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Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
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Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs Qing Tian a,1, Pengcheng Liu a,*,1, Yuwei Jiao b, Aifang Bie b, Jing Xia b, Baozhu Li b, Yang Liu a a b
School of Energy Resources, China University of Geosciences, Beijing, 100083, PR China Research Institute of Petroleum Exploration and Development of CNPC, Beijing, 100082, PR China
article info
abstract
Article history:
The paper developed a new constant-rate solution for non-planar asymmetric fracture
Received 17 February 2017
connected to a vertical wellbore in hydrocarbon reservoirs. The non-planar asymmetric
Received in revised form
fracture has been modeled by combination of two unequal length wings with varied
20 March 2017
intersection angles. The type curves were drawn in the case of varied fracture conduc-
Accepted 18 April 2017
tivities, non-planar angles, and asymmetric factors. The results indicated that fracture
Available online xxx
conductivity, non-planar angle, and asymmetric factor have coefficient effects on the pressure response, which makes the dimensionless wellbore pressure and pressure de-
Keywords:
rivative curves deviate from the conventional type curves of vertical fractured wells. The
Hydrocarbon reservoir
dimensionless wellbore pressure and pressure derivative curves are divided into five flow
Pressure transient behavior
periods. The non-planar angle and asymmetry factor have their unique influences on the
Non-planar asymmetric fracture
shape of type curves during different periods under the different fracture conductivities.
Finite conductivity fracture
The termination of bilinear flow and formation linear flow happened increasingly early
Fracture wing
with the decrease of non-planar angle or the increase of asymmetry factor. As the fracture conductivity increases, the effect of non-planar angle on the curves moves from the early flow period to the late flow period while the trend of effect of asymmetric factor is quite opposite. The new method lays the foundation of the pressure analysis of the complicated fracture system in developing hydrocarbon reservoirs. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Hydraulic-fracture technologies have been extensively applied to improve oil well productivity in hydrocarbon reservoirs. In order to understand hydraulic-fracture behaviors and characteristic parameters such as fracture conductivity and effective-fracture length, pressure transient analysis
(PTA) is widely used, which is treated as one of the most efficient diagnostic techniques in well test analysis. Many analytical/semi-analytical and numerical solutions have been reported for understanding the transient flow in fractured vertical wells in hydrocarbon reservoirs. Gringarten and Ramey first introduced source functions for transient pressure analysis of uniform flux fractures and infinite conductivity
* Corresponding author. E-mail address: [email protected] (P. Liu). 1 P.C. Liu and Q. Tian contributed equally to this work (Co-first authors). http://dx.doi.org/10.1016/j.ijhydene.2017.04.161 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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fracture wells [1]. Taking into account more realistic factors, Cinco-Ley et al. [2] extended the Green's functions and presented the transient pressure solutions for a well with a finiteconductivity vertical fracture in an infinite-acting hydrocarbon reservoir using a discretized fracture approach. Cinco-Ley and Samaniego [3] also proposed a bilinear flow model for analyzing early-time pressure data and type curves and for identifying all the flow regimes for wells intersected by finiteconductivity fracture. Furthermore, more general solutions were presented, which took into account the effects of wellbore storage and skin factor near wellbore [4]. For the sake of simplicity, most of the theoretical models used for transient pressure analysis of hydraulically fractured wells in hydrocarbon reservoirs were based on the assumption that the fractures are symmetric about the wellbore. The effects of the fracture asymmetry, which could be an important consideration for fracture characterizations and design, have only been reported in few literatures. Crawford and Landrum [5] first noticed and discussed the effect of fracture asymmetry. Narasimhan and Plen, Bennett et al. [6,7] utilized the numerical approach to analyzing the issue of asymmetry fractures. However, it was not until the ~ o and Fernando [8] first achieved the early 1990s, Resurreica comprehensive study of the pressure transient analysis of asymmetrically fractured wells, which presented a semianalytical solution (a graphical technique) to evaluate asymmetry of hydraulically fractured wells in an infinite reservoir. Following this work, Rodriguez [9] examined the effects of the asymmetry factor on the reciprocal of the rate of a fractured well producing hydrocarbon under constant wellbore pressure condition. Berumen et al. [10] also investigated the pressure behavior of wells intercepting asymmetric fractures of both infinite and finite conductivity under constant rates by numerical methods. Tiab [11] applied the TDS technique to evaluate fracture asymmetry of finite-conductivity fractured wells producing at a constant-rate. Wang et al. [12] extended the work to study the pressure transient behavior of asymmetric fractured wells in coal seam reservoirs. They also forward the solution of Rodriguez in Laplace domain and presented type curves considering the wellbore storage and skin effect [13]. Liu et al. [14] provided a new mathematical model to investigate the skin effect on the pressure response. However, the above models were based on the assumption that the fractures were coplanar vertical fractures. In fact, a number of micro-seismic and laboratorial studies showed that the complicated fracture network can be observed, and most of these fractures are neither symmetry nor planar. For example, in Barnett shale, some micro-seismic fractures mapping showed that hydraulic fracture treatments create fracture networks [15e17]. By observing fractured cores directly, it was found that hydraulic fractures are not single, symmetric and planar features in reservoirs; instead they have irregular branches [18]. For this reason, a numerical model of 3-Dimensional (3-D) non-planar hydraulic fracture growth in multiple-layer hydrocarbon reservoirs with descriptions of fluid and proppant transport was developed recently for optimal fracture design [19]. This new comprehensive simulator was developed by numerical calculation, which is difficult to determine the quantitative effect of each factor on pressure response. It must be pointed that the
methods of numerical simulation and experiment are much more convenient than those analytical methods to study fluid flow behaviors for some complicated cases [20,21]. As can be seen from the literature review above, few researchers presented the model to describe the pressure response of asymmetry non-planar fractures with the finite conductivity. The object of this paper is to establish a new analytic model for pressure analysis of a finite conductivity fracture system, including both the asymmetry and the nonplanar effects. The effects of influence factors on pressure response were analyzed quantitatively in detail as well. For the sake of simplicity, just two fracture wings are presented to illustrate the new method. Additionally, the fracture model and fracture dimensionless definitions were established based on the length of a fracture wing. By coupling the reservoir flow model and fracture flow model, we can obtain the pressure transient response of non-planar asymmetrical fractures connected to vertical wellbores in hydrocarbon reservoirs.
Physical model In this section, basic mathematical models are introduced for pressure transient analysis of fractured wells in hydrocarbon reservoirs. Fig. 1 presents a common vertical fractured well consisting of two wings. The basic assumptions are as follows. 1) The hydrocarbon reservoir is isotropic, homogeneous with impermeable upper and lower boundaries. The hydrocarbon reservoir is an infinite slab of constant thickness h, of constant porosity 4 and of constant permeability k. The hydrocarbon reservoir contains a slightly compressible single-phase fluid with constant compressibility ct and constant viscosity m. The flow in the formation and fracture is assumed to obey Darcy's law. 2) The hydrocarbon reservoir is fully penetrated by finiteconductivity vertical fractures. The fracture has an angle
Fig. 1 e Schematic of a non-planar asymmetric fractured well in infinite hydrocarbon reservoir.
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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q with x axis. The wing is of length Lf, of constant width wf and of constant permeability kf (Fig. 2). The asymmetric fractures are defined with different angles (Dq) between two fractures (Fig. 3). 3) The well is produced at constant rate Q. Additionally, it is assumed that the flow from the hydrocarbon reservoir to the wellbore sections between fractures is negligible compared with the flow from the hydrocarbon reservoir to the fractures. The hydrocarbon fluid enters the fracture at ~f ðx; y; tÞ and total flow rate of a each point with flow rate q wing is qfw. No hydrocarbon fluid is assumed to flow into fracture at the tip. Moreover, the flow in fracture is assumed incompressible [2].
3
fracture in the new approach are referred to the length of a fracture wing instead of the fracture half length (see Fig. 2). In the fracture model, the dimensionless fracture wing conductivity CfD, dimensionless rate q~fD and dimensionless rD are written as CfD ¼
kf wf ~fD ¼ q ~f Lf Q; rD ¼ r Lf 2½0; 1: ; q kLf
(5)
From Eq. (5), the dimensionless fracture conductivity alters with the fracture wing length changing. So an asymmetrically vertical fracture may have different conductivity for each fracture wing. Dimensionless flow rate of the fracture wing in the wellbore is stated as
Dimensionless definitions
. qfwD ¼ qfw Q;
The dimensionless reservoir pressure and fracture pressure are defined as
where Q is the sum of flow rate of all fracture wings, and qfw is the flow rate of one of the fracture wings.
2pkh pi p ; pD ¼ mQ
pfD ¼
2pkh pi pf mQ
(1)
The dimensionless time is given as tD ¼
k t; fmct x2f
(2)
Flow model in hydrocarbon reservoir The transient pressure behavior in hydrocarbon reservoirs can be studied by considering the fracture as a line source. For an isotropic reservoir, the fully penetrating vertical-fracture uniform-flux solution in Laplace domain can be written as [22]. x*wD þLfD =2
in which xf ¼
(6)
~fD $ spD ¼ q
Lf 1 þ Lf 2 : 2
(3)
Dimensionless definitions in the reservoir are as following: xD ¼ x xf ; yD ¼ y xf ; LfD ¼ Lf xf :
(4)
Unlike the previous methods developed in most literatures [1e4,8e13], fundamental dimensionless definitions of the
Z
K0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 du; s ðx*D uÞ2 þ y*D ywD
(7)
x*wD LfD =2
where parameters with the overline “-” are variables in Laplace space, while s is the Laplace variable. The above equation is available when the fracture wing is parallel to the x axis. For some cases that the fracture wing is not parallel to the x axis, and there is an angle q between the
Fig. 2 e (a) Top view of a non-planar asymmetric fractured well. (b) Location of a fracture wing in hydrocarbon reservoir. (c) Hydrocarbon flow inside a fracture wing.
Fig. 3 e Schematic of asymmetric fracture with different non-planar angle. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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fracture wing and x axis (see Fig. 2). The fracture wing can be transferred to be parallel to the x axis by coordinate transformation method.
where DrD ¼ n1 , rDi ¼ i DrD Dr2D , qfwD ¼ j Pnj DrDk ¼ k¼1 qfDk . In Laplace domain, Eq. (11) can be written as
Pnj
~
k¼1 qfDi
2p $D$sqfD : CfD
x*D ¼ xD cosðqÞ þ yD sinðqÞ;
(8a)
spwD spfD ¼
y*D ¼ yD cosðqÞ xD sinðqÞ;
(8b)
x*wD ¼ xwD cosðqÞ þ yD sinðqÞ;
(8c)
The matrix D is only related to the number of discretized segments of fracture wing and can be tabulated. For example, if the discretized number, nj, is set to be 2, we can obtain the matrix D
y*wD ¼ ywD cosðqÞ xD sinðqÞ;
(8d)
*
where the superscript “ ” denotes the coordinate after transformation. To solve the finite conductivity case, the fracture is divided into several uniform flux segments, and then by using superposition, the desired solution is obtained. If we divide the jth fracture into nj segments, the total segments can be summed two fractures: m¼
2 X
nj :
D22 ¼
(13)
0:1875 0:25 : 0:25 0:6875
(14)
Coupling models According to the continuity condition that the pressure and the flux must be continuous along the fracture surface, the following conditions must hold along the fracture wing (0 < rD 1)
(9)
pfD ðrD Þ ¼ pD ðrD ; 0Þ;
(15a)
We will have m þ 1 unknowns and get m þ 1 equations, m equations to express fracture pressure drop by convolution, the rest one equation restricts the total dimensionless flow rate to unit. In Laplace domain, the fracture pressure drop can be expressed as
qfD ðrD Þ ¼ qD ðrD ; 0Þ:
(15b)
j¼1
nj nj 2 X 2 X X X q spDi s Di $spDi ¼ sqDi $ ; spD ¼ LfDi LfDi j¼1 i¼1 j¼1 i¼1
From Fig. 2(b) and (c), the flow in the fracture can be described as the 1-D polar coordinate originating from the wellbore. Appendix A presented the detailed derivation for flow equation in a wing [2,23].
0
(11)
0
The j-th fracture wing is divided into nj equal length segments, Eq. (11) can be transformed into the following discretized equation. The pressure of the i-th segment can be written as " nj X 2p DrD ~fDi $ DrD q qfDk $ rDi $ CfD;j 8 k¼1
# i1 X DrD ~fDi $ DrD q þ ðrDi kDrD Þ 2 k¼1 " nj X 2p DrD $ qfDi qfDk $ rDi $ ¼ CfD;j 8 k¼1 # i1 Dr X D qfDi $ þ ðrDi kDrD Þ ; 2 k¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2D þ y2D :
(16)
In addition, the sum of flow rates must satisfy the unity condition, namely nj 2 X X
Flow model in fracture
ZrD Zv 2p ~fD ðuÞdudv : qfwD rD q CfD
rD ¼
(10)
where subscripts i and j indicate the i-th segment of j-th wing, whereas qfDi and LfDi are the flow rate and length of the i-th segment, respectively.
pwD pfD ðrD Þ ¼
It is noted that Eq. (15) is satisfied with the expression
j¼1
(17)
By solving Eqs. (10), (13), (15) and (17), the wellbore pressure and flow rate distribution solution in the Laplace space can be obtained, by Stehfest numerical algorithm, we can obtain the time domain solution [24].
Model validations To validate our results, the pressure response of a symmetry fracture and the pressure response of an asymmetry fracture were selected from the reported literatures [2,10]. Fig. 2 presents a vertical well stimulated by a fracture consisting of two wings with different lengths Lf1 and Lf2 (Lf1 þ Lf2 ¼ 2xf). As proposed in the literatures, an asymmetric factor, a, is introduced to measure the offsetting of the well about the center of the fracture [8e13],
a¼
pwD pfDi ðrDi Þ ¼
sqfDi ¼ 1:
i¼1
Lf 1 Lf 2 LfD1 LfD2 ¼ : Lf 1 þ Lf 2 LfD1 þ LfD2
(18)
In addition, the apparent fracture conductivity is also introduced according to the conventional definition of fracture conductivity [2]. CfDa ¼
kf wf : kxf
(19)
(12) For a bi-wing asymmetric fracture (see Fig. 3), we assume that each wing have the same fracture parametersdkf and wf.
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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From Eqs. (3) and (19), according to the different definitions of the dimensionless fracture conductivity, the relationship between CfDa and CfD can be expressed as CfDa ¼
2CfD1 CfD2 : CfD1 þ CfD2
(20)
Symmetry fracture with finite conductivity Specially, the asymmetry factor, a, is equal to 0, namely they are the symmetric fractures. Because of the equal wing length, the relation of different dimensionless fracture conductivities is CfDa ¼ CfD1 ¼ CfD2 :
(21)
We compare the dimensionless pressure presented in this paper with the semi-analytical solution obtained by Cinco-Ley in different dimensionless fracture conductivities [2] (see Fig. 4). It is shown that the new solution match exactly the Cinco-Ley's solution. The relative error is less than 0.36%.
Fig. 5 e Comparison of dimensionless pressure for a well with a fully penetrating, finite-conductivity symmetry fracture, CfDa ¼ 0.1e50, a ¼ 0.8.
Asymmetry fracture with finite conductivity
Results and discussion Another case is used to further validate the new method. In this case, we assume an asymmetry fracture consisting of two bi-wing fractures whose asymmetric factor, a, is set to be 0.8. According to the definitions of dimensionless fracture conductivity, we can get CfD1 ¼ CfDa 1:8;
(22a)
CfD2 ¼ CfDa 0:2:
(22b)
Fig. 5 illustrates the comparison of dimensionless pressure for a well with a fully penetrating, finite-conductivity vertical asymmetry fracture with the semi-analytical solution proposed by Berumen et al. [10]. From Fig. 5, the results obtained in this paper are consistent with other results in the literature [10], which verifies the results of this paper should be correct.
Effects of parameters including non-planar angle and asymmetric factor with different fracture conductivity conditions on the bottom hole pressure behavior and pressure derivative were all analyzed in detail. In the following graphs, NPA is the abbreviation of non-planar angle, and AF means asymmetry factor. For comparison, the scope of dimensionless time tD and dimensionless pressure pwD is set to be in the ranges of 107e103 and 104e101 for all curves, respectively. In this part, the dimensionless wellbore pressure and pressure derivative curves are divided into five flow periods, the pressure derivative curve with the slope of 1/4 is I-bilinear flow period, the curve with the slope of 1/2 is III-formation linear flow period, the level line is V-pseudoradial flow period, and the curves connecting the I to III, and III to V flow period, are transition flow periods.
Effects of non-planar angle
Fig. 4 e Comparison of dimensionless pressure for a well with a fully penetrating, finite-conductivity symmetry fracture, CfDa ¼ 0.2pe100p.
We define the non-planar angle of the two fracture wings as Dq (see Figs. 2 and 3). It should be noted that the case of Dq ¼ p is corresponding to the particular case of coplanar fracture. Several cases of fractured well were run to generate solutions for the dimensionless wellbore pressure and pressure derivative response, in terms of dimensionless fracture conductivity, asymmetry factor (a ¼ 0.2) and non-planar angle (Dq ¼ p/6, p/4, p/2, 3p/4, p) (Fig. 6). The effects of Dq on pressure depend on the values of CfDa. In the case of low fracture conductivity (CfDa 1), the value of the non-planar angle Dq mainly affects the curve shape of the early bilinear flow period (see Fig. 6(a) and (b)). The smaller the non-planar angle becomes, the earlier the bilinear flow period terminates and the bigger the pressure derivative becomes. Meanwhile, as the CfDa increases, the effect of Dq on pressure derivative occurs more and more lately and even wears off. It is shown that in the case of extremely low CfDa, like CfDa ¼ 0.1, the system may not exhibit the bilinear flow
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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Fig. 6 e Effect of the non-planar angle on the dimensionless wellbore pressure under constant hydrocarbon flow conditions, AF ¼ 0.2. (a) CfDa ¼ 0.1. (b) CfDa ¼ 1. (c) CfDa ¼ 5. (d) CfDa ¼ 10. (e) CfDa ¼ 50. (f) CfDa ¼ 100. (g) CfDa ¼ 500. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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Fig. 7 e Effect of the fracture asymmetry factor on the dimensionless wellbore pressure under constant hydrocarbon flow conditions, NPA ¼ p/4. (a) CfDa ¼ 0.1. (b) CfDa ¼ 1. (c) CfDa ¼ 5. (d) CfDa ¼ 10. (e) CfDa ¼ 50. (f) CfDa ¼ 100. (g) CfDa ¼ 500. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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period when Dq p/6 (see Fig. 6(a)). The absence of the bilinear flow is caused by the interference of the fracture wings. However, in the range of 1 < CfDa < 100, it is found that the pressure behavior and its derivative show quantitative deviations in the I-bilinear flow period, II-transition flow period and IV-transition flow period from the coplanar fracture case (Dq ¼ p), simultaneously. However, as the CfDa increases, the effect of Dq on pressure derivative occurs increasingly late and even wears off in I-bilinear flow period and II-transition flow period while the effect reinforces in the IV-transition flow period (see Fig. 6(c)e(e)). When it comes to high fracture conductivity (CfDa 100), the Dq mainly affects the curve shape of the IV-transition flow period (see Fig. 6(f) and (g)). When CfDa is no less than 50, the effect of Dq on I-bilinear flow period and II-transition flow period may not be observed (see Fig. 6(e)e(g)). The end of formation liner flow period occurs earlier and the value of pressure derivative enlarges when the non-planar angle becomes increasingly small. The effect of Dq on the pressure derivative curve reinforces as the CfDa increases. Generally, as the CfDa increases, the effect of the nonplanar angle Dq on pressure derivative curves moves from the period of I to period of IV. That is to say, the effect on the periods of I and II wears off and even disappears when the CfDa 50, while the effect on periods of III and IV reinforces.
Effects of asymmetric factor From Eqs. (3)e(5), if the fracture permeability, fracture width, and reservoir permeability are equal for both two fracture wings, the dimensionless fracture wing conductivity is inversely proportional to the dimensionless fracture wing length (LfD). From Eq. (18), it is found that as long as the length of each fracture wing is defined, the asymmetry factor and the ratio of dimensionless fracture wing conductivity can be determined. Fig. 7(a)e(g) show that the effects of asymmetry factor on pressure response depend on the values of CfDa. Note that all cases discussed in this section have a non-planar angle of p/4. With the same fracture conductivity, the pwD of complete asymmetric fracture (a ¼ 1) is always bigger than the values of other cases (a ¼ 0, 0.2, 0.5, 0.8). The pressure derivative curve of a ¼ 1 obviously differs from the other asymmetric cases (a ¼ 0.2, 0.5, 0.8), however, the differences wear off as the CfDa increases. For fractures of low fracture conductivity (CfDa 1), the curves of dimensionless pressure derivative are overlapped with varied asymmetry factors (Fig. 7(a) and (b)). The slope of one fourth, characteristic of bilinear flow, is presented and the bilinear flow ends by some extreme value in the convex curve of the pressure derivative. This convex curve is caused by the non-planar angle influence which we have discussed above. The asymmetry factor enlarges the pressure derivative value in the second half of the II-transitional flow period, and the effect occurs earlier as CfDa increases. For fractures with intermediate conductivity (1 < CfDa < 100), the asymmetry factor mainly affects the Ibilinear flow period, II-transition flow period and IV-transition flow period (Fig. 7(c)e(e)). The termination of bilinear flow happened increasingly early with the increase of asymmetry factor except the case of
complete asymmetry (a ¼ 1). The effect of asymmetry factor on II-transition flow period occurs earlier as CfDa increases, and if the asymmetry factor enlarges in the influenced period, then as a consequence, the pressure derivative increases. In IV-transition flow period, the effect strengthens as CfDa increases, whereas it should be noticed that the pressure derivative decreases as the asymmetry factor increases. An intersection of pressure derivative curves, connecting IItransition flow period and IV-transition flow period, occurs earlier as CfDa increases. Regarding high fracture conductivity (CfDa 100), the effect of the asymmetry factor on the pressure derivative curves is similar to that in intermediate fracture conductivity cases; meanwhile we should notice that an early terminal time of formation linear flow will be observed in the case of large asymmetry factor (see Fig. 7(f) and (g)). From the trend of the pressure curves, we can forecast that the effect of the asymmetry factor on the pressure response in II-transitional period will not be observed when CfDa is far beyond 500. Noticeably, according to the solution proposed by Berumen et al. [10], the pressure derivative curves of coplanar asymmetric fractures will not yield any deviations with varied asymmetric factors in IV-transitional period under the same CfDa value. When CfDa 5, the pressure derivative curves in IVtransitional period results from the comprehensive influence of non-planar angle and asymmetric factor (Fig. 7(c)e(g)).
Conclusions 1) Some new dimensionless definitions based on a fracture wing are introduced to investigate the pressure response of finite conductivity fracture in hydrocarbon reservoir. The best advantage of the new definitions is that some complicated fractures can be modeled by different combinations of fracture wing, such as multiple radial fractures and multi-fractured horizontal wells. 2) The results of this model can fit well to the previous study when in some special cases, which indicates that the proposed model is feasible. 3) A series of type curves were drawn under a wide range of non-planar and asymmetric conditions, which can be used to determine the non-planar angle, asymmetric factor, and fracture conductivity in the application of well testing. 4) The termination of bilinear flow and formation linear flow happened increasingly early with the decrease of nonplanar angle or the increase of asymmetry factor. As the fracture conductivity increases, the effect of non-planar angle on the curves moves from the early flow period to the late flow period while the trend of effect of asymmetric factor is quite opposite.
Acknowledgements This research was conducted with partial financial support from the Science and Technology Special Funds of China for 2016ZX05015-002 the contribution of which and the permission for publication is gratefully acknowledged.
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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Nomenclature CfD CfDa tD CD pwD pD dpD pfD S rDi a q Dq s pD pwD pfD ~fD q ct k p pi pf Q ~f q qfw m h f r t xf Lf wf u K0(x)
pf ðr; t ¼ 0Þ ¼ pi ; 0 < r < Lf
dimensionless fracture conductivity dimensionless apparent fracture conductivity dimensionless time dimensionless wellbore storage coefficient dimensionless well bottom pressure dimensionless pressure dimensionless pressure derivative dimensionless fracture pressure skin factor midpoint of the i segment fracture asymmetry factor intersection angle of fracture wing and x axis non-planar angle time variable in Laplace domain, dimensionless the dimensionless pressure pD in Laplace domain bottom pressure pwD in Laplace domain dimensionless fracture pressure pfD in Laplace domain ~fD in Laplace domain dimensionless flow rate q total compressibility, 1/atm effective permeability, mD pressure, atm initial formation pressure, atm fracture pressure, atm total rate in wellbore, cm3/s rate of per unit fracture length from formation, cm3/ s flow rate of a fracture wing in the wellbore, cm3/s fluid viscosity, cp formation thickness, cm porosity, fraction radial distance in polar coordinate, cm time variable, s fracture half length, cm fracture wing length, cm width of the fracture, cm integral variable Modified Bessel function (2nd kind, zero order)
Boundary conditions kf hwf vpf $ ¼ qfw m vr r¼0
(A-3)
and vpf ¼0 vr r¼Lf
(A-4)
We can obtain the dimensionless equation according to the dimensionless transformation. The fluid flow equation can be written as v2 pfD 2p q~ ¼ 0 CfD fD vr2D
(A-5)
The initial condition pfD ðrD ; tD ¼ 0Þ ¼ 0; 0 < rD < 1
(A-6)
The boundary conditions vpfD 2p ¼ qfwD ; 0 < qfwD 1 CfD vrD rD ¼0
(A-7)
and vpfD ¼0 vrD rD ¼1
(A-8)
Integrating Eq. (A-5) from 0 to rD with respect to rD vpfD ðrD Þ vpfD ðrD ¼ 0Þ 2p ¼ CfD vrD vrD
ZrD
q~fD ðuÞdu
(A-9)
0
Substituting Eq. (A-7) into Equ. (A-9) yields vpfD ðrD Þ 2p ¼ CfD vrD
ZrD
q~fD ðuÞdu
0
2p qfwD CfD
(A-10)
Integrating Eq. (A-10) from 0 to rD with respect to rD 2p pfD ðrD Þ pfD ð0Þ ¼ CfD
Special subscripts f fracture property D dimensionless w wellbore property * coordinate transformation
(A-2)
ZrD Zv 0
~fD ðuÞdudv q
0
2p qfwD rD CfD
(A-11)
In the wellbore pwD ¼ pfD ð0Þ
(A-12)
Rearranging Eq. (A-11) yields
Appendix A. Model of fluid flow in a wing The fracture wing is homogeneous, finite and slab, with height (h), wing length (Lf) and width (wf). Luo and Tang presented the solution [23]. The steady-state flow in the fracture can be described by the following equation [2]. ~ v2 pf m q f ¼ 0; 0 < r < Lf þ k wf h vr2 The initial condition
(A-1)
0 1 ZrD Zv 2p @ ~fD ðuÞdudvA qfwD rD q pwD pfD ðrD Þ ¼ CfD 0
(A-13)
0
references
[1] Gringarten AC, Ramey HJ. The use of source and Green's function in solving unsteady-flow problems in reservoirs. SPE J 1973;13(5):285e96.
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[2] Cinco-Ley H, Samaniego VF, Dominguez AN. Transient pressure behavior for a well with a finite-conductivity vertical fracture. SPE J 1978;18(4):253e64. [3] Cinco-Ley H, Samaniego VF. Transient pressure analysis for fractured wells. J Pet Technol 1981;33(9):1749e66. [4] Cinco-Ley H, Meng HZ. Pressure transient analysis of wells with finite conductivity vertical fractures in double porosity reservoirs. SPE-18172-MS. 1988. [5] Crawford PB, Landrum BL. Effect of unsymmetrical vertical fractures on production capacity. SPE-433-G. 1955. [6] Narasimhan TN, Palen WA. A purely numerical approach to a well intercepting a vertical fracture. SPE-7983-MS. 1979. [7] Bennett CO, Rosato ND, Reynolds AC, Raghavan R. Influence of fracture heterogeneity and wing length on the response of vertically fractured wells. SPE J 1983;23(2):219e30. ~ o CES, Fernando R. Transient rate behavior of [8] Resurreica finite-conductivity asymmetrically fractured wells producing at constant pressure. SPE-22657-MS. 1991. [9] Rodriguez F. Evaluation of fracture asymmetry of finiteconductivity fractured wells. SPE J 1992;7(2):233e9. [10] Berumen S, Tiab D, Rodriguez F. Constant rate solutions for a fractured well with an asymmetric fracture. J Pet Sci Eng 2000;25(1):49e58. [11] Tiab D. Analysis of derivative data of hydraulically fractured wells by the Tiab's direct synthesis technique. J Pet Sci Eng 2005;49(1):1e21. [12] Wang L, Wang XD, Li JQ, Wang JH. Simulation of pressure transient behavior for asymmetrically finite-conductivity fractured wells in coal reservoirs. Transp Porous Med 2013;97(3):353e72. [13] Wang L, Wang XD. Type curves analysis for asymmetrically fractured wells. J Energy Resour Technol 2014;136(2):1e8. [14] Liu PC, Li WH, Xia J, Jiao YW, Bie AF. Derivation and application of mathematical model for well test analysis
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with variable skin factor in hydrocarbon reservoirs. AIP Adv 2016;6(6):1e11. Fisher MK, Davison BM, Goodwin AK, Fielder EO, Buckler WS, Steinberger NP. Integrating fracture mapping technologies to optimize stimulations in the Barnett shale. SPE-77441-MS. 2002. Fisher MK, Heinze JR, Harris CD, Davison BM, Wright CA, Dunn KP. Optimizing horizontal completion techniques in the Barnett shale using microseismic fracture mapping. SPE90051-MS. 2004. Mayerhofer MJ, Lolon EP, Youngblood JE, Heinze JR. Integration of microseismic fracture mapping results with numerical fracture network 207 production modeling in the Barnett shale. SPE-102103-MS. 2006. Germanovich LN, Astakhov DK, Mayerhofer JS, Ring LM. Hydraulic fracture with multiple segments I. Observations an model formulation. Int J Rock Mech Min Sci 1997;34(3e4):97 (e1ee19). Wong SW, Xu GS. Interaction of multiple hydraulic fractures in horizontal wells. SPE-163982-MS. 2013. Liu PC, Zhang XK. Enhanced oil recovery by CO2eCH4 flooding in low permeability and rhythmic hydrocarbon reservoir. Int J Hydrogen Energ 2015;40(37):12849e53. Sakoda N, Onoue K, Kuroki T, Shinzato K, Kohno M, Monde M, et al. Transient temperature and pressure behavior of highpressure 100 MPa hydrogen during discharge through orifices. Int J Hydrogen Energ 2016;41(38):17169e74. Ozkan E, Raghavan R. New solutions for welltest analysis problems: part1-analytical considerations. SPE J 1991;6(3):359e68. Luo WJ, Tang CF. A semianalytical solution of a vertical fractured well with varying conductivity under non-Darcyflow condition. SPE J 2015;20(5):1028e40. Stehfest H. Numerical inversion of laplace transform. Comm ACM 1970;13(1):47e9.
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Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs Qing Tian a,1, Pengcheng Liu a,*,1, Yuwei Jiao b, Aifang Bie b, Jing Xia b, Baozhu Li b, Yang Liu a a b
School of Energy Resources, China University of Geosciences, Beijing, 100083, PR China Research Institute of Petroleum Exploration and Development of CNPC, Beijing, 100082, PR China
article info
abstract
Article history:
The paper developed a new constant-rate solution for non-planar asymmetric fracture
Received 17 February 2017
connected to a vertical wellbore in hydrocarbon reservoirs. The non-planar asymmetric
Received in revised form
fracture has been modeled by combination of two unequal length wings with varied
20 March 2017
intersection angles. The type curves were drawn in the case of varied fracture conduc-
Accepted 18 April 2017
tivities, non-planar angles, and asymmetric factors. The results indicated that fracture
Available online xxx
conductivity, non-planar angle, and asymmetric factor have coefficient effects on the pressure response, which makes the dimensionless wellbore pressure and pressure de-
Keywords:
rivative curves deviate from the conventional type curves of vertical fractured wells. The
Hydrocarbon reservoir
dimensionless wellbore pressure and pressure derivative curves are divided into five flow
Pressure transient behavior
periods. The non-planar angle and asymmetry factor have their unique influences on the
Non-planar asymmetric fracture
shape of type curves during different periods under the different fracture conductivities.
Finite conductivity fracture
The termination of bilinear flow and formation linear flow happened increasingly early
Fracture wing
with the decrease of non-planar angle or the increase of asymmetry factor. As the fracture conductivity increases, the effect of non-planar angle on the curves moves from the early flow period to the late flow period while the trend of effect of asymmetric factor is quite opposite. The new method lays the foundation of the pressure analysis of the complicated fracture system in developing hydrocarbon reservoirs. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Hydraulic-fracture technologies have been extensively applied to improve oil well productivity in hydrocarbon reservoirs. In order to understand hydraulic-fracture behaviors and characteristic parameters such as fracture conductivity and effective-fracture length, pressure transient analysis
(PTA) is widely used, which is treated as one of the most efficient diagnostic techniques in well test analysis. Many analytical/semi-analytical and numerical solutions have been reported for understanding the transient flow in fractured vertical wells in hydrocarbon reservoirs. Gringarten and Ramey first introduced source functions for transient pressure analysis of uniform flux fractures and infinite conductivity
* Corresponding author. E-mail address: [email protected] (P. Liu). 1 P.C. Liu and Q. Tian contributed equally to this work (Co-first authors). http://dx.doi.org/10.1016/j.ijhydene.2017.04.161 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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fracture wells [1]. Taking into account more realistic factors, Cinco-Ley et al. [2] extended the Green's functions and presented the transient pressure solutions for a well with a finiteconductivity vertical fracture in an infinite-acting hydrocarbon reservoir using a discretized fracture approach. Cinco-Ley and Samaniego [3] also proposed a bilinear flow model for analyzing early-time pressure data and type curves and for identifying all the flow regimes for wells intersected by finiteconductivity fracture. Furthermore, more general solutions were presented, which took into account the effects of wellbore storage and skin factor near wellbore [4]. For the sake of simplicity, most of the theoretical models used for transient pressure analysis of hydraulically fractured wells in hydrocarbon reservoirs were based on the assumption that the fractures are symmetric about the wellbore. The effects of the fracture asymmetry, which could be an important consideration for fracture characterizations and design, have only been reported in few literatures. Crawford and Landrum [5] first noticed and discussed the effect of fracture asymmetry. Narasimhan and Plen, Bennett et al. [6,7] utilized the numerical approach to analyzing the issue of asymmetry fractures. However, it was not until the ~ o and Fernando [8] first achieved the early 1990s, Resurreica comprehensive study of the pressure transient analysis of asymmetrically fractured wells, which presented a semianalytical solution (a graphical technique) to evaluate asymmetry of hydraulically fractured wells in an infinite reservoir. Following this work, Rodriguez [9] examined the effects of the asymmetry factor on the reciprocal of the rate of a fractured well producing hydrocarbon under constant wellbore pressure condition. Berumen et al. [10] also investigated the pressure behavior of wells intercepting asymmetric fractures of both infinite and finite conductivity under constant rates by numerical methods. Tiab [11] applied the TDS technique to evaluate fracture asymmetry of finite-conductivity fractured wells producing at a constant-rate. Wang et al. [12] extended the work to study the pressure transient behavior of asymmetric fractured wells in coal seam reservoirs. They also forward the solution of Rodriguez in Laplace domain and presented type curves considering the wellbore storage and skin effect [13]. Liu et al. [14] provided a new mathematical model to investigate the skin effect on the pressure response. However, the above models were based on the assumption that the fractures were coplanar vertical fractures. In fact, a number of micro-seismic and laboratorial studies showed that the complicated fracture network can be observed, and most of these fractures are neither symmetry nor planar. For example, in Barnett shale, some micro-seismic fractures mapping showed that hydraulic fracture treatments create fracture networks [15e17]. By observing fractured cores directly, it was found that hydraulic fractures are not single, symmetric and planar features in reservoirs; instead they have irregular branches [18]. For this reason, a numerical model of 3-Dimensional (3-D) non-planar hydraulic fracture growth in multiple-layer hydrocarbon reservoirs with descriptions of fluid and proppant transport was developed recently for optimal fracture design [19]. This new comprehensive simulator was developed by numerical calculation, which is difficult to determine the quantitative effect of each factor on pressure response. It must be pointed that the
methods of numerical simulation and experiment are much more convenient than those analytical methods to study fluid flow behaviors for some complicated cases [20,21]. As can be seen from the literature review above, few researchers presented the model to describe the pressure response of asymmetry non-planar fractures with the finite conductivity. The object of this paper is to establish a new analytic model for pressure analysis of a finite conductivity fracture system, including both the asymmetry and the nonplanar effects. The effects of influence factors on pressure response were analyzed quantitatively in detail as well. For the sake of simplicity, just two fracture wings are presented to illustrate the new method. Additionally, the fracture model and fracture dimensionless definitions were established based on the length of a fracture wing. By coupling the reservoir flow model and fracture flow model, we can obtain the pressure transient response of non-planar asymmetrical fractures connected to vertical wellbores in hydrocarbon reservoirs.
Physical model In this section, basic mathematical models are introduced for pressure transient analysis of fractured wells in hydrocarbon reservoirs. Fig. 1 presents a common vertical fractured well consisting of two wings. The basic assumptions are as follows. 1) The hydrocarbon reservoir is isotropic, homogeneous with impermeable upper and lower boundaries. The hydrocarbon reservoir is an infinite slab of constant thickness h, of constant porosity 4 and of constant permeability k. The hydrocarbon reservoir contains a slightly compressible single-phase fluid with constant compressibility ct and constant viscosity m. The flow in the formation and fracture is assumed to obey Darcy's law. 2) The hydrocarbon reservoir is fully penetrated by finiteconductivity vertical fractures. The fracture has an angle
Fig. 1 e Schematic of a non-planar asymmetric fractured well in infinite hydrocarbon reservoir.
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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q with x axis. The wing is of length Lf, of constant width wf and of constant permeability kf (Fig. 2). The asymmetric fractures are defined with different angles (Dq) between two fractures (Fig. 3). 3) The well is produced at constant rate Q. Additionally, it is assumed that the flow from the hydrocarbon reservoir to the wellbore sections between fractures is negligible compared with the flow from the hydrocarbon reservoir to the fractures. The hydrocarbon fluid enters the fracture at ~f ðx; y; tÞ and total flow rate of a each point with flow rate q wing is qfw. No hydrocarbon fluid is assumed to flow into fracture at the tip. Moreover, the flow in fracture is assumed incompressible [2].
3
fracture in the new approach are referred to the length of a fracture wing instead of the fracture half length (see Fig. 2). In the fracture model, the dimensionless fracture wing conductivity CfD, dimensionless rate q~fD and dimensionless rD are written as CfD ¼
kf wf ~fD ¼ q ~f Lf Q; rD ¼ r Lf 2½0; 1: ; q kLf
(5)
From Eq. (5), the dimensionless fracture conductivity alters with the fracture wing length changing. So an asymmetrically vertical fracture may have different conductivity for each fracture wing. Dimensionless flow rate of the fracture wing in the wellbore is stated as
Dimensionless definitions
. qfwD ¼ qfw Q;
The dimensionless reservoir pressure and fracture pressure are defined as
where Q is the sum of flow rate of all fracture wings, and qfw is the flow rate of one of the fracture wings.
2pkh pi p ; pD ¼ mQ
pfD ¼
2pkh pi pf mQ
(1)
The dimensionless time is given as tD ¼
k t; fmct x2f
(2)
Flow model in hydrocarbon reservoir The transient pressure behavior in hydrocarbon reservoirs can be studied by considering the fracture as a line source. For an isotropic reservoir, the fully penetrating vertical-fracture uniform-flux solution in Laplace domain can be written as [22]. x*wD þLfD =2
in which xf ¼
(6)
~fD $ spD ¼ q
Lf 1 þ Lf 2 : 2
(3)
Dimensionless definitions in the reservoir are as following: xD ¼ x xf ; yD ¼ y xf ; LfD ¼ Lf xf :
(4)
Unlike the previous methods developed in most literatures [1e4,8e13], fundamental dimensionless definitions of the
Z
K0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 du; s ðx*D uÞ2 þ y*D ywD
(7)
x*wD LfD =2
where parameters with the overline “-” are variables in Laplace space, while s is the Laplace variable. The above equation is available when the fracture wing is parallel to the x axis. For some cases that the fracture wing is not parallel to the x axis, and there is an angle q between the
Fig. 2 e (a) Top view of a non-planar asymmetric fractured well. (b) Location of a fracture wing in hydrocarbon reservoir. (c) Hydrocarbon flow inside a fracture wing.
Fig. 3 e Schematic of asymmetric fracture with different non-planar angle. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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fracture wing and x axis (see Fig. 2). The fracture wing can be transferred to be parallel to the x axis by coordinate transformation method.
where DrD ¼ n1 , rDi ¼ i DrD Dr2D , qfwD ¼ j Pnj DrDk ¼ k¼1 qfDk . In Laplace domain, Eq. (11) can be written as
Pnj
~
k¼1 qfDi
2p $D$sqfD : CfD
x*D ¼ xD cosðqÞ þ yD sinðqÞ;
(8a)
spwD spfD ¼
y*D ¼ yD cosðqÞ xD sinðqÞ;
(8b)
x*wD ¼ xwD cosðqÞ þ yD sinðqÞ;
(8c)
The matrix D is only related to the number of discretized segments of fracture wing and can be tabulated. For example, if the discretized number, nj, is set to be 2, we can obtain the matrix D
y*wD ¼ ywD cosðqÞ xD sinðqÞ;
(8d)
*
where the superscript “ ” denotes the coordinate after transformation. To solve the finite conductivity case, the fracture is divided into several uniform flux segments, and then by using superposition, the desired solution is obtained. If we divide the jth fracture into nj segments, the total segments can be summed two fractures: m¼
2 X
nj :
D22 ¼
(13)
0:1875 0:25 : 0:25 0:6875
(14)
Coupling models According to the continuity condition that the pressure and the flux must be continuous along the fracture surface, the following conditions must hold along the fracture wing (0 < rD 1)
(9)
pfD ðrD Þ ¼ pD ðrD ; 0Þ;
(15a)
We will have m þ 1 unknowns and get m þ 1 equations, m equations to express fracture pressure drop by convolution, the rest one equation restricts the total dimensionless flow rate to unit. In Laplace domain, the fracture pressure drop can be expressed as
qfD ðrD Þ ¼ qD ðrD ; 0Þ:
(15b)
j¼1
nj nj 2 X 2 X X X q spDi s Di $spDi ¼ sqDi $ ; spD ¼ LfDi LfDi j¼1 i¼1 j¼1 i¼1
From Fig. 2(b) and (c), the flow in the fracture can be described as the 1-D polar coordinate originating from the wellbore. Appendix A presented the detailed derivation for flow equation in a wing [2,23].
0
(11)
0
The j-th fracture wing is divided into nj equal length segments, Eq. (11) can be transformed into the following discretized equation. The pressure of the i-th segment can be written as " nj X 2p DrD ~fDi $ DrD q qfDk $ rDi $ CfD;j 8 k¼1
# i1 X DrD ~fDi $ DrD q þ ðrDi kDrD Þ 2 k¼1 " nj X 2p DrD $ qfDi qfDk $ rDi $ ¼ CfD;j 8 k¼1 # i1 Dr X D qfDi $ þ ðrDi kDrD Þ ; 2 k¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2D þ y2D :
(16)
In addition, the sum of flow rates must satisfy the unity condition, namely nj 2 X X
Flow model in fracture
ZrD Zv 2p ~fD ðuÞdudv : qfwD rD q CfD
rD ¼
(10)
where subscripts i and j indicate the i-th segment of j-th wing, whereas qfDi and LfDi are the flow rate and length of the i-th segment, respectively.
pwD pfD ðrD Þ ¼
It is noted that Eq. (15) is satisfied with the expression
j¼1
(17)
By solving Eqs. (10), (13), (15) and (17), the wellbore pressure and flow rate distribution solution in the Laplace space can be obtained, by Stehfest numerical algorithm, we can obtain the time domain solution [24].
Model validations To validate our results, the pressure response of a symmetry fracture and the pressure response of an asymmetry fracture were selected from the reported literatures [2,10]. Fig. 2 presents a vertical well stimulated by a fracture consisting of two wings with different lengths Lf1 and Lf2 (Lf1 þ Lf2 ¼ 2xf). As proposed in the literatures, an asymmetric factor, a, is introduced to measure the offsetting of the well about the center of the fracture [8e13],
a¼
pwD pfDi ðrDi Þ ¼
sqfDi ¼ 1:
i¼1
Lf 1 Lf 2 LfD1 LfD2 ¼ : Lf 1 þ Lf 2 LfD1 þ LfD2
(18)
In addition, the apparent fracture conductivity is also introduced according to the conventional definition of fracture conductivity [2]. CfDa ¼
kf wf : kxf
(19)
(12) For a bi-wing asymmetric fracture (see Fig. 3), we assume that each wing have the same fracture parametersdkf and wf.
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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5
From Eqs. (3) and (19), according to the different definitions of the dimensionless fracture conductivity, the relationship between CfDa and CfD can be expressed as CfDa ¼
2CfD1 CfD2 : CfD1 þ CfD2
(20)
Symmetry fracture with finite conductivity Specially, the asymmetry factor, a, is equal to 0, namely they are the symmetric fractures. Because of the equal wing length, the relation of different dimensionless fracture conductivities is CfDa ¼ CfD1 ¼ CfD2 :
(21)
We compare the dimensionless pressure presented in this paper with the semi-analytical solution obtained by Cinco-Ley in different dimensionless fracture conductivities [2] (see Fig. 4). It is shown that the new solution match exactly the Cinco-Ley's solution. The relative error is less than 0.36%.
Fig. 5 e Comparison of dimensionless pressure for a well with a fully penetrating, finite-conductivity symmetry fracture, CfDa ¼ 0.1e50, a ¼ 0.8.
Asymmetry fracture with finite conductivity
Results and discussion Another case is used to further validate the new method. In this case, we assume an asymmetry fracture consisting of two bi-wing fractures whose asymmetric factor, a, is set to be 0.8. According to the definitions of dimensionless fracture conductivity, we can get CfD1 ¼ CfDa 1:8;
(22a)
CfD2 ¼ CfDa 0:2:
(22b)
Fig. 5 illustrates the comparison of dimensionless pressure for a well with a fully penetrating, finite-conductivity vertical asymmetry fracture with the semi-analytical solution proposed by Berumen et al. [10]. From Fig. 5, the results obtained in this paper are consistent with other results in the literature [10], which verifies the results of this paper should be correct.
Effects of parameters including non-planar angle and asymmetric factor with different fracture conductivity conditions on the bottom hole pressure behavior and pressure derivative were all analyzed in detail. In the following graphs, NPA is the abbreviation of non-planar angle, and AF means asymmetry factor. For comparison, the scope of dimensionless time tD and dimensionless pressure pwD is set to be in the ranges of 107e103 and 104e101 for all curves, respectively. In this part, the dimensionless wellbore pressure and pressure derivative curves are divided into five flow periods, the pressure derivative curve with the slope of 1/4 is I-bilinear flow period, the curve with the slope of 1/2 is III-formation linear flow period, the level line is V-pseudoradial flow period, and the curves connecting the I to III, and III to V flow period, are transition flow periods.
Effects of non-planar angle
Fig. 4 e Comparison of dimensionless pressure for a well with a fully penetrating, finite-conductivity symmetry fracture, CfDa ¼ 0.2pe100p.
We define the non-planar angle of the two fracture wings as Dq (see Figs. 2 and 3). It should be noted that the case of Dq ¼ p is corresponding to the particular case of coplanar fracture. Several cases of fractured well were run to generate solutions for the dimensionless wellbore pressure and pressure derivative response, in terms of dimensionless fracture conductivity, asymmetry factor (a ¼ 0.2) and non-planar angle (Dq ¼ p/6, p/4, p/2, 3p/4, p) (Fig. 6). The effects of Dq on pressure depend on the values of CfDa. In the case of low fracture conductivity (CfDa 1), the value of the non-planar angle Dq mainly affects the curve shape of the early bilinear flow period (see Fig. 6(a) and (b)). The smaller the non-planar angle becomes, the earlier the bilinear flow period terminates and the bigger the pressure derivative becomes. Meanwhile, as the CfDa increases, the effect of Dq on pressure derivative occurs more and more lately and even wears off. It is shown that in the case of extremely low CfDa, like CfDa ¼ 0.1, the system may not exhibit the bilinear flow
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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Fig. 6 e Effect of the non-planar angle on the dimensionless wellbore pressure under constant hydrocarbon flow conditions, AF ¼ 0.2. (a) CfDa ¼ 0.1. (b) CfDa ¼ 1. (c) CfDa ¼ 5. (d) CfDa ¼ 10. (e) CfDa ¼ 50. (f) CfDa ¼ 100. (g) CfDa ¼ 500. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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7
Fig. 7 e Effect of the fracture asymmetry factor on the dimensionless wellbore pressure under constant hydrocarbon flow conditions, NPA ¼ p/4. (a) CfDa ¼ 0.1. (b) CfDa ¼ 1. (c) CfDa ¼ 5. (d) CfDa ¼ 10. (e) CfDa ¼ 50. (f) CfDa ¼ 100. (g) CfDa ¼ 500. Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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period when Dq p/6 (see Fig. 6(a)). The absence of the bilinear flow is caused by the interference of the fracture wings. However, in the range of 1 < CfDa < 100, it is found that the pressure behavior and its derivative show quantitative deviations in the I-bilinear flow period, II-transition flow period and IV-transition flow period from the coplanar fracture case (Dq ¼ p), simultaneously. However, as the CfDa increases, the effect of Dq on pressure derivative occurs increasingly late and even wears off in I-bilinear flow period and II-transition flow period while the effect reinforces in the IV-transition flow period (see Fig. 6(c)e(e)). When it comes to high fracture conductivity (CfDa 100), the Dq mainly affects the curve shape of the IV-transition flow period (see Fig. 6(f) and (g)). When CfDa is no less than 50, the effect of Dq on I-bilinear flow period and II-transition flow period may not be observed (see Fig. 6(e)e(g)). The end of formation liner flow period occurs earlier and the value of pressure derivative enlarges when the non-planar angle becomes increasingly small. The effect of Dq on the pressure derivative curve reinforces as the CfDa increases. Generally, as the CfDa increases, the effect of the nonplanar angle Dq on pressure derivative curves moves from the period of I to period of IV. That is to say, the effect on the periods of I and II wears off and even disappears when the CfDa 50, while the effect on periods of III and IV reinforces.
Effects of asymmetric factor From Eqs. (3)e(5), if the fracture permeability, fracture width, and reservoir permeability are equal for both two fracture wings, the dimensionless fracture wing conductivity is inversely proportional to the dimensionless fracture wing length (LfD). From Eq. (18), it is found that as long as the length of each fracture wing is defined, the asymmetry factor and the ratio of dimensionless fracture wing conductivity can be determined. Fig. 7(a)e(g) show that the effects of asymmetry factor on pressure response depend on the values of CfDa. Note that all cases discussed in this section have a non-planar angle of p/4. With the same fracture conductivity, the pwD of complete asymmetric fracture (a ¼ 1) is always bigger than the values of other cases (a ¼ 0, 0.2, 0.5, 0.8). The pressure derivative curve of a ¼ 1 obviously differs from the other asymmetric cases (a ¼ 0.2, 0.5, 0.8), however, the differences wear off as the CfDa increases. For fractures of low fracture conductivity (CfDa 1), the curves of dimensionless pressure derivative are overlapped with varied asymmetry factors (Fig. 7(a) and (b)). The slope of one fourth, characteristic of bilinear flow, is presented and the bilinear flow ends by some extreme value in the convex curve of the pressure derivative. This convex curve is caused by the non-planar angle influence which we have discussed above. The asymmetry factor enlarges the pressure derivative value in the second half of the II-transitional flow period, and the effect occurs earlier as CfDa increases. For fractures with intermediate conductivity (1 < CfDa < 100), the asymmetry factor mainly affects the Ibilinear flow period, II-transition flow period and IV-transition flow period (Fig. 7(c)e(e)). The termination of bilinear flow happened increasingly early with the increase of asymmetry factor except the case of
complete asymmetry (a ¼ 1). The effect of asymmetry factor on II-transition flow period occurs earlier as CfDa increases, and if the asymmetry factor enlarges in the influenced period, then as a consequence, the pressure derivative increases. In IV-transition flow period, the effect strengthens as CfDa increases, whereas it should be noticed that the pressure derivative decreases as the asymmetry factor increases. An intersection of pressure derivative curves, connecting IItransition flow period and IV-transition flow period, occurs earlier as CfDa increases. Regarding high fracture conductivity (CfDa 100), the effect of the asymmetry factor on the pressure derivative curves is similar to that in intermediate fracture conductivity cases; meanwhile we should notice that an early terminal time of formation linear flow will be observed in the case of large asymmetry factor (see Fig. 7(f) and (g)). From the trend of the pressure curves, we can forecast that the effect of the asymmetry factor on the pressure response in II-transitional period will not be observed when CfDa is far beyond 500. Noticeably, according to the solution proposed by Berumen et al. [10], the pressure derivative curves of coplanar asymmetric fractures will not yield any deviations with varied asymmetric factors in IV-transitional period under the same CfDa value. When CfDa 5, the pressure derivative curves in IVtransitional period results from the comprehensive influence of non-planar angle and asymmetric factor (Fig. 7(c)e(g)).
Conclusions 1) Some new dimensionless definitions based on a fracture wing are introduced to investigate the pressure response of finite conductivity fracture in hydrocarbon reservoir. The best advantage of the new definitions is that some complicated fractures can be modeled by different combinations of fracture wing, such as multiple radial fractures and multi-fractured horizontal wells. 2) The results of this model can fit well to the previous study when in some special cases, which indicates that the proposed model is feasible. 3) A series of type curves were drawn under a wide range of non-planar and asymmetric conditions, which can be used to determine the non-planar angle, asymmetric factor, and fracture conductivity in the application of well testing. 4) The termination of bilinear flow and formation linear flow happened increasingly early with the decrease of nonplanar angle or the increase of asymmetry factor. As the fracture conductivity increases, the effect of non-planar angle on the curves moves from the early flow period to the late flow period while the trend of effect of asymmetric factor is quite opposite.
Acknowledgements This research was conducted with partial financial support from the Science and Technology Special Funds of China for 2016ZX05015-002 the contribution of which and the permission for publication is gratefully acknowledged.
Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Nomenclature CfD CfDa tD CD pwD pD dpD pfD S rDi a q Dq s pD pwD pfD ~fD q ct k p pi pf Q ~f q qfw m h f r t xf Lf wf u K0(x)
pf ðr; t ¼ 0Þ ¼ pi ; 0 < r < Lf
dimensionless fracture conductivity dimensionless apparent fracture conductivity dimensionless time dimensionless wellbore storage coefficient dimensionless well bottom pressure dimensionless pressure dimensionless pressure derivative dimensionless fracture pressure skin factor midpoint of the i segment fracture asymmetry factor intersection angle of fracture wing and x axis non-planar angle time variable in Laplace domain, dimensionless the dimensionless pressure pD in Laplace domain bottom pressure pwD in Laplace domain dimensionless fracture pressure pfD in Laplace domain ~fD in Laplace domain dimensionless flow rate q total compressibility, 1/atm effective permeability, mD pressure, atm initial formation pressure, atm fracture pressure, atm total rate in wellbore, cm3/s rate of per unit fracture length from formation, cm3/ s flow rate of a fracture wing in the wellbore, cm3/s fluid viscosity, cp formation thickness, cm porosity, fraction radial distance in polar coordinate, cm time variable, s fracture half length, cm fracture wing length, cm width of the fracture, cm integral variable Modified Bessel function (2nd kind, zero order)
Boundary conditions kf hwf vpf $ ¼ qfw m vr r¼0
(A-3)
and vpf ¼0 vr r¼Lf
(A-4)
We can obtain the dimensionless equation according to the dimensionless transformation. The fluid flow equation can be written as v2 pfD 2p q~ ¼ 0 CfD fD vr2D
(A-5)
The initial condition pfD ðrD ; tD ¼ 0Þ ¼ 0; 0 < rD < 1
(A-6)
The boundary conditions vpfD 2p ¼ qfwD ; 0 < qfwD 1 CfD vrD rD ¼0
(A-7)
and vpfD ¼0 vrD rD ¼1
(A-8)
Integrating Eq. (A-5) from 0 to rD with respect to rD vpfD ðrD Þ vpfD ðrD ¼ 0Þ 2p ¼ CfD vrD vrD
ZrD
q~fD ðuÞdu
(A-9)
0
Substituting Eq. (A-7) into Equ. (A-9) yields vpfD ðrD Þ 2p ¼ CfD vrD
ZrD
q~fD ðuÞdu
0
2p qfwD CfD
(A-10)
Integrating Eq. (A-10) from 0 to rD with respect to rD 2p pfD ðrD Þ pfD ð0Þ ¼ CfD
Special subscripts f fracture property D dimensionless w wellbore property * coordinate transformation
(A-2)
ZrD Zv 0
~fD ðuÞdudv q
0
2p qfwD rD CfD
(A-11)
In the wellbore pwD ¼ pfD ð0Þ
(A-12)
Rearranging Eq. (A-11) yields
Appendix A. Model of fluid flow in a wing The fracture wing is homogeneous, finite and slab, with height (h), wing length (Lf) and width (wf). Luo and Tang presented the solution [23]. The steady-state flow in the fracture can be described by the following equation [2]. ~ v2 pf m q f ¼ 0; 0 < r < Lf þ k wf h vr2 The initial condition
(A-1)
0 1 ZrD Zv 2p @ ~fD ðuÞdudvA qfwD rD q pwD pfD ðrD Þ ¼ CfD 0
(A-13)
0
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Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
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Please cite this article in press as: Tian Q, et al., Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.161