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Flow in multi-scale discrete fracture networks with stress sensitivity
Journal of Natural Gas Science and Engineering 35 (2016) 851e859
Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Flow in multi-scale discrete fracture networks with stress sensitivity Bin Liang a, Hanqiao Jiang a, Junjian Li a, * , Changcheng Gong a, Ruyi Jiang b, Yanli Pei a, Shiming Wei a a b
Key Laboratory for Petroleum Engineering of the Ministry of Education, China University of Petroleum, Beijing 102249, PR China Department of Science and Technology Management, China National Petroleum Corporation, Beijing 100007, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 March 2016 Received in revised form 7 September 2016 Accepted 8 September 2016 Available online 9 September 2016
For hydraulic fracturing in the shale reservoir, the existence of natural fractures has a great impact on the propagation of hydraulic fractures and the flow capacity of porous media. Our previous study mainly discussed the effect of different fracture parameters (i.e., orientation, aperture) on the fracture network permeability based on discrete fracture model and finite elements analyses (Liang et al., 2016). To further account for the in-situ condition, we now investigate the effect of stress sensitivity, i.e. the stress difference between hydraulic fractures and natural fractures, on flow behavior in multi-scale discrete fracture networks. A series of sensitivity experiments on both naturally and artificially fractured cores with the existence of proppants were carried out. Fracture models at different scale were established based on the corresponding mathematical fitting models, to study the effect of stress sensitivity on the seepage process. Results show that with the increase of confining pressure, the permeability of natural fractures decreases exponentially while the permeability of hydraulic fractures decreases more slowly following the cubic polynomial law. The stress sensitivity of natural fractures has more influence on the flow dynamics than that of hydraulic fractures, and this difference is subject to the fracture orientation, fracture length, and intersection relationship. When the fracture orientation is parallel to pressure gradient or the fracture can serve as part of the main channel, the stress sensitivity has great impact on the productivity of the fracture network. The drainage area around the hydraulic fractures are almost the same irrespective of the stress, but the drainage area of natural fractures changes significantly when considering stress sensitivity. Therefore, it is necessary to incorporate the stress effect on the flow conductivity of hydraulic and natural fractures while modeling complex fractured reservoirs. © 2016 Elsevier B.V. All rights reserved.
Keywords: Discrete fracture model Hydraulic fracture Natural fracture Stress sensitivity Multi-scale
1. Introduction With the growing challenge of conventional reservoir exploitation, producing gas or oil from shale reservoir has gradually become the focus of current energy industry (Wang and Krupnick, 2013). Unlike conventional reservoirs, fluid transport in naturally fractured reservoirs (NFR) is a complex process (Liang et al., 2016; Ren et al., 2015). Most NFR are comprised of natural fractures with arbitrary orientations, various apertures and different fracture patterns. NFR unfolds an upper degree of heterogeneity and diversity caused by discrete fractures compared to conventional porous media. The complexity of discrete fracture geometry and connectivity of site-specific fractured matrix make it hard to
* Corresponding author. College of petroleum engineering, China University of Petroleum, Beijing city Changping District Road No. 18, PR China. E-mail address: [email protected] (J. Li). http://dx.doi.org/10.1016/j.jngse.2016.09.025 1875-5100/© 2016 Elsevier B.V. All rights reserved.
characterize the flow behavior in NFR (Farayola et al., 2013). The seepage capacity of natural fractures is strong, so the mass transfer processes needs to be studied clearly. Traditional continuum model is insufficient to simulate such discrete fracture models because of its inability of depicting the mazy features of natural fractures. Therefore, multiphase simulation of the NFR with the complex fracture network is a challenging task in reservoir engineering (Reichenberger, 2003). The modeling of flow performance in discrete fracture media has been kept ongoing. Modeling the flow behavior of NFR can be roughly divided into three categories: dual porosity continuum models (DCM), discrete fracture network models (DFN) and discrete fracture models (DFM) (Sahimi, 2012). For the DCM, there are two mass balance equations describing fracture and matrix system, respectively (Warren and Root, 1963). Initially, flow through fractured porous media is simulated by using DCM. This approach suffers from some critical limitations despite of its
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simulation efficiency. Firstly, DCM cannot capture the complex structure of discrete fracture network, which may dominate the whole seepage process. Another disadvantage lies in the inaccurate evaluation of the transfer mechanism between matrix and fractures (Karimi-Fard et al., 2003). Unlike DCM, the DFN can address the mass transfer mechanism between matrix and fractures (Mi et al., 2014a). However, the DFN still has some limitations. The model takes the intersection of cracks as the basic research nodes, this brings the difficulty to describe the flow behavior in the matrix away from the research nodes. Compared with DFN, research nodes exist in both the matrix and the cracks in DFM. This modeling method can precisely capture the complexity feature of a discrete fracture medium. DFM model uses grid nodes to represent flow relationship between fracturefracture, fracture-matrix and matrix-matrix. It is necessary to generate unstructured grids adjusted to the distribution of discontinuous fractures, as shown in Fig. 1. The DFM has obtained considerable development in the last decade (Feuga and Peaudecerf, 1990; Gong et al., 2008). Besides discrete fracture modeling methods, the difference between the stress-sensitive conductivity of hydraulic and natural fractures also has a great influence on flow simulation in shale porous media. Hydraulic fracturing is a common technique for the tight reservoir development, which will generate a complex fracture system composed of both natural and hydraulic fractures. Obvious conductivity differences can be detected between the natural fractures and hydraulic fractures, and the pressure sensitivity of natural fractures is rather strong (Jianzheng, 2002; Ren et al., 2016). For hydraulic fractures, the conductivity capacity will be greatly improved due to the existence of proppants (Davies and Kuiper, 1988), as show in Fig. 2. The proppants provide effective resistance to ensure the effective conductivity of hydraulic fractures. Many mathematical models have been proposed to characterize stress-sensitive conductivity, but they are subject to numerous assumptions (Gao et al., 2012). There are also some experimental studies on the stress sensitivity of fracture conductivity (Phueakphum and Fuenkajorn, 2014). The purpose of this paper is to study the flow conductivity of hydraulic fractures and natural fractures under stress effect in multi-scale discrete fracture networks with stress sensitivity. For better understanding the flow dynamics, three discrete fracture models with different scale (disjoint fracture scale, fracture network scale and reservoir scale) have been established
considering the pressure sensitivity. The whole paper is organized as following; firstly, experiments on the pressure sensitivity of permeability are conducted on both artificially-fractured and naturally-fractured cores. Permeability fitting models of the hydraulic and natural fracture are then obtained. Based on the conductivity models, discrete fracture models are established by using DFM and the finite element method. The simulation results are finally presented in three aspects: pressure distribution, streamline analysis, and velocity field. 2. Model definition and introduction The discrete fracture model consists of two parts, the matrix and the facture network. Flow in matrix follows Darcy's law while flow in fracture belongs to modified Darcy's law. The whole flow process includes flow from matrix to matrix, matrix to fracture, fracture to matrix, and fracture to fracture. These flow processes will be presented in the following results. The whole flow area is divided into a series of flow units. The flow area is gridded by non-structured mesh using triangulation algorithm, and the center of each grid is regarded as a research node. The crack is equivalent to an internal boundary with a unique tangential method algorithm to define it. This can describe the crack behavior without a large number of dense and tiny grid elements. Unstructured mesh can efficiently split the matrix and fracture elements, which can provide a guarantee for the accurate calculation of mass transfer between any two mobile units. Based on these grids, the finite element method is used to implement the simulation process. 2.1. DFM description The basic model is 2D. The first set of mode is a square with constant pressure at left and right sides. The second set of models is a square with a production well and an injection well with constant pressure at the two diagonal corners. The flow in the model is single phase flow. The parameters of the model are in Table 1. Fig. 3 consists of pressure distribution, streamline distribution, and velocity field distribution. Considering the pressure distribution map, it is obvious that the existence of cracks will make the pressure contour complicated. Through the pressure contour we can find in which direction pressure drop is the most, and how the cracks affect the pressure distribution. Streamline describe the velocity direction of different fluid particles in the flow field, the
Fig. 1. The unstructured grid and the node of DFM (Liang et al., 2016).
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Fig. 2. The closing up of different types of fractures.
Table 1 Basic model data. Variables
Description
Parameter values
P0 Pinj Pw
Initial pressure Injection pressure Production pressure Fluid density Dynamic viscosity Matrix porosity Fracture porosity, Matrix permeability Fluid compressibility Matrix compressibility Fracture thickness
1 105Pa P0 0.1P0 1 kg/m3 1 104 Pa s 0.1 0.25 0.1 mD 2 1011 1/Pa 1 109 1/Pa e
r m
εp εf k
cf cp df
greater the streamline density the greater the flow velocity. Many streamlines wriggle through the cracks, thus the influence of crack on the seepage can be seen directly. The black arrows represent the flow velocity in fracture while the red arrows represent the surface velocity. The greater flow velocity means the greater arrow size. The flow velocity in the crack is described by the size of the arrows. In other word, the importance of different fractures can be
distinguished by the size of the velocity arrows. The distribution of red arrows indicate that velocity is the highest near the wellbore, fluid accumulates into crack along the main flow direction, and fluid first flows through the cracks then penetrates into the matrix towards production well. The greater the arrow density around the vicinity of the crack is, the greater the contribution of the fracture to the seepage flow is. From these results we can find DFM model can accurately describe the seepage process in discrete fracture media. The fluid flows from the matrix into the nearest cracks, and then spread into the matrix at the other end of cracks. Based on this theory, scientific study of the crack parameter sensitivity on permeability is conducted as follows where pressure distribution map, streamline distribution map and velocity field map are combined together. 2.2. The mathematical model The mathematical model has been described previously (Liang et al., 2016). The seepage mathematical model is divided into the matrix model and the fracture model. Flow in the matrix satisfies Darcy's law. Because the permeability of the fracture is larger than
Fig. 3. Field distribution.
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rS
vp þ V$ðruÞ ¼ Qm vt
(3)
The seepage mathematical model in the crack also contains several parts. The motion equation in the crack is written as:
kf u ¼ VT P
m
(4)
Permeability in individual fracture is written as:
kf ¼
d2f 12
(5)
The comprehensive seepage governing equation in the crack is described as Equation (6), which can represent the pressure sensitivity of the aperture with the pressure change.
df
v rεf þ VT $ df ru ¼ df Qm vt
(6)
The mathematical model is capable to represent the pressure sensitivity of the aperture with the pressure change. Fig. 4. Artificial fractured core sample with proppants.
3. Stress sensitivity experiments the matrix, modified Darcy's law is applied to describe the flow in the fracture (COMSOL Multiphysics Model library, Earth Science Module, 2015). The motion equation in the matrix block can be described as (Mi et al., 2014b):
k
m ¼ Vp m
(1)
Porous media and fluids are compressible, and the storage coefficient equation of the matrix is:
S ¼ εp cf þ 1 εp cp
(2)
The continuity equation of fluid in the matrix is:
3.1. Core sample preparations Fig. 4 is a photo of the artificial fractured core sample with proppants. Ceramic particles are used as the proppant with diameter of 212e380um. Modified acrylate adhesive (HL - 302) is used to fix the core halves. On the cut surface, there are two adhesive belts, and a layer of ceramic particles is shown on the left part. Yingrui Bai previously put forward a fractured core processing method (Bai et al., 2015). In our study, this method is improved to provide a more repeatable fractured core preparing process. The preparation procedure can be briefly described as follows: (1) Making the homogeneous cylindrical cores with a diameter of 254 mm and a length of 50 mm from natural outcrop rocks with average permeability of 0.3 mD. (2) Cutting each cylindrical core
Fig. 5. Experimental apparatus for stress sensitivity experiment.
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core, putting a layer of particles in the rest of the region after the two mixture belts are finished, then the two halves should be tightly fixed together and the bottom surface of the fractured core should be sealed with adhesive tape. The core should be left alone for 30 min before the next step. (6) Setting the artificially-fractured core vertically with the sealed face down, dumping ceramic particles to the fracture until no more particles could enter.
3.2. Experimental procedure
Fig. 6. Fitting curves of fracture permeability.
into two halves along the longitudinal direction, the cut surface should be fully grinded. (3) Drying the core by placing the core in the constant temperature box at 60 centigrade for 24 h (4) Mixing ceramic particles and modified acrylate adhesive uniformly, daubing the mixture alongside the edges of the cut surface. To ensure constant fracture width, only a layer of ceramic particles should appear in the mixture belt on the cut surface. For natural fractured core, the width of the mixture should be kept as 2 mm while the width should be 5 mm for hydraulic fracture core. (5) For naturally-fractured core, the two halves should be fixed together after the two mixture belts are finished; for artificially-fractured
The stress sensitivity of both naturally-fractured core and artificially-fractured core are measured under different confining pressures. The experiment setup is shown as Fig. 5. The ISCO pump provides constant injection rate, the pressure sensor detects the pressure dynamics, and the data display system will show the change of the injection pressure. The pressure tensor and data display system can be replaced by putting a precision pressure gauge on the Six-way valve. The confining pressure pump supplies adequate confining pressure to prevent inter-porosity flow between core surface and core gripper. The permeability will be calculated according to Darcy's law. Due to the high conductivity of fractured core, the injection rate is kept at 30 ml/min and the confining pressure changes from 3 MPa to 40 MPa. When a specific confining pressure is set, the injection pressure will fluctuate for a while and the final steady data will be taken as the injection pressure. Fig. 6 shows the experimental results. By changing the confining pressure, the fractured cores will show different conductivity
Fig. 7. Seepage field maps with constant pressure boundary.
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Fig. 8. Seepage field maps with different pressure gradients.
With the increase of confining pressure, the permeability of natural fractures decreases exponentially while the fitting curve of hydraulic fracture is in a cubic polynomial manner. In the following discrete fracture networks, stress-sensitive permeability models of hydraulic fractures and natural fractures are based on the above conclusions.
used to study the stress sensitivity of horizontal cracks. Cracks are believed to have no pressure sensitivity in the vertical direction. Pressure sensitivity is not considered in Case 1 while it is considered in case 2. The upper three horizontal cracks are hydraulic cracks, the lower three cracks are natural fractures. Case 3 and case 4 are used to study the stress sensitivity of the cracks in the vertical direction. There is no pressure sensitivity in the horizontal direction. No pressure sensitivity is added in case 3. The upper five cracks in case 4 are hydraulic cracks, the lower five vertical cracks are natural fractures. Fig. 7 shows the flow field maps of the above cases. According to the seepage field maps of case 1 and case 2, the influence of fracture stress sensitivity in horizontal direction is obvious. The black arrow represents the velocity vector of flow in the crack, and the bigger the arrow, the greater the flow velocity. The flow rate in natural fractures in case 2 was significantly reduced. Due to the existence of proppants, the hydraulic fractures still maintained a high conductivity capacity. Same law can be obtained from the distribution of velocity vector field and the streamline. The crack endpoints gathered a crowd of red arrows, which means the fracture absorbs large amount of fluid from the adjacent matrix. According to the seepage field maps of case 3 and case 4, we can find that the pressure sensitivity of the crack has little effect on the flow field in the vertical direction for both natural and artificial fractures. One of the main reasons is that when the cracks are perpendicular to the direction of the pressure gradient, the cracks are almost useless.
4. Results and discussion
4.2. Fracture network case
4.1. Disjoint fracture case
In this part, the model geometry is a square with production well and injection well at the two diagonal corners. Both of the wells are operating at constant pressure. The discrete fracture network consists of two sets of orthogonal fractures where the
capacity. Permeability is actually a direct embodiment of the conductivity capability. The permeability fitting model of naturally fractured core is
Knatural ¼ 5:4575DP 1:638
(7)
The permeability fitting model of artificially fractured core is
Khyraulic ¼ 0:00002DP 3 þ 0:0023DP 2 0:0788DP þ 1:0264 (8) According to equation (5), the mathematical model of the natural fracture aperture can be written as:
dnatural ¼ 8:0926DP 0:819
(9)
The mathematical model of hydraulic fracture aperture can be expressed as:
dhyraulic ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:00024DP 3 þ 0:0276DP 2 0:9456DP þ 12:3168 (10)
In this group of studies, the boundary pressure of the left side is 10 MPa and that of the right side is 1 MPa. Case 1 and case 2 are
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Fig. 9. The pressure distribution of different models.
horizontal cracks are natural cracks and the vertical cracks are hydraulic cracks. Flow field maps under different pressure gradients are shown in Fig. 8. Since the length of the model is only 1 m, we assume that each model is at a constant pressure gradient. When the pressure gradient is 0, there is no need to consider stress sensitivity. From the streamline distribution, we can see most of the fluid flow occurs in the fracture network rather than in the matrix, as shown by the red arrow distribution. Also the pressure drop of the fracture network is low as shown by the isobars. This reveals that fracture network has a very strong conductivity. From the black arrow distribution, we can see that the flow velocity in the natural and artificial fractures is high, thus the conductivity capacity in these part is strong accordingly. That is, the natural fractures and hydraulic fractures formed a dominant flow passage. When the pressure gradient increases to 5 MPa, natural fracture conductivity decreases greatly. Flow in matrix increases, as shown by the increasing red arrows in the matrix and isobars across the fracture network. Meanwhile, the seepage ability of fracture network decreases obviously. When the pressure gradient increases to 10 MPa, the black arrow which describes the crack velocity is hard to find, which indicates that the conductivity capacity of natural fractures is badly weakened. The fracture flow channel is destroyed, and the seepage ability of the fracture network is greatly reduced. When the pressure gradient is further increased to 15 MPa, the fracture conductivity is further reduced, but the change
has become not obvious. The hydraulic fracture still maintained a high flow capacity as shown by the black arrows. 4.3. Reservoir scale case This case is based on the geometric model built by Mi (Mi et al., 2014b). A 1000 m 250 m shale gas reservoir model with 60 natural fractures and a horizontal well is established. The five vertical fractures are hydraulic fractures. There is no fluid flow across the matrix block walls and the left and right walls are no flow boundary, although the flow through the edge of fracture is considered. The top matrix edge is no flow boundary while the down boundary is a wellbore with constant pressure control of 1 105Pa. Fig. 9 shows the fractured shale gas reservoir pressure distribution without and with pressure sensitivity respectively. When there is no pressure sensitivity, the fracture permeability is maintained at initial permeability. When considering pressure sensitivity, the fracture permeability is calculated according to equations (7) and (8). The first model shows the pressure distribution after producing two days with no pressure sensitivity, and the second model shows the stress-sensitivity considered case. Comparing pressure distribution of these models, we can find that the influence of pressure sensitivity is obvious. The drainage area is smaller when considering stress sensitivity. Moreover, the pressure sensitivity of natural
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B. Liang et al. / Journal of Natural Gas Science and Engineering 35 (2016) 851e859
Fig. 10. The pressure sensitivity effect on the production dynamic characteristics.
fractures has greater effect than that of hydraulic fractures. The pressure distribution around the hydraulic fractures is almost the same, but the difference is obvious when it comes to natural fractures. Comparing magnified area 1 and 2, we can find that if the natural fractures cannot serve as part of the fracture network, the pressure is almost insensible to the change of the fracture conductivity. When considering stress sensitivity, the conductivity of natural fracture will be smaller, so the area controlled by natural fractures will generate less shale gas and the pressure will be higher than the situation with no stress sensitivity. Comparing magnified area 3 and 4, we can also find that the area that has great difference is the area controlled by fracture network while there is not obvious change in the area merely controlled by hydraulic fracture. The seepage capacity of the fracture network is more susceptible to the stress sensitivity of fracture conductivity. So it is necessary to separately consider the stress-sensitive conductivity of hydraulic fractures and natural fractures while modeling complex fractured reservoirs. The production rate and the cumulative production is shown in Fig. 10, both of which indicate the fracture stress sensitivity have a significant effect on the shale gas transport and production at early depletion development stage. During this process, the pressure of the reservoir is large enough, so the fracture network seepage capacity is more sensible to fracture conductivity. When considering pressure sensitivity, both natural fractures and hydraulic fractures have smaller conductivity and lower gas production.
5. Conclusions The permeability of natural fractures decreases exponentially with the increase of confining pressure. In the contrast, the permeability decrease of hydraulic fractures follows the cubic polynomial law. The latter is a much slower process because the embedded proppants in the hydraulic fractures greatly lessens the impact of the stress changes on the permeability reduction and maintain its conductivity well. For disjoint fracture case, the apparent permeability of fractured porous media is largely affected by the stress sensitivity. The stress sensitivity of natural fractures has more influence on the model permeability than that of hydraulic fractures. When the fracture is parallel to the pressure gradient direction, the stress sensitivity has a great influence on the seepage field. However, when the fracture direction is perpendicular to the direction of the pressure gradient, the stress sensitivity has no obvious influence on the seepage field. This holds true for both hydraulic fractures and natural fractures. In the discrete fracture network, the stress sensitivity of fracture conductivity has a great influence on the seepage process. The
hydraulic fracture is the major contributors of the network apparent permeability under higher pressure gradient. The conductivity of the fracture network will decrease rapidly when considering pressure sensitivity. If the fractures cannot serve as part of connected network, the fracture conductivity is almost insensible to the change of the pressure. At the reservoir scale, when considering fracture stress sensitivity the drainage area is smaller than no-sensitivity case. Specifically, the drainage areas of hydraulic fractures is almost independent of the stress field, but the drainage area of natural fractures decreases a lot when considering stress sensitivity. It is, therefore, necessary to treat the stress-sensitive conductivity of hydraulic and natural fractures separately while modeling complex fractured reservoirs.
Future work The future work will focus on 3D DFM modeling and the generation of random discrete fracture network. Acknowledgement The financial support from the National Natural Science Foundation of China (Grant 51404280) is acknowledged. Nomenclature and units p0
r
p pinj p0 Qm u mf c
m
εp εf k kf
cf cp
S df
Initial pressure, Pa Fluid density, kg/m3 Pressure, Pa Injection pressure, Pa Bottom hole pressure, Pa Source, m3/s Flow rate, m/s Mass flux, Kg/(m$s) Correct factor, 1 Dynamic viscosity, Pa$s Matrix porosity, 1 Fracture porosity, 1 Matrix permeability, mD Fracture permeability, mD Fluid compressibility, 1/Pa Matrix compressibility, 1/Pa Matrix storage coefficient, 1/Pa Fracture thickness, m
References Bai, Y., Wei, F., Xiong, C., et al., 2015. Effects of fracture and matrix on propagation behavior and water shut-off performance of a polymer gel. Energy & Fuels 29 (9), 1624e1629. COMSOL, 2015. Multiphysics Model Library, Earth Science Module. Version 5.0. Davies, D.R., Kuiper, T.O.H., 1988. Fracture conductivity in hydraulic fracture stimulation. J. petroleum Technol. 40 (05), 550e552. Farayola, K.K., Olaoye, A.T., Adewuyi, A., 2013, August. Petroleum reservoir characterisation for fluid with yield stress using finite element analyses. In: SPE Nigeria Annual International Conference and Exhibition. Society of Petroleum Engineers. Feuga, B., Peaudecerf, P., 1990. Modeling fracture ow with a stochastic discrete fracture network: calibration and validation. 1. The ow model. Water Resour. Res. 26 (3), 479489 (Crassoulis). Gao, Y., Lv, Y., Wang, M., et al., 2012. New mathematical models for calculating the proppant embedment and fracture conductivity. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Gong, B., Karimi-Fard, M., Durlofsky, L.J., 2008. Upscaling discrete fracture characterizations to dual-porosity, dual-permeability models for efficient simulation of flow with strong gravitational effects. SPE J. 13 (01), 58e67. Jianzheng, Y., 2002. Study on the Stress Sensitivity of Fractured Reservoir. Southwest Petroleum Institute.
B. Liang et al. / Journal of Natural Gas Science and Engineering 35 (2016) 851e859 Karimi-Fard, M., Durlofsky, L.J., Aziz, K., 2003, January. An efficient discrete fracture model applicable for general purpose reservoir simulators. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers. Liang, B., Jiang, H., Li, J., et al., 2016. A systematic study of fracture parameters effect on fracture network permeability based on discrete-fracture model employing Finite Element Analyses. J. Nat. Gas Sci. Eng. 28, 711e722. Mi, L., Jiang, H., Li, J., 2014a. The impact of diffusion type on multiscale discrete fracture model numerical simulation for shale gas. J. Nat. Gas Sci. Eng. 20, 74e81. Mi, L., Jiang, H., Li, J., Li, T., Tian, Y., 2014b. The investigation of fracture aperture effect on shale gas transport using discrete fracture model. J. Nat. Gas Sci. Eng. 21, 631e635. Phueakphum, D., Fuenkajorn, K., 2014. Laboratory assessment of fracture permeability under normal and shear stresses[C]//isrm international Symposium-8th
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Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Flow in multi-scale discrete fracture networks with stress sensitivity Bin Liang a, Hanqiao Jiang a, Junjian Li a, * , Changcheng Gong a, Ruyi Jiang b, Yanli Pei a, Shiming Wei a a b
Key Laboratory for Petroleum Engineering of the Ministry of Education, China University of Petroleum, Beijing 102249, PR China Department of Science and Technology Management, China National Petroleum Corporation, Beijing 100007, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 March 2016 Received in revised form 7 September 2016 Accepted 8 September 2016 Available online 9 September 2016
For hydraulic fracturing in the shale reservoir, the existence of natural fractures has a great impact on the propagation of hydraulic fractures and the flow capacity of porous media. Our previous study mainly discussed the effect of different fracture parameters (i.e., orientation, aperture) on the fracture network permeability based on discrete fracture model and finite elements analyses (Liang et al., 2016). To further account for the in-situ condition, we now investigate the effect of stress sensitivity, i.e. the stress difference between hydraulic fractures and natural fractures, on flow behavior in multi-scale discrete fracture networks. A series of sensitivity experiments on both naturally and artificially fractured cores with the existence of proppants were carried out. Fracture models at different scale were established based on the corresponding mathematical fitting models, to study the effect of stress sensitivity on the seepage process. Results show that with the increase of confining pressure, the permeability of natural fractures decreases exponentially while the permeability of hydraulic fractures decreases more slowly following the cubic polynomial law. The stress sensitivity of natural fractures has more influence on the flow dynamics than that of hydraulic fractures, and this difference is subject to the fracture orientation, fracture length, and intersection relationship. When the fracture orientation is parallel to pressure gradient or the fracture can serve as part of the main channel, the stress sensitivity has great impact on the productivity of the fracture network. The drainage area around the hydraulic fractures are almost the same irrespective of the stress, but the drainage area of natural fractures changes significantly when considering stress sensitivity. Therefore, it is necessary to incorporate the stress effect on the flow conductivity of hydraulic and natural fractures while modeling complex fractured reservoirs. © 2016 Elsevier B.V. All rights reserved.
Keywords: Discrete fracture model Hydraulic fracture Natural fracture Stress sensitivity Multi-scale
1. Introduction With the growing challenge of conventional reservoir exploitation, producing gas or oil from shale reservoir has gradually become the focus of current energy industry (Wang and Krupnick, 2013). Unlike conventional reservoirs, fluid transport in naturally fractured reservoirs (NFR) is a complex process (Liang et al., 2016; Ren et al., 2015). Most NFR are comprised of natural fractures with arbitrary orientations, various apertures and different fracture patterns. NFR unfolds an upper degree of heterogeneity and diversity caused by discrete fractures compared to conventional porous media. The complexity of discrete fracture geometry and connectivity of site-specific fractured matrix make it hard to
* Corresponding author. College of petroleum engineering, China University of Petroleum, Beijing city Changping District Road No. 18, PR China. E-mail address: [email protected] (J. Li). http://dx.doi.org/10.1016/j.jngse.2016.09.025 1875-5100/© 2016 Elsevier B.V. All rights reserved.
characterize the flow behavior in NFR (Farayola et al., 2013). The seepage capacity of natural fractures is strong, so the mass transfer processes needs to be studied clearly. Traditional continuum model is insufficient to simulate such discrete fracture models because of its inability of depicting the mazy features of natural fractures. Therefore, multiphase simulation of the NFR with the complex fracture network is a challenging task in reservoir engineering (Reichenberger, 2003). The modeling of flow performance in discrete fracture media has been kept ongoing. Modeling the flow behavior of NFR can be roughly divided into three categories: dual porosity continuum models (DCM), discrete fracture network models (DFN) and discrete fracture models (DFM) (Sahimi, 2012). For the DCM, there are two mass balance equations describing fracture and matrix system, respectively (Warren and Root, 1963). Initially, flow through fractured porous media is simulated by using DCM. This approach suffers from some critical limitations despite of its
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simulation efficiency. Firstly, DCM cannot capture the complex structure of discrete fracture network, which may dominate the whole seepage process. Another disadvantage lies in the inaccurate evaluation of the transfer mechanism between matrix and fractures (Karimi-Fard et al., 2003). Unlike DCM, the DFN can address the mass transfer mechanism between matrix and fractures (Mi et al., 2014a). However, the DFN still has some limitations. The model takes the intersection of cracks as the basic research nodes, this brings the difficulty to describe the flow behavior in the matrix away from the research nodes. Compared with DFN, research nodes exist in both the matrix and the cracks in DFM. This modeling method can precisely capture the complexity feature of a discrete fracture medium. DFM model uses grid nodes to represent flow relationship between fracturefracture, fracture-matrix and matrix-matrix. It is necessary to generate unstructured grids adjusted to the distribution of discontinuous fractures, as shown in Fig. 1. The DFM has obtained considerable development in the last decade (Feuga and Peaudecerf, 1990; Gong et al., 2008). Besides discrete fracture modeling methods, the difference between the stress-sensitive conductivity of hydraulic and natural fractures also has a great influence on flow simulation in shale porous media. Hydraulic fracturing is a common technique for the tight reservoir development, which will generate a complex fracture system composed of both natural and hydraulic fractures. Obvious conductivity differences can be detected between the natural fractures and hydraulic fractures, and the pressure sensitivity of natural fractures is rather strong (Jianzheng, 2002; Ren et al., 2016). For hydraulic fractures, the conductivity capacity will be greatly improved due to the existence of proppants (Davies and Kuiper, 1988), as show in Fig. 2. The proppants provide effective resistance to ensure the effective conductivity of hydraulic fractures. Many mathematical models have been proposed to characterize stress-sensitive conductivity, but they are subject to numerous assumptions (Gao et al., 2012). There are also some experimental studies on the stress sensitivity of fracture conductivity (Phueakphum and Fuenkajorn, 2014). The purpose of this paper is to study the flow conductivity of hydraulic fractures and natural fractures under stress effect in multi-scale discrete fracture networks with stress sensitivity. For better understanding the flow dynamics, three discrete fracture models with different scale (disjoint fracture scale, fracture network scale and reservoir scale) have been established
considering the pressure sensitivity. The whole paper is organized as following; firstly, experiments on the pressure sensitivity of permeability are conducted on both artificially-fractured and naturally-fractured cores. Permeability fitting models of the hydraulic and natural fracture are then obtained. Based on the conductivity models, discrete fracture models are established by using DFM and the finite element method. The simulation results are finally presented in three aspects: pressure distribution, streamline analysis, and velocity field. 2. Model definition and introduction The discrete fracture model consists of two parts, the matrix and the facture network. Flow in matrix follows Darcy's law while flow in fracture belongs to modified Darcy's law. The whole flow process includes flow from matrix to matrix, matrix to fracture, fracture to matrix, and fracture to fracture. These flow processes will be presented in the following results. The whole flow area is divided into a series of flow units. The flow area is gridded by non-structured mesh using triangulation algorithm, and the center of each grid is regarded as a research node. The crack is equivalent to an internal boundary with a unique tangential method algorithm to define it. This can describe the crack behavior without a large number of dense and tiny grid elements. Unstructured mesh can efficiently split the matrix and fracture elements, which can provide a guarantee for the accurate calculation of mass transfer between any two mobile units. Based on these grids, the finite element method is used to implement the simulation process. 2.1. DFM description The basic model is 2D. The first set of mode is a square with constant pressure at left and right sides. The second set of models is a square with a production well and an injection well with constant pressure at the two diagonal corners. The flow in the model is single phase flow. The parameters of the model are in Table 1. Fig. 3 consists of pressure distribution, streamline distribution, and velocity field distribution. Considering the pressure distribution map, it is obvious that the existence of cracks will make the pressure contour complicated. Through the pressure contour we can find in which direction pressure drop is the most, and how the cracks affect the pressure distribution. Streamline describe the velocity direction of different fluid particles in the flow field, the
Fig. 1. The unstructured grid and the node of DFM (Liang et al., 2016).
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Fig. 2. The closing up of different types of fractures.
Table 1 Basic model data. Variables
Description
Parameter values
P0 Pinj Pw
Initial pressure Injection pressure Production pressure Fluid density Dynamic viscosity Matrix porosity Fracture porosity, Matrix permeability Fluid compressibility Matrix compressibility Fracture thickness
1 105Pa P0 0.1P0 1 kg/m3 1 104 Pa s 0.1 0.25 0.1 mD 2 1011 1/Pa 1 109 1/Pa e
r m
εp εf k
cf cp df
greater the streamline density the greater the flow velocity. Many streamlines wriggle through the cracks, thus the influence of crack on the seepage can be seen directly. The black arrows represent the flow velocity in fracture while the red arrows represent the surface velocity. The greater flow velocity means the greater arrow size. The flow velocity in the crack is described by the size of the arrows. In other word, the importance of different fractures can be
distinguished by the size of the velocity arrows. The distribution of red arrows indicate that velocity is the highest near the wellbore, fluid accumulates into crack along the main flow direction, and fluid first flows through the cracks then penetrates into the matrix towards production well. The greater the arrow density around the vicinity of the crack is, the greater the contribution of the fracture to the seepage flow is. From these results we can find DFM model can accurately describe the seepage process in discrete fracture media. The fluid flows from the matrix into the nearest cracks, and then spread into the matrix at the other end of cracks. Based on this theory, scientific study of the crack parameter sensitivity on permeability is conducted as follows where pressure distribution map, streamline distribution map and velocity field map are combined together. 2.2. The mathematical model The mathematical model has been described previously (Liang et al., 2016). The seepage mathematical model is divided into the matrix model and the fracture model. Flow in the matrix satisfies Darcy's law. Because the permeability of the fracture is larger than
Fig. 3. Field distribution.
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rS
vp þ V$ðruÞ ¼ Qm vt
(3)
The seepage mathematical model in the crack also contains several parts. The motion equation in the crack is written as:
kf u ¼ VT P
m
(4)
Permeability in individual fracture is written as:
kf ¼
d2f 12
(5)
The comprehensive seepage governing equation in the crack is described as Equation (6), which can represent the pressure sensitivity of the aperture with the pressure change.
df
v rεf þ VT $ df ru ¼ df Qm vt
(6)
The mathematical model is capable to represent the pressure sensitivity of the aperture with the pressure change. Fig. 4. Artificial fractured core sample with proppants.
3. Stress sensitivity experiments the matrix, modified Darcy's law is applied to describe the flow in the fracture (COMSOL Multiphysics Model library, Earth Science Module, 2015). The motion equation in the matrix block can be described as (Mi et al., 2014b):
k
m ¼ Vp m
(1)
Porous media and fluids are compressible, and the storage coefficient equation of the matrix is:
S ¼ εp cf þ 1 εp cp
(2)
The continuity equation of fluid in the matrix is:
3.1. Core sample preparations Fig. 4 is a photo of the artificial fractured core sample with proppants. Ceramic particles are used as the proppant with diameter of 212e380um. Modified acrylate adhesive (HL - 302) is used to fix the core halves. On the cut surface, there are two adhesive belts, and a layer of ceramic particles is shown on the left part. Yingrui Bai previously put forward a fractured core processing method (Bai et al., 2015). In our study, this method is improved to provide a more repeatable fractured core preparing process. The preparation procedure can be briefly described as follows: (1) Making the homogeneous cylindrical cores with a diameter of 254 mm and a length of 50 mm from natural outcrop rocks with average permeability of 0.3 mD. (2) Cutting each cylindrical core
Fig. 5. Experimental apparatus for stress sensitivity experiment.
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core, putting a layer of particles in the rest of the region after the two mixture belts are finished, then the two halves should be tightly fixed together and the bottom surface of the fractured core should be sealed with adhesive tape. The core should be left alone for 30 min before the next step. (6) Setting the artificially-fractured core vertically with the sealed face down, dumping ceramic particles to the fracture until no more particles could enter.
3.2. Experimental procedure
Fig. 6. Fitting curves of fracture permeability.
into two halves along the longitudinal direction, the cut surface should be fully grinded. (3) Drying the core by placing the core in the constant temperature box at 60 centigrade for 24 h (4) Mixing ceramic particles and modified acrylate adhesive uniformly, daubing the mixture alongside the edges of the cut surface. To ensure constant fracture width, only a layer of ceramic particles should appear in the mixture belt on the cut surface. For natural fractured core, the width of the mixture should be kept as 2 mm while the width should be 5 mm for hydraulic fracture core. (5) For naturally-fractured core, the two halves should be fixed together after the two mixture belts are finished; for artificially-fractured
The stress sensitivity of both naturally-fractured core and artificially-fractured core are measured under different confining pressures. The experiment setup is shown as Fig. 5. The ISCO pump provides constant injection rate, the pressure sensor detects the pressure dynamics, and the data display system will show the change of the injection pressure. The pressure tensor and data display system can be replaced by putting a precision pressure gauge on the Six-way valve. The confining pressure pump supplies adequate confining pressure to prevent inter-porosity flow between core surface and core gripper. The permeability will be calculated according to Darcy's law. Due to the high conductivity of fractured core, the injection rate is kept at 30 ml/min and the confining pressure changes from 3 MPa to 40 MPa. When a specific confining pressure is set, the injection pressure will fluctuate for a while and the final steady data will be taken as the injection pressure. Fig. 6 shows the experimental results. By changing the confining pressure, the fractured cores will show different conductivity
Fig. 7. Seepage field maps with constant pressure boundary.
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Fig. 8. Seepage field maps with different pressure gradients.
With the increase of confining pressure, the permeability of natural fractures decreases exponentially while the fitting curve of hydraulic fracture is in a cubic polynomial manner. In the following discrete fracture networks, stress-sensitive permeability models of hydraulic fractures and natural fractures are based on the above conclusions.
used to study the stress sensitivity of horizontal cracks. Cracks are believed to have no pressure sensitivity in the vertical direction. Pressure sensitivity is not considered in Case 1 while it is considered in case 2. The upper three horizontal cracks are hydraulic cracks, the lower three cracks are natural fractures. Case 3 and case 4 are used to study the stress sensitivity of the cracks in the vertical direction. There is no pressure sensitivity in the horizontal direction. No pressure sensitivity is added in case 3. The upper five cracks in case 4 are hydraulic cracks, the lower five vertical cracks are natural fractures. Fig. 7 shows the flow field maps of the above cases. According to the seepage field maps of case 1 and case 2, the influence of fracture stress sensitivity in horizontal direction is obvious. The black arrow represents the velocity vector of flow in the crack, and the bigger the arrow, the greater the flow velocity. The flow rate in natural fractures in case 2 was significantly reduced. Due to the existence of proppants, the hydraulic fractures still maintained a high conductivity capacity. Same law can be obtained from the distribution of velocity vector field and the streamline. The crack endpoints gathered a crowd of red arrows, which means the fracture absorbs large amount of fluid from the adjacent matrix. According to the seepage field maps of case 3 and case 4, we can find that the pressure sensitivity of the crack has little effect on the flow field in the vertical direction for both natural and artificial fractures. One of the main reasons is that when the cracks are perpendicular to the direction of the pressure gradient, the cracks are almost useless.
4. Results and discussion
4.2. Fracture network case
4.1. Disjoint fracture case
In this part, the model geometry is a square with production well and injection well at the two diagonal corners. Both of the wells are operating at constant pressure. The discrete fracture network consists of two sets of orthogonal fractures where the
capacity. Permeability is actually a direct embodiment of the conductivity capability. The permeability fitting model of naturally fractured core is
Knatural ¼ 5:4575DP 1:638
(7)
The permeability fitting model of artificially fractured core is
Khyraulic ¼ 0:00002DP 3 þ 0:0023DP 2 0:0788DP þ 1:0264 (8) According to equation (5), the mathematical model of the natural fracture aperture can be written as:
dnatural ¼ 8:0926DP 0:819
(9)
The mathematical model of hydraulic fracture aperture can be expressed as:
dhyraulic ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:00024DP 3 þ 0:0276DP 2 0:9456DP þ 12:3168 (10)
In this group of studies, the boundary pressure of the left side is 10 MPa and that of the right side is 1 MPa. Case 1 and case 2 are
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Fig. 9. The pressure distribution of different models.
horizontal cracks are natural cracks and the vertical cracks are hydraulic cracks. Flow field maps under different pressure gradients are shown in Fig. 8. Since the length of the model is only 1 m, we assume that each model is at a constant pressure gradient. When the pressure gradient is 0, there is no need to consider stress sensitivity. From the streamline distribution, we can see most of the fluid flow occurs in the fracture network rather than in the matrix, as shown by the red arrow distribution. Also the pressure drop of the fracture network is low as shown by the isobars. This reveals that fracture network has a very strong conductivity. From the black arrow distribution, we can see that the flow velocity in the natural and artificial fractures is high, thus the conductivity capacity in these part is strong accordingly. That is, the natural fractures and hydraulic fractures formed a dominant flow passage. When the pressure gradient increases to 5 MPa, natural fracture conductivity decreases greatly. Flow in matrix increases, as shown by the increasing red arrows in the matrix and isobars across the fracture network. Meanwhile, the seepage ability of fracture network decreases obviously. When the pressure gradient increases to 10 MPa, the black arrow which describes the crack velocity is hard to find, which indicates that the conductivity capacity of natural fractures is badly weakened. The fracture flow channel is destroyed, and the seepage ability of the fracture network is greatly reduced. When the pressure gradient is further increased to 15 MPa, the fracture conductivity is further reduced, but the change
has become not obvious. The hydraulic fracture still maintained a high flow capacity as shown by the black arrows. 4.3. Reservoir scale case This case is based on the geometric model built by Mi (Mi et al., 2014b). A 1000 m 250 m shale gas reservoir model with 60 natural fractures and a horizontal well is established. The five vertical fractures are hydraulic fractures. There is no fluid flow across the matrix block walls and the left and right walls are no flow boundary, although the flow through the edge of fracture is considered. The top matrix edge is no flow boundary while the down boundary is a wellbore with constant pressure control of 1 105Pa. Fig. 9 shows the fractured shale gas reservoir pressure distribution without and with pressure sensitivity respectively. When there is no pressure sensitivity, the fracture permeability is maintained at initial permeability. When considering pressure sensitivity, the fracture permeability is calculated according to equations (7) and (8). The first model shows the pressure distribution after producing two days with no pressure sensitivity, and the second model shows the stress-sensitivity considered case. Comparing pressure distribution of these models, we can find that the influence of pressure sensitivity is obvious. The drainage area is smaller when considering stress sensitivity. Moreover, the pressure sensitivity of natural
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Fig. 10. The pressure sensitivity effect on the production dynamic characteristics.
fractures has greater effect than that of hydraulic fractures. The pressure distribution around the hydraulic fractures is almost the same, but the difference is obvious when it comes to natural fractures. Comparing magnified area 1 and 2, we can find that if the natural fractures cannot serve as part of the fracture network, the pressure is almost insensible to the change of the fracture conductivity. When considering stress sensitivity, the conductivity of natural fracture will be smaller, so the area controlled by natural fractures will generate less shale gas and the pressure will be higher than the situation with no stress sensitivity. Comparing magnified area 3 and 4, we can also find that the area that has great difference is the area controlled by fracture network while there is not obvious change in the area merely controlled by hydraulic fracture. The seepage capacity of the fracture network is more susceptible to the stress sensitivity of fracture conductivity. So it is necessary to separately consider the stress-sensitive conductivity of hydraulic fractures and natural fractures while modeling complex fractured reservoirs. The production rate and the cumulative production is shown in Fig. 10, both of which indicate the fracture stress sensitivity have a significant effect on the shale gas transport and production at early depletion development stage. During this process, the pressure of the reservoir is large enough, so the fracture network seepage capacity is more sensible to fracture conductivity. When considering pressure sensitivity, both natural fractures and hydraulic fractures have smaller conductivity and lower gas production.
5. Conclusions The permeability of natural fractures decreases exponentially with the increase of confining pressure. In the contrast, the permeability decrease of hydraulic fractures follows the cubic polynomial law. The latter is a much slower process because the embedded proppants in the hydraulic fractures greatly lessens the impact of the stress changes on the permeability reduction and maintain its conductivity well. For disjoint fracture case, the apparent permeability of fractured porous media is largely affected by the stress sensitivity. The stress sensitivity of natural fractures has more influence on the model permeability than that of hydraulic fractures. When the fracture is parallel to the pressure gradient direction, the stress sensitivity has a great influence on the seepage field. However, when the fracture direction is perpendicular to the direction of the pressure gradient, the stress sensitivity has no obvious influence on the seepage field. This holds true for both hydraulic fractures and natural fractures. In the discrete fracture network, the stress sensitivity of fracture conductivity has a great influence on the seepage process. The
hydraulic fracture is the major contributors of the network apparent permeability under higher pressure gradient. The conductivity of the fracture network will decrease rapidly when considering pressure sensitivity. If the fractures cannot serve as part of connected network, the fracture conductivity is almost insensible to the change of the pressure. At the reservoir scale, when considering fracture stress sensitivity the drainage area is smaller than no-sensitivity case. Specifically, the drainage areas of hydraulic fractures is almost independent of the stress field, but the drainage area of natural fractures decreases a lot when considering stress sensitivity. It is, therefore, necessary to treat the stress-sensitive conductivity of hydraulic and natural fractures separately while modeling complex fractured reservoirs.
Future work The future work will focus on 3D DFM modeling and the generation of random discrete fracture network. Acknowledgement The financial support from the National Natural Science Foundation of China (Grant 51404280) is acknowledged. Nomenclature and units p0
r
p pinj p0 Qm u mf c
m
εp εf k kf
cf cp
S df
Initial pressure, Pa Fluid density, kg/m3 Pressure, Pa Injection pressure, Pa Bottom hole pressure, Pa Source, m3/s Flow rate, m/s Mass flux, Kg/(m$s) Correct factor, 1 Dynamic viscosity, Pa$s Matrix porosity, 1 Fracture porosity, 1 Matrix permeability, mD Fracture permeability, mD Fluid compressibility, 1/Pa Matrix compressibility, 1/Pa Matrix storage coefficient, 1/Pa Fracture thickness, m
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