Numerical Simulation of the Nonlinear Flow Properties in Self-Affine Aperture-Based Fractures

In order to study the effect of fracture geometry on the nonlinear flow properties in aperture-based fractures, a fractal model based on the self-affinity is proposed to characterize the three-dimensional geometry of rough-walled fractures. By solving the N–S (Navier–Stokes) equation directly, the relationships between the Forchheimer-flow characteristics, fractal dimension, and standard deviation of the aperture have been obtained. The Forchheimer equation is validated to describe the nonlinear relationship between flow rate and pressure gradient. For lower flow rate, the influence of the fractal dimension almost can be ignored, but the linear coefficient increases and the hydraulic aperture decreases with increasing standard deviation of the aperture, respectively. For larger flow rate, the nonlinear coefficient increases with the growth of the standard deviation of the aperture and fractal dimension. Thus, an empirical relationship between the nonlinear coefficient, fractal dimension, and standard deviation of aperture is proposed. In addition, the critical Reynolds number decreases with the increase of the standard deviation of the aperture and the fractal dimension, and the numerical results are generally consistent with the experimental data.

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The effect of contact areas on seepage behavior in 3D rough fractures under normal stress

10.21203/rs.3.rs-322946/v1 ◽

2021 ◽

Author(s):

Man Li ◽

Xianshan Liu ◽

Yu Li ◽

Zelin Hou ◽

Shihao Qiao

Keyword(s):

Fractal Dimension ◽

Flow Behavior ◽

Nonlinear Coefficient ◽

Nonlinear Flow ◽

Normal Displacement ◽

Normal Stresses ◽

Contact Ratio ◽

Underground Engineering ◽

Brazilian Splitting ◽

Geometric Morphology

Abstract The investigation of fluid flow in fractured rocks is a key issue in underground engineering. Reservoir sandstones as a case study, three specimens with different roughness using the Brazilian splitting were scanned to get the geometric morphology and aperture distribution, and the fractal dimension was introduced to characterize the fracture roughness. The flow experiments through the rough fractures subjected to different normal stresses were conducted to analyze the influence of the fractal dimension and the contact ratio on the nonlinear flow behavior, proving that the Forchheimer equation could better describe the flow nonlinearity. A modified Bandis model based on the experiments was proposed to calculate the max normal displacement of fracture under different normal stresses. Besides, a new model to forecast the nonlinear coefficient B was developed depending on the fractal dimension and the contact ratio, and a semi-empirical equation was employed to describe the critical Reynolds number. The influence of contact on the seepage path is simulated by COMSOL.

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Fluid Flow Behavior of Sheared Rough Fractures Subjected to Different Stress State

Crystals ◽

10.3390/cryst11091055 ◽

2021 ◽

Vol 11 (9) ◽

pp. 1055

Author(s):

Min Wang ◽

Qifeng Guo ◽

Pengfei Shan ◽

Yakun Tian ◽

Bing Dai

Keyword(s):

Fluid Flow ◽

Normal Stress ◽

Contact Area ◽

Reverse Flow ◽

Flow Behavior ◽

Nonlinear Coefficient ◽

Flow Characteristics ◽

Shear Displacement ◽

Navier Stokes Equation ◽

Linear Coefficient

The hydraulic properties of fractures are greatly affected by the stress. Knowing the fluid flow behavior of fractures is of great importance to underground engineering construction and environmental safety. The main purpose of this paper is to study the fluid flow characteristics of rough fractures under different stress states. First, rough fracture surfaces were generated by using the corrected successive random addition (SRA) algorithm. Then, the sheared fracture models subjected to different stress condition were obtained under the boundary condition of constant normal stiffness (CNS). Finally, the hydraulic characteristics of the three-dimensional rough rock fractures were analyzed by numerically solving the full Navier–Stokes equation. It has been found that (1) the aperture of fractures all obeys the Gaussian distribution. The dilatancy effect is gradually obvious and aperture becomes larger with the increase of shear displacement. (2) When the initial normal stress increases, the contact area of fracture becomes larger and the reverse flow can be observed around the contact area. (3) The relationship between hydraulic gradient and flowrate exhibits nonlinearity which can be described by the Forchheimer’s law. The linear coefficient a and the nonlinear coefficient b gradually decrease with the increase of shear displacement and finally stabilize. The values of a and b are reduced by 1–2 and 1–3 orders of magnitude respectively during the shear. The critical Reynolds number increases with the increase of shear displacement and decrease as the initial normal stress increases.

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Experimental Investigation of Grout Nonlinear Flow Behavior through Rough Fractures

Processes ◽

10.3390/pr7100736 ◽

2019 ◽

Vol 7 (10) ◽

pp. 736 ◽

Cited By ~ 1

Author(s):

Yuhao Jin ◽

Lijun Han ◽

Changyu Xu ◽

Qingbin Meng ◽

Zhenjun Liu ◽

...

Keyword(s):

Flow Behavior ◽

Nonlinear Coefficient ◽

Nonlinear Flow ◽

Flow State ◽

Inertia Effect ◽

Linear Coefficient ◽

Three Stages ◽

Stress Dependent ◽

Nonlinear Flow Behavior ◽

Forchheimer’S Law

This research experimentally analyzed the impacts of various water cement (W/C) ratios of ultrafine cement grout material and normal loads FN applied to fractures on grout nonlinear flow behavior through a rough plexiglass fractured sample. An effective self-made apparatus was designed and manufactured to conduct the stress-dependent grout flow tests on the plexiglass sample containing rough fractures. At each W/C ratio, the grout pressure P increased from 0 to 0.9 MPa, and the normal loads FN ranged from 666.3 to 1467.8 N. The results of the experiments indicate that (1) the Forchheimer’s law can be used to express the results of grout nonlinear flow through rough fractures. Moreover, both nonlinear coefficient a and linear coefficient b in Forchheimer’s law decreased with the increase of the W/C ratio, but increased with the increase of the FN value. (2) For normalized transmissivity, with the increase of Re, the decline of the T/T0–Re curves means that the grout flow behavior through the fracture mainly went through three stages: the viscosity effect, then the weak inertia effect, and finally the strong inertia effect. The three stages showed that with the increase of Re, the grout flow state changed from linear to nonlinear. Moreover, with the increase of the W/C ratio, the Forchheimer coefficient β decreased. (3) At a given FN, the critical grout hydraulic gradient Jc decreased, but the critical Reynolds number Rec increased as the W/C ratio increased; at a given W/C ratio, Jc increased, but Rec decreased as FN increased.

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Quantitative Estimates of Nonlinear Flow Characteristics of Deformable Rough-Walled Rock Fractures with Various Lithologies

Processes ◽

10.3390/pr6090149 ◽

2018 ◽

Vol 6 (9) ◽

pp. 149 ◽

Cited By ~ 4

Author(s):

Qian Yin ◽

Lixin He ◽

Hongwen Jing ◽

Dong Zhu

Keyword(s):

Nonlinear Coefficient ◽

Flow Regimes ◽

Confining Pressure ◽

Nonlinear Flow ◽

Hydraulic Pressure ◽

Rock Fractures ◽

Flow Process ◽

Wide Range ◽

Fracture Closure ◽

Hydraulic Aperture

The existence of surface roughness, various contact conditions and the occurrence of flow nonlinearity make the flow process in natural rock fractures more complicated. To evaluate the fluid flow regimes in deformable rough-walled rock fractures, a great many hydromechanical tests were conducted on nine real fractures artificially produced from a wide range of lithological diversity. For fractures with a certain JRC (fracture roughness coefficient) value, the confining pressure varied from 5 to 20 MPa, and the hydraulic pressure was increased from 0.4 to 6.0 MPa. The experimental results display that (i) regression analyses of the raw experimental data indicate that the Forchheimer’s law provides a perfect description for flow process through the fractures. The coefficients of viscous and inertial pressure drops undergo a growth of 2–3 orders of magnitude with an increase in the confining pressure; (ii) the hydraulic aperture decreases by approximately 87.41–92.81% as the confining pressure increases, and experiences a decrease of 1.52–2.96 times with the JRC values. A power-law function is used to evaluate the hydraulic aperture as a function of the nonlinear coefficient. The nonlinear coefficient decreases with increasing hydraulic aperture; (iii) using Forchheimer equation, the critical Reynolds number Rec was successfully assessed by choosing E percentage (generally 10%) of the nonlinear effect as the critical value between the linear and nonlinear flow regimes. The obtained Rec steadily increases with increasing confining pressure, while it diminishes with the JRC values; and (v) the transmissivity decreases as the pressure gradient increases. Additionally, transmissivity also exhibits a decreasing trend with both the confining pressures and JRC values due to fracture closure and tortuous and channeling flow paths in rougher fractures, and the rate of its decrease for a smaller confining pressure (5, 10 MPa) is more significant.

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Analysis of Fracture Roughness Control on Permeability Using SfM and Fluid Flow Simulations: Implications for Carbonate Reservoir Characterization

Geofluids ◽

10.1155/2019/4132386 ◽

2019 ◽

Vol 2019 ◽

pp. 1-19 ◽

Cited By ~ 9

Author(s):

Miller Zambrano ◽

Alan D. Pitts ◽

Ali Salama ◽

Tiziano Volatili ◽

Maurizio Giorgioni ◽

...

Keyword(s):

Fluid Flow ◽

Reservoir Characterization ◽

Flow Simulation ◽

Carbonate Reservoir ◽

Single Fracture ◽

Parallel Plates ◽

Surface Mapping ◽

Navier Stokes Equation ◽

Hydraulic Aperture ◽

Fracture Modeling

Fluid flow through a single fracture is traditionally described by the cubic law, which is derived from the Navier-Stokes equation for the flow of an incompressible fluid between two smooth-parallel plates. Thus, the permeability of a single fracture depends only on the so-called hydraulic aperture which differs from the mechanical aperture (separation between the two fracture wall surfaces). This difference is mainly related to the roughness of the fracture walls, which has been evaluated in previous works by including a friction factor in the permeability equation or directly deriving the hydraulic aperture. However, these methodologies may lack adequate precision to provide valid results. This work presents a complete protocol for fracture surface mapping, roughness evaluation, fracture modeling, fluid flow simulation, and permeability estimation of individual fracture (open or sheared joint/pressure solution seam). The methodology includes laboratory-based high-resolution structure from motion (SfM) photogrammetry of fracture surfaces, power spectral density (PSD) surface evaluation, synthetic fracture modeling, and fluid flow simulation using the Lattice-Boltzmann method. This work evaluates the respective controls on permeability exerted by the fracture displacement (perpendicular and parallel to the fracture walls), surface roughness, and surface pair mismatch. The results may contribute to defining a more accurate equation of hydraulic aperture and permeability of single fractures, which represents a pillar for the modeling and upscaling of the hydraulic properties of a geofluid reservoir.

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Optimization of the Ratio of ω-6 Linoleic and ω-3 α-Linolenic Fatty Acids of Hemp Seed Oil with Jackknife and Bootstrap Resampling

Chemical Product and Process Modeling ◽

10.1515/cppm-2019-0028 ◽

2019 ◽

Vol 15 (1) ◽

Cited By ~ 1

Author(s):

Vibha Devi ◽

Shabina Khanam

Keyword(s):

Fatty Acids ◽

Standard Deviation ◽

Flow Rate ◽

Seed Oil ◽

Cannabis Sativa ◽

Bootstrap Resampling ◽

Hemp Seed ◽

Solvent Flow Rate ◽

The Mean ◽

Experimental Runs

Abstract In the present work, supercritical fluid extraction (SFE) of hemp (Cannabis sativa) seed oil at various ranges of SFE parameters is performed. These parameters and respective ranges are temperature (40–80) °C, pressure (200–350) bar, solvent (CO2) flow rate (5–15) g/min, particle size (0.43–1.02) mm and amount of co-solvent (ethanol) (0–10) % of solvent flow rate. Central composite design (CCD) suggests 32 experimental runs to perform through SFE. The obtained oil is analysed through gas chromatography to identify its fatty acids concentrations. The ratio of ω-6 linoleic and ω-3 α-linolenic fatty acids (ω-6/ω-3) is optimized through CCD to obtain the desired amount of 3:1 as this ratio is highly preferred for various health benefits. Ratio of ω-6/ω-3 is obtained in the range from 2.11 to 3.06:1 for all experimental runs. The effect of SFE parameters on this ratio is investigated. Further, cross-validation is peformed on the experimental data obtained for the concentrations of both fatty acids by jackknife and bootstrap resampling to authenticate the obtained data. Small value of standard deviation (~1), less standard error of the mean (SEM) (<0.8) and less variance coefficient (<0.11) confirms the validity of the obtained data. All the estimators’ values such as standard deviation, variance coefficients and SEM are observed in 95 % of confidence intervals.

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Learning The Fluid/Flow Properties Using Microfluidics

EPJ Web of Conferences ◽

10.1051/epjconf/201921510002 ◽

2019 ◽

Vol 215 ◽

pp. 10002

Author(s):

Pooria Hadikhani ◽

Navid Borhani ◽

S. Mohammad H. Hashemi ◽

Demetri Psaltis

Keyword(s):

Neural Networks ◽

Fluid Flow ◽

Microfluidic Device ◽

Flow Rate ◽

Deep Neural Networks ◽

Good Accuracy ◽

Flow Properties ◽

Flow Rates

Deep neural networks (DNN) are employed to measure the flow rate and the concentration of the liquid using the images of the droplets in a microfluidic device. The trained networks are able to measure flow rates and concentrations with good accuracy.

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Performance of Air-Cooled Heat Exchanger with Laminar, Transitional, and Turbulent Tube Flow

MATEC Web of Conferences ◽

10.1051/matecconf/201824002012 ◽

2018 ◽

Vol 240 ◽

pp. 02012

Author(s):

Dawid Taler

Keyword(s):

Nusselt Number ◽

Heat Exchanger ◽

Heat Flow ◽

Friction Factor ◽

Flow Rate ◽

Empirical Relationship ◽

Heat Flow Rate ◽

Transition Flow ◽

Water Side ◽

Wide Range

Some air-cooled heat exchangers, especially in air conditioning and heating installations, heat pumps, as well as car radiators, work in a wide range of loads when the liquid flow in the tubes can be laminar, transitional or turbulent. In this paper, a semi-empirical and empirical relationship for the Nusselt number on the liquid-side in the transitional and turbulent range was derived. The friction factor in the transition flow range Rew,trb ≤ Rew ≤ Rew,tre was calculated by linear interpolation between the values of the friction factor for Rew,trb =2,100 and Rew,tre =3,000. Based on experimental data for a car radiator, empirical heat transfer relationships for the air and water-side were found by using the least squares method. The water temperature at the outlet of the heat exchanger was calculated using P-NTU (effectiveness-number of transfer units) method. The heat flow rate from water to air was calculated as a function of the water flow rate to compare it with the experimental results. The theoretical and empirical correlation for the water-side Nusselt number developed in the paper were used when determining the heat flow rate. The calculation results agree very well with the results of the measurements.

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Semianalytical Model for Flow Behavior Analysis of Unconventional Reservoirs with Complex Fracture Distribution

Geofluids ◽

10.1155/2020/9394140 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Dayou Shi ◽

Zhengyuan Lu

Keyword(s):

Fractal Dimension ◽

Flow Rate ◽

Flow Behavior ◽

Great Influence ◽

Fracture Aperture ◽

Secondary Fracture ◽

Stimulated Reservoir Volume ◽

Fracture Spacing ◽

Reservoir Volume ◽

Aperture Distribution

The secondary fracture spacing/aperture has a great influence on flow behavior of multifractured horizontal wells. In this paper, the effects of secondary fracture spacing/aperture distribution on porosity/permeability of the stimulated reservoir volume are described. A new fractal composite flow model was proposed, which can comprehensively characterize the secondary fracture spacing/aperture heterogeneity in the stimulated reservoir volume. Results showed that the larger the fractal dimension of the secondary fracture aperture is, the greater the dimensionless flow rate will be. The smaller the fractal dimension of the secondary fracture spacing is, the smaller the dimensionless flow rate will be. The secondary fracture spacing distribution had a greater influence on flow behavior than the secondary fracture aperture distribution. If the heterogeneity of the secondary fracture spacing or aperture is not considered, the bottomhole pressure and production curve will change greatly. At last, the proposed model was applied to match the production data of two actual fields.

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Nonlinear Flow Rate Response to Pumping Frequency and Reduced Hemolysis in the Drastically Under-Occluded Pulsatile Roller Pump

Artificial Organs ◽

10.1111/aor.12731 ◽

2016 ◽

Vol 41 (2) ◽

pp. 178-185 ◽

Cited By ~ 2

Author(s):

Choon Hwai Yap ◽

Chang Quan Lai ◽

Ivan Guang Hui Loh ◽

Thaddaeus Zhongren Ong

Keyword(s):

Flow Rate ◽

Nonlinear Flow ◽

Roller Pump

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