Fractal Fracture Network Research Articles

The Inversion Method of Shale Gas Effective Fracture Network Volume Based on Flow Back Data—A Case Study of Southern Sichuan Basin Shale

Fracture network fracturing is pivotal for achieving the economical and efficient development of shale gas, with the connectivity among fracture networks playing a crucial role in reservoir stimulation effectiveness. However, flow back data that reflect fracture network connectivity information are often ignored, resulting in an inaccurate prediction of the effective fracture network volume (EFNV). The accurate calculation of the EFNV has become a key and difficult issue in the field of shale fracturing. For this reason, the accurate shale gas effective fracture network volume inversion method needs to be improved. Based on the flow back characteristics of fracturing fluids, a tree-shaped fractal fracture flow back mathematical model for inversion of EFNV was established and combined with fractal theory. A genetic algorithm workflow suitable for EFNV inversion of shale gas was constructed based on the flow back data after fracturing, and the fracture wells in southern Sichuan were used as an example to carry out the EFNV inversion. The reliability of the inversion model was verified by testing production, cumulative gas production, and microseismic results. The field application showed that the inversion method proposed in this paper can obtain tree-shaped fractal fracture network structure parameters, fracture system original pressure, matrix gas breakthrough pressure, fracture compressibility coefficient, reverse imbibition index, equivalent main fracture half length, and effective initial fracture volume (EIFV). The calculated results of the model belong to the same order of magnitude as those of the HD model and Alkouh model, and the model has stronger applicability. This research has important theoretical guiding significance and field application value for improving the accuracy of the EFNV calculation.

A multi-physics coupled multi-scale transport model for CO2 sequestration and enhanced recovery in shale formation with fractal fracture networks

CO2 sequestration and enhanced gas recovery (CS-EGR) of shale formation involve complex multi-physics couplings, which are still not precisely captured by recent models. In this regard, a modified model fully considering the fractal characterization of fracture networks and multi-physics coupled transport behaviors in multi-scale shale formation is established. The fractal fracture networks are depicted based on the L-system theory with microseismic events (MSE), and embedded into the numerical modeling by an automatic fitting algorithm. As a development from the traditional transport model, the formulas among the pressure field, thermal field, and the molecular diffusion of binary gases are constructed based on the Chapman-Enskog theory. The effect of tortuosity and surface diffusion on gas transport efficiency can reach up to 5 %. The morphology parameters of induced fractures affect gas migration mainly by changing the size of SRV and the space between adjacent induced fractures, with their influence on CH4 production potentially as high as 20 %. Meanwhile, there are improved CH4 production, energy efficiency, and CO2 storage amounts, when the injection parameters are optimized. By optimizing the injection conditions (temperature, pressure), CH4 production can be increased by about 8 %.

Study on the Movement of Pulverized Coal Particles in Fractal Fracture Network.

Coalbed methane (CBM) exploitation is a complex multiphase flow process. With the development of CBM extraction technology, it is recognized that the migration of reservoir fluid in the fracture network is affected by the matrix fracture structure, fracture width, and pulverized coal particles plugging, therefore the fluid-particle coupling problem has received increasing attention. In this paper, six groups of fracture models with different fractal dimensions are established using the Weierstrass-Mandelbrot function, and the fluid-particle coupling phenomena in fractures with different roughness and aperture are simulated using an immersed boundary-lattice Boltzmann method (IB-LBM), by comparing the numerical simulation results with the theoretical analysis, it is proved that the IB-LBM numerical method has high accuracy and effectiveness. The effects of fractal dimension, particle size, and particle concentration on the plugging effect were investigated. The results show that the plugging effect on the fracture is enhanced with the increase of the pulverized coal particles content. When the concentration and size of pulverized coal particles are the same, the migration path of pulverized coal particles in complex fractures is more tortuous and more likely to be plugged. It may lead to a shift in the main seepage network of the fracture and the formation of new seepage channels. The research in this paper provides a data basis for the formation and plugging pattern of pulverized coal particles in coal seam fractures, which can provide a basis for the control of pulverized coal particles concentration in CBM drainage.

Production Decline Analysis of Tight Conglomerate Reservoirs with Small Well Spacing, Based on the Fractal Characteristics of Fracture Networks

The conglomerate matrix and fracture propagation are special in tight conglomerate reservoir with small well spacing. In this article, the fractal propagation characteristics of the fracture network in conglomerate reservoirs are described by experiment and a micro-mathematical model. According to the core slice, the conglomerate reservoir matrix presents the multi-modal pore structure, described as the “pseudo-dual-media” model. Given the above, the unsteady seepage mathematical model, comprehensively considering the fractal fracture network, stress sensitivity of main fractures, and threshold pressure gradient of the reservoir matrix, was developed and analytically solved. The Blasingame type curves for production decline analysis were plotted, and the sensitive parameters were analyzed. The field application was performed for validation. The research results show that the fractal dimension decides the complexity of the fracture network distribution. As it increases, the unsteady flow occurs earlier, and the boundary flow is delayed. The anomalous diffusion exponent represents the smoothness of crude oil migration and a higher value leads to higher resistance to oil migration and larger pressure drawdown for the same production rate. The growth of the threshold pressure gradient within a certain range can result in a localized downward shift of the type curves. The field application in a conglomerate oil reservoir showed that the presented model presents a fitting accuracy 10% higher than that of the conventional SRV model and has high reliability and precision for the production performance evaluation of the small-well-spacing development of tight conglomerate reservoirs.

A Fractional Step Method to Solve Productivity Model of Horizontal Wells Based on Heterogeneous Structure of Fracture Network

The existing productivity models of staged fractured horizontal wells in tight oil reservoir are mainly linear flow models based on the idealized dual-medium fracture network structure, which have a certain limitation when applied to the production prediction. Aiming at the difficulty in describing the shape of the complex fractal fracture network, a two-dimensional heterogeneous structure model of the fracture network is proposed in this paper. Considering the deformation characteristics of porous media and the characteristic of non-Darcy fluid flow, a three-zone steady-state productivity model with the combination of radial and linear flow is established. To eliminate strong nonlinear characteristics of the mathematical model, a fractional step method is employed to deduce the production formulas of staged fractured horizontal wells under infinite and finite conductivity fractures. The established productivity model is verified with the actual data of three horizontal wells in different blocks of S oilfield, and the error between the model calculation results and the actual production data is less than 4%. The analysis results of productivity sensitive factors show that production of horizontal wells is primarily influenced by the reservoir physical properties and fracturing parameters. The steady-state productivity model established in this study can be applied to effectively predict the average production of a horizontal well in stable stage of production, and it has theoretical and practical application value for improving the development effect of tight oil reservoir.

Evaluating Flow in Fractal-Fracture Networks: Effect of Variable Aperture

Abstract. While fractal models are often employed for describing the geometry of fracture networks, a constant aperture is mostly assigned to all the fractures when such models are flow simulated. In nature however, almost all fracture networks exhibit variable aperture values and it is this fracture aperture that controls the conductivity of individual fractures as described by the well-known cubic-law. It would therefore be of practical interest to investigate flow patterns in a fractal-fracture network where the apertures scale in accordance to their position in the hierarchy of the fractal. A set of synthetic fractal-fracture networks and two well-connected natural fracture maps that belong to the same fractal system are used for this purpose. A set of dominant sub-networks are generated from a given fractal-fracture map by systematically removing the smaller fracture segments with narrow apertures. The connectivity values of the fractal-fracture networks and their respective dominant sub-networks are then computed. Although a large number of fractures with smaller aperture are eliminated, no significant decrease is seen in the connectivity of the dominant sub-networks. A streamline simulator based on Darcy's law is used for flow simulating the fracture networks, which are conceptualized as two-dimensional fracture continuum models. A single high porosity value is assigned to all the fractures. The permeability assigned to fractures within the continuum model is based on their aperture values and there is nearly no matrix porosity and permeability. The recovery profiles and time-of-flight plots for each network and its dominant sub-networks at different time steps are compared. The results from both the synthetic networks and the natural data show that there is no significant decrease in fluid recovery in the dominant sub-networks compared to their respective parent fractal-fracture networks. It may therefore be concluded that in the case of such hierarchical fractal-fracture systems with scaled aperture, the smaller fractures do not significantly contribute to connectivity or fluid flow. In terms of decision making, this result will aid geoscientists and engineers in identifying only those fractures that ultimately matter in evaluating the flow recovery, thus building models that are computationally less expensive while being geologically realistic.

A smart productivity evaluation method for shale gas wells based on 3D fractal fracture network model

The generation method of three-dimensional fractal discrete fracture network (FDFN) based on multiplicative cascade process was developed. The complex multi-scale fracture system in shale after fracturing was characterized by coupling the artificial fracture model and the natural fracture model. Based on an assisted history matching (AHM) using multiple-proxy-based Markov chain Monte Carlo algorithm (MCMC), an embedded discrete fracture modeling (EDFM) incorporated with reservoir simulator was used to predict productivity of shale gas well. When using the natural fracture generation method, the distribution of natural fracture network can be controlled by fractal parameters, and the natural fracture network generated coupling with artificial fractures can characterize the complex system of different- scale fractures in shale after fracturing. The EDFM, with fewer grids and less computation time consumption, can characterize the attributes of natural fractures and artificial fractures flexibly, and simulate the details of mass transfer between matrix cells and fractures while reducing computation significantly. The combination of AMH and EDFM can lower the uncertainty of reservoir and fracture parameters, and realize effective inversion of key reservoir and fracture parameters and the productivity forecast of shale gas wells. Application demonstrates the results from the proposed productivity prediction model integrating FDFN, EDFM and AHM have high credibility.

Pressure-Transient Behavior of Double-Porosity Reservoirs with Transient Interporosity Transfer with Fractal Matrix Blocks

SummaryDouble-porosity/naturally fractured reservoir models have traditionally been used to represent the flow and pressure behavior for highly fractured carbonate reservoirs. Given that unconventional reservoirs such as shale-oil/gas reservoirs might not be considered to be multiporosity media, the use of the traditional/classical “double-porosity” models might not be adequate (or appropriate). The recent development of anomalous diffusion models has opened the possibility of adapting double-porosity models to estimate reservoir (and related) parameters for unconventional reservoirs. The primary objective of this work is to develop and demonstrate analytical reservoir models that provide (possible) physical explanations for the anomalous diffusion phenomenon.The models considering anomalous diffusion in reservoirs with Euclidean shape are developed using a convolved (i.e., time-dependent) version of Darcy's law. The use of these models can yield a power-law (straight-line) behavior for the pressure and/or rate performance, similar to the fractal reservoir models. The main advantage of using anomalous diffusion models compared with models considering fractal geometry is the reduction from two parameters (i.e., the fractal dimension and the conductivity index) to only one parameter (i.e., the anomalous diffusion exponent). However, the anomalous diffusion exponent does not provide information regarding the geometry or spatial distribution of the reservoir properties.To provide an alternative explanation for the anomalous diffusion phenomenon in petroleum reservoirs, we have developed double-porosity models considering matrix blocks with fractal geometry and fracture networks with either radial or fractal fracture networks. The flows inside the matrix blocks and the fractal fracture network assume that Darcy’s law is valid in its space-dependent (fractal) form, whereas the classical version of Darcy’s law is assumed for the radial-fracture-network case. The transient interporosity transfer is modeled using the classical convolution schemes given in the literature.We have defined the matrix blocks to be “infinite-acting” to represent the nano/micropermeability of shale reservoir. For the system defined by a fractal fracture network and infinite-acting fractal matrix blocks, we have investigated the influence of the fractal parameters (both matrix and fracture network) in the pressure- and rate-transient performance behaviors. We have defined the flow periods that can be observed in these sorts of systems and we have developed analytical solutions for pressure-transient analysis. We demonstrate that the use of the convolved version of Darcy’s law results in a model very similar to the diffusivity equation for double-porosity systems (which incorporates transient interporosity flow).In performing this work, we establish the following observations/conclusions derived from our new solutions:We find that the assumption of a well producing at variable rate (time-dependent inner-boundary condition) has a more-significant effect on the pressure (and derivative) functions and obscures the effects of the properties of the reservoir. We demonstrate that the anomalous-diffusion-phenomena model proposed for unconventional reservoirs can be directly related to the multiporosity concept model. Pressure and pressure-derivative responses can be used in the diagnosis of flow periods and in the evaluation/estimation of reservoir parameters in unconventional reservoirs.

Clustering, Connectivity and Flow Responses of Deterministic Fractal-Fracture Networks

Abstract. It is well known that fracture networks display self-similarity in many cases and the connectivity and flow behavior of such networks are influenced by their respective fractal dimensions. In the past, the concept of lacunarity, a parameter that quantifies spatial clustering, has been implemented by one of the authors in order to demonstrate that a set of seven nested natural fracture maps belonging to a single fractal system, but of different visual appearances, have different clustering attributes. Any scale-dependency in the clustering of fractures will also likely have significant implications for flow processes that depend on fracture connectivity. It is therefore important to address the question as to whether the fractal dimension alone serves as a reasonable proxy for the connectivity of a fractal-fracture network and hence, its flow response or, if it is the lacunarity, a measure of scale-dependent clustering, that may be used instead. The present study attempts to address this issue by exploring possible relationships between the fractal dimension, lacunarity and connectivity of fractal-fracture networks. It also endeavors to study the relationship between lacunarity and fluid flow in such fractal-fracture networks. A set of deterministic fractal-fracture models generated at different iterations and, that have the same theoretical fractal dimension are used for this purpose. The results indicate that such deterministic synthetic fractal-fracture networks with the same theoretical fractal dimension have differences in their connectivity and that the latter is fairly correlated with lacunarity. Additionally, the flow simulation results imply that lacunarity influences flow patterns in fracture networks. Therefore, it may be concluded that at least in synthetic fractal-fracture networks, rather than fractal dimension, it is the lacunarity or scale-dependent clustering attribute that controls the connectivity and hence the flow behavior.

Semi-analytical modeling of transient pressure behaviour for fractured horizontal wells in a tight formation with fractal-like discrete fracture network

In this work, theoretical models have been formulated, validated, and applied to evaluate transient pressure behaviour of a horizontal well with fractal-like multistage fractures in a tight formation. A coupled matrix-fracture flow model is proposed based on the superposition principle. More specifically, a slab source function was developed to describe the transient fluid flow behaviour from matrix to fracture, while a numerical solution was derived to describe the fluid flow behaviour in the complex fracture network. Then, a semi-analytical method is applied to solve such coupled models by discretizing hydraulic fracture into small segments. Furthermore, type curves are generated to examine the effect of the fractal fracture network. Sensitivity analysis of some major parameters (e.g., iteration number, deviation angle, and branch number) has been conducted to investigate their effects on the transient pressure responses. It is found that the fractal fracture network will greatly affect the transient pressure behaviour. The deviation of the pressure derivative curve from the case with the traditional bi-wing fracture can be used to quantify the complexity and the conductivity of the fracture network, while the existence of the complex fracture network will greatly improve the well performance in the intermediate time period.

Starting pressure head gradient and flow of Bingham plastics through a scaled fractal fracture network

In this study, a new approach is developed to determine the starting pressure head gradient and flowrate of a Bingham plastic through a fractal fracture network. The trace length of fracture is assumed to follow a power-law fractal distribution. The aperture of the fracture is related to the trace length by a scaling relationship. Flow in each fracture is treated as rectangular channel-type flow. A dimensionless flowrate ratio is defined by the average fracture flowrate of Bingham plastic over the flowrate of Newtonian fluid in the smallest fracture. The impact of the input parameters that define the fractal characteristics of the network, the pressure head gradient condition and the Bingham plastic properties is quantified and discussed. It is found that the flowrate ratio significantly decreases with the fractal dimension of trace length as the contrast of fracture lengths in the fractal network decreases with the fractal dimension. The flowrate significantly increases with the maximum over minimum trace length ratio in the network. The scaling exponent dramatically enhances the flowrate in the scaled fractal network. The dimensionless flowrate ratio increases with the pressure head gradient in a non-linear fashion. When the Bingham number is small enough, the dimensionless flowrate ratio does not depend on the pressure head gradient. After flows in more fractures have been activated due to the increase in the pressure head gradient, the Bingham plastic flowrate dramatically increases with the pressure head gradient compared to the Newtonian fluid counterpart.

Flow of power-law fluid through fractal discrete fracture network

In this study, a new approach to determine the flowrate of a power-law fluid through a fractal fracture network where fractures are parallel is developed. The aperture and trace length of fracture network can follow separate fractal distributions. The flowrate through the fractal network is determined through mean field approximation from the integration of flowrates in individual fractures over the aperture range of the fractal network. Three dimensionless parameters are defined to investigate the relative effects of a power-law fluid in an individual fracture and in a fractal fracture network. The analytical expression for the ratio of flowrate of power-law fluid over Newtonian fluid is developed. The effects of fractal dimension of aperture, power-law fluid index of fluid and pressure head gradient parameter on the flowrate are analyzed and discussed. The results demonstrate that for a shear-thinning fluid, increase in the fractal dimension of aperture reduces the flowrate ratio of power-law fluid over Newtonian fluid, while the reverse is true for a shear-thickening fluid. A large fractal dimension of aperture amplifies the effect of power-law fluid index of fluid when the pressure head gradient is low. The flowrate ratio of the power-law fluid over Newtonian fluid in the network increases quickly for the shear-thinning fluid and decreases for the shear-thickening fluid with the increase in the pressure head gradient parameter.

Impact of yield stress and fractal characteristics on the flow of Bingham fluid through fracture network

Flow process of a fluid with yield stress in fractured reservoirs is important in petroleum recovery. In this study, a new approach is proposed to study the combined effect of Bingham fluid properties and fractal characteristics on the flow of a Bingham fluid through a fractal fracture network. Both the aperture and trace length of fractures are distributed according to the power-law fractal distribution with distinct fractal dimensions. The relative effect on the flowrate of Bingham fluid in the network by the Bingham fluid properties and fractal characteristics compared to the Newtonian counterpart is examined. The results demonstrate that the flowrate of Bingham fluid through the network increases quickly with the pressure head gradient since large fractures are first activated which are responsible to quickly increase the Bingham fluid flowrate. The flowrate ratio of the Bingham fluid over Newtonian fluid decreases with the fractal dimension of aperture since the Bingham fluid flowrate decreases with increasing fractal dimension at a faster rate than the Newtonian fluid. A larger ratio of minimum over maximum apertures in the fractal network results in a larger flowrate ratio of the Bingham fluid over Newtonian fluid. The trace length fractal parameters do not affect the flowrate ratio of the Bingham fluid over the Newtonian fluid due to the fact that these parameters affect the actual flowrate for both types of fluid to the same extent.

Drained rock volume around hydraulic fractures in porous media: planar fractures versus fractal networks

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 bi-wing 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 single-fracture 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.

SEEPAGE PROPERTIES OF ROCK FRACTURES WITH POWER LAW LENGTH DISTRIBUTIONS

Fractures with power law length distributions abound in nature such as carbonate oil and gas reservoirs, sandstone, hot dry rocks, etc. The fluid transport properties and morphology characterization of fracture networks have fascinated numerous researchers to investigate for several decades. In this work, the analytical models for fracture density and permeability are extended from fractal fracture network to general fracture network with power law length distributions. It is found that the fracture density is related to the power law exponents [Formula: see text] and the area porosity [Formula: see text] of fracture network. Then, a permeability model for the fracture length distribution with general power law exponent [Formula: see text] and the power law exponent [Formula: see text] for fracture length versus aperture is proposed based on the well-known cubic law in individual fracture. The analytical expression for permeability of fractured networks is found to be a function of power law exponents [Formula: see text], area porosity [Formula: see text] of fracture network, and the micro-structural parameters (maximum fracture length [Formula: see text], fracture azimuth [Formula: see text] and fracture dip angle [Formula: see text]). The present model may shed light on the mechanism of seepage in fracture networks with power law length distributions.

Non-linear flow reduction factor and effective permeability of fractal fracture network

In this study, we investigate flow behaviors in a subsurface reservoir with fractal fracture network when both laminar and non-linear flows may occur in different fractures. The objective is to investigate how the non-linear flows in some fractures affect overall network permeability. A new non-linear flow reduction factor is developed to quantify the overall effective permeability by two separate components related to fractal fracture network features and flow characteristics respectively. The non-linear flow reduction factor takes into account flowrate reduction due to non-linear effect that is related to the hydraulic gradient. The other component is mainly related to the network characteristics representing the permeability of the fractal fracture network. The critical Reynolds number is explicitly used to distinguish non-linear flow from laminar flow in individual fractures. The effects of network and flow characteristics, including fractal dimension, scaling coefficient and exponent, maximum fracture trace length, and hydraulic gradient, on the non-linear flow ratio and effective permeability are examined. Both the laminar flow effective permeability and non-linear flow effective permeability decrease with the increase in hydraulic gradient. After scaling by the non-linear flow reduction factor, the scaled effective permeabilities vary much less significantly with the hydraulic gradient compared with the non-scaled ones, especially for the non-linear flow portion. Although it is proposed based on the idea of non-linear flows, the newly developed factor can also capture the laminar flow effective permeability reduction due to the reduction in the number of fractures where laminar flows occur.

Scaling of Fracture Patterns in Three Deep Boreholes and Implications for Constraining Fractal Discrete Fracture Network Models

We present a methodology for generating fractal fracture networks in one, two and three dimensions that respects the dual power-law model, in which the scaling characteristics are set by the two independent parameters: (1) the correlation dimension that pertains to separation of fracture centers, and (2) the length exponent that governs the distribution of fracture lengths. Synthetic fracture distributions were generated to evaluate the stereological relationships between the scaling parameters of 2D and 3D networks, and the scaling of fracture intersection points along a scanline through the network. The results showed that it is not possible to estimate the 2D and 3D fractal scaling parameters of the correlation dimension from the 1D correlation dimension of fracture spacing from scanlines through the network, even if the length exponent is known a priori. Synthetic 1D distributions of fracture spacing of known correlation dimension were used as a benchmark to test the consistency of estimates of fractal dimension derived from box-counting, two-point correlation, and power-law fitting. The results showed that the correlation dimension obtained from the two-point correlation method provided the most stable and reliable estimate of the fractal dimension of fractures on 1D scanlines or boreholes. Application of the two-point correlation function to the observed fracture distributions along three deep boreholes in crystalline rock at Basel, Switzerland and Soultz-sous-Forets, France showed that the distribution was fractal over more than two orders of magnitude in scale, and in all cases the fractal dimensions was in the range 0.86–0.88. Similar results were obtained for fracture sets of common orientation within the wells, although the fractal dimension ranged between 0.65 and 0.75. This constitutes strong evidence that fracturing in rock masses penetrated by the wells follows a fractal organization.

Impact of scaling break and fracture orientation on effective permeability of fractal fracture network

Fractures and fracture networks determine the permeability of many natural rocks. In this study, new solutions are developed to calculate the effective permeability of fractal fracture network with non-universal scaling of fracture aperture and trace length. One component of the permeability tensor is derived by assuming that the hydraulic gradient is perpendicular to the trace direction. A new solution is also derived to examine the reduction in the effective permeability of fracture network due to the influence of random fracture orientation. The fracture lengths in the network are assumed to follow the fractal scaling law. The orientation dispersion of factures in the fracture network varies according to the Fisher distribution. The effective permeability is derived by treating the fractal fracture network as a porous medium and using Darcy’s law. Results demonstrate that the effective permeability decreases with the scaling exponent before the break when the scaling break occurs early. However, the effective permeability increases with the scaling exponent before the break if the scaling break occurs late. The effective permeability increases with the scaling exponent after the break. The extent of increase varies with the location of the scaling break. If the scaling break occurs early, the increase in the effective permeability is more significant. The effective permeability of fractal fracture network increases with the location of scaling break. The reduction in the effective permeability due to fracture orientation dispersion varies from 2/3 to 0 depending on the degree of orientation dispersion as quantified by the precision parameter.

Effective aperture and orientation of fractal fracture network

Distribution of fracture length has been shown to exhibit fractal characteristics. In this study, we proposed a new idea of effective fracture in terms of aperture and orientation for a fractal fracture network and developed solutions of effective aperture in relation to fractal characteristics, flow behavior and hydraulic gradient. Another main point is that the use of Darcy’s law is not required. The effective fracture approach takes into account the scenario that the entire fractal fracture network may effectively behave similarly to a bundle of fractures where flows may be non-linear. The effective fracture approach explicitly distinguishes laminar and turbulent flows based on the size of effective fracture. To do so, an iterative procedure is developed to determine the effective fracture aperture. Results demonstrate that the effective aperture only slightly varies with hydraulic gradient, which illustrates that the idea of effective aperture can be used in the mixture of laminar and turbulent flows. The effective aperture decreases with the scaling exponent, increases linearly with the maximum fracture length, increases significantly with the minimum over maximum fracture length ratio, and increases with the scaling coefficient. The effective orientation shows expected decrease as the precision parameter of the fracture orientation increases.

Flow model of a multi-stage hydraulic fractured horizontal well based on tree-shaped fractal fracture networks

Multi-stage hydraulic fractured horizontal wells (MHFHWs) are used to develop low permeability reservoirs. A flow model of the MHFHW is proposed based on tree-shaped fractal fracture networks (TFFNs) considering the tree-shaped fractal network and the hydraulic theory of fracture. Then, a physical model of the MHFHW is established by several TFFNs, including the physical model, generating rules, and numbering rules of the TFFN. A coupling mathematical model is presented by combining the flow and seepage models for the TFFN of the MHFHW. The iterative method is used to solve the mathematical model, and the distributions of pressure, flow rate, and seepage rate in the MHFHW are obtained. The proposed model provides a new method to simulate the flow of the MHFHW considering the tree shape and the fractal characteristics of fractures in the MHFHW.