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Transport in Porous Media

, Volume 19, Issue 1, pp 37–66 | Cite as

Dispersion in consolidated sandstone with radial flow

  • W. Kwok
  • R. E. Hayes
  • H. A. Nasr-El-Din
Article

Abstract

This paper presents some experimental and theoretical results for dispersion processes occurring in consolidated Berea sandstone with radial flow geometry. A comprehensive review of the derivation and application of several analytical solutions is also presented. The Galerkin finite element method is applied to solve the advection-dispersion equation for unidimensional radial flow.

Individual and combined effects of mechanical dispersion and molecular diffusion are examined using velocity-dependent dispersion models. Comparison of simulated results with experimental data is made. The effect of flow rates is examined. The results suggest that a linear dispersion model,D=αu, whereD is the dispersion coefficient,u the velocity andα a constant, is not a good approximation despite its wide acceptance in the literature. The most suitable mathematical formulation is given by an empirical form of\(D = D_0 + \mathop \alpha \limits^` u^m\), whereDois the molecular diffusion coefficient. For the range of Péclet number (Pe=vd/Dm,wherev is the characteristic velocity,d the characteristic length andDmthe molecular diffusion coefficient in porous media) examined (Pe=0.5 to 285), a power constant ofm=1.2 is obtained which agrees with the value reported by some other workers for the same regime.

From the results of experiments and numerical modelling, the effect of mobility ratios (defined as the ration of viscosities of displaced and displacing fluids) on dispersion is found to be negligible, provided that the ratio is favourable.

Key words

dispersion sandstone radial flow 

Nomenclature

a

proportionality constant, Eqn. (4)

A

Flow coefficient,Q/2πhΦ, m2/s

b

distance travelled by the advective front, (Qt/πh)1/2, m

c

concentration in bulk fluid, mol/l

C

concentration in the effuent solution measured or calculated atr=ro,mol/l

Co

concentration in the injected solution, mol/l

d

characteristic length of unconsolidated porous medium, m

dp

characteristic length of consolidated porous medium, m

D

longitudinal dispersion coefficient, m2/s

Dm

molecular diffusion coefficient in porous media, m2/s

Do

effective molecular diffusion coefficient in porous medium, m2/s

Dz

axial diffusion coefficient, m2/s

F

formation resistivity

h

core thickness, m

m

power constant, Eqn. (4)

M

mobility ratio,Μd/Μi

Pe

Péclet number,vd/Dm

Peh

element Péclet number,vh/D

Q

volumetric flow rate, ml/h

r

radial distance measured from the centre, m

ro

radial distance at the exit, m

rw

radial distance at the wellbore, m

Ro

fictitious radius of the simulated core, m

t

time, s

t50

time at whichC/Co=0.5, s

u

interstitial velocity, m/s

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}\)

interstitial velocity vector, m/s

uinj

Interstitial velocity at the injection wellbore, m/s

uz

axial component of interstitial velocity, m/s

w

variable defined in Equation (15), 2/3 (ρ2/3ρ w 2/3 )

Greek Symbols

α

dispersivity coefficient, m

\(\mathop \alpha \limits^`\)

equivalent dispersivity coefficient, m(2−m) s−(1−m)

δ

standard deviation. Eqn. (11)

Μd

viscosity of the displaced fluid, mPa·s

Μi

viscosity of the injected fluid, mPa·s

ρ

dimensionless radial distance,r/α

ρw

dimensionless radius of the injection wellbore,rw/α

ρs

bulk density of solid, gR/m R 3

σ

inhomogeneity factor

Τ

dimensionless time,At/α2

v

characteristic velocity, m/ s

Φ

porosity

χ

function defined for plotting on an arithmetic probability paper, 2A(tt50)/r2

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • W. Kwok
    • 1
  • R. E. Hayes
    • 1
  • H. A. Nasr-El-Din
    • 1
    • 2
  1. 1.Department of Chemical EngineeringUniversity of AlbertaEdmontonCanada
  2. 2.Petroleum Recovery InstituteCalgaryCanada

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