- Email: [email protected]
Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs
international journal of hydrogen energy xxx (xxxx) xxx
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs Yun Wu a,b, Zhen Huang a,b,*, Kui Zhao a, Wei Zeng a, Qixiong Gu a, Rui Zhang c a
School of Resources and Environment Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China b School of Earth Sciences and Engineering, Nanjing University, Nanjing, 210023, China c Kunming Survey, Design and Research Institute CO. LTD of Creec, Kunming, 650200, China
highlights A
unsteady
graphical abstract
seepage
solution
model for hydraulic fracturing was improved. Variations of water pressure and hydraulic gradient around wells were analysed. Hydraulic fracturing mechanism around
wells
under
unsteady
seepage was revealed.
article info
abstract
Article history:
This paper establishes an analytical model to study the influence mechanism of hydraulic
Received 16 December 2019
fracturing around vertical wellbores under the unsteady seepage in hydrocarbon reser-
Received in revised form
voirs, deduces the analytical solution of water pressure and hydraulic gradient of the
27 January 2020
model, and compares the law of water pressure and hydraulic gradient changing with time
Accepted 28 January 2020
with the results of numerical simulation. The results confirm the accuracy of these
Available online xxx
analytical solutions. The variation laws of water pressure and hydraulic gradient in the sample under unsteady seepage are analysed by using the COMSOL Multiphysics software.
Keywords:
The results show that: the increasing rate and amplitude of water pressure decrease with
Hydraulic fracturing
the distance of water inlet, however, hydraulic gradient near the water inlet is the largest
Unsteady seepage
and decreasing with the distance. In order to better understand the mechanism of hy-
Analytical solution
draulic fracturing of rock mass, we studied the influence of permeability and water in-
Numerical simulation
jection pressure on water pressure and hydraulic gradient of rock mass. The results show
Hydrocarbon reservoirs
that: large permeability coefficient and high hydraulic gradient will increase the probability of rock mass hydraulic fracturing. The permeability and hydraulic gradient of rock mass is
* Corresponding author. School of Resources and Environment Engineering, Jiangxi University of Science and Technology, Ganzhou, 341000, China. E-mail addresses: [email protected], [email protected] (Z. Huang). https://doi.org/10.1016/j.ijhydene.2020.01.222 0360-3199/© 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
2
international journal of hydrogen energy xxx (xxxx) xxx
important factor in determining whether the rock has hydraulic fracturing. The distribution law of water pressure and hydraulic gradient in rock mass under unsteady seepage provides important reference and basis in hydrogen developing reservoirs. © 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Fossil energy has long been one of the main energy structures in China, occupying a significant proportion [1]. However, burning fossil energy produces a series of environmental problems; for example, the gases released can cause greenhouse effects and seriously damage ecosystems. Therefore, the development of clean energy has become a major research topic to be solved. As is known to all, the hydrogen energy is a kind of secondary energy, which needs to be generated by hydrocarbon reservoirs, as one of the most important clean energy sources and has thus have attracted global attention [2]. Hydraulic fracturing technology plays an important role in increasing the output of hydrocarbon reservoirs [3e10]. In the process of developing hydrocarbon reservoirs by hydraulic fracturing (see Fig. 1), a series of technical problems are involved. For example, the process of water injection into boreholes is actually a problem of coupling the seepage field with a stress field when water is injected, the rocks around the borehole wall may crack and fail under the action of highpressure water and the internal stress of the rock mass around the borehole wall will redistribute. The redistribution of stress changes the hydraulic properties of the rock mass, and the coupling of the seepage field and stress field is a very complex scientific problem [11e20]. In order to understand hydraulic characteristics parameters in rock mass such as water pressure and hydraulic gradient during the hydraulic fracturing, which is treated as one of the effective techniques in hydrocarbon reservoirs [21e23]. Many researchers have adopted analytical model to examine and obtain the change rule of pressure transient of rock mass under the action of hydraulic fracturing in the hydrocarbon reservoirs [24,25]. Wang et al. [26]established a semi-analytical model to study pressure transient behavior for asymmetrically fractured wells in hydrocarbon reservoirs, and the results can predict production performance and enable sufficient test analysis in the development of dualpermeability organic compound reservoir of hydrogen and carbon. Xue et al. [27] presented a comprehensive analytical model to discuss the transient flow behavior through the horizontal fractures and flow characteristics. Wang et al. [28] combined superposition, Laplace integral transformation, Stehfest numerical inverse algorithm, and the perturbation technique to build a model, deduce the analytical solution of the model. Guppy et al. [29] established an analytical model to study the effect of non-Darcy flow on the pressure transient behavior of finite conductivity vertical fractures. Tian et al. [30] developed a new constant-rate solution for non-planar asymmetric fracture connected to a vertical wellbore in hydrocarbon reservoirs.
All the above researches are carried out from the perspective of mathematical model, however, with the development of computer technology, many scholars use numerical simulation software to examine and obtain the change rule of hydraulic characteristics of rock mass under the action of hydraulic fracturing in the hydrocarbon reservoirs. For example, Xue et al. [31] used COMSOL Multiphysics to study the productivity of fractured wells in the organic hydrogen and carbon reservoirs. Guo et al. [32] applied ABAQUS to investigate fracture propagation of zipper and synchronous fracturing in hydrogen energy. Liu et al. [33] established a model of radial holes combined with hydraulic fracturing to directional propagation of hydraulic fracture in fossil hydrogen energy. At present, there are few studies on the hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs under unsteady seepage. In this paper, we established a simple model of unsteady seepage flow and adopted the numerical simulation software COMSOL Multiphysics coupling to discuss the unsteady seepage of rock mass under the action of the internal water pressure and hydraulic gradient through the specific example in the hydrocarbon reservoirs, then, we discussed the effect of different permeability coefficient and injection pressure on the hydraulic fracturing and revealed the mechanism of hydraulic fracturing of rock mass under the condition of unsteady seepage. The research results can provide a reference for the pressure distribution laws connected to vertical wellbores in the hydrocarbon reservoirs.
Fig. 1 e Schematic diagram of hydraulic fracturing using vertical wellbores in hydrocarbon reservoirs.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
international journal of hydrogen energy xxx (xxxx) xxx
Establishment of the unsteady seepage model In order to study the hydraulic characteristics of rock mass in the hydrocarbon reservoirs, we need to establish a model under the condition of unsteady seepage and assume that the rock mass is isotropic. Fig. 2 presents a schematic diagram of seepage and water pressure distribution near the injection borehole. The water flow will appear a process of gradual oozes around the injection borehole. In addition, when water is injected into the borehole, the water pressure in the borehole increases to P0 instantaneously and remains stable, i.e. f(0,t) ¼ Pt. The water pressure at R from the center of the borehole is the original pressure Pt, i.e. f(R,t) ¼ Pt. A simplified one-dimensional model can be established to investigate the seepage state at any position in a rock mass with time t (x away from the borehole). In addition, this paper assumes that the seepage state in the initial state of rock mass is equal, that is, f(x,0) ¼ Pt. In the presented one-dimensional model, the equation of seepage movement of a section in the rock mass can be expressed as[34]： vp v2 p ¼ d2 2 vt vx
(1)
Here, d is a parameter related to the permeability coefficient (k) and the storage capacity (Sr) of rock mass by the following formula: d2 ¼
k Sr
(2)
To determine the Fourier transform solution, the following new functions are introduced: uðx; tÞ ¼ f ðx; tÞ f ðx; 0Þ ð0 x R; t 0Þ
vuðx; tÞ v2 uðx; tÞ ¼ d2 vt vx2
(4)
According to the model conditions, the following definitions are applied to the presented formula: 8 < uðx; 0Þ ¼ f ðx; 0Þ f ðx; 0Þ ¼ 0 ð0; x RÞ uð0; tÞ ¼ f ð0; tÞ f ð0; 0Þ ¼ P0 Pt : uðR; tÞ ¼ f ðR; tÞ f ðR; 0Þ ¼ 0 ðt 0Þ
(5)
The finite Fourier sinusoidal transformation of variable x can be obtained as follows: uðx; tÞ ¼
2 ∞ 2ðP0 Pt Þ X 1 np np k sin x 1 exp t p n R R Sr n¼1
(6)
The water pressure in the borehole increases from Pt to P0 instantaneously during the water injection test. When the injection time (t) is equal to 0, the water pressure in the surrounding rock mass remains Pt f ðx; 0Þ ¼ Pt
(7)
Combining Eqs. (3), (6) and (7) yields the following solution: p ¼ f ðx; tÞ ¼ uðx; tÞ þ f ðx; 0Þ ¼ P0 þ
2 ∞ np 2ðP0 Pt Þ X 1 np k sin x 1 exp t p n R R Sr n¼1
(8)
Eq. (8) is an analytical formula for calculating water pressure at any position and time during hydraulic fracturing in the hydrocarbon reservoirs. Because the hydraulic gradient is the ratio of head loss along the seepage path to seepage distance, the hydraulic gradient J can be obtained by derivation Eq. (8) with respect to x.
(3)
Then, substituting Eq. (3) into Eq. (1) yields the following:
3
J¼
2 ∞ np 2ðP0 Pt Þ X np k x 1 exp cos t R R R Sr n¼1
(9)
Fig. 2 e Schematic diagrams of (a) flow around the injection borehole during fluid injection, (b) cross-section, and (c) pressure curve around the injection borehole.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
4
international journal of hydrogen energy xxx (xxxx) xxx
Table 1 e Summary of the calculation parameters used in the numerical analysis. Parameters Length (m) Wide (m) Permeability coefficient, k (m/s) Dynamic viscosity of fluid, m (Pa$s) Compressibility of fluid, Xf (m$s2/kg) Compressibility of solid, Xs (m$s2/kg) Unit storage, Sr (m1) Injection pressure, p0 (MPa) Initial water pressure, pt (MPa)
Value 3 2 1 107 1 103 4.4 1010 1 108 1 103 5 2
Fig. 3 e Water pressure contours of the numerical calculation.
Calculation examples To verify the correctness of the analytical model of vertical wellbore pressure under unsteady seepage in the hydrocarbon reservoirs, the variation of water pressure and hydraulic gradient in rock mass under unsteady seepage is discussed through a specific example in this paper, and the validity of the model is verified by numerical simulation. The numerical simulation software COMSOL Multiphysics is used for coupling analyzing multiple physical quantities. The size of the rock mass is 2 3 m (width length), and the permeability
coefficient (k)is 1 107 m/s, and the specific calculation parameters are shown in Table 1. The free mesh generator of the software is used to divide the sample into 10,016 element meshes. The upper and lower boundaries of the sample are set to zero flux and are impermeable. The constant fluid pressures P0 and Pt are then applied to the left and right boundaries, respectively. The calculation time step increases according to the equal step length of 100 s, and the initial value is 1s and the termination time is 5 104s (that is, 14 h). Fig. 3 shows the water pressure distribution in the rock mass obtained by COMSOL Multiphysics software. The results show that the water pressure in the calculated sample decreases gradually from the left to the right. In addition, the three following monitoring points were established in the numerical model: 0.2R (i.e.0.6 m), 0.5R (i.e. 1.5 m) and 0.8R (i.e. 2.4 m) from the left intake boundary. Figs. 4 and 5 show the curves of water pressure and hydraulic gradient at the three monitoring points obtained by theoretical calculation of analytical model and numerical simulation, respectively. It can be seen from the two figures that the results calculated by these two methods are similar, as the calculation time increases, the coincidence degree also increases, which verifies the correctness of the analytical solution. Fig. 4 shows that when the water inlet is injected, the water pressure of the three monitoring points in the rock sample increases gradually with the increase of time, and the water pressure then stabilises. In addition, as the monitoring point from the intake increases, the increase rate and amplitude of water pressure decrease, indicating that as the distance from the borehole increases, the probability of hydraulic fracturing decreases. These results are consistent with the actual situation. Fig. 5 shows that at the position of 0.2R from the intake, a large hydraulic gradient appears at the beginning of injection in a short time and then the hydraulic gradient tends to be stable as time increases (Fig. 5a). However, farther away from the inlet (x ¼ 0.5Rand x ¼ 0.8R), the hydraulic gradient gradually increases with time and eventually reaches a stable state (Fig. 5(b, c)). The main reason for this difference is that the water pressure on the inlet increases rapidly in a very short period of time during injection, and there is no time to permeate outwards. Therefore, this could lead to a high hydraulic gradient on the side near the inlet. Hydraulic fracturing is easy to occur under the action of high hydraulic gradient, and a new seepage channel is generated, leading to the seepage failure of water-resisting surrounding the rock. As water penetrates into the surrounding rock mass, the
Fig. 4 e Comparison of analytical and numerical simulation results of water pressure distribution. Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
international journal of hydrogen energy xxx (xxxx) xxx
5
Fig. 5 e Comparison of analytical and numerical simulation results of hydraulic gradient variation.
hydraulic gradient near the water inlet gradually decreases while the hydraulic gradient in the surrounding rock mass gradually increases.
Results and discussion The presented unsteady seepage analytical model was validated through numerical results, the variation of water pressure and hydraulic gradient around the well under different conditions is of great significance for us to understand the unsteady seepage in the hydrocarbon reservoirs. Fig. 6(a) shows the variation of water pressure in the sample with time and distance from the inlet. The figure shows that as the distance from the inlet increases, the magnitude of the increase in water pressure decreases. In addition, as the time increases, the water pressure at each position in the sample gradually increases and shows an approximately linear distribution along the direction of the length. Fig. 6(b) shows the variation of the hydraulic gradient at different positions and times in the sample, with the increase of time, the hydraulic gradient within a certain range of the inlet gradually decreases with time, the hydraulic gradient away from the inlet gradually increases with time and the hydraulic gradient at each position reaches the same stable value. To investigate the effect of rock mass permeability on water pressure and hydraulic gradient, the variation of water pressure and hydraulic gradient of four different samples at x ¼ 0.1R was calculated by the presented example. Fig. 7(a) shows the variation laws of water pressure with time of the rock mass with different permeability coefficients. The results show that as the permeability coefficient of the rock mass increases, the rate of water flow penetration and water pressure increased, and a shorter time was needed when the water pressure became stable. In contrast, stabilising the water pressure takes more time. The result shows that the percolation instability of the surrounding rock is more likely to occur under the action of high water pressure. In addition, the results show that high permeability rapidly leads to water pressure penetration into the surrounding rock mass. The variation law of hydraulic gradient of rock mass with different permeability coefficients is presented in Fig. 7(b). As the rock mass permeability coefficient increases, the hydraulic gradient at the position of the near inlet increases to a large
value in a shorter time and then gradually decreases with time and eventually up to a stable value. In contrast, as the permeability of rock mass decreases, the hydraulic gradient increases and the time required for stability increases as well. For example, by comparing two samples with different permeability (k ¼ 1 107 m/s and k ¼ 1 109 m/s), the results show that the time required for the samples with high permeability to reach stability is significantly shorter than
Fig. 6 e Variations of (a) water pressure p and (b) hydraulic gradient J with time t and distance from injection borehole x/R.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
6
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 7 e Variations of (a) water pressure p and (b) hydraulic gradient J with time t and hydraulic conductivity k for x ¼ 0.1R.
Fig. 8 e Relationship between pressurization rate and critical breakdown water pressure [36]. that of the samples with low permeability. The results indicate that for low permeability and a slower increase rate of water pressure, the water pressure on the water inlet side penetrates more slowly into the surrounding area and the
Fig. 9 e Variations of (a) water pressure p and (b) hydraulic gradient J with time t and injection pressure p0 for x ¼ 0.5R.
time for hydraulic gradient to stabilise increases, increasing the probability of hydraulic splitting at this location. We need to point out that the above analysis is based on the inlet water pressure can reach P0 instantaneously and can be kept stable, but the actual situation is that when the water is injected into the borehole, the pressure in the borehole gradually increases. As the permeability of the borehole wall rock mass decreases, the faster the pressure in the borehole increases, that is, the higher the pressure increase rate. Song et al. [35] found that the critical fracture pressure of rock samples increases with the increased pressure rise rate (dp/dt) through experiment, as shown in Fig. 8. The results show that as the permeability of the rock decreases, the rate of pressure increases in the borehole increases and the critical fracture pressure of the rock also increase. In contrast, as the permeability of the rock increases, the rate of the pressure increase in the borehole decreases. At a greater pressure, the critical fracture pressure of the rock decreases [36]. Therefore, the permeability is an important factor that determines whether the rock has seepage failure. When the permeability of rock increases, the probability of hydraulic fracturing also increases.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
international journal of hydrogen energy xxx (xxxx) xxx
In addition, to explore the effect of water injection pressure on the water pressure and hydraulic gradient distribution in the rock mass, the water injection pressure P0 (that is, 3 MPa, 5 MPa and 6 MPa) is selected for calculation. Fig. 9(a) shows the variation laws of water pressure in the rock mass with time under different water injection pressures. The figure shows that as the water injection pressure increases, the water pressure increases and gradually becomes larger. Fig. 9(b) shows the variation law of the hydraulic gradient of the rock mass under different water injection pressures. In addition, as the water injection pressure increases, the hydraulic gradient increases and the rate of increase increases. Consequently, the results show that as the water pressure increases, the risk of hydraulic fracturing increases, which is consistent with the actual situation.
Conclusions In this paper, by establishing a one-dimensional model under the condition of unsteady seepage, the temporal and spatial evolution law of water pressure and hydraulic gradient in the process of hydraulic fracturing is studied, and the mechanism of hydraulic fracturing of rock mass is revealed. The main conclusions are as follows: (1) A comprehensive analytical solutions model is established for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs. The validity of the analytical model is verified by numerical simulation. (2) The varieties law of water pressure and hydraulic gradient in different positions of the model is studied by analytical solution and numerical simulation respectively. As the distance from the inlet increases, the water pressure decreases, while the hydraulic gradient increases the maximum value in a short time, and then decreases to a stable value. This shows that near the inlet, high water pressure and hydraulic gradient are most likely to lead to hydraulic fracturing of rock mass. (3) The effect of different permeability coefficient and water injection pressure on the water pressure and hydraulic gradient distribution in the rock mass were discussed in detail. The results show that as the water injection pressure and permeability increase, risk of hydraulic fracturing also increases. The occurrence of high water pressure or a high hydraulic gradient in rock mass could cause hydraulic fracturing.
Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (41702326, 41602294), the Innovative Experts, Long-term Program of Jiangxi Province (jxsq2018106049), and the Program for Excellent Young Talents, JXUST.
7
references
[1] Su FQ, Itakura KI, Deguchi G, Ohga K. Monitoring of coal fracturing in underground coal gasification by acoustic emission techniques. Appl Energy 2017;189:142e56. [2] Dincer I, Acar C. Smart energy solutions with hydrogen options. Int J Hydrogen Energy 2018;43(18):8579e99. [3] Liu PC, Li WH, Hao MQ, Wu YB, Li XL. Quantitative evaluation of factors affecting foamy oil recovery in the development of heavy hydrocarbon reservoirs. Int J Hydrogen Energy 2016;41(35):15624e31. [4] Liu PC, Zhang XK. Enhanced oil recovery by CO2 -CH4 flooding in low permeability and rhythmic hydrocarbon reservoir. Int J Hydrogen Energy 2015;40(37):12849e53. [5] Sakoda N, Onoue K, Kuroki T, Shinzato K, Kohno M, Monde M, et al. Transient temperature and pressure behavior of high pressure 100 MPa hydrogen during discharge through orifices. Int J Hydrogen Energy 2016;41(38):17169e74. [6] Wu Q, Xu Y, Wang XQ, Wang TF, Zhang SL. Volume fracturing technology of unconventional reservoirs: connotation, design optimization and implementation. Petrol Explor Dev 2012;39(3):377e84. [7] Wang H, Liao X, Zhao X, Zhao D, Liao C. Advances of technology study of stimulated reservoir volume in unconventional reservoir. Special Oil Gas Reservoirs 2014;21(2):8e15. [8] Li YH, Wang CJ, Shi LH, Guo WY. Application and development of drilling and completion of the ultrashortradius radial well by high pressure jet flow techniques. SPE; 2000. p. 64756. [9] Tiab D. Analysis of derivative data of hydraulically fractured wells by the Tiab's direct synthesis technique. J Pet Sci Eng 2005;49:1e21. [10] Gringarten AC, Ramey Jr HJ. Unsteady-state pressure distributions created by a well with a single horizontal fracture. Soc Petrol Eng J 1974;257:413e26. [11] Luo ZF, Zhang NL, Zhao LQ, Yao LM, Liu F. Seepage-stress coupling mechanism for intersections between hydraulic fractures and natural fractures. J Petrol Sci Eng 2018;171:37e47. [12] Ju Y, Wang YL, Xu B, Chen JL, Yang YM. Numerical analysis of the effects of bedded interfaces on hydraulic fracture propagation in tight multilayered reservoirs considering hydro-mechanical coupling. J Petrol Sci Eng 2019;178:356e75. [13] Huang B, Liu J. Experimental investigation of the effect of bedding planes on hydraulic fracturing under true triaxial stress. Rock Mech Rock Eng 2017;50(10):2627e43. [14] Chen H, Zhao Z, Sun J. Coupled hydro-mechanical model for fractured rock masses using the discontinuous deformation analysis. Tunn Undergr Space Technol 2013;38:506e16. [15] Li N, Zhang SC, Zou YS, Ma XF, Wu S, Zhang YN. Experimental analysis of hydraulic fracture growth and acoustic emission response in a layered formation. Rock Mech Rock Eng 2018;51(4):1047e62. [16] Liu P, Ju Y, Ranjith PG, Zheng ZM, Chen JL. Experimental investigation of the effects of heterogeneity and geostress difference on the 3D growth and distribution of hydrofracturing cracks in unconventional reservoir rocks. J Nat Gas Sci Eng 2016;35:541e54. [17] Huang Z, Jiang ZQ, Fu JJ, Cao DT. Experimental measurement on the hydraulic conductivity of deep low-permeability rock. Arab J Geo Sci 2015;8:5389e90. [18] Zhu YWS, Zhang TT. Permeability of the coal seam floor rock mass in a deep mine based on in-situ water injection tests. Mine Water Environ 2018;37:724e33.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
8
international journal of hydrogen energy xxx (xxxx) xxx
[19] Huang Z, Zeng W, Wu Y, Li SJ, Zhao K. Experimental investigation of fracture propagation and inrush characteristics in tunnel construction. Nat Hazards 2019;97:193e210. [20] Huang Z, Jiang ZQ, Zhu SY, Qian ZW, Cao DT. Characterizing the hydraulic conductivity of rock formations between deep coal and aquifers using injection tests. Int J Rock Mech Min Sci 2014;71:12e8. [21] Bourdet D, Gringarten AC. Determination of fissure volume and block size in fractured reservoirs by type-curve analysis. In: Paper SPE 9293, 55th society of petroleum engineering annual fall technical conference and exhibition, Dallas, Texas; 1980. p. 21e4. [22] Guo JC, Nie RS, Jia YL. Dual permeability flow behavior for modeling horizontal well production in fractured-vuggy carbonate reservoirs. J Hydrol 2012;464e465:281e93. [23] Lee ST, Brockenbrough JR. A new approximate analytic solution for finite-conductivity vertical fractures. SPE Form Eval 1986. https://doi.org/10.2118/12013-PA. [24] Yao S, Zeng F, Liu H, Zhao G. A semi-analytical model formulti-stage fractured horizontal wells. J Hydrol 2013;507:201e12. [25] Wang H, Liao X, Zhao X, Zhao D, Liao C. Advances of technology study of stimulated reservoir volume in unconventional reservoir. Special Oil Gas Reservoirs 2014;21(2):8e15. [26] Wang L, Dai C, Li X, Chen X, Xia Z. Pressure transient analysis for asymmetrically fractured wells in dual-permeability organic compound reservoir of hydrogen and carbon. Int J Hydrogen Energy 2018:1e8. [27] Xue L, Chen X, Wang L. Pressure transient analysis for fluid flow through horizontal fractures in shallow organic compound reservoir of hydrogen and carbon. Int J Hydrogen Energy 2018:1e9. [28] Wang H, Ran QQ, Liao XW. Pressure transient responses study on the hydraulic volume fracturing vertical well in
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
stress-sensitive tight hydrocarbon reservoirs. Int J Hydrogen Energy 2017;42:18343e9. Guppy KH, Cinco-Ley H, Ramey Jr HJ, Samaniego VF. NonDarcy flow in wells with finite-conductivity vertical fractures. Soc Petrol Eng J 1982;22(5):681e98. Tian Q, Liu PC, Jiao YW, Bie AF, Xia J, Li BZ. Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs. Int J Hydrogen Energy 2017;42:18146e55. Xue Y, Teng T, Dang FN, Ma ZY, Wang SH. Productivity analysis of fractured wells in reservoir of hydrogen and carbon based on dual-porosity medium model. Int J Hydrogen Energy 2019. https://doi.org/10.1016/ j.ijhydene.2019.11.146. Guo TK, Wang XZ, Li Z, Gong FC, Lin Q, Qu ZQ, Lv W, Tian QZ, Xie ZS. Numerical simulation study on fracture propagation of zipper and synchronous fracturing in hydrogen energy development. Int J Hydrogen Energy 2019;44:5270e85. Liu XQ, QuZQ GuoTK, Tian QZ, Lv W, Xie ZS, Chu CB. An innovative technology of directional propagation of hydraulic fracture guided by radial holes in fossil hydrogen energy development. Int J Hydrogen Energy 2018;44:5286e302. Wang Y, Yan Q. Primary research of influence of unsteady seepage processes on hydraulic fracturing in rock mass. Chin J Rock Mech Eng 2012;31(10):2016e21. Song I, Suh M, Won KS, Haimson B. A laboratory study of hydraulic fracturing breakdown pressure in tablerock sandstone. Geosci J 2001;5(3):263e71. Yang TH, Liu HL, Zhu WC, Meng ZP, Wang R. Modification and application of critical water outburst coefficient in coalmine based on the concept of effective stress. Chin J Rock Mech Eng 2011;30(sup.2):4012e8.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs Yun Wu a,b, Zhen Huang a,b,*, Kui Zhao a, Wei Zeng a, Qixiong Gu a, Rui Zhang c a
School of Resources and Environment Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China b School of Earth Sciences and Engineering, Nanjing University, Nanjing, 210023, China c Kunming Survey, Design and Research Institute CO. LTD of Creec, Kunming, 650200, China
highlights A
unsteady
graphical abstract
seepage
solution
model for hydraulic fracturing was improved. Variations of water pressure and hydraulic gradient around wells were analysed. Hydraulic fracturing mechanism around
wells
under
unsteady
seepage was revealed.
article info
abstract
Article history:
This paper establishes an analytical model to study the influence mechanism of hydraulic
Received 16 December 2019
fracturing around vertical wellbores under the unsteady seepage in hydrocarbon reser-
Received in revised form
voirs, deduces the analytical solution of water pressure and hydraulic gradient of the
27 January 2020
model, and compares the law of water pressure and hydraulic gradient changing with time
Accepted 28 January 2020
with the results of numerical simulation. The results confirm the accuracy of these
Available online xxx
analytical solutions. The variation laws of water pressure and hydraulic gradient in the sample under unsteady seepage are analysed by using the COMSOL Multiphysics software.
Keywords:
The results show that: the increasing rate and amplitude of water pressure decrease with
Hydraulic fracturing
the distance of water inlet, however, hydraulic gradient near the water inlet is the largest
Unsteady seepage
and decreasing with the distance. In order to better understand the mechanism of hy-
Analytical solution
draulic fracturing of rock mass, we studied the influence of permeability and water in-
Numerical simulation
jection pressure on water pressure and hydraulic gradient of rock mass. The results show
Hydrocarbon reservoirs
that: large permeability coefficient and high hydraulic gradient will increase the probability of rock mass hydraulic fracturing. The permeability and hydraulic gradient of rock mass is
* Corresponding author. School of Resources and Environment Engineering, Jiangxi University of Science and Technology, Ganzhou, 341000, China. E-mail addresses: [email protected], [email protected] (Z. Huang). https://doi.org/10.1016/j.ijhydene.2020.01.222 0360-3199/© 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
2
international journal of hydrogen energy xxx (xxxx) xxx
important factor in determining whether the rock has hydraulic fracturing. The distribution law of water pressure and hydraulic gradient in rock mass under unsteady seepage provides important reference and basis in hydrogen developing reservoirs. © 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Fossil energy has long been one of the main energy structures in China, occupying a significant proportion [1]. However, burning fossil energy produces a series of environmental problems; for example, the gases released can cause greenhouse effects and seriously damage ecosystems. Therefore, the development of clean energy has become a major research topic to be solved. As is known to all, the hydrogen energy is a kind of secondary energy, which needs to be generated by hydrocarbon reservoirs, as one of the most important clean energy sources and has thus have attracted global attention [2]. Hydraulic fracturing technology plays an important role in increasing the output of hydrocarbon reservoirs [3e10]. In the process of developing hydrocarbon reservoirs by hydraulic fracturing (see Fig. 1), a series of technical problems are involved. For example, the process of water injection into boreholes is actually a problem of coupling the seepage field with a stress field when water is injected, the rocks around the borehole wall may crack and fail under the action of highpressure water and the internal stress of the rock mass around the borehole wall will redistribute. The redistribution of stress changes the hydraulic properties of the rock mass, and the coupling of the seepage field and stress field is a very complex scientific problem [11e20]. In order to understand hydraulic characteristics parameters in rock mass such as water pressure and hydraulic gradient during the hydraulic fracturing, which is treated as one of the effective techniques in hydrocarbon reservoirs [21e23]. Many researchers have adopted analytical model to examine and obtain the change rule of pressure transient of rock mass under the action of hydraulic fracturing in the hydrocarbon reservoirs [24,25]. Wang et al. [26]established a semi-analytical model to study pressure transient behavior for asymmetrically fractured wells in hydrocarbon reservoirs, and the results can predict production performance and enable sufficient test analysis in the development of dualpermeability organic compound reservoir of hydrogen and carbon. Xue et al. [27] presented a comprehensive analytical model to discuss the transient flow behavior through the horizontal fractures and flow characteristics. Wang et al. [28] combined superposition, Laplace integral transformation, Stehfest numerical inverse algorithm, and the perturbation technique to build a model, deduce the analytical solution of the model. Guppy et al. [29] established an analytical model to study the effect of non-Darcy flow on the pressure transient behavior of finite conductivity vertical fractures. Tian et al. [30] developed a new constant-rate solution for non-planar asymmetric fracture connected to a vertical wellbore in hydrocarbon reservoirs.
All the above researches are carried out from the perspective of mathematical model, however, with the development of computer technology, many scholars use numerical simulation software to examine and obtain the change rule of hydraulic characteristics of rock mass under the action of hydraulic fracturing in the hydrocarbon reservoirs. For example, Xue et al. [31] used COMSOL Multiphysics to study the productivity of fractured wells in the organic hydrogen and carbon reservoirs. Guo et al. [32] applied ABAQUS to investigate fracture propagation of zipper and synchronous fracturing in hydrogen energy. Liu et al. [33] established a model of radial holes combined with hydraulic fracturing to directional propagation of hydraulic fracture in fossil hydrogen energy. At present, there are few studies on the hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs under unsteady seepage. In this paper, we established a simple model of unsteady seepage flow and adopted the numerical simulation software COMSOL Multiphysics coupling to discuss the unsteady seepage of rock mass under the action of the internal water pressure and hydraulic gradient through the specific example in the hydrocarbon reservoirs, then, we discussed the effect of different permeability coefficient and injection pressure on the hydraulic fracturing and revealed the mechanism of hydraulic fracturing of rock mass under the condition of unsteady seepage. The research results can provide a reference for the pressure distribution laws connected to vertical wellbores in the hydrocarbon reservoirs.
Fig. 1 e Schematic diagram of hydraulic fracturing using vertical wellbores in hydrocarbon reservoirs.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
international journal of hydrogen energy xxx (xxxx) xxx
Establishment of the unsteady seepage model In order to study the hydraulic characteristics of rock mass in the hydrocarbon reservoirs, we need to establish a model under the condition of unsteady seepage and assume that the rock mass is isotropic. Fig. 2 presents a schematic diagram of seepage and water pressure distribution near the injection borehole. The water flow will appear a process of gradual oozes around the injection borehole. In addition, when water is injected into the borehole, the water pressure in the borehole increases to P0 instantaneously and remains stable, i.e. f(0,t) ¼ Pt. The water pressure at R from the center of the borehole is the original pressure Pt, i.e. f(R,t) ¼ Pt. A simplified one-dimensional model can be established to investigate the seepage state at any position in a rock mass with time t (x away from the borehole). In addition, this paper assumes that the seepage state in the initial state of rock mass is equal, that is, f(x,0) ¼ Pt. In the presented one-dimensional model, the equation of seepage movement of a section in the rock mass can be expressed as[34]： vp v2 p ¼ d2 2 vt vx
(1)
Here, d is a parameter related to the permeability coefficient (k) and the storage capacity (Sr) of rock mass by the following formula: d2 ¼
k Sr
(2)
To determine the Fourier transform solution, the following new functions are introduced: uðx; tÞ ¼ f ðx; tÞ f ðx; 0Þ ð0 x R; t 0Þ
vuðx; tÞ v2 uðx; tÞ ¼ d2 vt vx2
(4)
According to the model conditions, the following definitions are applied to the presented formula: 8 < uðx; 0Þ ¼ f ðx; 0Þ f ðx; 0Þ ¼ 0 ð0; x RÞ uð0; tÞ ¼ f ð0; tÞ f ð0; 0Þ ¼ P0 Pt : uðR; tÞ ¼ f ðR; tÞ f ðR; 0Þ ¼ 0 ðt 0Þ
(5)
The finite Fourier sinusoidal transformation of variable x can be obtained as follows: uðx; tÞ ¼
2 ∞ 2ðP0 Pt Þ X 1 np np k sin x 1 exp t p n R R Sr n¼1
(6)
The water pressure in the borehole increases from Pt to P0 instantaneously during the water injection test. When the injection time (t) is equal to 0, the water pressure in the surrounding rock mass remains Pt f ðx; 0Þ ¼ Pt
(7)
Combining Eqs. (3), (6) and (7) yields the following solution: p ¼ f ðx; tÞ ¼ uðx; tÞ þ f ðx; 0Þ ¼ P0 þ
2 ∞ np 2ðP0 Pt Þ X 1 np k sin x 1 exp t p n R R Sr n¼1
(8)
Eq. (8) is an analytical formula for calculating water pressure at any position and time during hydraulic fracturing in the hydrocarbon reservoirs. Because the hydraulic gradient is the ratio of head loss along the seepage path to seepage distance, the hydraulic gradient J can be obtained by derivation Eq. (8) with respect to x.
(3)
Then, substituting Eq. (3) into Eq. (1) yields the following:
3
J¼
2 ∞ np 2ðP0 Pt Þ X np k x 1 exp cos t R R R Sr n¼1
(9)
Fig. 2 e Schematic diagrams of (a) flow around the injection borehole during fluid injection, (b) cross-section, and (c) pressure curve around the injection borehole.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
4
international journal of hydrogen energy xxx (xxxx) xxx
Table 1 e Summary of the calculation parameters used in the numerical analysis. Parameters Length (m) Wide (m) Permeability coefficient, k (m/s) Dynamic viscosity of fluid, m (Pa$s) Compressibility of fluid, Xf (m$s2/kg) Compressibility of solid, Xs (m$s2/kg) Unit storage, Sr (m1) Injection pressure, p0 (MPa) Initial water pressure, pt (MPa)
Value 3 2 1 107 1 103 4.4 1010 1 108 1 103 5 2
Fig. 3 e Water pressure contours of the numerical calculation.
Calculation examples To verify the correctness of the analytical model of vertical wellbore pressure under unsteady seepage in the hydrocarbon reservoirs, the variation of water pressure and hydraulic gradient in rock mass under unsteady seepage is discussed through a specific example in this paper, and the validity of the model is verified by numerical simulation. The numerical simulation software COMSOL Multiphysics is used for coupling analyzing multiple physical quantities. The size of the rock mass is 2 3 m (width length), and the permeability
coefficient (k)is 1 107 m/s, and the specific calculation parameters are shown in Table 1. The free mesh generator of the software is used to divide the sample into 10,016 element meshes. The upper and lower boundaries of the sample are set to zero flux and are impermeable. The constant fluid pressures P0 and Pt are then applied to the left and right boundaries, respectively. The calculation time step increases according to the equal step length of 100 s, and the initial value is 1s and the termination time is 5 104s (that is, 14 h). Fig. 3 shows the water pressure distribution in the rock mass obtained by COMSOL Multiphysics software. The results show that the water pressure in the calculated sample decreases gradually from the left to the right. In addition, the three following monitoring points were established in the numerical model: 0.2R (i.e.0.6 m), 0.5R (i.e. 1.5 m) and 0.8R (i.e. 2.4 m) from the left intake boundary. Figs. 4 and 5 show the curves of water pressure and hydraulic gradient at the three monitoring points obtained by theoretical calculation of analytical model and numerical simulation, respectively. It can be seen from the two figures that the results calculated by these two methods are similar, as the calculation time increases, the coincidence degree also increases, which verifies the correctness of the analytical solution. Fig. 4 shows that when the water inlet is injected, the water pressure of the three monitoring points in the rock sample increases gradually with the increase of time, and the water pressure then stabilises. In addition, as the monitoring point from the intake increases, the increase rate and amplitude of water pressure decrease, indicating that as the distance from the borehole increases, the probability of hydraulic fracturing decreases. These results are consistent with the actual situation. Fig. 5 shows that at the position of 0.2R from the intake, a large hydraulic gradient appears at the beginning of injection in a short time and then the hydraulic gradient tends to be stable as time increases (Fig. 5a). However, farther away from the inlet (x ¼ 0.5Rand x ¼ 0.8R), the hydraulic gradient gradually increases with time and eventually reaches a stable state (Fig. 5(b, c)). The main reason for this difference is that the water pressure on the inlet increases rapidly in a very short period of time during injection, and there is no time to permeate outwards. Therefore, this could lead to a high hydraulic gradient on the side near the inlet. Hydraulic fracturing is easy to occur under the action of high hydraulic gradient, and a new seepage channel is generated, leading to the seepage failure of water-resisting surrounding the rock. As water penetrates into the surrounding rock mass, the
Fig. 4 e Comparison of analytical and numerical simulation results of water pressure distribution. Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
international journal of hydrogen energy xxx (xxxx) xxx
5
Fig. 5 e Comparison of analytical and numerical simulation results of hydraulic gradient variation.
hydraulic gradient near the water inlet gradually decreases while the hydraulic gradient in the surrounding rock mass gradually increases.
Results and discussion The presented unsteady seepage analytical model was validated through numerical results, the variation of water pressure and hydraulic gradient around the well under different conditions is of great significance for us to understand the unsteady seepage in the hydrocarbon reservoirs. Fig. 6(a) shows the variation of water pressure in the sample with time and distance from the inlet. The figure shows that as the distance from the inlet increases, the magnitude of the increase in water pressure decreases. In addition, as the time increases, the water pressure at each position in the sample gradually increases and shows an approximately linear distribution along the direction of the length. Fig. 6(b) shows the variation of the hydraulic gradient at different positions and times in the sample, with the increase of time, the hydraulic gradient within a certain range of the inlet gradually decreases with time, the hydraulic gradient away from the inlet gradually increases with time and the hydraulic gradient at each position reaches the same stable value. To investigate the effect of rock mass permeability on water pressure and hydraulic gradient, the variation of water pressure and hydraulic gradient of four different samples at x ¼ 0.1R was calculated by the presented example. Fig. 7(a) shows the variation laws of water pressure with time of the rock mass with different permeability coefficients. The results show that as the permeability coefficient of the rock mass increases, the rate of water flow penetration and water pressure increased, and a shorter time was needed when the water pressure became stable. In contrast, stabilising the water pressure takes more time. The result shows that the percolation instability of the surrounding rock is more likely to occur under the action of high water pressure. In addition, the results show that high permeability rapidly leads to water pressure penetration into the surrounding rock mass. The variation law of hydraulic gradient of rock mass with different permeability coefficients is presented in Fig. 7(b). As the rock mass permeability coefficient increases, the hydraulic gradient at the position of the near inlet increases to a large
value in a shorter time and then gradually decreases with time and eventually up to a stable value. In contrast, as the permeability of rock mass decreases, the hydraulic gradient increases and the time required for stability increases as well. For example, by comparing two samples with different permeability (k ¼ 1 107 m/s and k ¼ 1 109 m/s), the results show that the time required for the samples with high permeability to reach stability is significantly shorter than
Fig. 6 e Variations of (a) water pressure p and (b) hydraulic gradient J with time t and distance from injection borehole x/R.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
6
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 7 e Variations of (a) water pressure p and (b) hydraulic gradient J with time t and hydraulic conductivity k for x ¼ 0.1R.
Fig. 8 e Relationship between pressurization rate and critical breakdown water pressure [36]. that of the samples with low permeability. The results indicate that for low permeability and a slower increase rate of water pressure, the water pressure on the water inlet side penetrates more slowly into the surrounding area and the
Fig. 9 e Variations of (a) water pressure p and (b) hydraulic gradient J with time t and injection pressure p0 for x ¼ 0.5R.
time for hydraulic gradient to stabilise increases, increasing the probability of hydraulic splitting at this location. We need to point out that the above analysis is based on the inlet water pressure can reach P0 instantaneously and can be kept stable, but the actual situation is that when the water is injected into the borehole, the pressure in the borehole gradually increases. As the permeability of the borehole wall rock mass decreases, the faster the pressure in the borehole increases, that is, the higher the pressure increase rate. Song et al. [35] found that the critical fracture pressure of rock samples increases with the increased pressure rise rate (dp/dt) through experiment, as shown in Fig. 8. The results show that as the permeability of the rock decreases, the rate of pressure increases in the borehole increases and the critical fracture pressure of the rock also increase. In contrast, as the permeability of the rock increases, the rate of the pressure increase in the borehole decreases. At a greater pressure, the critical fracture pressure of the rock decreases [36]. Therefore, the permeability is an important factor that determines whether the rock has seepage failure. When the permeability of rock increases, the probability of hydraulic fracturing also increases.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
international journal of hydrogen energy xxx (xxxx) xxx
In addition, to explore the effect of water injection pressure on the water pressure and hydraulic gradient distribution in the rock mass, the water injection pressure P0 (that is, 3 MPa, 5 MPa and 6 MPa) is selected for calculation. Fig. 9(a) shows the variation laws of water pressure in the rock mass with time under different water injection pressures. The figure shows that as the water injection pressure increases, the water pressure increases and gradually becomes larger. Fig. 9(b) shows the variation law of the hydraulic gradient of the rock mass under different water injection pressures. In addition, as the water injection pressure increases, the hydraulic gradient increases and the rate of increase increases. Consequently, the results show that as the water pressure increases, the risk of hydraulic fracturing increases, which is consistent with the actual situation.
Conclusions In this paper, by establishing a one-dimensional model under the condition of unsteady seepage, the temporal and spatial evolution law of water pressure and hydraulic gradient in the process of hydraulic fracturing is studied, and the mechanism of hydraulic fracturing of rock mass is revealed. The main conclusions are as follows: (1) A comprehensive analytical solutions model is established for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs. The validity of the analytical model is verified by numerical simulation. (2) The varieties law of water pressure and hydraulic gradient in different positions of the model is studied by analytical solution and numerical simulation respectively. As the distance from the inlet increases, the water pressure decreases, while the hydraulic gradient increases the maximum value in a short time, and then decreases to a stable value. This shows that near the inlet, high water pressure and hydraulic gradient are most likely to lead to hydraulic fracturing of rock mass. (3) The effect of different permeability coefficient and water injection pressure on the water pressure and hydraulic gradient distribution in the rock mass were discussed in detail. The results show that as the water injection pressure and permeability increase, risk of hydraulic fracturing also increases. The occurrence of high water pressure or a high hydraulic gradient in rock mass could cause hydraulic fracturing.
Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (41702326, 41602294), the Innovative Experts, Long-term Program of Jiangxi Province (jxsq2018106049), and the Program for Excellent Young Talents, JXUST.
7
references
[1] Su FQ, Itakura KI, Deguchi G, Ohga K. Monitoring of coal fracturing in underground coal gasification by acoustic emission techniques. Appl Energy 2017;189:142e56. [2] Dincer I, Acar C. Smart energy solutions with hydrogen options. Int J Hydrogen Energy 2018;43(18):8579e99. [3] Liu PC, Li WH, Hao MQ, Wu YB, Li XL. Quantitative evaluation of factors affecting foamy oil recovery in the development of heavy hydrocarbon reservoirs. Int J Hydrogen Energy 2016;41(35):15624e31. [4] Liu PC, Zhang XK. Enhanced oil recovery by CO2 -CH4 flooding in low permeability and rhythmic hydrocarbon reservoir. Int J Hydrogen Energy 2015;40(37):12849e53. [5] Sakoda N, Onoue K, Kuroki T, Shinzato K, Kohno M, Monde M, et al. Transient temperature and pressure behavior of high pressure 100 MPa hydrogen during discharge through orifices. Int J Hydrogen Energy 2016;41(38):17169e74. [6] Wu Q, Xu Y, Wang XQ, Wang TF, Zhang SL. Volume fracturing technology of unconventional reservoirs: connotation, design optimization and implementation. Petrol Explor Dev 2012;39(3):377e84. [7] Wang H, Liao X, Zhao X, Zhao D, Liao C. Advances of technology study of stimulated reservoir volume in unconventional reservoir. Special Oil Gas Reservoirs 2014;21(2):8e15. [8] Li YH, Wang CJ, Shi LH, Guo WY. Application and development of drilling and completion of the ultrashortradius radial well by high pressure jet flow techniques. SPE; 2000. p. 64756. [9] Tiab D. Analysis of derivative data of hydraulically fractured wells by the Tiab's direct synthesis technique. J Pet Sci Eng 2005;49:1e21. [10] Gringarten AC, Ramey Jr HJ. Unsteady-state pressure distributions created by a well with a single horizontal fracture. Soc Petrol Eng J 1974;257:413e26. [11] Luo ZF, Zhang NL, Zhao LQ, Yao LM, Liu F. Seepage-stress coupling mechanism for intersections between hydraulic fractures and natural fractures. J Petrol Sci Eng 2018;171:37e47. [12] Ju Y, Wang YL, Xu B, Chen JL, Yang YM. Numerical analysis of the effects of bedded interfaces on hydraulic fracture propagation in tight multilayered reservoirs considering hydro-mechanical coupling. J Petrol Sci Eng 2019;178:356e75. [13] Huang B, Liu J. Experimental investigation of the effect of bedding planes on hydraulic fracturing under true triaxial stress. Rock Mech Rock Eng 2017;50(10):2627e43. [14] Chen H, Zhao Z, Sun J. Coupled hydro-mechanical model for fractured rock masses using the discontinuous deformation analysis. Tunn Undergr Space Technol 2013;38:506e16. [15] Li N, Zhang SC, Zou YS, Ma XF, Wu S, Zhang YN. Experimental analysis of hydraulic fracture growth and acoustic emission response in a layered formation. Rock Mech Rock Eng 2018;51(4):1047e62. [16] Liu P, Ju Y, Ranjith PG, Zheng ZM, Chen JL. Experimental investigation of the effects of heterogeneity and geostress difference on the 3D growth and distribution of hydrofracturing cracks in unconventional reservoir rocks. J Nat Gas Sci Eng 2016;35:541e54. [17] Huang Z, Jiang ZQ, Fu JJ, Cao DT. Experimental measurement on the hydraulic conductivity of deep low-permeability rock. Arab J Geo Sci 2015;8:5389e90. [18] Zhu YWS, Zhang TT. Permeability of the coal seam floor rock mass in a deep mine based on in-situ water injection tests. Mine Water Environ 2018;37:724e33.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222
8
international journal of hydrogen energy xxx (xxxx) xxx
[19] Huang Z, Zeng W, Wu Y, Li SJ, Zhao K. Experimental investigation of fracture propagation and inrush characteristics in tunnel construction. Nat Hazards 2019;97:193e210. [20] Huang Z, Jiang ZQ, Zhu SY, Qian ZW, Cao DT. Characterizing the hydraulic conductivity of rock formations between deep coal and aquifers using injection tests. Int J Rock Mech Min Sci 2014;71:12e8. [21] Bourdet D, Gringarten AC. Determination of fissure volume and block size in fractured reservoirs by type-curve analysis. In: Paper SPE 9293, 55th society of petroleum engineering annual fall technical conference and exhibition, Dallas, Texas; 1980. p. 21e4. [22] Guo JC, Nie RS, Jia YL. Dual permeability flow behavior for modeling horizontal well production in fractured-vuggy carbonate reservoirs. J Hydrol 2012;464e465:281e93. [23] Lee ST, Brockenbrough JR. A new approximate analytic solution for finite-conductivity vertical fractures. SPE Form Eval 1986. https://doi.org/10.2118/12013-PA. [24] Yao S, Zeng F, Liu H, Zhao G. A semi-analytical model formulti-stage fractured horizontal wells. J Hydrol 2013;507:201e12. [25] Wang H, Liao X, Zhao X, Zhao D, Liao C. Advances of technology study of stimulated reservoir volume in unconventional reservoir. Special Oil Gas Reservoirs 2014;21(2):8e15. [26] Wang L, Dai C, Li X, Chen X, Xia Z. Pressure transient analysis for asymmetrically fractured wells in dual-permeability organic compound reservoir of hydrogen and carbon. Int J Hydrogen Energy 2018:1e8. [27] Xue L, Chen X, Wang L. Pressure transient analysis for fluid flow through horizontal fractures in shallow organic compound reservoir of hydrogen and carbon. Int J Hydrogen Energy 2018:1e9. [28] Wang H, Ran QQ, Liao XW. Pressure transient responses study on the hydraulic volume fracturing vertical well in
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
stress-sensitive tight hydrocarbon reservoirs. Int J Hydrogen Energy 2017;42:18343e9. Guppy KH, Cinco-Ley H, Ramey Jr HJ, Samaniego VF. NonDarcy flow in wells with finite-conductivity vertical fractures. Soc Petrol Eng J 1982;22(5):681e98. Tian Q, Liu PC, Jiao YW, Bie AF, Xia J, Li BZ. Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs. Int J Hydrogen Energy 2017;42:18146e55. Xue Y, Teng T, Dang FN, Ma ZY, Wang SH. Productivity analysis of fractured wells in reservoir of hydrogen and carbon based on dual-porosity medium model. Int J Hydrogen Energy 2019. https://doi.org/10.1016/ j.ijhydene.2019.11.146. Guo TK, Wang XZ, Li Z, Gong FC, Lin Q, Qu ZQ, Lv W, Tian QZ, Xie ZS. Numerical simulation study on fracture propagation of zipper and synchronous fracturing in hydrogen energy development. Int J Hydrogen Energy 2019;44:5270e85. Liu XQ, QuZQ GuoTK, Tian QZ, Lv W, Xie ZS, Chu CB. An innovative technology of directional propagation of hydraulic fracture guided by radial holes in fossil hydrogen energy development. Int J Hydrogen Energy 2018;44:5286e302. Wang Y, Yan Q. Primary research of influence of unsteady seepage processes on hydraulic fracturing in rock mass. Chin J Rock Mech Eng 2012;31(10):2016e21. Song I, Suh M, Won KS, Haimson B. A laboratory study of hydraulic fracturing breakdown pressure in tablerock sandstone. Geosci J 2001;5(3):263e71. Yang TH, Liu HL, Zhu WC, Meng ZP, Wang R. Modification and application of critical water outburst coefficient in coalmine based on the concept of effective stress. Chin J Rock Mech Eng 2011;30(sup.2):4012e8.
Please cite this article as: Wu Y et al., Unsteady seepage solutions for hydraulic fracturing around vertical wellbores in hydrocarbon reservoirs, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.222