Semi-analytical Approach to Modeling Forchheimer Flow in Porous Media at Meso- and Macroscales

Abstract

Darcy’s law (which states that a fluid flow rate is directly proportional to the pressure gradient) is shown to be accurate in a rather narrow range of flow velocities. Numerous studies show that at low pressure gradients gas slippage effect occurs, which gives overestimated flow rates compared to Darcy’s law. At higher flow rates, Darcy’s law is usually replaced by the Forchheimer equation which accounts for inertial forces including a quadratic term in the flow rate. Darcy’s and Forchheimer’s laws and the problem of detecting transitions between their ranges of applicability are discussed in this study. Analysis of experimental data shows that deviation from Darcy’s law is governed by the Forchheimer number, which is defined by the authors as a product of tortuosity and Reynolds number. The use of the Forchheimer number and semi-analytical approaches enables us to describe non-Darcy flow as a simple universal equation valid for any flow geometry. Comparison of the experimental data with predictions based on a semi-analytical model shows excellent agreement for a wide range of reservoir properties.

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Abbreviations

\({D}_{\mathrm{p}}\) :

Particles diameter, m

\({D}_{\mathrm{t}}\) :

Throat diameter, m

\(E\) :

Non-Darcy effect

\(F\) :

Formation resistivity factor

\(\mathrm{Fo}\) :

Forchheimer number

\({\mathrm{Fo}}_{\mathrm{D}}\) :

Forchheimer number related to Darcy flow conditions

\({\mathrm{Fo}}_{\mathrm{c}}\) :

Critical Forchheimer number

\({\mathrm{Fo}}_{\mathrm{c exp}}\) :

Experimentally measured critical Forchheimer number

\({\mathrm{Fo}}_{\mathrm{c sim}}\) :

Critical Forchheimer number obtained during the simulation

\(P\) :

Pressure, Pa

\({P}_{0}\) :

Standard pressure, Pa

\({P}_{1}\) :

Inlet pressure, Pa

\({P}_{2}\) :

Outlet pressure, Pa

\({P}_{\mathrm{D}}\) :

Pressure related to Darcy flow conditions, Pa

\({P}_{\mathrm{F}}\) :

Pressure related to Forchheimer flow conditions, Pa

\({P}_{\mathrm{e}}\) :

External boundary pressure, Pa

\({P}_{\mathrm{w}}\) :

Bottomhole pressure, Pa

\(\mathrm{Re}\) :

Reynolds number

\({\mathrm{Re}}_{\mathrm{D}}\) :

Reynolds number related to Darcy flow conditions

\({\mathrm{Re}}_{\mathrm{F}}\) :

Reynolds number related to Forchheimer flow conditions

\({\mathrm{Re}}_{\mathrm{c}}\) :

Critical Reynolds number

\(d\) :

Mean pore diameter, m

\({d}_{\mathrm{eqv}}\) :

Equivalent pore diameter, m

\(k\) :

Permeability, mD

\(q\) :

Gas flow rate, m3/d

\({q}_{\mathrm{D}}\) :

Volumetric flow rate related to Darcy flow conditions, m3/d

\({q}_{\mathrm{Dm}}\) :

Mass flow rate related to Darcy flow conditions, kg/d

\({q}_{\mathrm{F}}\) :

Volumetric flow rate related to Forchheimer flow conditions, m3/d

\({q}_{\mathrm{Fm}}\) :

Mass flow rate related to Forchheimer flow conditions, kg/d

\(r_{\text{e}}\) :

External drainage radius, m

\({r}_{\mathrm{w}}\) :

Wellbore radius, m

\(u\) :

Velocity, m/s

\({u}_{\mathrm{D}}\) :

Volumetric velocity related to Darcy flow conditions, m/s

\({u}_{\mathrm{Dm}}\) :

Mass velocity related to Darcy flow conditions, kg/(s m2)

\({u}_{\mathrm{F}}\) :

Volumetric velocity related to Forchheimer flow conditions, m/s

\({u}_{\mathrm{Fm}}\) :

Mass velocity related to Forchheimer flow conditions, kg/(s m2)

\(h\) :

Formation thickness, m

\(\mathrm{grad} \,P\) :

Pressure gradient, Pa/m

\(r\) :

Pore throat radius in Table 2, m

\(x\) :

Linear coordinate, m

\(R2\) :

Determination coefficient

\({B}_{1}\) :

Constant in Eq. 20

\(a\) :

Constants in Table 1

\(b\) :

Constants in Table 1

\(c\) :

Constant in Eq. 4

\({c}^{^{\prime}}\) :

Constant in Eq. 5

\(a,b,c\) :

Coefficients in Eq. 34

\({a}_{1},{b}_{1},{c}_{1}\) :

Coefficients in Eq. 57

\(\alpha \) :

Constant in Eq. 20

\(\beta \) :

Non-Darcy coefficient, m1

\(\gamma \) :

Weak inertia factor

\(\lambda \) :

New turbulent flow correlation factor, ft

\(\mu \) :

Viscosity, Pa s

\(\rho \) :

Density, kg/m3

\({\rho }_{0}\) :

Density at standard conditions, kg/m3

\(\tau \) :

Tortuosity

\(\phi \) :

Porosity, fraction

\(\mathrm{D}\) :

Related to Darcy flow conditions

\(\mathrm{F}\) :

Related to Forchheimer flow conditions

\(\mathrm{c}\) :

Critical

\(\mathrm{e}\) :

External

\(\mathrm{eqv}\) :

Equivalent

\(\mathrm{exp}\) :

Experiment

\(\mathrm{m}\) :

Mass

\(\mathrm{p}\) :

Particles

\(\mathrm{sim}\) :

Simulation

\(\mathrm{t}\) :

Throat

\(\mathrm{w}\) :

Well

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All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by ABZ and ATG The first draft of the manuscript was written by ABZ and ATG, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. ABZ conceived the study and contributed to project administration, supervision, and validation. ABZ and ATG curated the data, performed the formal analysis, and contributed to investigation, methodology, resources, and writing—review and editing. Funding acquisition is not applicable. ATG contributed to software, visualization, and writing—original draft.

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Zolotukhin, A.B., Gayubov, A.T. Semi-analytical Approach to Modeling Forchheimer Flow in Porous Media at Meso- and Macroscales. Transp Porous Med 136, 715–741 (2021). https://doi.org/10.1007/s11242-020-01528-4

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Keywords

  • Forchheimer’s law
  • Non‐Darcy coefficient
  • Reynolds number
  • Forchheimer number
  • Tortuosity
  • Permeability
  • Porosity