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1
Wang, Yuan, Yu-long Niu, and Qiang Feng. "Study on the REV Size of Fractured Rock in the Non-Darcy Flow Based on the Dual-Porosity Model." Geofluids 2018 (2018): 1–22. http://dx.doi.org/10.1155/2018/7535927.
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For the problem of whether the representative elementary volume (REV) obtained in the Darcy flow is also applicable to the case of the non-Darcy flow, the study on the REV size within the non-Darcy flow is proposed tentatively. The concept of the REV in the non-Darcy flow is based on the definition of the REV. According to the determination of the REV in the Darcy flow, the intrinsic permeability k and non-Darcy coefficient β are used simultaneously for the determination of the REV in the non-Darcy flow. The pore pressure cohesive element (PPCE) is developed with the subroutine in ABAQUS. Then the simulation method of the Darcy and non-Darcy flow in the fractured rock mass is built using the PPCE. The proposed plan is examined through the comparison with existing research results. It is validated that this technic is efficient and accurate in simulating the Darcy and non-Darcy flow in the fractured rock mass. Combined with fracture networks generated by Monte Carlo Simulation technique, the PPCE is applied to the study on the REV size. Both conditions of the Darcy and non-Darcy flow are simulated for comparison. The simulation results of this model show that the REV of the non-Darcy flow is inconsistent with the REV of the Darcy flow, and the REV of the non-Darcy flow is more significant than the REV of the Darcy flow. The intrinsic permeability k tensors obtained in the Darcy flow and the non-Darcy flow are basically the same.
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Di Nucci, Carmine, and Daniele Celli. "From Darcy Equation to Darcy Paradox." Fluids 7, no. 4 (March 22, 2022): 120. http://dx.doi.org/10.3390/fluids7040120.
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This theoretical paper focuses on the single-phase fluid flow through a granular porous medium. The emphasis is on the Darcy flow regime (without free boundary) of a linear viscous fluid in a saturated, deformable, homogeneous porous medium. The approach is developed at the Darcy scale (also referred to as macroscale or phenomenological scale). Within this framework, some discrete aspects of the flow model are highlighted, the governing equations are revisited, the thermodynamic state functions are reconsidered, and the Darcy paradox is presented. The Darcy paradox is illustrated for the isoshoric-isothermal flow of a viscous fluid in the liquid state, in a homogenous porous medium. After some remarks about the intrinsic assumption of this kind of flow, the governing equations are reduced to a well-known parabolic equation. According to this equation, infinitesimal pressure disturbances diffuse at an infinite speed. To remove this paradox, a mathematical model, based on the elementary scales method, is employed.
3
Yang, Bin, Tianhong Yang, Zenghe Xu, Honglei Liu, Wenhao Shi, and Xin Yang. "Numerical simulation of the free surface and water inflow of a slope, considering the nonlinear flow properties of gravel layers: a case study." Royal Society Open Science 5, no. 2 (February 2018): 172109. http://dx.doi.org/10.1098/rsos.172109.
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Groundwater is an important factor of slope stability, and 90% of slope failures are related to the influence of groundwater. In the past, free surface calculations and the prediction of water inflow were based on Darcy's law. However, Darcy's law for steady fluid flow is a special case of non-Darcy flow, and many types of non-Darcy flows occur in practical engineering applications. In this paper, based on the experimental results of laboratory water seepage tests, the seepage state of each soil layer in the open-pit slope of the Yanshan Iron Mine, China, were determined, and the seepage parameters were obtained. The seepage behaviour in the silt layer, fine sand layer, silty clay layer and gravelly clay layer followed the traditional Darcy law, while the gravel layers showed clear nonlinear characteristics. The permeability increases exponentially and the non-Darcy coefficient decreases exponentially with an increase in porosity, and the relation among the permeability, the porosity and the non-Darcy coefficient is investigated. A coupled mathematical model is established for two flow fields, on the basis of Darcy flow in the low-permeability layers and Forchheimer flow in the high-permeability layers. In addition, the effect of the seepage in the slope on the transition from Darcy flow to Forchheimer flow was considered. Then, a numerical simulation was conducted by using finite-element software (FELAC 2.2). The results indicate that the free surface calculated by the Darcy–Forchheimer model is in good agreement with the in situ measurements; however, there is an evident deviation of the simulation results from the measured data when the Darcy model is used. Through a parameter sensitivity analysis of the gravel layers, it can be found that the height of the overflow point and the water inflow calculated by the Darcy–Forchheimer model are consistently less than those of the Darcy model, and the discrepancy between these two models increases as the permeability increases. The necessity of adopting the Darcy–Forchheimer model was explained. The Darcy–Forchheimer model would be applicable in slope engineering applications with highly permeable rock.
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Lai, Bitao, Jennifer L. Miskimins, and Yu-Shu Wu. "Non-Darcy Porous-Media Flow According to the Barree and Conway Model: Laboratory and Numerical-Modeling Studies." SPE Journal 17, no. 01 (October 19, 2011): 70–79. http://dx.doi.org/10.2118/122611-pa.
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Summary This paper presents the results of our new experimental studies conducted for high flow rates through proppant packs, which show that the Barree and Conway (2004) flow model is capable of overcoming limitations of the Forchheimer non-Darcy equation at very high flow rates. To quantify the non-Darcy flow behavior using the Barree and Conway model, a numerical model is developed to simulate non-Darcy flow. In addition, an analytical solution is presented for steady-state linear non-Darcy flow and is used to verify the numerical-simulation results. The numerical model incorporates the Barree and Conway model into a general-purpose reservoir simulator for modeling multidimensional, single-phase non-Darcy flow in porous and fractured media and supplements the laboratory findings. The numerical model is then used to perform sensitivity analysis of the Barree and Conway flow model's parameters and to investigate transient behavior of non-Darcy flow at an injection well.
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Sefidgar, Mostafa, M. Soltani, Kaamran Raahemifar, and Hossein Bazmara. "Effect of Fluid Friction on Interstitial Fluid Flow Coupled with Blood Flow through Solid Tumor Microvascular Network." Computational and Mathematical Methods in Medicine 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/673426.
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A solid tumor is investigated as porous media for fluid flow simulation. Most of the studies use Darcy model for porous media. In Darcy model, the fluid friction is neglected and a few simplified assumptions are implemented. In this study, the effect of these assumptions is studied by considering Brinkman model. A multiscale mathematical method which calculates fluid flow to a solid tumor is used in this study to investigate how neglecting fluid friction affects the solid tumor simulation. The mathematical method involves processes such as blood flow through vessels and solute and fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. The sprouting angiogenesis model is used for generating capillary network and then fluid flow governing equations are implemented to calculate blood flow through the tumor-induced capillary network. Finally, the two models of porous media are used for modeling fluid flow in normal and tumor tissues in three different shapes of tumors. Simulations of interstitial fluid transport in a solid tumor demonstrate that the simplifications used in Darcy model affect the interstitial velocity and Brinkman model predicts a lower value for interstitial velocity than the values that Darcy model predicts.
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Fu, Xiang, Xiang Fang Li, Shi Qing Cheng, Liang Huang, and Xiang Rong Nie. "Pressure Behavior of a Coupling Model with Variable Permeability Effect." Applied Mechanics and Materials 152-154 (January 2012): 364–68. http://dx.doi.org/10.4028/www.scientific.net/amm.152-154.364.
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It is indicated that the fluid flow does not abide by the Darcy law in certain low permeability reservoirs. The results of pressure date analysis in these low permeability reservoirs would be inaccurate when generalized using Darcy flow models. To improve analysis results, based on the concept of variable permeability effect, the dual non-Darcy coupling flow model is established in this paper. The iterative algorithm is adopted to solve the differential set of equations. This work also describes the pressure behaviors by pressure response curves.
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Fu, Xiang, Xiang Fang Li, Shi Qing Cheng, Liang Huang, and Xiang Rong Nie. "Pressure Behavior of a Coupling Model with Variable Permeability Effect." Applied Mechanics and Materials 152-154 (January 2012): 689–93. http://dx.doi.org/10.4028/www.scientific.net/amm.152-154.689.
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It is indicated that the fluid flow does not abide by the Darcy law in certain low permeability reservoirs. The results of pressure date analysis in these low permeability reservoirs would be inaccurate when generalized using Darcy flow models. To improve analysis results, based on the concept of variable permeability effect, the dual non-Darcy coupling flow model is established in this paper. The iterative algorithm is adopted to solve the differential set of equations. This work also describes the pressure behaviors by pressure response curves.
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Aryanti, N., Y. Bindar, and I. G. Wenten. "Two Dimentional Numerical Models Of Hollow Fiber Membrane Contactor." REAKTOR 6, no. 2 (June 19, 2017): 77. http://dx.doi.org/10.14710/reaktor.6.2.77-84.
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Membrane contactor is separation processing unit using membrane as a contacting device. The major advantage of membrane contactor relies on its high contact area compared to conventional scrubber. One of the important applications of membrane contactor is to reduce emission of acid gases. In this work, modeling of membrane contactor is conductedto describe concentration distribution along fiber length used to predict effective fiber length by solving mass conservation equation. Solving of mass conservation equation required information of fluid flow distribution obtained by solving continuity and momentum equation simultaneously. The finite volume method is used to obtain the solution. Modeling of fluid flow was carried out by adding Darcy`s and Brinkman-Darcy flow models into Navier-Stokes equation. The momentum and continuity equation are solved for two-dimentional cylindrical coordinate. The result of velocity profile at axial direction were validated with Pangrle et.al. (1992) experimental data. The comparison shows that consideration using Brinkman-Darcy flow model give agood agreement with experimental data in which maximal axial velocity achieved is 0,047 m/s for this model and 0,05 m/s for experimental data.the concentration profile at radial direction using Darcy and Brickman-Darcy flow models have also been investigated. Furthermore, concentration profile at axial direction using the both two flow models indicate a decrease of concentration along fiber length. The comparison between models and experimental data by Subhakti and Azmier (1997) agree very closely to the Brinkman- Darcy flow model. The prediction of effective fiber length was conducted based on minimum economical flux oe\f membrane contactor. The calculation gives the effective fiber length obtained is 0.19 m at gas concentration, gas flow rate, and sorbent concentration of 0.02 mol/L, 0.8 m/s and 0.256 M respectively.Keywords : modeling, membrane contactor, Darcy, Brinkman-Darcy
9
Hdhiri, Najib, and Brahim Ben Beya. "Numerical study of laminar mixed convection flow in a lid-driven square cavity filled with porous media." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 4 (April 3, 2018): 857–77. http://dx.doi.org/10.1108/hff-04-2016-0146.
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Purpose The purpose of this study is to produce a numerical model capable of predicting the mixed convection flows in a rectangular cavity filled with a porous medium and to analyze the effects of several parameters on convective flow in porous media in a differentially heated enclosure. Design/methodology/approach The authors used the finite volume method. Findings The authors predicted and analyzed the effects of Richardson number, Darcy number, porosity values and Prandtl number in heat transfer and fluid flow. On other hand, the porosity and Richardson number values lead to reducing the heat transfer rate of mixed convection flow in a porous medium. Originality/value A comparison between Darcy–Brinkman–Forchheimer model and Darcy–Brinkman model is discussed and analyzed. The authors finally conclude that the Darcy–Brinkman model overestimates the heat transfer rate.
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Yan, Liang Dong, Zhi Juan Gao, and Feng Gang Dai. "Effective use Model of Low Permeability Oil Reservoir." Advanced Materials Research 753-755 (August 2013): 53–57. http://dx.doi.org/10.4028/www.scientific.net/amr.753-755.53.
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The throat of low permeability oil reservoir is narrow and small, the reservoir fluid flow resistance is big, and with the start-up pressure gradient, compare with medium and high permeability reservoir fluid flow, the characteristics are obviously different in performance for non-darcy flow at low speed. This kind of oil field reservoir started in the process of mining scope is small, the degree of use and the development effect is low. To solve these problems, this paper established considering start-up pressure gradient of the new unstable seepage flow mathematical model of non-darcy radial flow which the analytical solution and the productivity equation is deduced, established the effective radius of the use of low permeability reservoirs, and systemicly researched the calculation method of area well pattern of different types of non-darcy seepage.
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Zhang, Feng, and Daoyong Yang. "Determination of Fracture Conductivity in Tight Formations With Non-Darcy Flow Behavior." SPE Journal 19, no. 01 (August 28, 2013): 34–44. http://dx.doi.org/10.2118/162548-pa.
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Summary In this paper, a mathematical model has been developed and successfully applied to accurately determine the fracture conductivity in tight formations with non-Darcy flow behavior. A new non-Darcy flow number is first defined to account for the effect of characteristic length in a hydraulic fracture. A semianalytical method is then applied to solve the newly formulated mathematical model by discretizing the fracture into small segments, assuming that there exists unsteady flow between the adjacent segments. The newly developed model has been validated by simplifying it to the traditional Forchheimer (i.e., non-Darcy) model and by performing numerical simulation with a reservoir simulator as well. The pressure response and its corresponding derivative type curves have been reproduced to examine non-Darcy flow behavior under different fracture conductivities. Both relative minimum permeability and characteristic length are found to impose a negative effect on the fracture conductivity. Compared with relative minimum permeability, characteristic length is a strong function dominating the non-Darcy flow behavior in the fractures. It is obvious that the fracture conductivity can be accurately determined when non-Darcy flow behavior in the fracture network is taken into account.
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Zhang, Longjun, Daolun Li, Lei Wang, and Detang Lu. "Simulation of Gas Transport in Tight/Shale Gas Reservoirs by a Multicomponent Model Based on PEBI Grid." Journal of Chemistry 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/572434.
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The ultra-low permeability and nanosize pores of tight/shale gas reservoir would lead to non-Darcy flow including slip flow, transition flow, and free molecular flow, which cannot be described by traditional Darcy’s law. The organic content often adsorbs some gas content, while the adsorbed amount for different gas species is different. Based on these facts, we develop a new compositional model based on unstructured PEBI (perpendicular bisection) grid, which is able to characterize non-Darcy flow including slip flow, transition flow, and free molecular flow and the multicomponent adsorption in tight/shale gas reservoirs. With the proposed model, we study the effect of non-Darcy flow, length of the hydraulic fracture, and initial gas composition on gas production. The results show both non-Darcy flow and fracture length have significant influence on gas production. Ignoring non-Darcy flow would underestimate 67% cumulative gas production in lower permeable gas reservoirs. Gas production increases with fracture length. In lower permeable reservoirs, gas production increases almost linearly with the hydraulic fracture length. However, in higher permeable reservoirs, the increment of the former gradually decreases with the increase in the latter. The results also show that the presence of CO2in the formation would lower down gas production.
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Li, Juhua, and Chen Chen. "Numerical Simulation of the Non-Darcy Flow Based on Random Fractal Micronetwork Model for Low Permeability Sandstone Gas Reservoirs." Geofluids 2020 (October 9, 2020): 1–9. http://dx.doi.org/10.1155/2020/8884885.
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Darcy’s law is not suit for describing high velocity flow in the near wellbore region of gas reservoirs. The non-Darcy coefficient β of the Forchheimer’s equation is a main parameter for the evaluation of seepage capacity in gas reservoirs. The paper presented a new method to calculate β by performing gas and con-water flow simulations with random 3D micropore network model. Firstly, a network model is established by random fractal method. Secondly, based on the network simulation method of non-Darcy flow in the literature of Thauvin and Mohanty, a modified model is developed to describe gas non-Darcy flow with irreducible water in the porous medium. The model was verified by our experimental measurements. Then, we investigated the influence of different factors on the non-Darcy coefficient, including micropore structure (pore radius and fractal dimension), irreducible water saturation ( S wi ), tortuosity, and other reservoir characteristics. The simulation results showed that the value of the non-Darcy coefficient decreases with the increase in all: the average pore radius, fractal dimension, irreducible water saturation, and tortuosity. The non-Darcy coefficients obtained by the fractal method of microparameters are estimated more precisely than the conventional methods. The method provides theoretical support for the productivity prediction of non-Darcy flow in gas reservoirs.
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Xu, Hui, Nannan Liu, Yan Chen, Yapeng Tian, Zhenghuai Guo, Wanjun Jiang, and Yanfeng He. "A Novel Equivalent Numerical Simulation Method for Non-Darcy Seepage Flow in Low-Permeability Reservoirs." Energies 15, no. 22 (November 14, 2022): 8505. http://dx.doi.org/10.3390/en15228505.
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The low permeability and submicron throats in most shale or tight sandstone reservoirs have a significant impact on microscale flow. The flow characteristics can be described with difficultly by the conventional Darcy flow in low-permeability reservoirs. In particular, the thickness of the boundary layer is an important factor affecting the formation permeability, and the relative permeability curve obtained under conventional conditions cannot accurately express the seepage characteristics of porous media. In this work, the apparent permeability and relative permeability were calculated by using non-Darcy-flow mathematical modeling. The results revealed that the newly calculated oil–water relative permeability was slightly higher than that calculated by the Darcy seepage model. The results of the non-Darcy flow based on the conceptual model showed that the area swept by water in non-Darcy was smaller than that in Darcy seepage. The fingering phenomenon and the high bottom hole pressure in the non-Darcy seepage model resulted from the larger amount of injected water. There was a large pressure difference between the injection and production wells where the permeability changed greatly. A small pressure difference between wells resulted in lower variation of permeability. Consequently, the non-Darcy simulation results were consistent with actual production data.
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Jiang, Liwu, Tongjing Liu, and Daoyong Yang. "A Semianalytical Model for Predicting Transient Pressure Behavior of a Hydraulically Fractured Horizontal Well in a Naturally Fractured Reservoir With Non-Darcy Flow and Stress-Sensitive Permeability Effects." SPE Journal 24, no. 03 (March 20, 2019): 1322–41. http://dx.doi.org/10.2118/194501-pa.
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Summary Non-Darcy flow and the stress-sensitivity effect are two fundamental issues that have been widely investigated in transient pressure analysis for fractured wells. The main object of this work is to establish a semianalytical solution to quantify the combined effects of non-Darcy flow and stress sensitivity on the transient pressure behavior for a fractured horizontal well in a naturally fractured reservoir. More specifically, the Barree-Conway model is used to quantify the non-Darcy flow behavior in the hydraulic fractures (HFs), while the permeability modulus is incorporated into mathematical models to take into account the stress-sensitivity effect. In this way, the resulting nonlinearity of the mathematical models is weakened by using Pedrosa's transform formulation. Then a semianalytical method is applied to solve the coupled nonlinear mathematical models by discretizing each HF into small segments. Furthermore, the pressure response and its corresponding derivative type curve are generated to examine the combined effects of non-Darcy flow and stress sensitivity. In particular, stress sensitivity in HF and natural-fracture (NF) subsystems can be respectively analyzed, while the assumption of an equal stress-sensitivity coefficient in the two subsystems is no longer required. It is found that non-Darcy flow mainly affects the early stage bilinear and linear flow regime, leading to an increase in pressure drop and pressure derivative. The stress-sensitivity effect is found to play a significant role in those flow regimes beyond the compound-linear flow regime. The existence of non-Darcy flow makes the effect of stress sensitivity more remarkable, especially for the low-conductivity cases, while the stress sensitivity in fractures has a negligible influence on the early time period, which is dominated by non-Darcy flow behavior. Other parameters such as storage ratio and crossflow coefficient are also analyzed and discussed. It is found from field applications that the coupled model tends to obtain the most-reasonable matching results, and for that model there is an excellent agreement between the measured and simulated pressure response.
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Hagoort, Jacques. "An Improved Model for Estimating Flow Impairment by Perforation Damage." SPE Journal 12, no. 02 (June 1, 2007): 235–44. http://dx.doi.org/10.2118/98137-pa.
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Summary In this work, we present two simple formulas for the skin of a perforated well caused by perforation damage: one for the reduction in permeability, and one for the increase in non-Darcy flow coefficient (beta factor). They are based on the inflow performance of a single perforation obtained by means of a prolate-spheroidal flow model. This model rigorously accounts for the flow convergence toward a perforation, especially near the tip of the perforation. It provides a more realistic description of the inflow than a radial flow model, the basis for the existing skin formulas proposed by McLeod (1983). In the case of perforations with a large aspect ratio and a thin damaged zone, the formula for the skin due to permeability reduction reduces to McLeod's formula. The formula for the non-Darcy skin yields a significantly larger skin than predicted by the radial flow model, up to a factor 1.4 for large aspect ratios. Finally, we demonstrate that perforated wells are much more liable to non-Darcy flow than openhole wells, in particular if the perforations are severely damaged. Introduction Oil and gas wells are commonly completed with production casing cemented in place and perforated to enable fluids to enter the wellbore. The perforations are created by perforating guns and have the form of straight elongated and circular holes that stick into the formation perpendicular to the wall of the wellbore. The perforation holes are surrounded by a damaged zone of crushed and compacted rock. Typically, a perforation has a diameter of approximately a quarter-in., a length of a few up to more than a dozen inches and a crushed zone thickness of up to 1 in. It has been long recognized that perforation damage may drastically impair the flow efficiency of a perforated well. Not only is this caused by a lower permeability in the crushed zone, but also by a higher inertial resistance coefficient (non-Darcy flow coefficient), which is particularly important for prolific, high-rate gas wells. Customarily, the inflow performance of a perforated well is described by the radial openhole inflow formula, in which the effect of the perforations (e.g. geometry, shot density, phasing, and perforation damage) is included as a pseudo skin (Bell et al. 1995). The current model for estimating the Darcy and non-Darcy skins due to perforation damage was proposed by McLeod(1983). In this model the perforation is represented by an open circular cylinder surrounded by a concentric crushed zone of reduced permeability and enhanced non-Darcy flow coefficient, and the inflow into this cylinder is assumed to be radial, perpendicular to its axis.
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Song, Hong Qing, Ming Yue, Wei Yao Zhu, Dong Bo He, and Huai Jian Yi. "Formation Pressure Analysis of Water-Bearing Tight Gas Reservoirs with Unsteady Low-Velocity Non-Darcy Flow." Advanced Materials Research 201-203 (February 2011): 399–403. http://dx.doi.org/10.4028/www.scientific.net/amr.201-203.399.
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Porous media containing water is the prerequisite of existence of threshold pressure gradient (TPG) for gas flow. Based on theory of fluid mechanics in porous medium considering TPG, the non-Darcy flow mathematical model is established for formation pressure analysis of water-bearing tight gas reservoirs. It could provide semi-analytic solution of unsteady radial non-Darcy flow. According to the solution of unsteady radial non-Darcy flow, an easy and accurate calculation method for formation pressure analysis is presented. It can provide theoretical foundation for development design of water-bearing tight gas reservoirs. The analysis of calculation results demonstrates that the higher TPG is, the smaller formation pressure of water-bearing tight gas reservoirs spreads. In the same output, the reservoir sweep of non-Darcy gas flow is larger than that of non-Darcy liquid flow. And the pressure drop near wellbore is smaller than that of non-Darcy liquid flow, which is different from Darcy flow.
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Trilok, G., and N. Gnanasekaran. "Comparison of Numerical Models of Flow and Heat Transfer Through Porous Medium in a Vertical Channel." IOP Conference Series: Earth and Environmental Science 850, no. 1 (November 1, 2021): 012023. http://dx.doi.org/10.1088/1755-1315/850/1/012023.
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Abstract Porous medium modelling technique has opened up ways for number of numerical studies to investigate the performance of many devices that involve heat exchanging process. Such modelling technique not only avoids huge cost and time as compared to experimental analysis but also makes computationally less time-consuming as in case of numerical simulation by exact geometry modelling of porous materials. In this regard the present paper analyses two different thermal models namely local thermal equilibrium model and local thermal non equilibrium model along with two different flow models namely Darcy flow model and Darcy extended Forchheimer model. Suitability of the mentioned models in predicting heat transfer through metal foam and wire mesh porous medium is examined subjected to variations in structural aspects of the porous medium that could be primarily represented by variation in porosity and pore density. For this purpose, a vertical channel subjected to constant heat flux capable of housing porous medium reported in literature is numerically modelled and air flow is numerically simulated through the channel. A variety of structural configuration (combination of different porosity and pore density) of the mentioned porous media are considered and among the mentioned flow and thermal models, best suited models for predicting flow and heat transfer through such medium are identified with appropriate justifications. It is revealed from the present study that, Darcy-Forchheimer and LTNE models are best suited to predict flow and heat transfer through porous media than the basic Darcy and LTE models.
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Chen, Jie, Shuyu Sun, and Zhangxin Chen. "Coupling Two-Phase Fluid Flow with Two-Phase Darcy Flow in Anisotropic Porous Media." Advances in Mechanical Engineering 6 (January 1, 2014): 871021. http://dx.doi.org/10.1155/2014/871021.
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This paper reports a numerical study of coupling two-phase fluid flow in a free fluid region with two-phase Darcy flow in a homogeneous and anisotropic porous medium region. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the anisotropic porous medium region. A Robin-Robin domain decomposition method is used for the coupled Navier-Stokes and Darcy system with the generalized Beavers-Joseph-Saffman condition on the interface between the free flow and the porous media regions. Obtained results have shown the anisotropic properties effect on the velocity and pressure of the two-phase flow.
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Akanni, Olatokunbo O., Hisham A. Nasr-El-Din, and Deepak Gusain. "A Computational Navier-Stokes Fluid-Dynamics-Simulation Study of Wormhole Propagation in Carbonate-Matrix Acidizing and Analysis of Factors Influencing the Dissolution Process." SPE Journal 22, no. 06 (October 4, 2017): 2049–66. http://dx.doi.org/10.2118/187962-pa.
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Summary This study demonstrates the application of an alternative numerical-simulation approach to effectively describe the flow field in a two-scale carbonate-matrix-acidizing model. The modified model accurately captures the dissolution regimes that occur during carbonate-matrix acidizing. Sensitivity tests were performed on the model to compare the output with experimental observations and previous two-scale models in the literature. A nonlinear reaction-kinetics model for alternative acidizing fluids is also introduced. In this work, the fluid-field flow is described by the Navier-Stokes momentum approach instead of Darcy's law or the Darcy-Brinkman approach used in previous two-scale models. The present model is implemented by means of a commercial computational-fluid-dynamics (CFD) package to solve the momentum, mass-conservation, and species-transport equations in Darcy scale. The software is combined with functions and routines written in the C programming language to solve the porosity-evolution equation, update the pore-scale parameters at every timestep in the simulation, and couple the Darcy and pore scales. The output from the model simulations is consistent with experimental observations, and the results from the sensitivity tests performed are in agreement with previously developed two-scale models with the Darcy approach. The simulations at very-high injection rates with this model require less computational time than models developed with the Darcy approach. The results from this model show that the optimal injection rate obtained in laboratory coreflood experiments cannot be directly translated for field applications because of the effect of flow geometry and medium dimensions on the wormholing process. The influence of the reaction order on the optimal injection rate and pore volumes (PVs) of acid required to reach breakthrough is also demonstrated by simulations run to test the applicability of the model for acids with nonlinear kinetics in reaction with calcite. The new model is computationally less expensive than previous models with the Darcy-Brinkman approach, and simulations at very-high injection rates with this model require less computational time than Darcy-based models. Furthermore, the possibility of extending the two-scale model for acid/calcite reactions with more-complex chemistry is shown by means of the introduction of nonlinear kinetics in the reaction equation.
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Shao, Jianli, Qi Zhang, Wenbin Sun, Zaiyong Wang, and Xianxiang Zhu. "Numerical Simulation on Non-Darcy Flow in a Single Rock Fracture Domain Inverted by Digital Images." Geofluids 2020 (June 27, 2020): 1–13. http://dx.doi.org/10.1155/2020/8814327.
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The influence of rock seepage must be considered in geotechnical engineering, and understanding the fluid flow in rock fractures is of great concern in the seepage effect investigation. This study is aimed at developing a model for inversion of rock fracture domains based on digital images and further study of non-Darcy flow. The visualization model of single rock fracture domain is realized by digital images, which is further used in flow numerical simulation. We further discuss the influence of fracture domain geometry on non-Darcy flow. The results show that it is feasible to study non-Darcy flow in rock fracture domains by inversion based on digital images. In addition, as the joint roughness coefficient (JRC) increases or the fracture aperture decreases, distortion of the fluid flow path increases, and the pressure gradient loss caused by the inertial force increases. Both coefficients of the Forchheimer equation decrease with increasing fracture aperture and increase with increasing JRC. Meanwhile, the critical Reynolds number tends to decrease when JRC increases or the fracture aperture decreases, indicating that the fluid tends to non-Darcy flow. This work provides a reference for the study of non-Darcy flow through rock fractures.
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Deng, Jia, Wei Yao Zhu, Qian Ma, and Huai Jian Yi. "Study on the Unstable Seepage Flow Model and the Productivity Characteristics in Tight Gas Heterogeneity Reservoir." Advanced Materials Research 868 (December 2013): 465–72. http://dx.doi.org/10.4028/www.scientific.net/amr.868.465.
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In this paper, considering on the real threshold pressure gradient (TPG), the mathematical model for low velocity non-Darcy unstable gas flow was established for the tight gas reservoir. Based on the low velocity non-Darcy unstable gas flow model, the gas flow model in heterogeneity formation was developed, and the analytical solution was obtained on the condition of constant production for inner boundary. Then we calculated the model numerically by Matlab programming. Its analyzed that the pressure distribution and the influencing factors such as production rate, producing time, reservoir thickness and the threshold pressure gradient. We compared the heterogeneity formation with the homogeneous formation by the heterogeneity low velocity non-Darcy unstable gas flow model, which shows great difference and illustrate the significance of research on heterogeneity.
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WANG, FUYONG, ZHICHAO LIU, JIANCHAO CAI, and JIAN GAO. "A FRACTAL MODEL FOR LOW-VELOCITY NON-DARCY FLOW IN TIGHT OIL RESERVOIRS CONSIDERING BOUNDARY-LAYER EFFECT." Fractals 26, no. 05 (October 2018): 1850077. http://dx.doi.org/10.1142/s0218348x18500779.
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Flow in nanoscale pore-throats of tight oil reservoirs is strongly affected by boundary-layers, and exhibits low-velocity non-Darcy flow phenomena. The relationship between flow velocity and pressure gradient is highly nonlinear and difficult to be modeled mathematically. This paper proposed a low-velocity non-Darcy flow model which can account for boundary-layer effect in tight oil reservoirs. First, a modified Hagen–Poiseuille equation coupled with boundary-layer effect in a single capillary tube was derived. Then, assuming pores in tight formations following fractal distribution, an analytical expression of nonlinear correlation between flow velocity and pressure gradient in fractal porous media was developed. Finally, the proposed model was validated with experiment data, and parameters influencing low-velocity non-Darcy flow were quantitatively evaluated. The research results show that the decreasing boundary-layer thickness with the increase pressure gradient is the main reason of low-velocity non-Darcy flow in tight oil reservoirs. Our model can effectively describe the nonlinear relationship between flow velocity and pressure gradient. The relationship between threshold pressure gradient (TPG) and pseudo threshold pressure gradient (PTPG) can also be predicted using our model. Fluid viscosity has great impact on nonlinear flow behavior, and with fluid viscosity increasing TPG and PTPG increase significantly. TPG is the function of fluid type, fluid viscosity and maximum pore diameter, and decreases exponentially with the increasing maximum pore size.
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Lei, Xiao, Mu Wang Wu, Feng Bo Zhang, Shuang Qi Liu, Yao Quan Xiang, Wei Xing Lan, and Hai Yang Yu. "Variable-Permeability Well Test Model and Pressure Response of Non-Darcy Flow in Low-Permeability Reservoirs." Applied Mechanics and Materials 644-650 (September 2014): 5097–100. http://dx.doi.org/10.4028/www.scientific.net/amm.644-650.5097.
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Considering the variable-permeability effect of non-Darcy flow in low-permeability reservoirs, this work set up a two-dimensional model with boundaries. The implicit finite difference algorithm was employed to solve the well test model, and typical curves in log-log scale by considering the effects of variable-permeability and non-Darcy flow were obtained. The results show that, in reservoirs with impermeable boundary, the upturned part of the pressure derivative curve caused by boundary effect will be covered by that of non-Darcy effect, indicating boundary effect is postponed to appear. The more of the non-Darcy effects result in less obvious upturn of the typical curves. The upturn of typical curves and the effect of boundary are enhanced by increasing the numbers of impermeable boundaries or by decreasing the distance from the well to the boundary.
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Xue, Yi, Yang Liu, Faning Dang, Jia Liu, Zongyuan Ma, Lin Zhu, and Hongwei Yang. "Assessment of the Nonlinear Flow Characteristic of Water Inrush Based on the Brinkman and Forchheimer Seepage Model." Water 11, no. 4 (April 24, 2019): 855. http://dx.doi.org/10.3390/w11040855.
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Underground fault water inrush is a hydrogeological disaster that frequently occurs in underground mining and tunnel construction projects. Groundwater may pour from an aquifer when disasters occur, and aquifers are typically associated with fractured rock formations. Water inrush accidents are likely to occur when fractured rock masses are encountered during excavation. In this study, Comsol Multiphysics, cross-platform multiphysics field coupling software, was used to simulate the evolution characteristics of water flow in different flow fields of faults and aquifers when water inrush from underground faults occurs. First, the Darcy and Brinkman flow field nonlinear seepage models were used to model the seepage law of water flow in aquifers and faults. Second, the Forchheimer flow field was used to modify the seepage of fluid in fault-broken rocks in the Brinkman flow field. In general, this phenomenon does not meet the applicable conditions of Darcy’s formula. Therefore, the Darcy and Forchheimer flow models were coupled in this study. Simulation results show that flow behavior in an aquifer varies depending on fault permeability. An aquifer near a fault is likely to be affected by non-Darcy flow. That is, the non-Darcy effect zone will either increase or decrease as fault permeability increases or decreases. The fault rupture zone that connects the aquifer and upper roadway of the fault leads to fault water inrush due to the considerably improved permeability of the fractured rock mass.
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Cui, Changzhi, and Kyosuke Ono. "Theoretical and Experimental Investigation of an Externally Pressurized Porous Annular Thrust Gas Bearing and Its Optimal Design." Journal of Tribology 119, no. 3 (July 1, 1997): 486–92. http://dx.doi.org/10.1115/1.2833524.
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Static and dynamic characteristics of an externally pressurized porous annular thrust gas bearing (PATGB), which has a thin restricted surface layer, are investigated by numerical analysis and experiment. In the analysis, it is assumed that the fluid flow obeys Darcy’s law in the porous material, restricted with Darcy’s restrictor (Darcy-Darcy model) or orifice restrictor (Darcy-Orifice model) in the surface layer. From experimental investigation, it is found that the theoretical results calculated by the Darcy-Darcy model agree with the experimental data better than those of the Darcy-Orifice model. Based on the Darcy-Darcy model, the unique relationships among the design parameters, which can provide the maximum damping ratio, were derived as functions of feeding parameter under the conditions of allowable static stiffness and the local minimum dynamic stiffness. Considering the dimensionless mass of the body supported by the bearing, an optimal design method is proposed to maximize the damping ratio at the natural frequency, while maintaining the required stiffness in the low frequency region.
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Skrzypacz, Piotr, and Dongming Wei. "Solvability of the Brinkman-Forchheimer-Darcy Equation." Journal of Applied Mathematics 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/7305230.
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The nonlinear Brinkman-Forchheimer-Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy’s range.
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Cui, Shuheng, Jie Kong, Hongwei Yu, Cheng Chen, and Junlei Wang. "Revising Transient-Pressure Solution for Vertical Well Intersected by a Partially Penetrating Fracture with Non-Darcy Flow Effect." Geofluids 2020 (November 26, 2020): 1–17. http://dx.doi.org/10.1155/2020/8884750.
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The principle purpose of this work is to formulate an accurate mathematical model to evaluate the transient pressure behavior of a well intercepted by a partially penetrating vertical fracture (PPVF) with non-Darcy flow effect. Fracture conductivity is taken into account by coupling the three-dimensional flow in reservoir and the two-dimensional flow within fracture; the Barree-Conway model is incorporated into the model to analyze non-Darcy flow behavior in fracture, which leads to the nonlinearity of the governing equations. A high-effective iterative algorithm using a combined technique of fracture-panel discretization and dimension transform is developed to render the nonlinear equations amenable to analytical linear treatment. On the basis of the solutions, the pressure response and its derivative type curves were generated to identify the evolution of flow regimes with time. Furthermore, the influences of fracture conductivity, penetration ratio, and non-Darcy characteristic parameters on pressure response are investigated. The results show that PPVF exhibits five typical flow regimes, and analytical solutions for each flow regime are similar to that for a fully penetrating vertical fracture (FPVF) that can be correlated with the penetration ratio and apparent conductivity. The non-Darcy flow effect is found to have more significant effect on the low and moderate conductivity, especially in early-stage flow regimes. When the penetration ratio is smaller than 0.5, the pressure behavior exhibit a more remarkable variation with penetration ratio. This study provides a better insight into understanding the influence of non-Darcy flow on flow regime identification.
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Ansari, Shahab U., Masroor Hussain, Sarvat M. Ahmad, Ahmar Rashid, and Suleman Mazhar. "STABILIZED MIXED FINITE ELEMENT METHOD FOR TRANSIENT DARCY FLOW." Transactions of the Canadian Society for Mechanical Engineering 41, no. 1 (March 2017): 85–97. http://dx.doi.org/10.1139/tcsme-2017-1006.
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Darcy flow is a steady-state model for laminar flow of a fluid through a porous medium. The present work proposes an extended model of laminar Darcy flow by introducing dynamic pressure and velocity to the classical formulation. The solution of the proposed time-space model is attained by discretizing the problem with a stabilized mixed Galerkin method in space and a forward Euler method in time. The resulting matrix equation is well-posed and is solved using the conjugate gradient (CG) method. The error analysis of the numerical solutions confirms convergence to the actual model.
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Lage, J. L., and B. V. Antohe. "Darcy’s Experiments and the Deviation to Nonlinear Flow Regime." Journal of Fluids Engineering 122, no. 3 (April 26, 2000): 619–25. http://dx.doi.org/10.1115/1.1287722.
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Many important technological and natural processes involving flow through porous media are characterized by large filtration velocity. It is important to know when the transition from the linear flow regime to the quadratic flow regime actually occurs to obtain accurate models for these processes. By interpreting the quadratic extension of the original Darcy equation as a model of the macroscopic form drag, we suggest a physically consistent parameter to characterize the transition to quadratic flow regime in place of the Reynolds number, Re. We demonstrate that an additional data set obtained by Darcy, and so far ignored by the community, indeed supports the Darcy equation. Finally, we emphasize that the cubic extension proposed in the literature, proportional to Re3 and mathematically valid only for Re≪1, is irrelevant in practice. Hence, it should not be compared to the quadratic extension experimentally observed when Re⩾O1.[S0098-2202(00)01703-X]
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Wanjing, Luo, and Tang Changfu. "A Semianalytical Solution of a Vertical Fractured Well With Varying Conductivity Under Non-Darcy-Flow Condition." SPE Journal 20, no. 05 (October 20, 2015): 1028–40. http://dx.doi.org/10.2118/178423-pa.
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Summary Fracture distributions (simple or complex fractures), fracture-conductivity heterogeneity (uniform or varying conductivity along the fracture), and flow regimes inside the fracture (Darcy or non-Darcy flow) are the three main issues that have been widely investigated for transient-pressure analysis of vertical fracture systems. In this study, we focus on the latter two issues by proposing a semianalytical solution to discuss the transient-pressure behaviors of a varying-conductivity fracture under non-Darcy-flow condition. First, a general fracture-flow equation is established for the uniform-/varying-conductivity fracture under Darcy/non-Darcy flow. Second, for the case of a varying-conductivity fracture, a dimension transformation and an unequal-length-discretization model are proposed to obtain the pressure solution. Then, the transient-pressure response for the case of non-Darcy flow in the fracture can be also obtained by use of an iterative procedure in each timestep in the Laplace domain. It is shown that results from our solutions agree very well with those reported in the literature (Guppy et al. 1982; Poe et al. 1992). Third, the transient-pressure behaviors of the varying-conductivity fracture under Darcy- and non-Darcy-flow condition are discussed in detail. Results show that non-Darcy flow in the fracture mainly reduces the effective conductivity and the transient-pressure curve follows the curve of an equivalently constant conductivity except for the case of extremely small conductivities. The pressure behaviors of varying-conductivity fractures depend on the value of average conductivity, the distribution of conductivity along the fracture, and the maximum-minimum-conductivity ratio. The presence of the varying conductivity not only affects the effective conductivity in the early and late times, but also changes the shape of the pressure curve, especially for the high-conductivity fracture in the early time. It is very difficult to accurately estimate the fracture parameters by well test for most of the cases of varying conductivities under non-Darcy flow in the fracture.
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Mustafa, Irfan, Abuzar Ghaffari, Tariq Javed, and Javeria Nawaz Abbasi. "Numerical Examination of Thermophysical Properties of Cobalt Ferroparticles over a Wavy Surface Saturated in Non-Darcian Porous Medium." Journal of Non-Equilibrium Thermodynamics 45, no. 2 (April 26, 2020): 109–20. http://dx.doi.org/10.1515/jnet-2019-0019.
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AbstractIn this study, the effect of magnetic field on an incompressible ferrofluid flow along a vertical wavy surface saturated in a porous medium is investigated. Ferrofluid is made by incorporating magnetic particles, in this case cobalt, at the nanoscale level into a base fluid. For the study of porous medium two well-known models, namely, Darcy and non-Darcy, are used. The mathematical model in terms of governing partial differential equations which are based on conservation laws in mechanics according to the assumption is developed, and this model is converted into a dimensionless form by suitable transformations. Due to the complex non-linear partial differential equations, the numerical solution is calculated by using an implicit finite difference scheme. The impact of involved parameters, namely, magnetic parameter, nanoparticle volume fraction parameter, the amplitude of the wavy surface, and the Grashof number, on Nusselt and average Nusselt numbers are studied through graphs and tables.The results show that for large values of the magnetic parameter, both the Nusselt number and the average Nusselt number decrease in ferrofluid flow. The value of the Nusselt number in the Darcy model is higher than the value of the Nusselt number in the non-Darcy model.
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Juncu, Gheorghe. "Brinkman – Forchheimer – Darcy Flow Past an Impermeable Sphere Embedded in a Porous Medium." Analele Universitatii "Ovidius" Constanta - Seria Matematica 23, no. 3 (November 1, 2015): 97–112. http://dx.doi.org/10.1515/auom-2015-0050.
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Abstract The flow past an impermeable sphere embedded in a fluid saturated porous medium was studied numerically considering valid the Brinkman-Forchheimer-Darcy (or Brinkman-Hazen-Dupuit-Darcy) model. The flow is viscous, laminar, axisymmetric, steady and incompressible. The porous medium is isotropic, rigid and homogeneous. The stream function - vorticity equations were solved numerically in spherical coordinates system by a multigrid method. The influence of the Darcy number and Forchheimer term on the velocities field was investigated for two boundary conditions on the surface of the sphere: slip and no - slip.
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Knabner, Peter, and Jean E. Roberts. "Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture." ESAIM: Mathematical Modelling and Numerical Analysis 48, no. 5 (August 13, 2014): 1451–72. http://dx.doi.org/10.1051/m2an/2014003.
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XIAO, BOQI, YONGHUI LIU, HANXIN CHEN, XUBING CHEN, and GONGBO LONG. "A NOVEL FRACTAL SOLUTION FOR LAMINAR FLOW RESISTANCE IN ROUGHENED CYLINDRICAL MICROCHANNELS." Fractals 28, no. 06 (September 2020): 2050097. http://dx.doi.org/10.1142/s0218348x20500978.
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In this work, a novel fractal model for the laminar flow in roughened cylindrical microchannels is proposed. The average height of rough elements is derived using the fractal theory. The effects of relative roughness on the friction factor and the Poiseuille number are discussed. It is found that the Darcy friction factor and the Poiseuille number increase with the increase in the relative roughness in the cylindrical microchannel. Besides, it is observed that the Darcy friction factor decreases with the increase in the Reynolds number. Each parameter of the proposed model has a clear physical meaning. The present model can properly reveal some mechanisms that affect the laminar flow in roughened cylindrical microchannels. The present model improves the understanding of the physical mechanisms of fluid flows through roughened cylindrical microchannels. Our model predictions are compared with the existing experimental data, and good agreement can be found.
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Sakamoto, H., and F. A. Kulacki. "Buoyancy Driven Flow in Saturated Porous Media." Journal of Heat Transfer 129, no. 6 (September 24, 2006): 727–34. http://dx.doi.org/10.1115/1.2717937.
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Measurements are reported of heat transfer coefficients in steady natural convection on a vertical constant flux plate embedded in a saturated porous medium. Results show that heat transfer coefficients can be adequately determined via a Darcy-based model, and our results confirm a correlation proposed by Bejan [Int. J. Heat Mass Transfer. 26(9), 1339–1346 (1983)]. It is speculated that the reason that the Darcy model works well in the present case is that the porous medium has a lower effective Prandtl number near the wall than in the bulk medium. The factors that contribute to this effect include the thinning of the boundary layer near the wall and an increase of effective thermal conductivity.
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Ettefagh, J., K. Vafai, and S. J. Kim. "Non-Darcian Effects in Open-Ended Cavities Filled With a Porous Medium." Journal of Heat Transfer 113, no. 3 (August 1, 1991): 747–56. http://dx.doi.org/10.1115/1.2910627.
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The importance and relevance of non-Darcian effects associated with the buoyancy driven convection in open-ended cavities filled with fluid-saturated porous medium is analyzed in this work. Several different flow models for porous media, such as Brinkman-extended Darcy, Forchheimer-extended Darcy, and generalized flow models, are considered. The significance of inertia and boundary effects, and their crucial influence on the prediction of buouancy-induced flow and heat transfer in open-ended cavities, are investigated. Analysis is made on the proper choice of parameters that can fully determine the criteria for the range of validity of Darcy’s law in this type of configuration. Critical values of the inertial parameter, Λcrit, below which, for any given modified Rayleigh number, the Darcy flow model breaks down, have been investigated. It is shown that the critical value of the inertial parameter depends on the modified Rayleigh number and that this critical value increases as Ra* increases. It is also observed that for higher modified Rayleigh number, the deviation from a Darcian formulation appears at Darcy numbers greater than 1×10−4. The Prandtl number effects on convective flow and heat transfer are shown to be quite significant for small values of Pr. The Prandtl number effects are reduced significantly for higher values of the Prandtl number.
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Liu, Piyang, Xiaoxia Ren, Liang Kong, and Jun Yao. "Three-dimensional simulation of acidizing process in carbonate rocks using the Darcy–Forchheimer framework." Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles 75 (2020): 48. http://dx.doi.org/10.2516/ogst/2020035.
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Acidizing is an economical and effective practice to remove the near wellbore damage, which is performed by injecting acid into the formation through the wellbore. The injected acid dissolves the rock, by which the permeability nearby the wellbore can be improved. For a carbonate reservoir, the injected acid dissolves some of the minerals and some narrow and long channels, named wormholes, are formed then. These wormholes can bypass the damaged zone and hence improve the productivity of the well. The process for acid dissolving rocks involves complex physicochemical change, including the chemical reactions at the pore scale and the fluid flow at Darcy scale. In this paper, a 3-D reactive flow model with non-Darcy framework is developed based on the two-scale continuum model, and is solved by using the finite volume method. Five types of dissolution patterns, named face dissolution, conical wormhole, wormhole, ramified wormhole, and uniform dissolution, are obtained as the injection velocity increases. The effect of non-Darcy flow on dissolution pattern and breakthrough volume is analyzed. It is found that there is no effect of non-Darcy on dissolution structure and breakthrough volume when the injection velocity is very low. However, when the injection velocity is very high, the generated wormhole has more branches when using the Forchheimer equation than using the Darcy equation. Moreover, the optimal injection velocity is found to be the same whether considering the non-Darcy flow or not.
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Wu, Zhongwei, Chuanzhi Cui, Japan Trivedi, Ning Ai, and Wenhao Tang. "Pressure Analysis for Volume Fracturing Vertical Well considering Low-Velocity Non-Darcy Flow and Stress Sensitivity." Geofluids 2019 (November 20, 2019): 1–10. http://dx.doi.org/10.1155/2019/2046061.
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In general, there is stress sensitivity damage in tight reservoirs and fractures. Furthermore, the flow in tight reservoirs is the low-velocity non-Darcy flow. Currently, few researches of pressure analysis for volume fracturing vertical well are conducted simultaneously considering the low-velocity non-Darcy flow and stress sensitivity. In the paper, a novel flow model of a volume fractured vertical well is proposed and solved numerically. Firstly, the threshold pressure gradient, permeability modulus, and experimental data are, respectively, utilized to characterize the low-velocity non-Darcy flow, matrix stress sensitivity, and fracture stress sensitivity. Then, a two-region composite reservoir is established to simulate the vertical well with volume fracturing. After that, the logarithm meshing method is used to discrete the composite reservoir, and the flow model is solved by the method of finite difference and IMPES. Finally, the model verification is conducted, and the effects of the low-velocity non-Darcy flow and stress sensitivity on the pressure and pressure derivative are analyzed. The six flow regimes are identified by the dimensionless pressure and pressure derivative curve. They are, respectively, the fracture linear flow regime, early transition flow regime, radial flow regime, crossflow regime, advanced transition flow regime, and boundary controlling flow regime. The stress sensitivity and threshold pressure gradient have a great effect on the dimensionless pressure and pressure derivative. With the increase of reservoir stress sensitivity, the pressure and pressure derivative are upward at the advanced transition flow and boundary controlling regimes. However, the pressure and pressure derivative are downward at the advanced transition flow and boundary controlling regimes when the fracture sensitivity increases. An increase in the threshold pressure gradient results in a high dimensionless pressure and pressure derivative. This work reveals the effects of low-velocity non-Darcy flow and stress sensitivity on pressure and provides a more accurate reference for reservoir engineers in pressure analysis when developing a tight reservoir by using the volume fracturing vertical well.
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Tang, Yula, Turhan Yildiz, Erdal Ozkan, and Mohan G. Kelkar. "Effects of Formation Damage and High- Velocity Flow on the Productivity of Perforated Horizontal Wells." SPE Reservoir Evaluation & Engineering 8, no. 04 (August 1, 2005): 315–24. http://dx.doi.org/10.2118/77534-pa.
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Summary A comprehensive semianalytical model has been built to investigate the effects of drilling and perforating damage and high-velocity flow on the performance of perforated horizontal wells. The model incorporates the additional pressure drop caused by formation damage and high-velocity flow into a semianalytical coupled wellbore/reservoir model. The reservoir model considers the details of flow in the vicinity of the wellbore, including 3Dconvergent flow into individual perforations, flow through the damaged zone around the wellbore and the crushed zone around the perforation tunnels, and non-Darcy flow in the near-wellbore region. The wellbore flow model includes the effect of frictional pressure drop. Both oil and gas wells are considered. The expressions provided in this paper for additional pressure losses caused by perforating damage, drilling damage, and high-velocity flow can be used to optimize perforating parameters and decompose the total skin into its components (perforation pseudoskin, damage skin, and non-Darcy skin). Introduction The performance of oil and gas wells may be influenced by the simultaneous effect of mechanical skin, high-velocity (non-Darcy) skin, and completion pseudoskin factors. The skin factors caused by formation damage and perforating damage constitute the mechanical-skin factor. The extra pressure drop caused by high-velocity flow is known as the rate-dependent or non-Darcy flow factor. Compared to an ideal open hole, the wells with completions and other geometries such as perforations, slotted liner, or partial penetration may experience additional pressure loss or gain. The additional pressure change caused by wellcompletion and geometry is quantified in terms of pseudoskin factor. The combined effects of all the skin factors lead to a total skin factor that maybe estimated from pressure-transient data. The total skin factor, however, is not simply the sum of the individual skin components, and the computation of the individual skin components is not straightforward (the interaction between the individual components of total skin is nonlinear). Many studies have concentrated on the effects of formation damage and high-velocity (non-Darcy) flow on well performance. For perforated vertical wells, McLeod's analytical model has been a widely accepted approximation to account for the additional pressure drop caused by formation damage and high-velocity flow. Karakas and Tariq presented a semianalytical model to predict the pseudoskin and productivity of perforated vertical wells with formation damage. The models suggested by McLeod and Tariq, however, may not work for selectively completed wells in which the flux distribution may be nonuniform. An example of this case is selectively perforated horizontal wells. Tang et al. presented models for horizontal wells completed with slottedliners or perforations. The additional pressure drop in the vicinity of the wellbore because of formation damage, perforating, flow convergence, and high-velocity flow was included in their models in the form of a total-skinterm. The existing horizontal-well models are not capable of explicitly relating the skin factor to the physical parameters controlling the additional pressure drop around the wellbore. In addition, the interplay between the skin and flux distribution and its impact on the productivity of perforated horizontal wells have not been discussed, especially for selectively perforated horizontal wells. Non-Darcy flow effect in perforated horizontal wells is another topic that has not been addressed adequately in the literature. In this study, we present a semianalytical model to predict the productivity of perforated horizontal wells under the influence of formation damage, perforating damage, and high-velocity flow. The nonlinear interaction between the individual skin components is accurately represented in the model. The model is applicable to both single-phase oil and gas wells (the pseudo pressure concept is used to extend the oil-flow model to the gas wells). Using the model, the combined effects of formation damage, the crushed zone around the perforation tunnels, and the high-velocity flow on the horizontal-well performance have been investigated in detail. The completion and damage parameters controlling the well productivity were identified through sensitivity studies.
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Guo, Zhanwei, and Jincheng Shi. "Structural stability for the Darcy model in double diffusive convection flow with Magnetic field effect." AIMS Mathematics 7, no. 9 (2022): 16366–86. http://dx.doi.org/10.3934/math.2022894.
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<abstract><p>In this paper, we consider the Darcy model with magnetic field affect which is used to describe the double diffusive flow of a fluid containing a solute. Using the energy estimate methods, we derive the prior bounds of the solutions. By using these a prior bounds, the continuous dependence of the solutions to Darcy model on the magnetic coefficient and the boundary parameter is established.</p></abstract>
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Huang, Ting, Xiao Guo, and Kun Wang. "Nonlinear Seepage Model of Gas Transport in Multiscale Shale Gas Reservoirs and Productivity Analysis of Fractured Well." Journal of Chemistry 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/349507.
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Shale is abundant in nanoscale pores, so gas flow in shales cannot be simply represented by Darcy formula anymore. It is crucial to figure out the influence of gas flow in nano/micro pores on actual productivity, which can provide basic theories for optimizing parameters and improving the gas production from engineering perspective. This paper considers the effects of slippage and diffusion in nanoscale based on Beskok-Karniadakis (BK) equation, which can be applicable for different flow regimes including continuum flow, slip flow, transition flow, and free-molecule flow. A new non-Darcy equation was developed based on the analysis of effects of high order terms of BK equation on permeability correction factor. By using the conformal transformation principle and pressure coupling method, we established the productivity formula of fractured well (infinite and limited conductivity) satisfying mass variable seepage flowing in fractures. The simulation results have been compared with field data and influencing parameters are analyzed thoroughly. It is concluded that slippage effect affects gas production of fractured well when wellbore pressure is less than 15 MPa, and the effects of slippage and diffusion have greater influence on gas production of fractured well for reservoir with smaller permeability, especially when permeability is at nano-Darcy scale.
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Wang, Jingyu, Jian Yang, Long Li, Pei Qian, and Qiuwang Wang. "Numerical simulation and circuit network modelling of flow distributions in 2-D array configurations." Thermal Science 22, no. 5 (2018): 1987–98. http://dx.doi.org/10.2298/tsci171230256w.
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Packing configuration is widely used in chemical industries such as chemical re-action and chromatograph where the flow distribution has a significant effect on the performance of heat and mass transfer. In the present paper, numerical simulation is carried out to investigate the fluid-flow in three 2-D array configurations including in-line array, staggered array and hexagonal array. Meanwhile, a simplified equivalent circuit network model based on the Voronoi tessellation is proposed to simulate the flow models. It is found that firstly, the local Reynolds number could be used as a criterion to determine the flow regime. Flow with maximum local Reynolds number less than 40 could be regarded as Darcy flow. Secondly, the flow pattern can be well represented by the network model in the range of Darcy flow with the determination method of hydraulic resistance pro-posed in the present paper.
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Krotkiewski, Marcin, Ingeborg S. Ligaarden, Knut-Andreas Lie, and Daniel W. Schmid. "On the Importance of the Stokes-Brinkman Equations for Computing Effective Permeability in Karst Reservoirs." Communications in Computational Physics 10, no. 5 (November 2011): 1315–32. http://dx.doi.org/10.4208/cicp.290610.020211a.
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AbstractCavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models. Herein, we consider the problem of computing the effective permeability of rock samples based on high-resolution 3D CT scans containing millions of voxels. We use the Stokes-Brinkman equations in the entire domain, covering regions of free flow governed by the Stokes equations, porous Darcy flow, and transitions between them. The presence of different length scales and large (ten orders of magnitude) contrasts in permeability leads to highly ill-conditioned linear systems of equations, which are difficult to solve. To obtain a problem that is computationally tractable, we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models. We find that, in terms of effective permeability, the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability. All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows, depending on if there do or do not exist percolating free-flow regions. The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.
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Hamdan, Mohammad. "An empirical correlation for isothermal parallel plate channel completely filled with porous media." Thermal Science 17, no. 4 (2013): 1061–70. http://dx.doi.org/10.2298/tsci120419015h.
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This study reports a simple empirical correlation for friction factor and Nusselt number for laminar, steady state, hydraulically and thermally fully developed flow in isothermal parallel plate channel completely filled with porous media. The study is carried out using a finite difference numerical analysis. The Darcy-Brinkman-Forchheimer model is used to model the flow inside the porous media. The empirical correlations are developed to relate friction factor and Nusselt number to Darcy and Forchheimer coefficient.
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Gruais, Isabelle, and Dan Poliševski. "Thermal flows in fractured porous media." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 3 (May 2021): 789–805. http://dx.doi.org/10.1051/m2an/2020087.
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We consider the thermal flow problem occuring in a fractured porous medium. The incompressible filtration flow in the porous matrix and the viscous flow in the fractures obey the Boussinesq approximation of Darcy-Forchheimer law and respectively, the Stokes system. They are coupled by the Saffman’s variant of the Beavers–Joseph condition. Existence and uniqueness properties are presented. The use of the energy norm in describing the Darcy-Forchheimer law proves to be appropriate. In the ε-periodic framework, we find the two-scale homogenized system which governs their asymptotic behaviours when ε → 0 and the Forchheimer effect vanishes. The limit problem is mainly a model of two coupled thermal flows, neither of them being incompressible.
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Civan, F., C. S. S. Rai, and C. H. H. Sondergeld. "Determining Shale Permeability to Gas by Simultaneous Analysis of Various Pressure Tests." SPE Journal 17, no. 03 (August 16, 2012): 717–26. http://dx.doi.org/10.2118/144253-pa.
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Summary A model-assisted analysis is presented of pressure-pulse-transmission data obtained under different pressure conditions with core plugs of shale-gas formations. Applications and validations for steady-state and transient-state laboratory tests are provided. Best-estimate values of the intrinsic permeability and tortuosity at a reference condition and the Langmuir volume and pressure are determined by matching the solution of a modified Darcy model to several pressure-pulse-transmission flow tests with core samples simultaneously. The data-interpretation model considers the prevailing characteristics of the apparent permeability under the various flow regimes involving gas flow through extremely low-permeability core samples. Further, the present fully pressure-dependent shale-and gas-property formulation allows for model-assisted extrapolation from the reference conditions to field conditions once the unknown model parameters have been estimated under laboratory conditions. The improved method provides a better match to the measurements of the pressure tests than previous models, which assume only Darcy flow.
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Eroglu, Fatma G., Songul Kaya, and Leo G. Rebholz. "POD-ROM for the Darcy–Brinkman equations with double-diffusive convection." Journal of Numerical Mathematics 27, no. 3 (September 25, 2019): 123–39. http://dx.doi.org/10.1515/jnma-2017-0122.
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Abstract This paper extends proper orthogonal decomposition reduced order modeling to flows governed by double diffusive convection, which models flow driven by two potentials with different rates of diffusion. We propose a reduced model based on proper orthogonal decomposition, present a stability and convergence analyses for it, and give results for numerical tests on a benchmark problem which show it is an effective approach to model reduction in this setting.
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WU, JINSUI, DEZHI HU, WENJUN LI, and XIN CAI. "A REVIEW ON NON-DARCY FLOW — FORCHHEIMER EQUATION, HYDRAULIC RADIUS MODEL, FRACTAL MODEL AND EXPERIMENT." Fractals 24, no. 02 (June 2016): 1630001. http://dx.doi.org/10.1142/s0218348x16300014.
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In this paper, a brief summary of some of the relevant models for non-Darcy flow is reviewed, they mainly include seven models, among them three models are based on the average hydraulic radius, two are based on fractal geometry theory and technique, one is based on first principle, one is based on experimental measurement. Each model has its own advantages and disadvantages.
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Naderi, Masoud, Hossein Afzalimehr, Ayoub Dehghan, Nader Darban, Mohammad Nazari-Sharabian, and Moses Karakouzian. "Field Study of Three–Parameter Flow Resistance Model in Rivers with Vegetation Patch." Fluids 7, no. 8 (August 22, 2022): 284. http://dx.doi.org/10.3390/fluids7080284.
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Bed shear stress in coarse–bed rivers with vegetation patches is one of the challenging parameters in hydraulic engineering, mechanical engineering, fluvial morphology, and environmental studies. Based on this necessity, in this study, the values of bed shear stress in four reaches of rivers in Iran were estimated and compared using the methods of boundary layer characteristics, logarithmic law, and Darcy–Weisbach. Data collection in this study started in February 2021 and ended in April 2021. Estimation of flow resistance is a key factor in many numerical and physical models. In order to obtain a reasonable evaluation of this factor, it is necessary to measure and calculate the key variables of resistance to flow. Accordingly, the experimental design in this study includes surveying operations, velocity measurement, and sampling of bed sediments. The results show that due to bed forms, vegetation patches, and variations of flow depth and grain size in the river, the universal velocity distribution law (the log law) may not be suitable to estimate the shear velocity, which is a key parameter of flow resistance. This calls for more justifiable methods which are not affected by near–the–bed conditions. Accordingly, a three–parameter flow resistance model is presented, which shows an average error of 17%, indicating the accuracy of the model. The investigation of 71 measured velocity profiles shows the occurrence of the Dip phenomenon in the velocity profiles near the vegetation patches. However, by moving away from the vegetation patches, the effect of this phenomenon is decreased, and the profiles illustrate an S–shaped distribution. The results show that the relative differences between the logarithmic law and Darcy–Weisbach methods compared to the boundary layer characteristics method (BLCM) are equal to 87% and 39%, respectively, indicating a more reasonable agreement between the Darcy–Weisbach method and the boundary layer characteristics method. This is due to the application of key parameters of the boundary layer theory to calculate shear velocity by BLCM. However, to simplify data collection in the field, the application of the Darcy–Weisbach method is suggested.