Drained rock volume around hydraulic fractures in porous media: planar fractures versus fractal networks
Abstract
This study applies the Lindenmayer system based on fractal theory to generate synthetic fracture networks in hydraulically fractured wells. The applied flow model is based on complex analysis methods, which can quantify the flow near the fractures, and being gridless, is computationally faster than traditional discrete volume simulations. The representation of hydraulic fractures as fractals is a more realistic representation than planar biwing fractures used in most reservoir models. Fluid withdrawal from the reservoir with evenly spaced hydraulic fractures may leave dead zones between planar fractures. Complex fractal networks will drain the reservoir matrix more effectively, due to the mitigation of stagnation flow zones. The flow velocities, pressure response, and drained rock volume (DRV) are visualized for a variety of fractal fracture networks in a singlefracture treatment stage. The major advancement of this study is the improved representation of hydraulic fractures as complex fractals rather than restricting to planar fracture geometries. Our models indicate that when the complexity of hydraulic fracture networks increases, this will suppress the occurrence of dead flow zones. In order to increase the DRV and improve ultimate recovery, our flow models suggest that fracture treatment programs must find ways to create more complex fracture networks.
Keywords
Hydraulic fractures Drained rock volume Dead zones Fractals Branched fractures1 Introduction
The massive shift in US oil and gas production, after the Millennium turn, from conventional to unconventional reservoirs, has seen the hydraulic fracturing of production wells become a crucial aspect of completion engineering. The productivity of shale wells is now primarily based on how effectively hydraulic fractures help to provide new pathways for flow toward the wells from the reservoir matrix with ultralow permeability. A proper understanding of the creation of hydraulic fractures and modeling of fluid flow near these fractures is needed for improvement in both the early well productivity and the ultimate recovery factor. The engineering of hydraulic fractures in unconventional hydrocarbon plays is a rapidly evolving art. Industry has moved to reduce fracture spacing from over 100 ft in 2010, to 50 ft in 2014, and less than 20 ft in 2018. The fracture spacing is designed using estimations of geomechanical rock properties from pilot wells in combination with fracture propagation models.
The earliest attempts to compare hydraulic fracture patterns may be traced back to Warpinski et al. (1994), but today there is still no consensus regarding the relative merits of the various fracture propagation modeling platforms. The American Rock Mechanics Association (ARMA) has recently initiated seven benchmark tests for 20 participating models (Han 2017) with the intent to showcase recognized physics of hydraulic fracturing. Most platforms for modeling hydraulic fracture propagation are based on assumed homogeneous rock properties, which therefore uniquely favor the formation of planar, subparallel hydraulic fractures (Parsegov et al. 2018).
The likelihood of complex fracture networks being created by the fracture treatment process (rather than mutually subparallel planar fractures) is further supported by evidence from microseismic monitoring (Fisher et al. 2002; Maxwell et al. 2002). In fact, most microseismic clouds generated during fracturing jobs show a poor correlation with the assumed planar, subparallel fractures. Therefore, we assume that the creation of complex hydraulic fracture networks may be more representative for many fractured or treated wells, especially those that possess a network of natural fractures due to stress regimes varying over geological time. Such conditions are typical of most unconventional shale plays under exploration and development. Consequently, the use of planar hydraulic fractures for modeling reservoir depletion may not always appropriately account for the actual reservoir attributes. The subsequent use of such oversimplified planar fracture geometries in flow models leads to unreliable calculations of important reservoir attributes such as the drained rock volume (DRV) and flaws in the associated pressure response.
Current fracture representation methods that try to capture fracture complexity include discrete fracture network models and the unconventional fracture model (Weng et al. 2011; Zhou et al. 2012) and are reviewed in Sect. 3.1. These established fracture geometry models use block centered grids typically coupled with finitedifference discretization flow models, including compositional flow models to simulate reservoir performance (Yu et al. 2017). The drawback of these finitedifference schemes is that they can be computationally intensive due to the necessity of fine meshing, especially at the fracture intersections. Other methods to model flow in fractured porous media include semianalytical models to simulate and analyze the pressure change for complex well interference systems (Yu et al. 2016). The suitability of the dualporosity flow model (Warren and Root 1963) for low permeability reservoirs has been questioned (Cai et al. 2015). Further work has led to the development of triple porosity models to model flow in fractured reservoirs (Sang et al. 2016). Zhou et al. (2012) proposed a semianalytical solution for flow in a complex hydraulic fracture network model, which combined an analytical reservoir solution with a numerical solution on discretized fracture panels. The present study applies the analytical CAM flow model (Weijermars et al. 2016, 2017a, b, 2018), which is computationally efficient, while being able to accurately model the flow near fractal fractures such as those observed in field tests (Raterman et al. 2017).
The flow analysis in this study uses branched fractals for describing the complex fracture networks that are present in the subsurface. A variety of branched fractal fracture networks are imported into a drainage model based on complex analysis methods (CAM) to determine the flow response and pressure changes in the reservoir, for a given fracture geometry and fracture surface area. The major effect observed due to increasing fractal nature and branching of the fracture network (as outlined later in this study) is that the extent of dead zones between hydraulic fracture stages is suppressed. Instead, a more diffuse network of fractures drains the matrix between the fracture initiation points spaced by the perforation zones. Depending on the geometry of hydraulic fractures, an otherwise nonfractured matrix with negligible spatial variation in permeability can be drained more or less effectively. Future work will need to determine when hydraulic fractures will develop as fractal networks. While the jury is still out on the prominent geometry of hydraulic fractures (planar vs. fractal), the models developed in the present study consider the effect on drained rock volume in a systematic investigation of hydraulic fracture geometry ranging from planar to multibranched, higherorder fractals. The present study breaks new ground by modeling the flow around fractal fracture networks in porous media. The results have implications for fracture treatment designs required to maximize the drained rock volume.
2 Natural examples of hydraulic fractures
The precise natural pressure responsible for the injection of the hydraulic veins is unknown, but the pressure has exceeded the strength of the rock and was large enough to open the fractures at several km burial depth, thus being in the order of 100 MPa. The fluid was injected into the fractures as well as into a pervasive system of microcracks connected to the main fractures. Based upon the splaying of the fractures, one may reconstruct the provenance of the fracture propagation (van Harmelen and Weijermars 2018). Local heterogeneities in elastic properties may create conditions favoring the nucleation of fracture bifurcation points. More work is needed to determine the critical conditions required for creating fractal fracture networks in hydraulic fracture treatment programs.
Slabs like those shown in Fig. 3a may serve as a natural analog for flow into hydraulic fractures in shale reservoirs, with the limitation that shale may have different elastic moduli, different petrophysics, grain sizes and most crucially, the fracture aperture width from hydraulic fracturing which is smaller than that in our natural analog presented here. Hydraulic fracture apertures in shale reservoirs are thought to be in the range of 1–5 mm with the majority of created fracture apertures being less than 2 mm (Gale et al. 2014; Zolfaghari et al. 2016; Arshadi et al. 2017). Natural fracture networks created in the rocks of Bidasar due to hydrothermal activity in the earth’s crust bears similarity to manmade hydraulic fracture networks that require the use of high pressure fluids and proppants by fleets of pumps and trucks.
3 Fractures and fractal theory
3.1 Prior models of complex hydraulic fractures
3.1.1 Fracture propagation and fracture flow models
Various attempts have been made by researchers to develop new models to better represent complex hydraulic fracture network systems, in both geomechanical fracture propagation models and in production forecasting based on flow models in fractured reservoirs. For example, the geomechanical unconventional fracture model (UFM) was developed to simulate the propagation of complex fractures in formations with preexisting natural fractures (Weng et al. 2011). The UFM simulates the propagation, deformation, and fluid flow in a complex network of fractures. The model seeks to solve a system of equations governing parameters such as fracture deformation, height growth, fluid flow, and proppant transport, while considering the effect of natural fractures by using an analytical crossing model. The Wiremesh model consists of a fracture network with two orthogonal sets of parallel and uniformly spaced fractures (Xu et al. 2010; Meyer and Bazan 2011). Given fracture spacing, mechanical properties of the formation layers and pumping parameters, this shale fracturing simulator can be used to predict the growth of the hydraulic fracture network. Benefits of the Wiremesh model come in the form of increased surface area of the fracture network and mechanical interaction of fractures but are still only an approximation of the network’s complexity. Limitations of this model include not being able to directly link preexisting natural fractures to the hydraulic fracture network with regard to the fracture spacing used and that the network geometry is assumed to be elliptical in shape and thus symmetric. These assumptions do not always fit with fracture geometry indicated by microseismic data. Alternative modeling attempts sought to create a complex fracture network by finding a full solution to the coupled elasticity and fluid flow equations using 2D plane strain conditions (Zhang et al. 2007). Other studies presented a complex fracture network capable of predicting the interaction of hydraulic fractures with natural fractures but did not consider fluid flow and proppant transport (Olson and Taleghani 2009).
Flow models of fractured reservoirs have also advanced by upscaling a discrete fracture network (DFN) model into a dualporosity reservoir model or by enhancing the permeability of stimulated reservoir areas (Zhou et al. 2012). The fundamental discrete fracture network (DFN) solution methodology is based on satisfying continuity, mass conservation, constitutive relationships, and momentum equations (Meyer and Bazan 2011). For fracture representation in this method, each fracture panel had to be manually input with specific fracture parameters thus requiring prior knowledge of hydraulic fracture orientation. The model also assumes the intersection of individual planar fractures to create the complex fracture network with drained area represented by pressure depletion plots. These DFN are created using stochastic simulations based on probabilistic density functions of geometric parameters of fracture sets relating to fracture density, location, orientation and sizes based on measurements from field outcrops or borehole images. DFN requires an extremely fine grid at the scale of the fractures leading to complicated gridding and for multistage wells with large fracture numbers is very computationally expensive.
Recent advancements with DFN have now led to the embedded discrete fracture model (EDFM). EDFM allows for complex fractures to be implemented in conventionally structured matrix grids without using local grid refinement (Yu and Sepehrnoori 2018). EDFM can be thought of as a hybrid approach where the dualporosity model is used for the smaller and mediumsize fractures, and the DFN is used to model larger fractures (Li and Lee 2008). Advantages of EDFM include the use of a structured grid to represent the matrix and fractures. EDFM was initially used for planar 2D cases but has developed to model in 3D (Moinfar et al. 2014). Though EDFM has overcome some of the problems of the traditional DFM method, it can still be computationally expensive in complexly fractured reservoirs.
3.1.2 Fracture geometry models
Beyond the modeling attempts outlined above to recreate and describe complex fracture networks, work has been done by various authors to characterize the created fracture complexity based on field data. Zolfaghari et al. (2016) proposed the use of flowback salinity data to help characterize the fracture network complexity. The shape of the flowback curves is used to define the aperture size distribution (ASD) for a particular well. A narrow ASD is correlated with a simple fracture network, while a wider ASD is believed to match a fracture network that is more dendritic and complex in nature. Zolfaghari et al. (2017) looked at correlating total ions produced from chemical flowback to estimate fracture surface area for two wells that was validated against rate transient analysis (RTA) values. Based on these results, the authors postulated that greater production from one well was due to the larger fracture area calculated. This larger fracture area was attributed to a more complex fracture network in the subsurface, but there was no indication of potential fracture geometry. Another attempt to characterize fracture complexity utilizes tracer flowback data. Li et al. (2016) made use of tracer flowback data to characterize fracture morphology into three general categories. Based on the tracer breakthrough curve (BTC) the hydraulic fractures are roughly classified as microfractures, large fractures, and their mix. These methods allow for qualitative descriptions of the subsurface fracture network but do not allow for quantitative description in terms of surface area of the complex fracture network in contact with the reservoir matrix or fracture network geometry.
The majority of fracture flow methods attempt to introduce discrete fractures to model explicitly the elastic fracture propagation, subsequent flow and evacuation of fluid from the reservoir. The importance of accounting for fracture network complexity is apparent from production and pressure transient responses (Jones et al. 2013). Properly modeling the complexity of the fracture network is crucial for accurate history matching in these reservoirs. In addition to the discrete fracture models based on geomechanical failure modes, another potential approach to model fracture complexity uses fractal geometry. Fractals have long been used to model naturally occurring phenomena including petroleum reservoir and subsurface properties and equations (Berta et al. 1994; Cossio et al. 2012). Early work by Katz and Thompson (1985) and Pandey et al. (1987) showed that fracture propagation in nature was not irregular and could be represented by various fractal models. Building forward on this work AlObaidy et al. (2014) and Wang et al. (2015) approached the fracture network problem by creating branched fractal models to capture fracture network complexity.
3.2 Fractal theory
Fractal theory was first put forth by Mandelbrot (1979) as “a workable geometric middle ground between the excessive geometric order of Euclid and the geometric chaos of general mathematics”. A fractal was defined by Mandelbrot as a rough or fragmented geometric shape that can be split into parts each of which is a reducedsize copy of the whole. For an object to be termed a fractal, it must possess some noninteger (fractal) dimension (Frame et al. 2012). If this fractal dimension is an integer, we can obtain normal Euclidean geometry such as lines, triangles and regular polygons. Cossio et al. (2012) put into simple terms that a property of a given system can be termed a fractal if its seemingly chaotic, and unpredictable behavior with respect to time and space can be captured in a simple powerlaw equation. One of the basic principles underlying fractal geometry is the concept of selfsimilarity at various levels. If one zooms in on the represented object, a natural repetition of patterns and properties can be observed.
The abundance of fractals in our natural environment ranges from the fractal nature of coastlines to the growth and bifurcation of trees and plants. The use of fractals allows one to make mathematical sense from seemingly random and chaotic processes. Early use of fractals in petroleum engineering began with the work of Katz and Thompson (1985) to represent pore spaces in sandstone cores. The use of fractal theory to represent the pore space was verified by its accurate prediction of the core porosity. We now extend this approach of fractals to model complex hydraulic fracture networks in a reservoir with assumed parameters.
 1.
Fractal distance (d) controls the extending distance of the fractal fractions, (can be thought of as a basic repeating pattern) and closely relates to the halflength of the fractures created.
 2.
Deviation angle (α) controls the orientation of the fracture branching once deviation from the base fracture pattern occurs and relates to the area of the stimulated reservoir.
 3.
Number of iterations (i) controls the growth complexity of the fracture network or in other words fracture network density. This parameter relates to the multilevel feature of the fractal branches; during each iteration, the fractal fractures will branch from the original nodes following the given generating rules to construct that part of the network.
 4.
Growth of the bifurcation of the fractures and irregular propagation mode of a complex fracture network are subject to fractal rules, which are an implicit means to account for geomechanical heterogeneities (Wang et al. 2015, 2017, 2018).
Parameters used for creation of different fracture geometries
Fracture model  F length, ft  G length, ft  Branching angle, degrees  Created fracture halflength x_{f}, ft  Created fractal network span, ft 

Planar  400  –  –  400.0  – 
Firstgeneration fractal  100  100  10  398.5  34.70 
Secondgeneration fractal  40  40  10  398.2  69.04 
Thirdgeneration fractal  18  15  10  391.1  89.44 
4 Flow models
4.1 Complex analysis method (CAM) tool
Traditional applications of CAM in subsurface flow models make use of integral solutions to model streamlines for steady state flows (Muskat 1949; Strack 1989; Sato 2015). A fundamental expansion of the CAM modeling tool is the application of Eulerian particle tracking of timedependent flows, which was first explored in Weijermars (2014) and then benchmarked against numerical reservoir simulations in Weijermars et al. (2016).
Most current studies use numerical reservoir simulation to create pressure depletion plots as a proxy for the drained regions in the reservoir after production. CAM can determine the drained rock volume (DRV) by constructing timeofflight contours to the well based on Eulerian particle tracking taking into account the changing velocity field (Weijermars et al. 2017a, b). This approach provides accurate determinations of the DRV (Parsegov et al. 2018) with the added benefit of identifying flow stagnation zones. Such stagnation zones or “dead zones” are defined as regions of zero flow velocity (Weijermars et al. 2017a, b), which create undrained areas that can be targeted for refracturing (Weijermars and Alves 2018; Weijermars and van Harmelen 2018). Another added advantage of CAM models is their infinite resolution at the fracture scale due to the method being gridless and meshless, resulting also in faster computational times compared to numerical simulations.
The basic premise of the CAM solution is placing the produced fluid volume back into the reservoir to determine the areas drained and the pressure response corresponding to this fluid placement. From replacing production into the reservoir based on history matching using decline curve analysis, the corresponding pressure depletion is obtained by simply reversing the signs of the values on the pressure scale from positive to negative (Weijermars et al. 2017b). For the pressure depletion plots later in this study, the spatial pressure change \( \Delta P\left( {z,t} \right) \) is shown.
4.2 Flux allocation and production modeling
This study assumes a synthetic production well of 8000 ft horizontal length and 80 transverse fractures with 100ft spacing between them. This gives a total distance covered by the fractures of 7900 ft, leaving an untreated distance of 100 ft between the heel of the well and the first hydraulic fracture of the treatment plan. The flow simulation starts with a single fracture, using a base case model with a singleplanar fracture, expanded with branched iteration models of the fracture geometry. The fracture trees initiating from single perforations are then expanded to multiple fractal systems for fracture stages with variations in complexity to observe the impacts on the DRV, velocity field, and pressure field. By assuming symmetry about the wellbore, we initially look at only one half of the fracture (halflength x_{f}) to determine the effects on the flow velocities and pressure depletion for different fracture geometry models.
Current fracture propagation models that use simple planar fractures have the ability to predict proppant placement density which due to uneven placement can create zones of higher fracture conductivity (Parsegov et al. 2018). Though work has been done on proppant placement in complex fracture networks (Shrivastava and Sharma 2018) as we assume infinite fracture conductivity in our fractal network, uneven proppant placement is not considered in this model.
Production data from a typical Wolfcamp well used in a companion study (Parsegov et al. 2018) were used to produce a history matched type curve based on decline curve analysis. To match the production decline, the Duong decline method was used and found to give a total cumulative production over 30 years that is in line with forecasted EUR for wells in the Wolfberry play, Midland Basin under which the Wolfcamp Formation falls. Forecasts give an ultimate per well recovery estimated at 100,000–140,000 barrels of oil equivalent (Hamlin and Baumgardner 2012). The well used Duong decline parameters resulting in a cumulative production forecast of 102,069 bbls after a productive well life of 30 years.
Reservoir parameters used for modeling
Porosity n  Permeability k, μD  Water–oil ratio (WOR)  Formation volume factor B  Viscosity µ, cP  Residual oil saturation R_{o}  Fracture height H, ft 

0.05  1  4.592  1.05  1  0.20  75 
4.3 Drained rock volume (DRV)
Initially, we expected that a larger fractal dimension with more surface area would increase the injectivity of the matrix and require lower pressures to evacuate the reservoir fluid. Our models, however, show that once a constant total fluid production is used the overall pressure change remains the same regardless of the fracture network complexity. The models confirm the expectation that more complex fractal networks cause smaller lateral drained areas away from the fractures with greater local pressure variations. The reason for the localized pressure depletion peaks is that denser fracture networks with the same injectivity per fracture length will locally remove more fluid molecules from the matrix, thus resulting in larger pressure depletion locally.
The hydraulic fractal network is created and applied using an effective method of investigation by first modeling a small section of the horizontal wellbore. Because we use the method of fractals, a small sample of the well system should in fact be representative of the much larger drainage behavior of the well. This modeling strategy will also be beneficial in terms of computational and modeling time. Once the flow and pressure response have been determined based on individual fractal networks with increasing complexity, the investigation is extended to multiple fractal networks to investigate the possible effects of flow interference in fractured wells with numerous stages. Using this method both symmetrical and asymmetrical networks are modeled to determine changes in drained areas and flow response. The impact of fractal network complexity on reduction in flow stagnation zones is investigated to help determine the ideal fracture geometry to increase overall recoveries.
4.4 Model validation
5 Results
5.1 Fractal network creation
The Lindenmayer (Lsystem) rewriting system based on fractals is used to construct numerous branching fractal networks. This system defines a complex object by replacing parts of the initial object according to given rewriting rules. The Lsystem, combined with information on fractal network geometry, fractal distance (d), deviation angle (α), and iteration number (i), allows the defining of rules for creating the overall network. A systematic workflow to investigate the effect of fractal network complexity is laid out in the subsequent sections.
The network structure is defined by a simple string or axiom using variables ‘F’ and ‘G’. Using these variables, branching is represented by the use of square brackets with the ‘+’ and ‘−’ symbols denoting either clockwise or anticlockwise branching angles. The iteration number gives the replacement rules, changing the branching complexity and is referred to as different fractal generations. A simple fractal code written in MATLAB from the M2TUM group from the TU Munich was modified for our purpose of fractal network generation in 2D (available at http://m2matlabdb.ma.tum.de/author_list.jsp).

Symmetrical axiom rule = ‘F [+ G] [− G] F F [+ G] [− G]’.
5.2 Drainage by single symmetrical fractal networks
Comparison of various parameters for different symmetrical fracture geometry
Fracture model  Maximum velocity, ft/month  Maximum pressure change, psi*  Fracture surface area, ft^{2} 

Planar fracture  0.9477  1.3939 × 10^{6}  6.000 × 10^{4} 
Firstgeneration fractal  1.1088  1.4547 × 10^{6}  10.501 × 10^{4} 
Secondgeneration fractal  1.0087  1.4286 × 10^{6}  20.403 × 10^{4} 
Thirdgeneration fractal  1.0979  1.5035 × 10^{6}  37.040 × 10^{4} 
Drained areas are outlined by the timeofflight contours inferred from particle tracking, based on the production allocation due to the selected fracture strengths (Fig. 8d). Results for a planar fracture geometry show equal drainage around the entire fracture. As more complex fractal networks are simulated, the results show the total drained area stays constant (regardless of fracture complexity as a constant production is used). However, the DRV regions are not distributed equally around the fracture segments in the network, leading to some small undrained areas between the branches of the fractal network.
5.3 Drainage by single asymmetrical fractal networks

Asymmetrical axiom rule = ‘F [− G] F F [+ G] [− G]’.
5.4 Interference effects of multiple fractal networks
The base case models the flow response of three planar fractures and shows that with the given fracture halflength and fracture spacing, extremely low flow velocities occur between the central and outer fractures (Fig. 11a, left column). Flow stagnation zones are identified by velocity lows. These stagnation zones create areas in the reservoir that are left undrained due to the interference effect of the multiple fractures. The only way to drain these areas would be refracturing into the stagnation zones. The pressure depletion plot (Fig. 11b, left column) shows the largest pressure drop occurs between the fractures; however, this coincides with our lowest flow velocities and stagnation zones. This reinforces the idea put forward in Weijermars et al. 2017b that the pressure plots are poor proxies to recognize the reservoir areas drained by the fractures. The drained region after 30 years is visualized by the timeofflight contours to the fractures (Fig. 11c, left column) and shows the majority of the drained area is at the outer fractures where we also have the highest flow velocities. Flow interference between the fractures creates the stagnation zones that lead to undrained rock volumes.
The second scenario investigates the response to three symmetrical secondgeneration fractal networks (Fig. 11, center column). Slower velocities are again found between the branched fractal areas but for this case are confined to a smaller area. This in turn means that branched networks create smaller stagnation zones, than with the planar fractures and thus the fractal network should be conducive to drain more of the reservoir space effectively (Fig. 11c, center column). Better drainage coverage from the fractal network means less refractures are needed between the initial fractures. For branching fractal networks, too small a fracture spacing will result in draining the same reservoir areas due to overlapping fractal networks creating an inefficient drainage process.
A third scenario looks at a central symmetrical fractal network flanked by two asymmetrical fractal networks (Fig. 11, right column). Again, the areas of highest velocity occur at the periphery of the fractures with the slowest flow between the fractal networks. From the various simulations, there is a clear correlation between higher fractal network complexity and suppression in the areal extent of flow stagnation zones. Reduction in stagnation zones in turn means more efficient drainage of our rock and smaller undrained areas between fracture stages.
5.5 Multiple fulllength fractal networks
6 Discussion
The true nature of hydraulic fracture geometries created in the subsurface during fracture treatment programs is still not properly resolved. Most fracture propagation models result in fractures that generate as simple planar features due to ease of modeling and the lack of inclusion of mechanical heterogeneity in such models. Meanwhile, numerous experimental and field observations show that planar fractures are too simple an assumption and they are more likely to exist as branching fracture networks. What is beyond doubt is that differences in the fracture geometry will have a distinct impact on the outcome of production forecasting models and history matching the actual production rates, drained areas and estimated ultimate recovery. Previous analytical solutions have looked at flow into parallel planar fracture arrays (Zhou et al. 2012; Yu et al. 2017) but failed to consider the effect on flow when fracture geometries are nonplanar. Our method takes into account variable fracture geometries and visualizes the flow interference of fractal fracture networks. Highresolution visualizations of velocity and DRV areas are presented, which may substantially contribute to improve our current understanding of the flow process in hydraulically fractured reservoirs. The use of pressure depletion plots as proxies for drained rock volume is unreliable as has been highlighted in prior studies (Weijermars and van Harmelen 2018; Khanal and Weijermars 2019). In low permeability reservoirs, there occurs a distinct mismatch between the depth of pressure investigation and drained rock volume growth (Weijermars and Alves 2018), which is why the determination of the tracking of the timeofflight of drained fluid to the hydraulic fractures of a well is required to delineate the DRV more accurately.
6.1 Interference effects
The effect of fracture geometry on flow interference was investigated using a fractal fracture network description in combination with the complex analysis methods (CAM) to model drainage patterns and the resulting DRV near hydraulic fractures. Several series of simulations were conducted to determine the impact on drained areas and flow velocities when the fracture geometry varies, starting from a singleplanar fracture and evolving up to thirdgeneration branching fractals. For greater fractal network complexity, the local area drained away from each individual fracture segment becomes smaller as compared to the area of drained regions near a singleplanar fracture. The difference occurs because fractals have a larger fracture surface area and we are putting back a constant amount of produced fluid (via the principle of flow reversal) in both the single and fractal models. Consequently, the fractal network shows more variations in flow velocities and pressure depletion peaks as compared to a planar fracture. These extreme changes in velocity lead to uneven drainage by the fracture network with the possibility of small undrained areas due to stagnation points occurring between the branches.
A planar fracture geometry based on our model’s fracture spacing and halflength creates stagnation surfaces leading to relatively large undrained areas between the fractures. In contrast, the fractal network geometry shows a reduction in the effect and areal extent of the stagnation zones (as seen from a comparison of the velocity and drained area plots, Fig. 11), due to a decrease in the interference effect on flow. The position of flow separation surfaces separating the drainage regions of individual fractures is controlled by the ratio of the fracture length and fracture spacing (Weijermars et al. 2018). When the fracture spacing is greater than a quarter of the fracture length, the flow stagnation points occur midway between the individual fractures. For complex fractal networks, each fracture branch has a smaller length compared to a singleplanar fracture. The smaller fracture branch lengths mean less flow interference will occur for an otherwise constant fracture cluster spacing.
6.2 Pressure depletion
Results show (Fig. 8c) that when the fracture surface area increases due to the more complex fractal networks, the average reservoir pressure change remains the same. One might expect that a greater fracture surface area to place fluid back into the reservoir model would result in smaller overall pressure changes. However, pressure peaks and lows show a larger spread where the fracture network complexity increases. The local variation in the pressure response is affected mostly by the fracture density. From the pressure plots (Fig. 11b), one can observe that areas with the highest fracture density give pressure contour depletion peaks. The current model uses a prefracture matrix permeability of 1 µD giving pressure changes in the magnitude of 10^{6} psi (Fig. 14). When the permeability is changed to an afterfracture permeability of 1 mD, the pressure change magnitude drops to the range of 10^{3} psi, which is in line with field observations. We assume this afterfracture permeability change is due to the creation of a network of microfractures in the rock that is termed the enhanced afterfracture permeability region.
6.3 Model limitations
One aspect that the current model does not consider is the effect of various fractal iterations on fracture conductivity. Beyond the concept of fracture conductivity decreasing with time due to partial fracture closure following reservoir pressure decline (Daneshy 2005), as we create successive iterations, each new branch will be less conductive due to fracture width reduction and the lesser ability for proppant placement. In the current model, all fractures are given a constant flux, whereas in reality, the shorter distal fracture branches may have a smaller aperture and consequently less proppant placement, which may suppress fluid flux. The use of microproppant to help prop these smaller secondary and microfracture networks can retain fracture conductivity and is a field currently under research (Kim et al. 2018). The impact of fracture closure with time can be looked at in future work by the addition of a parameter to further decrease strength of flux into the fractal network. Water blockage to flow due to imbibed water during the fracturing job and subsequent soaking period is also not accounted for. Another crucial point is that the current model ensured there was no overlapping of fractal branches either within a stage or by multiple stages. This may not always be true in nature, and with very low current fracture spacing, there is a possibility of these fractal networks crossing. The possible crossing of the fractal networks from sequential fracture clusters can result in communication between stages that is regularly seen in the field (Barree and Miskimins 2015; Li et al. 2016).
6.4 Practical implications
The impact of fractal fracture geometries on the DRV and stagnation zones is investigated in this study. Our models indicate that when the complexity of hydraulic fracture networks increases, this will suppress the occurrence of dead zones. In order to increase the DRV and boost the associated well productivity (and thus improve ultimate recovery), our models suggest that fracture treatment programs must find ways to create more complex fracture networks. The generation of such complex fracture networks is currently not included in concurrent fracture treatment design models, which limit the fracture development to mutually parallel planes. Because observational evidence from field experiments suggests that hydraulic fractures in hydrocarbon wells range from planar to multibranched fractals (Huang and Kim 1993; Raterman et al. 2017), fracture treatment propagation models need to be modified to more realistically account for the development of complex fracture geometries that predictably follows from local geomechanical heterogeneities at the grain scale of rocks. The complex fracture geometry and fracture crossing provide a valid alternative explanation for the fact that tracer readings may overlap across fracture stages, which some commercial fracture propagation models presently attribute to the occurrence of longitudinal fractures parallel to the wellbore (Barree and Miskimins 2015).
7 Conclusions
 1.
A complex fracture network enhances the drained rock volume via two mechanisms. The first is that with more complex networks, the overall fracture surface area increases resulting in larger access to fluid stored in the reservoir matrix rock. The second mechanism is the suppression of stagnant flow zones when the complexity of the hydraulic fracture network increases.
 2.
Hydraulic fracture treatment programs should stimulate the creation of bifurcating fractures as approximated by our fractal model. By reducing stagnant flow regions, the DRV will more effectively drain the reservoir. This will lead to improved drainage between the fractures, which will increase the estimated ultimate recovery from hydrocarbon wells.
 3.
Using CAM, we are able to visualize in high resolution the effects of various fractal network geometries on flow and pressure response in the reservoir. We highlighted the fact that pressure plots, commonly used as proxies for drainage patterns, are poor proxies for the actual DRV. The DRV can be more accurate predicted using streamline tracking and timeofflight contouring, as shown in our study.
 4.
For planar fractures, stagnation zones in a threefracture cluster occur close to the outer fractures, typically when the fracture spacing is less than a quarter of the fracture length (Fig. 11, left panel).
 5.
Once fracture complexity is introduced in the form of fractal networks, the effect of the branching fractures leads to suppression of the flow stagnation areas, allowing for more efficient drainage (Fig. 11, center panel). The velocity plots for the fractal networks show a larger spread in the local variation of velocity than for the planar fractures.
 6.
The highest velocities are still found at the periphery of the fractal networks for all cases. However, for asymmetrical fractal networks, there is a tendency for the highest pressure and velocity response to skew toward the areas of highest fracture density (Fig. 11, right panel).
 7.
It will be necessary to determine whether the creation of complex fracture networks in the subsurface is solely dependent on the reservoir matrix properties (presence of natural fractures or matrix heterogeneities) or if fractal networks can be created by applying specific techniques during the hydraulic fracturing process. This requires the application of better diagnostic tools including the refinement of microseismic techniques to properly define and monitor created fractal network geometry.
 8.
Improved capacity to engineer and model the propagation direction and control the generation of fractal geometries for hydraulic fractures are urgently needed in order to further increase the productivity of hydrocarbon wells by fracture treatment.
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