m and r = 0.96 m, and they close to the geothermal gradient line. 3.2. Numerical simulation of temperature distribution in water injection wellbore Fig. 8 is a simple model of mixed water and oil with three areas: wellbore, injection layers, adjacent formations, the radius of oil well is 0.1 m, the geothermal gradient is 0.025 ◦ C/m, ignore the influence of casing and cement, the initial formation temperature is 15 ◦ C when the depth z = 0 km. physical parameters and formation parameters are also given in Tables 2 and 3. Through RKHSM, numerical simulation program of downhole temperature distributions were given in the form of Figs. 9–12 when r = 0.09 m with different injection–production parameters (injection rate, injection temperature, injection time and oil thickness). Compared with the different injection–production parameters, the following analysis conclusions are given.
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
29
3.2.1. Effect of injection rate on the temperature profile The injection temperature is 70 ◦ C, temperature of numerical simulations are given in Fig. 9 at t = 1 h when the injection rate is 0.005 m/s, 0.01 m/s, 0.015 m/s. Fig. 9 shows that different injection rate cause different temperature profile. The temperature of injection layer is higher changed than others when νz = 0.005 m/s. The temperature in injection layer is obviously changed when the injection rate is small, because the fluid and borehole wall exchange the heat transfer sufficiently in the wellbore. 3.2.2. Effect of injection temperature on the temperature profile When injection rate νz = 0.005 m/s, the temperature of numerical simulation when injection temperature is 30 ◦ C, 45 ◦ C, 70 ◦ C are given in Fig. 10 at t = 1 h. Fig. 10 shows that different injection temperature cause different temperature profile, the temperature profile is less obvious when the injection temperature is close to the original formation temperature (42.5 ◦ C), the temperature profile is more obviously changed when the injection temperature is further deviated from the original formation temperature (30 ◦ C, 70 ◦ C). 3.2.3. Effect of injection time on the temperature profile When injection rate νz = 0.005 m/s, the injection temperature is 70 ◦ C, the temperature of numerical simulation when injection time is 0.1d, 1.5d, 10d are given in Fig. 11. Fig. 11 presents that different injection time cause different temperature profile. The temperature of injection layer is not higher changed than others when t = 0.1d, this is because the fluid and borehole wall are not exchange the heat transfer sufficiently in the wellbore. As the temperature of fluid and borehole wall reach consistent temperature when injection time is long, temperature profile cannot be distinguished. Therefore, reasonable injection time is good for temperature field of the recovery. 3.2.4. Effect of oil layer thickness on the injection temperature profile When injection rate νz = 0.005 m/s, the injection temperature is 70 ◦ C, injection time is t = 1 h, the temperature of numerical simulation when oil layer thickness is 2.5 m, 5 m, 7.5 m are given in Fig. 12. In Fig. 12, different oil layer thickness cause different temperature profile. When oil layer thickness is 7.5 m, the temperature of injection layer is higher changed than others. The temperature is high because the fluids are fully blended in the entrance of the injection layer. 4. Conclusions This paper, RKHSM which avoid having to practice Gram–Schmidt orthogonalization is used to solve a class of partial differential equations which can describe the temperature distribution in oil production and water injection well. According to the numerical simulation results of different radial distance and different injection–production conditions (injection rate, injection temperature, injection time, oil layer thickness) are carried out by mathematical 7.0, following conclusions are given: (1) When the oil wellbore production, different radial distance make different temperature distribution field: in the well axis, the temperature is higher than others; near borehole wall, the temperature is lower than that farther from the borehole wall. According to the change of temperature lines, the pay zones can be distinguished; the temperature is close to geothermal gradient line in the farthest place. (2) When the water injection wellbore production, different injection–production conditions make different temperature distribution field: when the injection rate is slow, the temperature in injection layer is obviously changed; when the injection temperature is further deviated from the original formation, the temperature profile is more obviously changed; when the injection time is short, the temperature of injection layer is obviously changed; when oil layer thickness is thick, the temperature of injection layer is obviously changed. The results of numerical simulation in this paper are reasonable and correspond to general knowledge. RKHSM can solve downhole temperature distribution feasibly and effectively. This paper provides reference for oil production and injection– production dynamic in future research. Acknowledgments The authors would like to express their thanks to the reviewers for their careful reading and helpful suggestions, which greatly improved the quality of the paper. This paper is supported by the Natural Science Foundation of China (no. 11361037), the Natural Science Foundation of Inner Mongolia (nos. 2013MS0109 and 2015MS0118), Project Application Technology Research And Development Foundation of Inner Mongolia (no. 20120312) and Autonomous Region PhD Research Innovation Project of Inner Mongolia (no. B20141012808).
30
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
References [1] H.J. Ramey, Well bore heat transmission, J. Pet. Technol. 4 (1962) 427–435. [2] C.J. Yao, G.L. Lei, Study of the factors impacting on wellbore temperature in high pour point oil production, Pet. Drill. Tech. 39 (5) (2011) 74–78. [3] W.C. Li, T. Qi, Research and application of wellbore temperature field models for thermal recovery well in offshore heavy oilfield, J. Southwest Pet. Univ. 34 (3) (2012) 105–110. [4] Y. Shi, Y.J. Song, H. Liu, Numerical simulation of downhole temperature distribution in producing oil wells, Appl. Geophys. 5 (4) (2008) 340–349. [5] M. Ren, H.L. Ma, Simulation of the characteristics of logging response in temperature field in wellbore, Chin. J. Eng. Geophys. 11 (1) (2014) 71–76. [6] S.G. Li, X.P. Li, Injection well shaft two-dimensional numerical simulation of the transient temperature field, Oil Gasfield Surf. Eng. 30 (1) (2011) 42–43. [7] M. Yang, Y.F. Meng, Effects of the radial temperature gradient and axial conduction of drilling fluid on the wellbore temperature distribution, Acta Phys. Sin. 62 (7) (2013) 1–9. [8] Z.S. Xiao, Y.J. Song, Temperature field model and numerical simulation of multilayer water injection well, Prog. Geophys. 20 (3) (2005) 801–807. [9] Y.L. Wang, L. Chao, Using reproducing kernel for solving a class of partial differential equation with variable-coefficients, Appl. Math. Mech. 29 (1) (2008) 129–137. [10] Y.L. Wang, L.J. Su, Using reproducing kernel for solving a class of singularly perturbed problems, Comput. Math. Appl. 61 (2011) 421–430. [11] Y.L. Wang, J. Pang, Efficient solution of a class of partial integro-differential equation in reproducing kernel space, Int. J. Comput. Math. 87 (14) (2010) 3196–3198. [12] Y.L. Wang, An efficient computational method for a class of singularly perturbed delay parabolic partial differential equation, Int. J. Comput. Math. 88 (16) (2011) 3496–3506. [13] B.Y. Wu, X.Y. Li, A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method, Appl. Math. Lett. 24 (2011) 156–159. [14] F.Z. Geng, S.P. Qian, Piecewise reproducing kernel method for singularly perturbed delay initial value problems, Appl. Math. Lett. 37 (2014) 67–71. [15] C.L. Temuer, G. Bluman, An algorithmic method for showing existence of nontrivial non-classical symmetries of partial differential equations without solving determining equations, J. Math. Anal. Appl. 411 (2014) 281–296. [16] B.S. Wu, X. Zhang, A model for downhole fluid and rock temperature prediction during circulation, Geothermics 50 (2014) 202–212. [17] M. Salari, M. Mohammadtabar, Numerical solutions to heat transfer of nanofluid flow over stretching sheet subjected to variations of nanoparticle volume fraction and wall temperature, J. Math. Anal. Appl. 35 (1) (2014) 63–72. [18] M. Mohammadi, R. Mokhtari, Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math. 235 (2011) 4003–4014. [19] R.K. Mohanty, S. Singh, A new two-level implicit discretization of o(k2 + kh2 + h4 ) for the solution of singularly perturbed two-space dimensional non-linear parabolic equations, J. Comput. Appl. Math. 208 (2007) 391–403.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Reproducing kernel method for numerical simulation of downhole temperature distribution Ming-Jing Du a, Yu-Lan Wang a,∗, Chao-Lu Temuer a,b a b
Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, PR China Department of Mathematics, Shanghai Maritime University, Shanghai 201306, PR China
a r t i c l e
i n f o
Keywords: Reproducing kernel Singularly partial differential equation Heat transfer Analytical solution
a b s t r a c t This paper research downhole temperature distribution in oil production and water injection using reproducing kernel Hilbert space method (RKHSM) for the first time. The aim of this paper is that using the highly accurate RKHSM can solve downhole temperature problems effectively. According to 2-D mathematical models of downhole temperature distribution, the analytical solution was given in a series expansion form and the approximate solution was expressed by n-term summation of reproducing kernel functions which initial and boundary conditions were selected properly. Numerical results of downhole temperature distribution with multiple pay zones, in which different radial distance and different injection–production conditions (such as injection rate, injection temperature, injection time, oil layer thickness), were carried out by mathematical 7.0, and numerical results correspond to general knowledge and show that use RKHSM to research downhole temperature distribution is feasible and effective. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Being informed of downhole temperature distribution is very important in oil producing and water injecting. The study of dynamic temperature log has been a very useful information in oil production field. Temperature log is used to search for pay zone at the beginning, later people find there is no difference in heat conduction performance between oil and water, soon they realized that it can evaluate the dynamic of produce well zone by measuring and analyzing temperature anomalies. At present, temperature logging is still mainly on qualitative analysis, the quantitative application temperature data is also still in a stage of development. Scholars at home and abroad have made a lot of numerical simulation of downhole research in the wellbore. It is worth mentioning that water injection is an effective and important measure adopted in oil field to keep formation energy, enhance crude oil recovery ratio, insure long-term high and stable production. With the deepening of oil field development and the extension of exploitation time, the viscosity of oil in the well become stickier and the production is reduced, lead to a phenomenon of halting production and a large number of stored oil in underground cannot be mined, but water injection can save this problem. By injecting water, pressure in oil layer is improved, which can guarantee the stability and persistence in crude oil production, and then realize the high oil field production [1–8,15–17]. Since the late 1930s, with the development of temperature measurement technology, numerical simulations of downhole temperature distributions for various kinds of wells have been previously developed by many researchers. Finite difference ∗
Corresponding author. Fax: +86 4716575863. E-mail address: [email protected] (Y.-L. Wang).
http://dx.doi.org/10.1016/j.amc.2016.10.036 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
20
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
0.8 T( r , z )
1.0 0.6 0.4 0.5 z
0.0 0.5 r 1.0
0.0
Fig. 1. Exact solution of the test.
method, finite element method and analytical method (semi-analytical method) have been used in downhole temperature field simulation. Ramay was the first to develop the theoretical solution for estimating well bore fluid temperature [1]. He presented an approximate solution of the wellbore heat transfer problem involving injection of hot or cold fluids. The solution provides an estimate of the fluid, tubing and casing temperatures. Based on Ramey theory, some scholars have calculated the distribution of underground temperature field and numerical simulation of downhole temperature distribution [2,3]. Due to the test of oil well temperature production has many complicated and influence factors, it has been a hot topic. Song and Shi simplify the complex well tube structure, they considered the heat transfers in the wellbore as continuous medium conduction and convection in a vertical tube while the well is producing, the heat transfers in the pay zones were treated as conduction and convection in porous media. The heat transfers surrounding adjacent formation were treated as conduction in porous media, they built the temperature field models [4] and used the alternating direction implicit method (ADI) to discrete downhole temperature field model; Ren used the full implicit form of finite difference method to discrete and solve the model [5]; Li used the finite difference method to solve the mathematical models [6]; Yang used the implicit finite difference method to solve the mathematical models [7]; Xiao used the alternating direction implicit method (ADI) and speedup method (on the radial shaft adopts evenly spaced, in stratum adopts logarithmic interval) to discrete temperature field model in water injection [8]. As we known, RKHSM is simple and effective. In recent years, there has been a growing interest use RKHSM to solve mathematical problems [9–14,18], but there is no one apply this method into oil well production problems. This paper try to study the numerical simulation of downhole temperature distribution in oil production and water injection with RKHSM. the numerical simulations are given in Figs. 4–7 and Figs. 9–12 by the mathematical 7.0, which are the downhole temperature distribution with multiple pay zones in different distance and different injection–production conditions (injection rate, injection temperature, injection time and oil layer thickness). The paper is organized as follows: Section 2 introduces the mathematical models of downhole temperature distribution and reproducing kernel Hilbert space method. Section 3 gives the numerical results and discussion. Section 4 is the conclusions of this paper. 2. Mathematical modeling 2.1. The mathematical model of well bore temperature field In the wellbore (r = 0), considering the heat transfers as continuous medium conduction and convection in a vertical tube, the temperature field model as [4]
∂ 2 (λl T ) ∂ 2 (λl T ) 1 ∂ (λl T ) ∂ (cl ρl T ) ∂ (cl ρl νz T ) + + = + . r ∂r ∂t ∂z ∂ z2 ∂ r2
(1)
At the axis(r = 0), the temperature field model as
∂ 2 (λl T ) ∂ 2 (λl T ) ∂ (cl ρl T ) ∂ (cl ρl νz T ) +2 = + . 2 ∂t ∂z ∂z ∂ r2
(2)
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
21
0.001 Error
1.0
0.000 −0.001 0.5 z
0.0 0.5 r 1.0
0.0
Fig. 2. Error of the test.
Fig. 3. A simple stratigraphic model of three areas.
In the pay zones or injection layers, considering the heat transfers as conduction and convection in porous media, the temperature field model as
∂ ((cρ )∗ T ) ∂ 2 (λc T ) ∂ 2 (λc T ) 1 ∂ (λc T ) ∂ (cl ρl νr T ) = + + − . ∂t r ∂r ∂r ∂ z2 ∂ r2
(3)
22
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
Fig. 4. The fluid temperature distribution in the well axis.
Fig. 5. Effect of different radial distance inside oil wellbore.
In adjacent formation, considering the heat transfers as conduction in porous media, the temperature field model as
∂ ((cρ )∗∗ T ) ∂ 2 (λr T ) ∂ 2 (λr T ) 1 ∂ (λr T ) = + + . ∂t r ∂r ∂ z2 ∂ r2
(4)
In the four partial differential equations, T(r, z) is the temperature, λl = Fw λw + (1 − Fw )λ0 is the thermal conductivity of mixed fluid inside the oil wellbore, λw , λ0 are the thermal conductivity of water and oil, Fw is water saturation. λc is the thermal conductivity of pay zones. λr is the thermal conductivity of adjacent formation. z is the depth of the oil wellbore, r is the radial distance. ρl cl = Fw ρw cw + (1 − Fw )ρ0 c0 , ρw , ρ0 are the density of water and oil, cw , c0 are the specific heat of water and oil. (cρ )∗ is equivalent specific heat of pay zones. (cρ )∗∗ is equivalent specific heat of adjacent formation. ν z is the velocity of fluid inside the oil wellbore, ν r is the velocity of fluid in pay zones. The initial temperature of the formation and wellbore is at the geothermal condition which can be written as T |t=0 = T0 = a + bz, a is the formation temperature when ∂ (λ T ) z = 0, b is the geothermal gradient, the boundary of adiabatic condition is ∂1r |r=0 = 0, the upper boundary of adiabatic condition is
∂ ( λ1 T ) ∂ ( λ1 T ) ∂ z |z=H = 0, the under boundary of adiabatic condition is ∂ z |z=0 = 0.
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
23
Fig. 6. Effect of different radial distance near borehole wall.
Fig. 7. Effect of different radial distance faraway from borehole wall.
In order to solving Eqs. (1)–(4), we introduce the reproducing kernel spaces. 2.2. Reproducing kernel Hilbert spaces method (RKHSM) In this paper, RKHSM is given to solve the four partial differential equations with prescribed initial and boundary conditions. There are several reproducing kernel spaces(RKHS): W21 [0, 1], W23 [0, 1], W23 [0, 1] W23 [0, 1], W21 [0, 1] W21 [0, 1]. (1) Space W21 [0, 1] = { f | f is one-variable absolutely continuous function, f ∈ L2 [0, 1]}. An inner product in W21 [0, 1] is given by
f (x ) , g(x )W21 = f (0 )g(0 ) +
1 0
f (x ) · g (x )dx ,
f (x ), g(x ) ∈ W21 [0.1].
The space W21 [0, 1] is a reproducing kernel space and its reproducing kernel is {1}
Rx ( y ) = (2) Space W23 [0, 1] =
1 + x,
y>x
1 + y,
x > y.
f | f, f , f are one-variable absolutely continuous functions, f (0 ) = f (0 ) = 0, f ∈ L2 [0, 1] .
24
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
Fig. 8. A simple model of mixed water and oil with three areas.
Fig. 9. Effect of different injection rate on the temperature profile.
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
Fig. 10. Effect of different injection temperature on the temperature profile.
Fig. 11. Effect of different injection time on the temperature profile.
An inner product in W23 [0, 1] is given by
f (x ) , g(x )W23 = f (0 )g (0 ) +
0
1
f (x ) · g (x )dx,
f (x ), g(x ) ∈ W23 [0.1].
25
26
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
The space W23 [0, 1] is a reproducing kernel space and its reproducing kernel is
{3}
Rx ( y ) =
y2 (−5xy2 + y3 + 10x2 (3 + y ))/120,
y > x,
x (x − 5x y + 30y + 10xy ))/120,
x > y.
2
3
2
2
2
(3) The reproducing kernel space W = W23 [0, 1] W23 [0, 1]
=
f (x, y )|
∂ n+m f (x, y ) are two-variable complete continuous functions, n = 0, 1, 2, ∂ xn ∂ ym
∂ p+q m = 0, 1, 2, f (x, y ) ∈ L2 (D ), p = 0, 1, 2, q = 0, 1, 2, f (x, 0 ) = f (x, 0 ) = f (0, y ) = f (0, y ) = 0 ∂ x p ∂ yq
=
f (x, y )|
∞ ∞
ci j g(1i ) (x ) · g(2j ) (y ) : ∀
i=1 j=1
∞ ∞
|ci j |2 < ∞
i=1 j=1
(i )
Where D = [0, 1] × [0, 1], and gk (k = 1, 2 ) is a complete orthogonal function system of W23 [0.1]. ∞ ∞ ( j) (l ) (i ) (k ) j=1 ci j g1 (x ) · g2 (y ) ∈ W, g(x, y ) = i=1 j=1 dkl g1 (x ) · g2 (y ) ∈ W. An inner product: f, g = {3} {3} ( j ) (l ) ∞ ∞ ¯ (i ) (k ) ¯ i=1 j=1 i=1 j=1 ci j dkl g1 , g1 1 g2 , g2 2 = i=1 j=1 ci j di j , so the reproducing kernel of W is R (ξ ,η ) (x, y ) = Rξ (x ) × {3} { 1 } {1} Rη (y ); By the same way, we continue get W1 = W21 [0, 1] W21 [0, 1], the reproducing kernel of W1 is R(ξ ,η ) (x, y ) = Rξ (x ) × {1}
For any f (x, y ) = ∞ ∞ ∞ ∞
∞ ∞ i=1
Rη ( y ).
2.3. Analytical solution and approximate solution ∂ 2 (λ T )
In order to solve Eq. (1), we let (LT )(r, z ) = ∂ z2l + (5), the solution of (1) turned into the solution of (5).
∂ 2 ( λl T ) ∂ (λ T ) ∂ (cl ρl T ) ∂ (cl ρl νz T ) + 1r ∂ rl , F (r, z ) = − , then (1) turned into ∂t ∂z ∂ r2
(LT )(r, z ) = F (r, z ),
(5)
Where L: W(D) → W1 (D) is bounded linear operator and L−1 is existent. W(D) is a RKHS with the repro{3} {1} ducing kernel R(ξ ,η ) (r, z ), W1 (D) is a RKHS with the reproducing kernel R(ξ ,η ) (r, z ). Then it can be noted that ϕi (r, z ) = R{(1r },z ) (r, z ), ψi (r, z ) = L∗ ϕi (r, z ), B = b, where b = [ψ1 (r, z ), ψ2 (r, z ), . . .]T , = [ζ1 , ζ2 , . . .]T , B = i
i
(Lψi (r, z )|(r,z)=(r j ,z j ) )i, j=1,2, ..., L∗ is the adjoint operator. If B−1 is existent, the matrix (Lζ j (r, z )|(r,z)=(ri ,zi ) )i, j=1,2,... is an identity matrix. According to Refs. [9–14], If {ri , zi }∞ is distinct densely points in D and L−1 is existent, then T (r, z ) = i=1 ∞ F ( r , z ) ζ ( r, z ) is an analytical solution of (5) , and the approximate solution is Tn (r, z ) = nj=1 F (r j , z j )ζ j (r, z ), j = j j j j=1 1, . . . , n [10,13,14].
·
Theorem 2.1. (Convergence analysis) If Tn (r, z ) −→ T (r, z )(n → ∞ ), then Tn (r, z) → T(r, z)(n → ∞). Proof. For any (r, z) ∈ D,
| Tn (r, z ) − T (r, z ) |=| Tn (ξ , η ) − T (ξ , η ), K(ξ ,η ) (r, z ) |≤ Tn (ξ , η ) − T (ξ , η ) K(ξ ,η ) (r, z ) .
(6)
then there exists C > 0 so that
| Tn (r, z ) − T (r, z ) | ≤ C Tn (ξ , η ) − T (ξ , η ) . ·
Since Tn (ξ , η ) −→ T (ξ , η )(n → ∞ ), hence, it holds Tn (r, z) → T(r, z)(n → ∞).
(7)
Theorem 2.2. (Error estimate) If for every given (r, z) ∈ [a, b] × [a, b], there is always {(ri , zi )}∞ satisfying r − ri = i=1 ∞ ∂ Kr (ξ ,η ) ∂ Kz (ξ ,η ) 1 M , then | T ( r, z ) − T ( r, z ) | < , where M = F ( r , z ) ζ ( r, z ) | + | . n j j j r= ζ z= θ j=n+1 n n ∂r ∂z
1 n,
z − zi =
Proof. Implying LT (ri , zi ) = LTn (ri , zi ), so we obtain
|LT (r, z ) − LTn (r, z )| = |LT (r, z ) − LT (ri , zi ) − [LTn (r, z ) − LTn (ri , zi )]|
(8)
Application reproducing kernel property, we have T (r, z ) = T (ξ , η ), K(r,z ) (ξ , η ), LT (r, z ) = T (ξ , η ), LK(r,z ) (ξ , η ) We also have
LT (r, z ) − LTn (r, z ) = LT (r, z ) − LT (ri , zi ) − [LTn (r, z ) − LTn (ri , zi )] = T (ξ , η ), LK(r,z ) (ξ , η ) − LK(ri ,zi ) (ξ , η ) − Tn (ξ , η ), LK(r,z ) (ξ , η ) − LK(ri ,zi ) (ξ , η ) = T (ξ , η ) − Tn (ξ , η ), LK(r,z ) (ξ , η ) − LK(ri ,zi ) (ξ , η )
(9)
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
27
Table 1 Numerical results. (r, z)
Approximate solution
Exact solution
Absolute error
(0.08, 0.08) (0.16, 0.16) (0.24, 0.24) (0.32, 0.32) (0.40, 0.40) (0.48, 0.48) (0.56, 0.56) (0.64, 0.64) (0.72, 0.72) (0.80, 0.80) (0.88, 0.88) (0.96, 0.96)
0.370239 0.373791 0.389475 0.406848 0.429949 0.45938 0.495906 0.540463 0.594175 0.658376 0.734607 0.824709
0.370239 0.373791 0.389479 0.406854 0.429947 0.459352 0.495822 0.540293 0.593907 0.658038 0.734332 0.824746
1.87642 × 10−8 2.68195 × 10−7 4.03275 × 10−6 5.52503 × 10−6 1.95128 × 10−6 2.87421 × 10−5 8.40834 × 10−5 1.6952 × 10−4 2.68744 × 10−4 3.38042 × 10−4 2.7483 × 10−4 3.68595 × 10−5
Table 2 Physical parameters. Category
Density/(kg/m3 )
Specific heat/(J/kg ·◦ C )
Thermal conductivity/(W/km ·◦ C )
Water Oil Sandstone matrix Mudstone skeleton
10 0 0 800 2650 2450
4220 1920 820 820
0.622 0.148 2.100 2.800
Moreover
|T (r, z ) − Tn (r, z )| = |L−1 [LT (r, z ) − LTn (r, z )]| ≤ |T (ξ , η ) − Tn (ξ , η ), L−1 LK(r,z ) (ξ , η ) − L−1 LK(ri ,zi ) (ξ , η )| ≤ T − Tn K(r,z ) (ξ , η ) − K(ri ,zi ) (ξ , η )
(10)
It is noted that we take norm of K(r,z ) (ξ , η ) − K(ri ,zi ) (ξ , η ) for variable (ξ , η). The function K(r, z) (ξ , η) is derived on (r, z) in the interval of [a, b] × [a, b]. So we have K(r,z ) (ξ , η ) − K(ri ,zi ) (ξ , η ) = ∂ Kr∂(ξr ,η ) |r=ζ (r − ri ) + ∂ Kz∂(ξz ,η ) |z=θ (z − zi ). Hence
∂ Kr (ξ , η ) ∂ Kz (ξ , η ) |T (r, z ) − Tn (r, z )| ≤ T − Tn |r=ζ (r − ri ) + |z=θ (z − zi ) ∂r ∂z ∞ 1 M ∂ K (ξ , η ) ∂ Kr (ξ , η ) = F (r j , z j )ζ j (r, z ) |r=ζ + r |z=θ n ≤ n ∂z j=n+1 ∂ r
(11)
2.4. Numerical test In order to show RKHSM is effective, the exact solution is used as a test to prove the high accuracy of the method. In Eq. (1), select the above conditions and boundary conditions, let λl = 1, D = (0, 1 ) × (0, 1 ), the exact solution of the equation is given T (r, z ) = e−1 · cosh r · coshz [19]. By mathematica 7.0, the number of nodes in this text is 12. Fig. 1 is the exact solution, Fig. 2 is the error, and Table 1 shows us the numerical results, exact solution T(r, z), and absolute errors at t = 1. 3. Numerical simulation results and discussion 3.1. Numerical simulation of downhole radial temperature distribution Fig. 3 is a simple stratigraphic model of three areas: wellbore, pay zones, adjacent formations, the lines in Fig. 3 correspond to the lines in Figs. 5–7. We use the common body structure of vertical oil well and assume that the fluid in the oil wellbore is the Newton fluid, the radius of oil well is 0.1 m, the geothermal gradient is 0.025 ◦ C/m, ignore the influence of casing and cement, the initial formation temperature is 15 ◦ C when the depth z = 0 km, the liquid producing capacity is 0.005 m/s. Table 2 shows us the physical parameters. Table 3 shows us the formation parameters. Through RKHSM, Numerical simulation results of downhole temperature distribution with multiple pay zones with different distance from producing oil wellbore are given in the form of Figs. 4–7 by mathematical 7.0, and the number of nodes is 12. Compared with the different radial distance, we get the following analysis conclusions.
28
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30 Table 3 Formation parameters. Horizon
z/(km )
Oil saturation/(%)
Water saturation/(%)
Porosity/(%)
Adjacent formation1 Pay zone1 Adjacent formation2 Pay zone2 Adjacent formation3
1.0 0 0–1.030 1.030–1.035 1.035–1.065 1.065–1.075 1.075–1.100
0.0 0.2 0.0 0.2 0.0
1.0 0.8 1.0 0.8 1.0
0.3 0.2 0.3 0.2 0.3
Fig. 12. Effect of different oil layer thickness on the temperature profile.
3.1.1. In the wellbore Fig. 4 is the fluid temperature distribution in the well axis which is the green line in Fig. 5, Fig. 5 is the temperature distribution inside oil wellbore. Obviously, there are two pay zones when r = 0.07 m, and the temperature in the well axis is higher than r = 0.07 m, this is because they are exchanged each other by the continuously heat fluid which come from underground when r = 0.07 m. 3.1.2. Near borehole wall Fig. 6 shows the temperature of near borehole wall is deviated from original temperature as the surrounding temperature has changed. Compared r = 0.16 m with r = 0.19 m, the closer distance from the borehole wall, the more temperature is changed. Because geothermal fluid constantly flowing come from far place in the pay zones, temperature near the pay zone is close to the geothermal temperature. 3.1.3. Faraway from borehole wall Fig. 7 presents the temperature of faraway from borehole wall, there is no difference when r = 0.76 m and r = 0.96 m, and they close to the geothermal gradient line. 3.2. Numerical simulation of temperature distribution in water injection wellbore Fig. 8 is a simple model of mixed water and oil with three areas: wellbore, injection layers, adjacent formations, the radius of oil well is 0.1 m, the geothermal gradient is 0.025 ◦ C/m, ignore the influence of casing and cement, the initial formation temperature is 15 ◦ C when the depth z = 0 km. physical parameters and formation parameters are also given in Tables 2 and 3. Through RKHSM, numerical simulation program of downhole temperature distributions were given in the form of Figs. 9–12 when r = 0.09 m with different injection–production parameters (injection rate, injection temperature, injection time and oil thickness). Compared with the different injection–production parameters, the following analysis conclusions are given.
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
29
3.2.1. Effect of injection rate on the temperature profile The injection temperature is 70 ◦ C, temperature of numerical simulations are given in Fig. 9 at t = 1 h when the injection rate is 0.005 m/s, 0.01 m/s, 0.015 m/s. Fig. 9 shows that different injection rate cause different temperature profile. The temperature of injection layer is higher changed than others when νz = 0.005 m/s. The temperature in injection layer is obviously changed when the injection rate is small, because the fluid and borehole wall exchange the heat transfer sufficiently in the wellbore. 3.2.2. Effect of injection temperature on the temperature profile When injection rate νz = 0.005 m/s, the temperature of numerical simulation when injection temperature is 30 ◦ C, 45 ◦ C, 70 ◦ C are given in Fig. 10 at t = 1 h. Fig. 10 shows that different injection temperature cause different temperature profile, the temperature profile is less obvious when the injection temperature is close to the original formation temperature (42.5 ◦ C), the temperature profile is more obviously changed when the injection temperature is further deviated from the original formation temperature (30 ◦ C, 70 ◦ C). 3.2.3. Effect of injection time on the temperature profile When injection rate νz = 0.005 m/s, the injection temperature is 70 ◦ C, the temperature of numerical simulation when injection time is 0.1d, 1.5d, 10d are given in Fig. 11. Fig. 11 presents that different injection time cause different temperature profile. The temperature of injection layer is not higher changed than others when t = 0.1d, this is because the fluid and borehole wall are not exchange the heat transfer sufficiently in the wellbore. As the temperature of fluid and borehole wall reach consistent temperature when injection time is long, temperature profile cannot be distinguished. Therefore, reasonable injection time is good for temperature field of the recovery. 3.2.4. Effect of oil layer thickness on the injection temperature profile When injection rate νz = 0.005 m/s, the injection temperature is 70 ◦ C, injection time is t = 1 h, the temperature of numerical simulation when oil layer thickness is 2.5 m, 5 m, 7.5 m are given in Fig. 12. In Fig. 12, different oil layer thickness cause different temperature profile. When oil layer thickness is 7.5 m, the temperature of injection layer is higher changed than others. The temperature is high because the fluids are fully blended in the entrance of the injection layer. 4. Conclusions This paper, RKHSM which avoid having to practice Gram–Schmidt orthogonalization is used to solve a class of partial differential equations which can describe the temperature distribution in oil production and water injection well. According to the numerical simulation results of different radial distance and different injection–production conditions (injection rate, injection temperature, injection time, oil layer thickness) are carried out by mathematical 7.0, following conclusions are given: (1) When the oil wellbore production, different radial distance make different temperature distribution field: in the well axis, the temperature is higher than others; near borehole wall, the temperature is lower than that farther from the borehole wall. According to the change of temperature lines, the pay zones can be distinguished; the temperature is close to geothermal gradient line in the farthest place. (2) When the water injection wellbore production, different injection–production conditions make different temperature distribution field: when the injection rate is slow, the temperature in injection layer is obviously changed; when the injection temperature is further deviated from the original formation, the temperature profile is more obviously changed; when the injection time is short, the temperature of injection layer is obviously changed; when oil layer thickness is thick, the temperature of injection layer is obviously changed. The results of numerical simulation in this paper are reasonable and correspond to general knowledge. RKHSM can solve downhole temperature distribution feasibly and effectively. This paper provides reference for oil production and injection– production dynamic in future research. Acknowledgments The authors would like to express their thanks to the reviewers for their careful reading and helpful suggestions, which greatly improved the quality of the paper. This paper is supported by the Natural Science Foundation of China (no. 11361037), the Natural Science Foundation of Inner Mongolia (nos. 2013MS0109 and 2015MS0118), Project Application Technology Research And Development Foundation of Inner Mongolia (no. 20120312) and Autonomous Region PhD Research Innovation Project of Inner Mongolia (no. B20141012808).
30
M.-J. Du et al. / Applied Mathematics and Computation 297 (2017) 19–30
References [1] H.J. Ramey, Well bore heat transmission, J. Pet. Technol. 4 (1962) 427–435. [2] C.J. Yao, G.L. Lei, Study of the factors impacting on wellbore temperature in high pour point oil production, Pet. Drill. Tech. 39 (5) (2011) 74–78. [3] W.C. Li, T. Qi, Research and application of wellbore temperature field models for thermal recovery well in offshore heavy oilfield, J. Southwest Pet. Univ. 34 (3) (2012) 105–110. [4] Y. Shi, Y.J. Song, H. Liu, Numerical simulation of downhole temperature distribution in producing oil wells, Appl. Geophys. 5 (4) (2008) 340–349. [5] M. Ren, H.L. Ma, Simulation of the characteristics of logging response in temperature field in wellbore, Chin. J. Eng. Geophys. 11 (1) (2014) 71–76. [6] S.G. Li, X.P. Li, Injection well shaft two-dimensional numerical simulation of the transient temperature field, Oil Gasfield Surf. Eng. 30 (1) (2011) 42–43. [7] M. Yang, Y.F. Meng, Effects of the radial temperature gradient and axial conduction of drilling fluid on the wellbore temperature distribution, Acta Phys. Sin. 62 (7) (2013) 1–9. [8] Z.S. Xiao, Y.J. Song, Temperature field model and numerical simulation of multilayer water injection well, Prog. Geophys. 20 (3) (2005) 801–807. [9] Y.L. Wang, L. Chao, Using reproducing kernel for solving a class of partial differential equation with variable-coefficients, Appl. Math. Mech. 29 (1) (2008) 129–137. [10] Y.L. Wang, L.J. Su, Using reproducing kernel for solving a class of singularly perturbed problems, Comput. Math. Appl. 61 (2011) 421–430. [11] Y.L. Wang, J. Pang, Efficient solution of a class of partial integro-differential equation in reproducing kernel space, Int. J. Comput. Math. 87 (14) (2010) 3196–3198. [12] Y.L. Wang, An efficient computational method for a class of singularly perturbed delay parabolic partial differential equation, Int. J. Comput. Math. 88 (16) (2011) 3496–3506. [13] B.Y. Wu, X.Y. Li, A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method, Appl. Math. Lett. 24 (2011) 156–159. [14] F.Z. Geng, S.P. Qian, Piecewise reproducing kernel method for singularly perturbed delay initial value problems, Appl. Math. Lett. 37 (2014) 67–71. [15] C.L. Temuer, G. Bluman, An algorithmic method for showing existence of nontrivial non-classical symmetries of partial differential equations without solving determining equations, J. Math. Anal. Appl. 411 (2014) 281–296. [16] B.S. Wu, X. Zhang, A model for downhole fluid and rock temperature prediction during circulation, Geothermics 50 (2014) 202–212. [17] M. Salari, M. Mohammadtabar, Numerical solutions to heat transfer of nanofluid flow over stretching sheet subjected to variations of nanoparticle volume fraction and wall temperature, J. Math. Anal. Appl. 35 (1) (2014) 63–72. [18] M. Mohammadi, R. Mokhtari, Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math. 235 (2011) 4003–4014. [19] R.K. Mohanty, S. Singh, A new two-level implicit discretization of o(k2 + kh2 + h4 ) for the solution of singularly perturbed two-space dimensional non-linear parabolic equations, J. Comput. Appl. Math. 208 (2007) 391–403.