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A continuum approach to high velocity flow in a porous medium

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Abstract

A general macroscopic linear momentum balance equation is derived in the form of a constitutive relation for high velocity fluid flow in a porous medium. It shows the nonlinearity in Forchheimer's formula for nonDarcy flow arising primarily from microscopic inertial phenomena, and expresses the inertial force in terms of the macroscopic velocity in an anisotropic and nonlinear manner. The point of departure is Euler's first law of motion, valid at any point in the fluid phase which is assumed to completely occupy the void space. The geometry of the void space, i.e., of the solid matrix, is taken as arbitrary. By introducing an alternative description of the microscopic kinematic field, namely deviations of the local velocity magnitudes and directions from the macroscopic values of these quantities separately, a general macroscopic momentum equation for fluid flow in a porous medium is obtained after averaging over a REV. From the general equation, most of the established relations for nonDarcy flow can be recovered as special cases. Explicit analytic expressions are obtained for the involved inertial coefficients from which the origin and nature of nonlinear (inertial) effects for high velocity flow in a porous medium is clearly demonstrated. It is also shown that the coefficient associated with the quadratic term for nonDarcy flow is not material and is a function of the macroscopic flow. Finally, some previous results are discussed and an extension of the derived equation to include higher-order nonlinear effects, with regard to the resistivity force, is proposed.

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Abbreviations

a :

coefficient in (1) and (67)

A f :

surface bounding the macroscopic fluid regionV f

A ff :

a part ofA f adjacent to the fluid

A fs :

a part ofA f adjacent to the solid

b :

coefficient in Equation (1) and (67)

C,C kl :

inertial coefficient tensor in Equation (57)

d,d kl :

deformation rate tensor

da :

microscopic infinitesimal surface area

dA f :

boundary surface of dV f

dA fs :

part of dA f adjacent to the solid

dA ff :

part of dA f adjacent to the fluid

dA x :

macroscopic surface element

dv :

microscopic infinitesimal volume element

dV :

a representative elementary volume (REV)

dV f :

volume occupied by the fluid within a REV

dV s :

volume occupied by the solid within a REV

dV x :

macroscopic volume element

e :

unit vector in the direction of the microscopic velocity with componentsα k

\(\hat e \equiv \hat \alpha _k i^k \) :

where\(\hat \alpha _k \) is defined in (22)

f,f k :

body force vector

i k :

unit vector in the direction of the coordinate axis 1, 2, 3

I :

unit tensor

Ĩ :

local inertia terms defined in (44)

I :

local inertia term defined in Equation (57)

J,J k :

driving force of motion defined in (49)

j :

integer with values ⩾ 1

¦k¦:

curvature magnitude defined in (52)

k *,k * kl :

coefficient tensor defined in (61)

K *,K * kl :

coefficient tensor defined in (59)

m :

real number

n :

microscopic surface normal unit vector for da over dA fs, oriented from the fluid to the solid phase

N :

macroscopic surface normal unit vector for dA x

p :

pressure

\(\tilde p\) :

macroscopic fluid pressure as defined in Hassanizadeh and Gray (1980)

P,P kl :

tensor defined in (44)

q,q k :

fluid mass flux

r :

microscopic position vector

r fs :

microscopic position vector on the surface dA fs

R,R kl :

resistivity coefficient tensor defined in (33)

\(\tilde R \equiv - R\) R *,R * kl :

coefficient tensor defined in (57)

R (i) :

tensors of order 3 and 4 defined in (81)

R′,R,\(\tilde R\) :

scalar coefficients defined in (85)

S,S kl :

tensor defined in (44)

\(\tilde S, \tilde S_{kl} \) :

tensor defined in (79)

t :

time

t,t k :

stress vector

t,t kl :

stress tensor

\(\tilde t, \tilde t_{kl} \) :

macroscopic fluid stress tensor as defined in Hassanizadeh and Gray (1980)

t *,t * kl :

stress tensor defined in (76)

T,T k :

resistivity force defined in (32)

\(\tilde T, \tilde T_{kl} \) :

dissipative part of\(\bar \rho {\rm T}\)

v,v k :

velocity vector

\(\hat v\) :

microscopic velocity deviation vector defined in (2)

\(\hat v\) :

deviation of the microscopic velocity magnitude defined in (21)

V :

a macroscopic portion of the porous region

V f :

part ofV occupied by the fluid

V s :

part ofV occupied by the solid

w k :

interphase surface velocity vector

x,x k :

macroscopic position vector

α k :

components ofe

\(\hat \alpha _k \) :

components ofê, defined in (22)

γ :

distribution function defined in (15)

δ kl :

Kronecker delta

δ(r -r fs):

Dirac delta function

δ k :

angle betweeni k ande

▽:

gradient operator, with respect to eitherr orx

▽ r , ▽ x :

gradient operator with respect tor, i.e.,x

ε :

porosity

θ :

temperature

κ,κ′ :

scalar quantities defined in (39)

λ :

viscosity coefficient

\(\tilde \mu \) :

macroscopic fluid viscosity coefficient as defined in Hassanizadeh and Gray (1980)

λ j :

coefficients defined in (86)

Λ, Λ kl :

tensor defined in (34)

μ :

viscosity coefficient

\(\lambda ^ \sim \) :

macroscopic fluid viscosity coefficient as defined in Hassanizadeh and Gray (1980)

π, π kl :

coefficient tensor defined in (46)

\(\tilde \Pi , \tilde \Pi _{kl} \) :

coefficient tensor defined in (46)

ϱ :

fluid mass density

Σ:

coefficient tensors of order 0, 2 and 3 defined in (89) and (91)

τ:

stress vector defined in (II4)

τ* :

stress vector defined in (75)

ψ :

an arbitrary quantity

∑:

summation

ε:

belongs to

U:

union

ψ :

deviating quantity

ψ :

macroscopic quantity

ψ, k ψ k ,l :

spatial derivation with respect to eitherr orx

〈 〉:

an averaging operator

( )T :

transposed

( )−1 :

inverse

ψ :

vector

ψ :

tensor

( )(i) :

symbolic notation for quantities defined in (81)

k, l, m, n :

take values 1 to 3 and denote Cartesian components of tensorial quantities

References

  • Ahmed, N. and Sunada, D. K., 1969, Nonlinear flow in porous media,J. Hydr. Div. ASCE 95, 1847–1857.

    Google Scholar 

  • Bachmat, Y., 1965, Basic transport coefficients as aquifer characteristics,IASH Symp. Hydr. of Fract. Rocks, Dubrovnik, pp. 63–75.

  • Barak, A. Z. and Bear, J., 1981, Flow at high Reynolds numbers through anisotropic porous media,Adv. Water Resour. 4, 54–66.

    Google Scholar 

  • Bear, J., 1972,Dynamics of Fluids in Porous Media, Elsevier, New York.

    Google Scholar 

  • Bedford, A. and Ingram, J. D., 1971, A continuum theory of fluid saturated porous media,J. Appl. Mech., March 1971, pp. 1–7.

  • Beran, M. J., 1968,Statistical Continuum Theories, Interscience, New York.

    Google Scholar 

  • Drew, D. A., 1971, Averaged field equations for two-phase media,Stud. Appl. Math. 50, 133–167.

    Google Scholar 

  • Eringen, C. A., 1967,Mechanics of Continua, Wiley, New York.

    Google Scholar 

  • Eringen, C. A. and Ingram, J. D., 1967, A continuum theory of chemically reacting media - II: Constitutive equations of reacting fluid mixtures,Int. J. Engng. Sci. 5, 289–322.

    Google Scholar 

  • Fulks, W. B., Guenther, R. B., and Roetman, E. L., 1971, Equations of motion and continuity for fluid flow in a porous medium,Acta Mechanica 12, 121–129.

    Google Scholar 

  • Gray, W. G. and Lee, P. C. Y., 1977, On the theorems of local volume averaging of multiphase systems,Int. J. Multiphase Flow 3, 333–340.

    Google Scholar 

  • Gray, W. G. and O'Neill, K., 1976, On the general equations for flow in porous media and their reduction to Darcys law,Water Resour. Res. 12, 148–154.

    Google Scholar 

  • Hannoura, A, A. and Barends, F. B. J., 1981, Non-Darcy flow: State of the art,Proc. Euromech 143, Delft, pp. 37–51.

  • Hassanizadeh, M. and Gray, W. G., 1979a, General conservation equations for multiphase systems: 1. Averaging procedure,Adv. Water Resour. 2, 131–144.

    Google Scholar 

  • Hassanizadeh, M. and Gray, W. G., 1979b, General conservation equations for multiphase systems: 2. Mass, momenta, energy and entropy equations,Adv. Water Resour. 2, 191–203.

    Google Scholar 

  • Hassanizadeh, M. and Gray, W. G., 1980, General conservation equations for multiphase systems: 3. Constitutive theory for porous media flow,Adv. Water Resour. 3, 25–40.

    Google Scholar 

  • Hubbert, M. K., 1940, The theory of ground water motion,J. Geol. 48, 785–944.

    Google Scholar 

  • Irmay, S., 1958, On the theoretical derivation of Darcys and Forehheimer formulas,J. Geophys. Res. 39, 702–707.

    Google Scholar 

  • Ishii, M., 1975,Thermo-Fluid Dynamics Theory of Two-Phase Flow, Eyrolles, Paris.

    Google Scholar 

  • Kenyon, D. E., 1976a, Thermostatics of solid-fluid mixtures,Arch. Rat. Mech. Anal. 62, 117–129.

    Google Scholar 

  • Kenyon, D. E., 1976b, The theory of an incompressible solid-fluid mixture,Arch. Rat. Mech. Anal. 62, 131–147.

    Google Scholar 

  • Morland, L. W., 1972, A simple constitutive theory for a fluid-saturated porous solid,J. Geophys. Res. 77, 890–900.

    Google Scholar 

  • Mueller, I., 1971,Entropy, Absolute Temperature and Coldness in Thermodynamics and Boundary Conditions in Porous Materials, ICMS, Courses and Lectures No. 76, Udine, Springer-Verlag, New York.

    Google Scholar 

  • Neuman, S. P., 1976, Theoretical derivation of Darcy's law,Acta Mechanica 25, 153–170.

    Google Scholar 

  • Nikolaevskii, V. N., 1959, Convective diffusion in porous media,Appl. Math. Mech. (PMM) 23, 1492–1503.

    Google Scholar 

  • Raats, P. A. C. and Klute, A., 1968, Transport in soils: the balance of momentum,Soils Sci. Soc. Am. Proc.,32, 452–456.

    Google Scholar 

  • Raats, P. A. C., 1971, The role of inertia in the hydrodynamics of porous media,Arch. Rat. Mech. Anal. 44, 267–283.

    Google Scholar 

  • Rabinovich, M. I., 1978, Stochastic self-oscillations and turbulence,Sov. Phys. Usp. 21, 443–469.

    Google Scholar 

  • Scheidegger, A. E., 1960,The Physics of Flow Through Porous Media, Univ. of Toronto Press.

  • Shapiro, A. M., 1981, Fractured porous media: equation development and parameter identification, Ph.D thesis, Dept. of Civil Engineering, Princeton Univ.

  • Slattery, J. C., 1969, Single-phase flow through porous media,AIChE J. November, 866–872.

  • Stoker, J. J., 1969,Differential Geometry, Wiley-Interscience, New York.

    Google Scholar 

  • Stokes, J., 1983, Field equations for flow through porous media and their relation to the micro-structure, Ph.D thesis, Dept. of Mechanics, RIT, Stockholm.

    Google Scholar 

  • Truesdell, C., 1965, On the foundations of Mechanics and energetics, in C. Truesdell (ed.),The Rational Mechanics of Materials, Gordon and Breach, New York, pp. 293–304.

    Google Scholar 

  • Whitaker, S., 1969, Advances in the theory of fluid motion in porous media,Ind. Eng. Chem. 61, 14–28.

    Google Scholar 

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Authors and Affiliations

  1. Water Resources Engineering, Royal Institute of Technology, S-10044, Stockholm, Sweden

    Vladimir D. Cvetković

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  1. Vladimir D. Cvetković
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Cvetković, V.D. A continuum approach to high velocity flow in a porous medium. Transp Porous Med 1, 63–97 (1986). https://doi.org/10.1007/BF01036526

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  • DOI: https://doi.org/10.1007/BF01036526

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