Permeability evolution induced by the precipitation of an advected solute: Effect of the microscopic and the macroscopic scales competition on the clogging mechanism

A model is proposed for coupling the one-dimensional transport of solute with surface precipitation kinetics which induces the clogging of an initially homogeneous porous medium. The aim is to focus the non-linear feedback effect between the transport and the chemical reaction through the permeability of the medium. A Lagrangian formulation, used to solve the coupled differential equations, gives semi-analytical expressions of the hydrodynamic quantities. A detailed analysis reveals that the competition between the microscopic and macroscopic scales controls the clogging mechanism, which differs depends on whether short or long times are considered. In order to illustrate this analysis, more quantitative results were obtained in the case of a second and zeroth order kinetic. It was necessary to circumvent the semi-analytic character of the solutions problem by successive approximation. A comparison with results obtained by simulations displays a good agreement during the most part of the clogging time.Nomenclature a(x, t) Capillary tube radius (L) - A _(aq) Chemical species in the aqueous phase - A _n(s) Chemical species of the solid phase - C(x, t) Aqueous concentration in a capillary tube (mole/L³] in the case of a permanent injection - mole/L³/L] in the case of an instantaneous injection) - C(x, t) C(x, t)/C ₀ Dimensionless aqueous concentration in a capillary tube - C ₀ Aqueous concentration imposed at the inlet and also initial concentration in an elementary volume of fluid (mole/L³) - C _i(t) Concentration in a fluid element i (mole/L³) - C(_R) (t, C_o) Aqueous concentration in a stirred reactor (mole/L³) - d_ij (t) Length belonging to the volume, inside a fluid element i, which interacts with a precipitate element j (L) - dM _ij(t) Mass exchange between a fluid element i and a precipitate element j (M) - dN ₀ Number of molecules in an elementary volume of fluid injected at the inlet of a capillary tube during dt ₀ - dN(x, t, C₀) Number of molecules in an elementary volume of fluid - dt ₀ Time injection of an elementary volume of fluid (T) - D(x, t) Dispersion coefficient (L²/T) - Da(t, x) Damköhler number - D _m Molecular diffusion coefficient (L²/T) - F(x, t) Advective flux (mole/L²/T) - k ₁ Kinetic constant of dissolution (mole/L³/T) - k ₂ Kinetic constant of precipitation (mole/L³]^{1 - n} /T) - k ₂ Kinetic constant of precipitation in the case of a zeroth order kinetics (mole/L³/T) - K(x, t) Permeability in a capillary tube (L²) - K(x, t) K (x, t)/K₀ Dimensionless permeability - k ₀ Permeability of a capillary tube at t = 0 (L²) - L Length of a capillary tube (L) - m Molecular weight of the reactive species (M/mole) - n Stochiometry of the chemical reaction and kinetic order of the precipitation reaction - P(x, t) Precipitate concentration in a capillary tube (mole/L³) - P _j(t) Concentration in a precipitate element j (mole/L³) - P(_r) (t, C_o) Precipitate concentration in a stirred reactor (mole/L³) - Pr(x, t) Local pressure in a capillary tube - (M/T²/L³) Pr(x, t) Pr(x, t)/Pr(x, 0) Dimensionless local pressure in a capillary tube - Q(t) Flow rate (L³/T) - Q(t) Q(t)U ₀/S₀ Dimensionless flow rate - R(x, t) Chemical flux between the aqueous and the solid phase in a capillary tube (mole/L³/T) - R _i(t) Chemical flux between an aqueous element i and the solid phase (mole/L³/T) - R _(R)t, C₀] Chemical flux between the aqueous and the solid phase in a stirred reactor (mole/L³/T) - S(x, t) Cross sectional area of a capillary tube accessible to the aqueous phase (L²) - S(x, t) S(x, t)/S₀ Dimensionless cross-sectional area - S ₀ Cross-sectional area of a capillary tube at t = 0 (L²) - tlim(x) Time at which the precipitation front concentration vanishes in the case of zeroth order kinetics (T) - t _max Time of maximum propagation of the precipitation front in the case of zeroth order kinetics (T) - tmin(x) Time at which the precipitation front arrives at x (T) - t _p L/U ₀. Time necessary for an elementary volume of fluid, moving with the velocity U ₀, to reach the oulet of the medium - t _U max Time of maximum value of the velocity field in the case of zeroth order kinetics (T) - t ₀ Time at which an elementary volume of fluid has left the inlet of a capillary tube (T) - t _0m (x, t) Time at which the last elementary volume of fluid has left the inlet of a capillary tube to reach x at a time lower or equal to t (T) - U(x, t) Fluid velocity (L/T) - U(x, t) U(x, t)/U ₀. Dimensionless fluid velocity - U _j(x, t) Fluid velocity defined from the precipitate element j (L/T) - U _l (t₀, t) Lagrangian fluid velocity (L/T) - U _l (t ₀, t) U _l (x, t)/U ₀. Dimensionless lagrangian fluid velocity - U ₀ Velocity of the fluid at t = 0 (L/T) - V _ij(t) Volume, inside a fluid element i, which interacts with a precipitate element j (L³) - x _i(t) Front position of the fluid element i (L) - x _j Front position of the precipitate element j (L) - X _front(t) Position of the precipitation front (L) - x _lim(t) Position of the precipitation front when the value of its concentration is zero (L) - xmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for high value of C ₀ (L) - Xmin (t) Position of the precipitation front (L) - x _inf* ^supmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for small value of C_o (L) Greek Symbols t Time step used during the numerical computation (T) - Pro Imposed pressure drop (M/L/T²) - Injection time of reactive species (T) - Density of the precipitate (M/L³) - Dynamic viscosity (M/L/T) - <Ri(t)> _infi ^supj Mean chemical flux between a precipitate element j and all the fluid elements i susceptible to interact with the precipitate element j (mole/L³/T)