Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity

Balankin, Alexander S.; Valdivia, Juan Carlos Rivas; Marquez, Jesús; Susarrey, Orlando; Solorio-Avila, Marco A.

doi:10.1016/j.physleta.2016.06.032

Cited by 36 publications

(13 citation statements)

References 54 publications

(111 reference statements)

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“…As was mentioned in Balankin et al [8], it is imperative to distinguish between three types of scale-invariant permeable media : 1) porous media in which the matrix is a fractal, whereas the pore space is non-fractal, (e.g. the Menger sponge and fractal aggregates) ; 2) fractally permeable media in which the pore space is a fractal, whereas the matrix is non-fractal, (e.g.…”

Section: Modelmentioning

confidence: 99%

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Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

Fellah

Ogam

et al. 2021

Wave Motion

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This paper provides a time domain model for the propagation of transient ultrasonic waves in a self-similar porous material having a rigid frame. This model is based on the formalism of Stillinger-Palmer-Stavrinou, which consists in modeling the fractal material as a porous medium with a non-integer dimensional space. This paper is devoted to the time-domain analytical calculus of the reection and transmission operators that are expressed in terms of Mittag-Leer functions. A sensitivity numerical study using ultrasonic reected and transmitted waves is performed, highlighting the eect of of the material's physical parameters (fractal dimension, tortuosity, viscous characteristic length and porosity) on the waveforms.

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Section: Modelmentioning

confidence: 99%

“…Many porous materials possess a self-similar structure exhibiting scale invariance over many length scales [6][7][8]. For such materials, classical homogenization methods become inapplicable because heterogeneities play an important role on almost all scales [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

Fellah

Ogam

et al. 2021

Wave Motion

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“…Considering the threshold pressure gradient, stress sensitive effect, and slippage effect, the nonlinear flow mathematical model of tight gas reservoirs can be derived through the generalized form of Darcy's Law [25,26].…”

Section: Mathematical Model For Nonlinear Flow Mechanismsmentioning

confidence: 99%

An Analytical Solution for Transient Productivity Prediction of Multi-Fractured Horizontal Wells in Tight Gas Reservoirs Considering Nonlinear Porous Flow Mechanisms

Wang

Wan

et al. 2020

Energies

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Multi-fractured horizontal wells (MFHW) is one of the most effective technologies to develop tight gas reservoirs. The gas seepage from tight formations in MFHW can be divided into three stages: early stage with high productivity, transitional stage with declined productivity, and final stage with stable productivity. Considering the characteristics and mechanisms of porous flows in different regions and at different stages, we derive three coupled equations, namely the equations of porous flow from matrix to fracture, from fracture to near wellbore region, and from new wellbore region to wellbore then an unstable productivity prediction model for a MFHW in a tight gas reservoir is well established. Then, the reliability of this new model, which considers the multi-fracture interference, is verified using a commercial simulator (CMG). Finally, using this transient productivity prediction model, the sensitivity of horizontal well's productivity to several relevant factors is analyzed. The results illustrate that threshold pressure gradient has the most significant influence on well productivity, followed by stress sensitivity, turbulence flow, and slippage flow. To summarize, the proposed model has demonstrated a potential practical usage to predict the productivity of multi-stage fractured horizontal wells and to analyze the effects of certain factors on gas production in tight gas reservoirs.Energies 2020, 13, 1066 2 of 20 of tight gas reservoirs has made considerable breakthrough and become the primary growth point for oil and gas production [1][2][3][4][5][6]. Because of its strong heterogeneity, complicated microscopic pore structure, and poor connectivity of the effective sand bodies, the porous flow mechanism of tight gas reservoirs is obviously distinct from those of conventional reservoirs. Moreover, because of poor reservoir physical properties, small discharging radius, lack of or low natural productivity, high development difficulty, and some other potential disadvantages, a MFHW with hydraulic stimulations is extensively used for developing tight gas reservoirs [7][8][9][10][11][12][13][14][15][16].To date, many scholars are dedicated to the transient productivity of fractured wells in unconventional hydrocarbon reservoirs, considering the different characteristics of nonlinear flows compared to multi-scale flows. The methods to calculate the transient productivity of fractured horizontal wells primarily include analytical and numerical solutions, among which the analytical solution includes the complex potential theory, conformal transformation, and the equivalent flow resistance method [14,17,18]. Research on the transient productivity of tight gas reservoirs is increasingly wide. Through an iterative algorithm, Zheng et al. [19] quantified the correlation between saturation and pressure of the infinitesimal coal. They used the Corey relative permeability model to describe relative gas/water permeability as a function of the pressure. By applying the inter-porosity flow equation based on a pseudo-pr...

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“…and for capillary tubes with unitary porosity, φ = 1. Where K τ , is a permeability ratio for permeable media with scale invariance in materials where a pre-fractal pore network exists [32]. It was shown that the equilibrium height x eq is the same for both, straight tubes and tubes with tortuous path.…”

Section: Capillary Rise On the Linear Fractalsmentioning

confidence: 99%

Fractal imbibition in Koch's curve-like capillar tubes

et al. 2018

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Fractal geometry eects in capillary imbibition process are studied.Capillary rise analysis in Koch's curve-like tubes were be carried out withiterations i = 0; 1; 2; 3; 4; 5. The behaviour was characterized in function oftime, fractal geometry and height of capillary rise. An geometrical relationshipfor fractal dimension of ow tortuosity (dr) in porous media is obtained.The analytical model of Lucas-Washburn-Cai to describe the capillary rise byspontaneus imbibition in tubes with deterministic fractal geometry is adjusted.The equilibrium height time as function of fractal dimension of ow tortuosity incapillary tubes with tortuous path is also derived.

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Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity

Cited by 36 publications

References 54 publications

Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

An Analytical Solution for Transient Productivity Prediction of Multi-Fractured Horizontal Wells in Tight Gas Reservoirs Considering Nonlinear Porous Flow Mechanisms

Fractal imbibition in Koch's curve-like capillar tubes

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