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Analytical Model for Fines Migration During Water Injection

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Abstract

Exact solution is derived for axisymmetric flow during water injection with fine particles detachment, migration, attachment and straining. The solution contains the so-called erosion front described as a weak discontinuity, behind which the mechanical equilibrium of attached particles holds and the dynamic attachment occurs ahead of the front. Introduction of a timely potential form for suspended concentration decreases the order of governing system allowing derivation of the erosion front trajectory. The analytical model describes the injectivity decline due to fines migration and reveals non-monotonic injection rate dependency of the well index.

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Abbreviations

\(A_{123}\) :

Hamaker constant \((\hbox {ML}^{2}\hbox {T}^{-2})\)

\(c\) :

Suspended particle concentration \((\hbox {L}^{-3})\)

\(C\) :

Dimensionless suspended particle concentration

\(D\) :

Erosion front velocity \((\hbox {LT}^{-1})\)

\(d\) :

Collision diameter (L)

\(f_{\mathrm{s}}\) :

Probability density function for colloid radius

\(f_{\mathrm{p}}\) :

Probability density function for pore radius

\(F\) :

Force \((\hbox {MLT}^{-2})\)

\(h\) :

Particle surface separation distance (L)

\(H\) :

Thickness of a rectangular pore channel (L)

\(J\) :

Impedance (normalized reciprocal of well index)

\(k\) :

Absolute permeability \((\hbox {L}^{2})\)

\(k_{\mathrm{B}}\) :

Boltzmann constant \((\hbox {ML}^{2}\hbox {T}^{-2}\hbox {K}^{-1})\)

\(l\) :

Characteristic wavelength (L)

\(l_{\mathrm{n}}\) :

Normal lever (L)

\(l_{\mathrm{d}}\) :

Tangential lever (L)

\(n_{\infty }\) :

Bulk number density of ions \((\hbox {L}^{-3})\)

\(p\) :

Pressure \((\hbox {MT}^{-2}\hbox {L}^{-1})\)

\(P\) :

Dimensionless pressure

PVI:

Pore volume injected (dimensionless unit for time T)

\(q\) :

Volumetric flow rate per unit of the reservoir production thickness \((\hbox {L}^{2}\hbox {T}^{-1})\)

\(r\) :

Radius (L)

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

Drainage radius of the well (L)

\(r_{\mathrm{s}}\) :

Radius of a particle (L)

\(S\) :

Dimensionless retained particle concentration

\(t\) :

Time (T)

\(T^{/}\) :

Dimensionless time coordinate of intersection point of characteristics with erosion front

\(U\) :

Darcy’s velocity in porous media \((\hbox {LT}^{-1})\)

\(u\) :

Dimensionless velocity

\(V\) :

Potential energy \((\hbox {ML}^{2}\hbox {T}^{-2})\)

\(U\) :

Dimensionless velocity

\(X\) :

Dimensionless radial coordinate

\(\beta \) :

Formation damage coefficient

\(\gamma \) :

Salinity

\(\Delta \) :

Difference between two values (pressure, retained concentration)

\(\varepsilon \) :

Erosion number (ratio between the torques of detaching and attaching forces)

\(\zeta \) :

Surface potential (mV)

\(\kappa \) :

Debye length \((\hbox {L}^{-1})\)

\(\lambda \) :

Dimensional filtration coefficient \((\hbox {L}^{-1})\)

\(\Lambda \) :

Dimensionless filtration coefficient

\(\mu \) :

Dynamic viscosity \((\hbox {ML}^{-1}\hbox {T}^{-1})\)

\(\rho \) :

Fluid density \((\hbox {ML}^{-3})\)

\(\sigma \) :

Concentration of retained particles \((\hbox {L}^{-3})\)

\(\phi \) :

Porosity

\(\chi \) :

Lifting force coefficient

\(\omega \) :

Drag force coefficient

cr:

Critical (for attached concentration and coordinate of erosion front)

i:

Initial conditions values (for suspended and retained concentrations, for erosion number)

e:

electrostatic (for force)

w:

Well (for radius, dimensionless radial coordinate and erosion number)

m:

Maximum value (for velocity and erosion number)

n:

Normal (for force)

p:

Pore (for pore radius and pore size distribution)

s:

Straining (for retained concentration, filtration coefficients and formation damage coefficient)

a:

Attachment (for retained concentration, filtration coefficients and formation damage coefficient)

0:

Initial value (for permeability and dimensionless radial coordinate)

d:

Drag (for force), damage (for reservoir radius and dimensionless radial coordinate)

g:

Gravitational (for force)

w:

Well

BR:

Born repulsion (for potential energy)

DLR:

Electrostatic double layer (for potential energy)

LVA:

London–van der Waal (for potential energy)

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Acknowledgments

The authors are grateful to Dr A Badalyan (The University of Adelaide) for assistance in modelling of the electrostatic interactions. Dr Zhenjiang You (The University of Adelaide) is gratefully acknowledged for fruitful discussions and for help in preparing the manuscript. The study is generously sponsored by the Australian Research Council and Santos Ltd.

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  1. University of Adela de, Adelaide, SA, Australia

    Pavel Bedrikovetsky & Noe Caruso

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  1. Pavel Bedrikovetsky
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Correspondence to Pavel Bedrikovetsky.

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Bedrikovetsky, P., Caruso, N. Analytical Model for Fines Migration During Water Injection. Transp Porous Med 101, 161–189 (2014). https://doi.org/10.1007/s11242-013-0238-7

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