# Naturally fractured hydrocarbon reservoir simulation by elastic fracture modeling

## Abstract

Accurate fluid flow simulation in geologically complex reservoirs is of particular importance in construction of reservoir simulators. General approaches in naturally fractured reservoir simulation involve use of unstructured grids or a structured grid coupled with locally unstructured grids and discrete fracture models. These methods suffer from drawbacks such as lack of flexibility and of ease of updating. In this study, I combined fracture modeling by elastic gridding which improves flexibility, especially in complex reservoirs. The proposed model revises conventional modeling fractures by hard rigid planes that do not change through production. This is a dubious assumption, especially in reservoirs with a high production rate in the beginning. The proposed elastic fracture modeling considers changes in fracture properties, shape and aperture through the simulation. This strategy is only reliable for naturally fractured reservoirs with high fracture permeability and less permeable matrix and parallel fractures with less cross-connections. Comparison of elastic fracture modeling results with conventional modeling showed that these assumptions will cause production pressure to enlarge fracture apertures and change fracture shapes, which consequently results in lower production compared with what was previously assumed. It is concluded that an elastic gridded model could better simulate reservoir performance.

### Keywords

Reservoir performance Discrete fracture model Naturally fractured reservoir History matching Elastic gridding## 1 Introduction

In this study, we try to increase realism of the fracture model by considering shape of the fracture and the variation of its properties through production and pressure regime change in fluid flow procedure. Fracture characteristics which change through time of production are included in elastic properties of fractures. The elastic gridding scheme was introduced in this study into the fracture modeling scheme to handle heterogeneity of a complex reservoir. This approach was then applied on a fractured limestone reservoir in southwest Iran. Results of the application of the proposed strategy for fracture modeling in the study field show production of more water in comparison with conventional fracture modeling result.

## 2 Fracture simulation methods for carbonate rocks

The intersection relationships of fractures are probably very complex in a realistic DFN model (Zhang 2015). Bisdom et al. (2016) investigated impact of in situ stress and outcrop fracture geometry on hydraulic apertures in reservoirs. They stated that each fracture network containing fractures is created of at least one fracture set, but is not necessarily limited to it. By proposing a fracture propagation model using multiple planar fractures with a mixed model, Jang et al. (2016) stated that there is a large discrepancy in reservoir volume stimulation, because of a number of intersections of fracture connectivity. Heffer and King (2006) introduced a spatial correlation function of fractures as displacement strain vectors using renormalization techniques in representation of stochastic tensor fields for strain modeling. Masihi and King (2007) applied this method to generate fracture networks based on the assumption that the elastic energy in the fractured media follows a Boltzmann distribution. Koike et al. (2012) used geostatistical fracture distribution and fracture orientation (strike and dip) in simulation of the fracture system to estimate the hydraulic conductivity. Bisdom et al. (2016) also proved that the fracture orientation and the associated hydraulic aperture distribution have stronger impact on equivalent permeability than length or spacing. Thus, spatial correlation of fractures is the most important parameter in any fracture networks model or gridding scheme. To make this correlation between fractures, various methods are introduced for assigning precise values from fracture characteristics to the model of fractures. Among them, various approaches of using outcrop fracture characteristics such as fracture spatial distribution, length, height, orientation, spacing and aperture are widely used for model regularization (Wilson et al. 2011; Hooker et al. 2012). Lapponi et al. (2011) used outcrop data to construct a 3D model in a dolomitized carbonate reservoir rock from the Zagros Mountains, southwest Iran. Lee et al. (2011) studied the spatial fracture intensity effect on hydraulic flow in fractured rock. They used outcrop for simulation of spatial fracture intensity distribution. They have built three spatial fracture intensity distribution models and showed that flow vectors are strongly affected by spatial fracture intensity. They also proved that the higher the fracture intensity, the higher flow velocity. Boro et al. (2014) presented a workflow to construct an upscaled fracture model based on outcrop studies in a carbonate platform. It is important to note that fractures in the outcrop might have been affected by surface processes like weathering and stress release. The overall analysis, however, can help to constrain possible scenarios on fracture populations that may be relevant to the subsurface reservoir. Malinouskaya et al. (2014) illustrated a method to rapidly estimate permeability of a fracture network, using fracture data from outcrops of a Jurassic carbonate ramp. The method proposed by Maffucci et al. (2015) used outcrop data combined with a discrete fracture network (DFN) model to increase the reliability of fracture system characterization in the case of limited data for carbonate rocks. Connectivity of fracture networks in carbonate rock is dependent on orientation, size distribution and densities of the different fracture sets. These parameters also define size of the blocks enveloped by fractures (i.e., the matrix block size), which is generally used to model transfer of fluids between the matrix and fractures (Wennberg et al. 2016). To incorporate interaction between the matrix and fractured media, Huang et al. (2014) divided the fractured porous medium into two non-overlapping subdomains. One domain has a continuum model in the rock matrix, and the other, in deep fractured and fissure zones, is described by a DFN model. Then, they coupled these domains to simulate groundwater flow in their case study. However, Boro et al. (2014) stated that in general, fracture intensities, apertures and their intrinsic permeability would have more significant impact on the permeability of the field. Fracture shape and orientations are more important in affecting connectivity. Bisdom et al. (2016) also showed the strong importance of fracture orientation and associated hydraulic aperture distribution on equivalent permeability. In general, it is widely believed that fluid flow is affected by heterogeneities at all scales, from millimeter scale (porosity) to kilometer scale (Shekhar et al. 2014). In the present work, field evidence of different solutions and fractures in limestone is used to certify the nature of elasticity of fractures through time of production.

The proposed strategy also exposes incorrect assumptions in conventional fracture modeling, which has a great impact on history matching studies. Afterward, the concept of allocating each fracture to a fracture set and subsequently to a fracture network was considered by comparing the formation outcrop and considering the elasticity nature of the fractures, which comes from formation fluid pressure and/or regional stress in modeling.

## 3 Elastic fracture modeling

Dennis et al. (2010) stated that the physical structure properties and complexity of fracture characterization both have a significant effect on fluid flow in fractured rock. Thus, they proposed that the fracture zone should be characterized fully before simulation. Wang et al. (2016) studied the flow stress damage and reservoir responses to injection rate under different DFN-connected configuration states. Their results proved significant influence of the hydraulic pressure flow on the properties of hydraulic fractures. Generally, Gan and Elsworth (2016) stated that in any fracture modeling for complex reservoirs, the upcoming assumptions should not be neglected: Fractures initiate from flaws and the process is controlled by the elastic stress around them, the material surrounding the flaws can be viewed as continuum media, and individual flaws are spaced widely enough so that stress anomalies associated with each do not overlap. These assumptions are necessary for analytical formulation and therefore will be retained in the analysis, although a slight degree of plasticity is allowed (Wang and Shahvali 2016). However, in any elastic gridding, it should be kept in mind that a fracture is initiated when the maximum stress concentration occurring on the critical flaw boundary reaches the strength of the material which surrounds the flaw. Fracture extension also occurs from the tensile and not the compressive stress concentrations under both tensile and compressive loading. Fan et al. (2012) stated that microcracks induced by the excess oil/gas pressure may propagate and form an interconnected fracture network. This indicates also that during production, different pressure regimes could change characteristics of fractures. Not only might they be closed in the case of pressure drop or reservoir depletion, but they also might be widened due to high production rates and excessive pressure of fluid flow to the walls of fractures or cracks. It also might create fractures which make connections between vugs, while cracks could be widened and/or become fractures. Fan et al. (2012) investigated mechanism of fracture propagation by a linear elastic model. They have shown that critical crack propagation takes place if the intensity of the induced stress reaches the fracture toughness of the reservoir rock. On the other hand, subcritical crack propagation occurs in the rock when the stress intensity has not reached the fracture toughness of the reservoir rock, but exceeds a threshold value, which is usually a fraction (e.g., 20%–50%) of fracture toughness of the reservoir rock. This conclusion states how important it is to consider fracture–matrix interaction and/or boundary condition of fractures in accurate flow simulation. Subsequently, Hassanzadeh and Pooladi-Darvish (2006) considered the time variability of the fracture boundary condition by the Laplace domain analytical solutions of the diffusivity equation for different geometries of fractures in constant fracture pressure through a large number of pressure steps. Guerriero et al. (2013) proposed an analytical model that could pave the way to a full numerical model allowing one to calculate the pore pressure within fractures, at several scales of observation, in a reasonable time. They also suggested that the model also allows one to obtain a better understanding of the hydraulic behavior of fractured porous rock. However, not only the pressure regime change, but other factors affecting fracture shape and apertures could be accounted for by considering elastic behavior of fractures in the model construction during gridding and throughout production history matching investigation.

Bisdom et al. (2016) stated relationships between the fracture geometrical parameters and some other parameters such as the stress applied in the medium. However, for this specific case, accurate relationship between degree of elasticity and fracture’s aperture would be defined only by core analyses in a wide range of applied stresses from pore fluid to the walls of fractures. However, previous studies have shown that this relationship is linear in a narrow range of applied stress (Bisdom et al. 2016).

However, in this study reservoir, the thermal fracturing is not planned in the master development plan of the field and the production history of the reservoir also shows some degree of overpressure fluid in the first periods of initial production, when accurate data were not available. Thus, neither the thermal fluid injection nor the fluid overpressure was considered here. The majority of fractures that we had in the study reservoir were of the type of fractures shown in Fig. 2.

The more the pressure regime changes in a short time, the more this will cause more change in the shape of fractures. Figure 4a shows a conventional model of fractures. Figure 4b, c illustrates shape change in the same fracture after high pressure regime change, modeled by elastic fracture modeling. Figure 4d also shows change in fracture aperture modeled by elastic fracture modeling. However, the most important parameter that changes the permeability of the reservoir rock is the dilation of fractures. Unlike other fracture parameters, the dilation degree of a fracture under stress cannot be described from core sample tests. Clearly, long fractures cause the core to fall apart during the experiment. Core recovery is also very poor in intensely fractured intervals, and the stress release when the core is taken to the surface will affect the observed apertures (Wennberg et al. 2016). Use of electrical image logs also suffers from uncertainty if the absolute aperture is large. This problem would be boosted if we want to accurately measure the dilation of the fracture. Therefore, it necessitates that the relationship between aperture dilation and applied stress is defined by numerical analysis and other permeability tests in various conditions. Taron et al. (2014) tested fracture dilation in a geothermal system.

However, Min et al. (2004) and Farahmand et al. (2015) derived relationships between aperture dilation and applied stress. Min et al. (2004) studied completely all cases and we used the results that they derived in their complete study. Min et al. (2004) have stated that the exact extent of shear dilations of fractures can only be identified through numerical experiments. Their experiments showed that on the one hand, equivalent permeability decreases with increase in stresses, when the differential stress is not large enough to cause shear dilation of fractures. On the other hand, the equivalent permeability increases with the increase in differential stresses, when the stress ratio was large enough to cause continued shear dilation of fractures. In this case, shear dilation is the dominating mechanism in characterizing the stress-dependent permeability. They have proved that the maximum contribution of dilation is more than one order of magnitude in permeability. Thus, we have used this role in our study for fracture dilation.

The pore fluid effects needed to calculate elastic properties of fluid-saturated rock could be obtained also in each study from the simulation case. However, in this study, we did not have enough data to derive an implicit model for elastic behavior simulation of a fracture. However, it is not only the matter of data, but it is about the matter of accurate and implicit relationship between the applied stress and strain of the medium and elastic properties and behavior of fractures. Thus, in this study, we used an explicit relationship between the stress applied to a fracture and the allowed degree of elasticity of the fracture. Consequently, we should define a maximum degree of change in shape that we consider for a fracture and the value of changes in curvature of the fracture. The former is defined based on the Bulk and/or Young’s modulus of the rock. For hard rocks, the lowest grade of change in shape is allowed and it is vice versa for soft rocks. The latter needs more explanation. At first, it should define whether curvature has any effect on permeability change or it is only a fracture shape change, without effect on permeability. According to Fig. 4, it changes the permeability only if both sides of the fracture experience convex curvature, which increases the permeability. In case of same convexity or concavity of fracture walls, no changes in permeability would happen. In this study, we assumed the first case for our fractures, as shown in Fig. 4d. The degree of curvature is defined based on the toughness of rock. Although the exact degree of convexity and curvature of the fracture wall should be defined by core sample test, it could be defined as a linear function of applied stress for medium to rocks.

## 4 The study reservoir

### 4.1 Petrophysical data

Limited cores were cut from the Asmari Formation for routine and special analyses. Total length of cores cut was 550 m out of which 421 m was recovered. The mean porosity of plugs cut from 421 m of cores in six wells was 9.8%, while 21% of samples have porosities less than 4% and only 3% of samples have porosity more than 20%, implying that the Asmari Formation is a low porosity reservoir (Hoseinzadeh et al. 2015). The median permeability of cores was calculated to be 0.43 mD with 60% of samples having permeability of less than one milli-darcy, indicating a very low permeable carbonate rock matrix. Production logging tools (PLT) logs were recorded in 15 oil wells and in 5 gas wells. Reviewing of the PLT log results indicates that distance between flowing intervals varies from 1 m to the maximum of 44 m. This indicates an active mechanism of fracture production in this field.

## 5 Zonation and fracture study

Classifying different types of fracture in the cores

Type of fracture | Open | Partly filled | Filled | Closed or hairline |
---|---|---|---|---|

Score | 1 | 2 | 3 | 4 |

Classifying different types of filling minerals in the cores

Filling mineral | Calcite | Dolomite | Anhydrite | Clay minerals |
---|---|---|---|---|

Code | 1 | 2 | 3 | 4 |

## 6 Elastic fracture model generation

### 6.1 Model initialization

*R*

_{w}is the formation water resistivity,

*R*

_{t}is the formation resistivity, and

*w*=

*f*(

*R*

_{w},

*R*, ø). The right side of Eq. (1) was derived from well logs and averaged for different regions and zones (subzones). To have a reasonable value for the right side of equation, a primary blocking step of the target formation was performed for averaging in each block. Subsequently, a weighted averaging based on the thickness of each zone/subzone was performed. Due to the large number of obtained maps with different blocking approaches and based on different interpretation of the averaged map result with the help of geological and hydrological data, the maps are not shown here. Then,

*S*

_{wi}was estimated from Eq. (1) for each simulation model elastic grid block based on its matrix porosity. The resulting

*S*

_{wi}values were introduced in the model as connate water saturation. To obtain different oil–water contacts (OWC), two aquifers and six equilibration regions were introduced into the model (Fig. 12). Two analytical aquifers were introduced instead of one aquifer, while the northwest part of the reservoir was completely disconnected from any analytical aquifer. Aquifer 2 is stronger than the other aquifers. The aquifers are separated by faults, while six OWC zones are controlled by the pressure regime of the field. The water saturation in the matrix also could affect the OWC in different zones.

### 6.2 Preliminary simulation

The study field’s production history is complicated, and the aquifer strength varies from the southeast to the northwest of the field. Thus, introducing the elastic behavior of fractures into the model was proposed here to realize the history matching of the reservoir production.

*x*direction by a factor of 10 and reducing fracture permeability in

*z*direction by a factor of 10 to compensate for the elasticity of microfracturing observed in the cores. Examination of elastic model runs showed that oil flow from matrix to fractures should be increased to match water and gas production from the oil wells. Elevating of this oil transfer was done by modifying the imbibition capillary pressure. Individual oil well modifications included changing capillary pressure curves assigned to the matrix grid blocks surrounding the well, varying the height of matrix blocks, modification of the pore volume of initially oil- and gas-saturated grid blocks and modification of the fracture permeability in

*x*,

*y*and

*z*directions. Figure 14 shows the conventional and elastic fracture water saturation model. As it can be seen on the elastic fracture model, wells located in the southeast (SE) of the field may produce more water. This is due to the increase in cracks and/or fracture apertures, which means previous small cracks were filled with water. Figure 15 shows elastic and conventional models of fracture oil saturation. Again, the southeast (SE) part of the field does not produce more oil after a while according to the elastic modeling results. This phenomenon is less observed for gas saturation. Figure 16 shows result of elastic and conventional fracture modeling. As a result, change in shape of fractures, (increasing aperture) would replace oil by gas in the vicinity of oil-gas contact. This is obvious in Fig. 16.

## 7 Conclusions

There are many problems in fractured reservoirs that may require new approaches in fracture modeling based on advanced concepts. Some problems cannot be simplified beyond a certain limit. The complexity, scale and uncertainty of natural systems are the primary reasons for most of the modeling difficulties which have to be handled.

The objective of this work was to develop a prototype workflow for history matching in naturally fractured reservoirs. This concern is also with the problem of generating more realistic computational grids for reservoir simulations. Here we face the particular problems connected to the complexity of the reservoir. The proposed strategy combines the accuracy of fracture modeling with the efficiency of elastic gridding. Elastic gridding is very simple, and in principle, the method should be well suited for weighing between mutually conflicting optimization criteria with highly nonlinear and discontinuous cost functions. The strategy was applied on a complex reservoir from southwest Iran. The strategy was observed to be reasonably effective in achieving field-level agreement in total oil and water production rates. Aperture increase in cracks will mean they will be filled by water, while it replaces gas by oil in the vicinity of the gas oil contact. Thus, elastic fracture modeling shows that more water will be produced compared to what was assumed by conventional fracture models, which do not take into account fracture properties change through production.

### References

- Abdollahie Fard I, Braathen A, Mokhtari M, Alavi A. Interaction of the Zagros fold–thrust belt and the Arabian-type, deep-seated folds in the Abadan plain and the Dezful embayment SW Iran. Pet Geosci. 2006. doi: 10.1144/1354-079305-706.Google Scholar
- Agar SM, Hampson GJ. Fundamental controls on flow in carbonates: an introduction. Pet Geosci. 2014. doi: 10.1144/petgeo2013-090.Google Scholar
- Bahrainian SS, Daneh Dezfuli A, Noghrehabadi A. Unstructured grid generation in porous domains for flow simulations with discrete-fracture network model. Transport Porous Media. 2015. doi: 10.1007/s11242-015-0544-3.Google Scholar
- Biryukov D, Kuchuk FJ. Transient pressure behavior of reservoirs with discrete conductive faults and fractures. Transport Porous Media. 2012. doi: 10.1007/s11242-012-0041-x.Google Scholar
- Bisdom K, Bertott G, Nick HM. The impact of in situ stress and outcrop-based fracture geometry on hydraulic aperture and upscaled permeability in fractured reservoirs. Techtonophysics. 2016. doi: 10.1016/j.tecto.2016.04.006.Google Scholar
- Boro H, Rosero E, Bertotti G. Fracture-network analysis of the Latemar Platform (northern Italy): integrating outcrop studies to constrain the hydraulic properties of fractures in reservoir models. Pet Geosci. 2014. doi: 10.1144/petgeo2013-007.Google Scholar
- Cai L, Ding DY, Wang C, Wu YS. Accurate and efficient simulation of fracture–matrix interaction in shale gas reservoirs. Transport Porous Media. 2015. doi: 10.1007/s11242-014-0437-x.Google Scholar
- Chen Y, Cai D, Fan Z, Li K, Ni J. 3D geological modeling of dual porosity carbonate reservoirs: a case from the Kenkiyak pre-salt oilfield Kazakhstan. Pet Explor Dev. 2008. doi: 10.1016/S1876-3804(08)60097-X.Google Scholar
- Dennis I, Pretorius J, Steyl G. Effect of fracture zone on DNAPL transport and dispersion: a numerical approach. Environ Earth Sci. 2010. doi: 10.1007/s12665-010-0468-8.Google Scholar
- Fan ZQ, Jin ZH, Johnson SE. Gas-driven subcritical crack propagation during the conversion of oil to gas. Pet Geosci. 2012. doi: 10.1144/1354-079311-030.Google Scholar
- Farahmand K, Baghbanan A, Shahriar K, Diederichs MS. Effect of fracture dilation angle on stress-dependent permeability tensor of fractured rock. In: 49th U.S. Rock mechanics/geomechanics symposium, San Francisco, 2015, ARMA-2015-542.Google Scholar
- Gan Q, Elsworth D. A continuum model for coupled stress and fluid flow in discrete fracture networks. Geomech Geophy Geo Energy Geo Resour. 2016. doi: 10.1007/s40948-015-0020-0.Google Scholar
- Guerriero V, Mazzoli S, Iannace A, Vitale S, Carravetta A, Strauss C. A permeability model for naturally fractured carbonate reservoirs. Mar Pet Geol. 2013. doi: 10.1016/j.marpetgeo.2012.11.002.Google Scholar
- Hassanzadeh H, Pooladi-Darvish M. Effects of fracture boundary conditions on matrix-fracture transfer shape factor. Transport Porous Media. 2006. doi: 10.1007/s11242-005-1398-x.Google Scholar
- Heffer KJ, King PR. Spatial scaling of effective modulus and correlation of deformation near the critical point of fracturing. Pure Appl Geophys. 2006. doi: 10.1007/978-3-7643-8124-010.Google Scholar
- Hoseinzadeh M, Daneshian J, Moallemi SA, Solgi A. Facies analysis and depositional environment of the Oligocene-Miocene Asmari Formation, Bandar Abbas hinterland Iran. Open J Geol. 2015. doi: 10.4236/ojg.2015.54016.Google Scholar
- Hooker JN, Gomez LA, Laubach SE, Gale JFW, Marrett R. Effects of diagenesis (cement precipitation) during fracture opening on fracture aperture-size scaling in carbonate rocks. Geol Soc Lond Spec Publ. 2012. doi: 10.1144/SP370.9.Google Scholar
- Huang T, Guo X, Chen F. Modeling transient flow behavior of a multiscale triple porosity model for shale gas reservoirs. J Nat Gas Sci Eng. 2015. doi: 10.1016/j.jngse.01.022.Google Scholar
- Huang Y, Zhou Z, Wang J, Dou Z. Simulation of groundwater flow in fractured rocks using a coupled model based on the method of domain decomposition. Environ Earth Sci. 2014. doi: 10.1007/s12665-014-3184-y.Google Scholar
- Jang A, Kim J, Ertekin T, Sung W. Fracture propagation model using multiple planar fracture with mixed mode in naturally fractured reservoir. J Pet Sci Eng. 2016. doi: 10.1016/j.petrol.02.015.Google Scholar
- Jiang J, Younis RM. Numerical study of complex fracture geometries for unconventional gas reservoirs using a discrete fracture-matrix model. J Nat Gas Sci Eng. 2015. doi: 10.1016/j.jngse.2015.08.013.Google Scholar
- Karimi-Fard M, Firoozabadi A. Numerical simulation of water injection in fractured media using discrete-fracture model and the Galerkin method. SPE Reserv Eval Eng. 2003. doi: 10.2118/83633-PA.Google Scholar
- Karimi-Fard M, Durlofsky LJ, Aziz K. An efficient discrete fracture model applicable for general-purpose reservoir simulators. SPE J. 2004. doi: 10.2118/88812-PA.Google Scholar
- Karimi-Fard M. Modeling tools for fractured systems: gridding, discretization, and upscaling. Stanford University. 2013. http://cees.stanford.edu/docs/KarimiFard13.
- Koike K, Liu C, Sanga T. Incorporation of fracture directions into 3D geostatistical methods for a rock fracture system. Environ Earth Sci. 2012. doi: 10.1007/s12665-011-1350-z.Google Scholar
- Kresse O, Weng XW, Gu HR, Wu RT. Numerical modeling of hydraulic fractures interaction in complex naturally fractured formations. Rock Mech Rock Eng. 2013. doi: 10.1007/s00603-012-0359-2.Google Scholar
- Lapponi F, Casini G, Sharp I, Blendinger W, Fernández N, Romaire I, Hunt D. From outcrop to 3D modelling: a case study of a dolomitized carbonate reservoir, Zagros Mountains Iran. Pet Geosci. 2011. doi: 10.1144/1354-079310-040.Google Scholar
- Lee CC, Lee CH, Yeh HF, Lin HI. Modeling spatial fracture intensity as a control on flow in fractured rock. Environ Earth Sci. 2011. doi: 10.1007/s12665-010-0794-x.Google Scholar
- Maffucci R, Bigi S, Corrado S, Chiodi A, Di Paolo L, Giordano G, Invernizzi C. Quality assessment of reservoirs by means of outcrop data and discrete fracture network models: the case history of Rosario de La Frontera (NW Argentina) geothermal system. Tectonophysics. 2015. doi: 10.1016/j.tecto.2015.02.016.Google Scholar
- Malinouskaya I, Thovert JF, Mourzenko VV, Adler PM, Shekhar R, Agar S, Rosero E, Tsenn M. Fracture analysis in the Amellago outcrop and permeability predictions. Pet Geosci. 2014. doi: 10.1144/petgeo2012-094.Google Scholar
- Masihi M, King PR. A correlated fracture network: modeling and percolation properties. Water Resour Res. 2007. doi: 10.1029/2006WR005331.Google Scholar
- Mi L, Jiang H, Li J, Li T, Tian Y. The investigation of fracture aperture effect on shale gas transport using discrete fracture model. J Nat Gas Sci Eng. 2014. doi: 10.1016/j.jngse.09.029.Google Scholar
- Min KB, Rutqvist J, Tsang CF, Jing L. Stress-dependent permeability of fractured rock masses: a numerical study. Int J Rock Mech Min Sci. 2004. doi: 10.1016/j.ijrmms.2004.05.005.Google Scholar
- Nie RS, Meng YF, Jia YL, Zhang FX, Yang XT, Niu XN. Dual porosity and dual permeability modeling of horizontal well in naturally fractured reservoir. Transport Porous Media. 2012. doi: 10.1007/s11242-011-9898-3.Google Scholar
- Presho M, Woc S, Ginting V. Calibrated dual porosity, dual permeability modeling of fractured reservoirs. J Pet Sci Eng. 2011. doi: 10.1016/j.petrol.2011.04.007.Google Scholar
- Salimi H, Bruining H. Upscaling in vertically fractured oil reservoirs using homogenization. Transport Porous Media. 2010. doi: 10.1007/s11242-009-9483-1.Google Scholar
- Sang G, Elsworth D, Miao X, Mao X, Wang J. Numerical study of a stress dependent triple porosity model for shale gas reservoirs accommodating gas diffusion in kerogen. J Nat Gas Sci Eng. 2016. doi: 10.1016/j.jngse.2016.04.044.Google Scholar
- Shekhar R, Sahni I, Benson G, et al. Modelling and simulation of a Jurassic carbonate ramp outcrop, Amellago, High Atlas Mountains. Morocco. Pet Geosci. 2014. doi: 10.1144/petgeo2013-010.Google Scholar
- Sherkati S, Letouzey J. Variation of structural style and basin evolution in the central Zagros (Izeh zone and Dezful Embayment) Iran. Mar Pet Geol. 2004. doi: 10.1016/j.marpetgeo.01.007.Google Scholar
- Soleimani M, Jodeiri-Shokri B. 3D static reservoir modeling by geostatistical techniques used for reservoir characterization and data integration. Environ Earth Sci. 2015. doi: 10.1007/s12665-015-4130-3.Google Scholar
- Soleimani M. Seismic imaging by 3D partial CDS method in complex media. J Pet Sci Eng. 2016a. doi: 10.1016/j.petrol.2016.02.019.Google Scholar
- Soleimani M. Seismic image enhancement of mud volcano bearing complex structure by the CDS method, a case study in SE of the Caspian Sea shoreline. Russ Geol Geophs. 2016b. doi: 10.1016/j.rgg.2016.01.020.Google Scholar
- Taron J, Hickman S, Ingebritsen SE, Williams C. Using a fully coupled, open-source THM simulator to examine the role of thermal stresses in shear stimulation of enhanced geothermal systems. In: 48th US Rock mechanics/geomechanics symposium held in Minneapolis, 2014, ARMA 14-7525.Google Scholar
- Wang Y, Shahvali M. Discrete fracture modeling using Centroidal Voronoi grid for simulation of shale gas plays with coupled nonlinear physics. Fuel. 2016. doi: 10.1016/j.fuel.2015.09.038.Google Scholar
- Wang Y, Li X, Tang CA. Effect of injection rate on hydraulic fracturing in naturally fractured shale formations: a numerical study. Environ Earth Sci. 2016. doi: 10.1007/s12665-016-5308-z.Google Scholar
- Wennberg OP, Casini G, Jonoud S, et al. The characteristics of open fractures in carbonate reservoirs and their impact on fluid flow: a discussion. Pet Geosci. 2016. doi: 10.1144/petgeo2015-003.Google Scholar
- Wilson CE, Aydin A, Karimi-Fard M, et al. From outcrop to flow simulation: constructing discrete fracture models from a LIDAR survey. AAPG Bull. 2011. doi: 10.1306/03241108148.Google Scholar
- Xia Y, Jin Y, Chen M. Comprehensive methodology for detecting fracture aperture in naturally fractured formations using mud loss data. J Pet Sci Eng. 2015a. doi: 10.1016/j.petrol.10.017.Google Scholar
- Xia Y, Jin Y, Chen M, et al. Hydrodynamic modeling of mud loss controlled by the coupling of discrete fracture and matrix. J Pet Sci Eng. 2015b. doi: 10.1016/j.petrol.2014.07.026.Google Scholar
- Yan X, Huang Z, Yao J, Li Y, Fan D. An efficient embedded discrete fracture model based on mimetic finite difference method. J Pet Sci Eng. 2016. doi: 10.1016/j.petrol.2016.03.013.Google Scholar
- Yoshida N, Levine JS, Stauffer PH. Investigation of uncertainty in CO2 reservoir models: a sensitivity analysis of relative permeability parameter values. Int J Greenh Gas Control. 2016. doi: 10.1016/j.ijggc.2016.03.008.Google Scholar
- Zhang QH. Finite element generation of arbitrary 3-D fracture networks for flow analysis in complicated discrete fracture networks. J Hydrol. 2015. doi: 10.1016/j.jhydrol.2015.08.065.Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.