10.17 Scalar_Diffusion_Model: Generalized Scalar Diffusion Model

SCALAR_DIFFUSION

Material_name = SCALAR_DIFFUSION

Material_set_number = mset , etc...

The generalized scalar diffusion model is defined as follows.

Note Variable Name Type Default Description

Material_set_number integer [1] Material set number Numat

Matrix_mass_density real [0.0] Matrix mass density

Matrix_compressibility real [0.0] Matrix compressibility

Grains_compressibility real [0.0] Grains compressibility

Fluid_mass_density real [0.0] Mass density (fluid phase)

Fluid_compressibility real [0.0] Compressibility_ [LT² / M]

Fluid_viscosity real [0.0] Viscosity [M / L T]

(1) Ideal_fluid list [off] Ideal fluid/gas option

on / off

Reference_pressure real [0.0] Reference pressure

Reference_temperature real [0.0] Reference temperature

Molecular_mass real [0.0] Molecular mass of fluid/gas

Porosity real [0.0] Porosity

Material_type list [linear] Material type

linear / nonlinear

Number_of_phases integer [1] Number of phases; Nphase

Max_number_of_data_points integer [0] Maximum number of data points used to

define relative permeability/capillary

pressure

(cont’d)

(cont’d)

Note Variable Name Type Default Description

• Permeability

Permeability list [none] Permeability

Type list [*] Form of permeability matrix:

isotropic / anisotropic if isotropic only k₁₁ need be specified.

(2) Name list [*] Name of permeability:

conductivity hydraulic conductivity [L / T]

mobility mobility [L³ T / M]

intrinsic intrinsic permeability [L²]

k_11 real [0.0] Permeability k₁₁

k_22 real [0.0] Permeability k₂₂

k_33 real [0.0] Permeability k₃₃

k_12 real [0.0] Permeability k₁₂

k_23 real [0.0] Permeability k₂₃

k_13 real [0.0] Permeability k₁₃

(3) Exponent_porosity real [0.0] Porosity exponent

• Diffusivity / Dispersivity

Diffusivity list [none] Diffusivity

Type list [*] Form of diffusivity matrix:

isotropic / anisotropic if isotropic only k₁₁ need be specified.

k_11 real [0.0] Diffusivity k₁₁

k_22 real [0.0] Diffusivity k₂₂

k_33 real [0.0] Diffusivity k₃₃

k_12 real [0.0] Diffusivity k₁₂

k_23 real [0.0] Diffusivity k₂₃

k_13 real [0.0] Diffusivity k₁₃

EXAMPLE

Scalar_Diffusion_Model /

material_type = linear /

material_name = scalar_diffusion

material_set_number = 1 /

mass_density = 1.e3 /

porosity = 0.30 /

compressibility = 1.e3-6 /

permeability /

type = isotropic /

name = conductivity /

k_11 = 1.60e-3

Notes /

(1) The equation of state is then of the form:

p =

where p = pressure; = mass density; R = fluid/gas constant (=8314 J/(kmol°K)); w = molecular mass; and T = temperature [°K]. Then, for instance for air:

w = 28.97 kg/kmol, and:

(2) Let k denote the intrinsic permeability (units: [L²]). Then

= hydraulic conductivity [L / T]

= mobility [L³ T / M]

where = viscosity [M / L T], = fluid mass density [M / L³]; and g = acceleration of gravity [L / T²].

(3) The permeability is function of porosity as:

where initial porosity, and porosity exponent.

10.17.1 Multi-Phase Fluid Flow

For multi-phase fluid flow problems the following material data must also be provided.

Note Variable Name Type Default Description

eos_options list [none] Equation of state options

PU_cmi

tough2

Peng_Robinson

dry_gas

reference_pressure real [0.0] reference pressure

reference_temperature real [0.0] reference temperature

matrix_mass_density real [0.0] Matrix mass density

matrix_compressibility real [0.0] Matrix compressibility

grains_compressibility real [0.0] Grains compressibility

(1) Relative_permeability list [*] Relative permeability formula

Touma_Vauclin

Linear

Power

Corey

Grant

perfect_mobility

Fatt_Klikoff

vanGenuchten_Mualem

Verma

Berea

Modified_Corey

Stone_3_phase

(1) rp_i real [0.0] Coefficient RP(i)

(cont’d)

(cont’d)

Note Variable Name Type Default Description

(2) Capillary_pressure list [none] Capillary pressure formula

Touma_Vauclin

Linear

Pickens

Trust

Milly

Leverett

vanGenuchten

none

(2) cp_i real [0.0] Coefficient CP(i)

Notes /

(1) Relative Permeability Functions

IRP = 0 Touma_Vauclin function

The relative permeability is assumed in this case to be given by a curve fit to the experimental data as:

formula_type = 1

formula_type = 2

where p_c = p₂ –p₁ = capillary pressure; p₀ = normalizing pressure;

and S_i = degree of saturation.

IRP = 1 linear function

k_rl increases linearly from 0 to 1 in the range RP(1) ≤ S_l ≤ RP(3);

k_rg increases linearly from 0 to 1 in the range RP(2) ≤ S_g ≤ RP(4).

Restrictions: RP(3) > RP(1); RP(4) > RP(2).

IRP = 2 Power function

k_rl = S_l**RP(1)

k_rg = 1.

IRP = 3 Corey’s curves (1954)

k_rl = Ŝ⁴

k_rg = (1 – Ŝ)² (1 – Ŝ²)

where Ŝ = (S_l – S_lr) / (1 – S_lr – S_gr)

with S_lr = RP(1); S_gr = RP(2)

Restrictions: RP(1) + RP(2) < 1.

IRP = 4 Grant’s curves (Grant, 1977)

k_rl = Ŝ⁴

k_rg = 1 - k_rl

where Ŝ = (S_l – S_lr) / (1 – S_lr – S_gr)

with S_lr = RP(1); S_gr = RP(2)

Restrictions: RP(1) + RP(2) < 1.

IRP = 5 all phases perfectly mobile

k_rg = k_rl = 1 for all saturations; no parameters

IRP = 6 functions of Fatt and Klikoff (1959)

k_rl = (S*)³

k_rg = (1 – S*)³

where S* = (S_l – S_lr) / (1 – S_lr)

with S_lr = RP(1).

Restriction: RP(1) < 1.

IRP = 7 van Genuchten-Mualem model (Mualem, 1976; van Genuchten, 1980)

Gas relative permeability can be chosen as one of the following two forms, the

second of which is due to Corey (1954)

subject to the restriction 0 ≤ k_rl, k_rg ≤ 1

Here, S* = (S_l – S_lr) / (S_ls – S_lr), Ŝ = (S_l – S_lr) / (1 – S_lr – S_gr)

Parameters: RP(1) =

RP(2) = S_lr

RP(3) = S_ls

RP(4) = S_gr

Notation: Parameter is m in van Genuchten’s notation, with m = 1 – 1/n;

parameter n is often written as .

IRP = 8 function of Verma et al. (1985)

k_rl = Ŝ³

k_rg = A + B Ŝ + C Ŝ²

where Ŝ = (S_l – S_lr) / (S_ls – S_lr),

Parameters as measured by Verma et al. (1985) for steam-water flow in an

unconsolidated sand:

Slr = RP(1) = 0.2	B = RP(4) = -1.7615
Sls = RP(2) = 0.895	C = RP(5) = 0.5089
A = RP(3) = 1.259

IRP = 12 modified Corey function

IRP = 14 Stone 3-phase model

a. Aqueous phase:

with irreducible aqueous phase saturation (typically )

m = exponent (typically m=3).

b. Liquid phase:

irreducible liquid phase saturation (typically .

c. Gas phase:

irreducible gas phase saturation (typically 0.01).

Parameters: RP(1) = m

RP(2) =

RP(3) =

RP(4) =

(2) Capillary Pressure Functions

ICP = 0 Touma_Vauclin function

The capillary pressure vs saturation is assumed in this case to be given by a van Genuchten-type curve fit to the experimental data as:

where P_cap = p₂ - p₁ = capillary pressure; p₀ = normalizing pressure;

and S_l = degree of saturation.

Parameters: S_lr = CP(1) S_ls = CP(2) n = CP(3) α = CP(4) p₀ = CP(5)

ICP = 1 linear function

Restriction: CP(3) > CP(2).

ICP = 2 function of Pickens et al. (1979)

with

A = (1 + S_l/S_l0) (S_l0 - S_lr) / (S_l0 + S_lr)

B = 1 – S_l/S_l0

where

P₀ = CP(1) S_lr = CP(2) S_l0 = CP(3) x = CP(4)

Restrictions: 0 < CP(2) < 1 ≤ CP(3); CP(4) ≠ 0

ICP = 3 TRUST capillary pressure (Narasimhan et al., 1978)

where

P₀ = CP(1) S_lr = CP(2) = CP(3) P_e = CP(4)

Restrictions: CP(2) ≥ 0; CP(3) ≠ 0

ICP = 4 Milly’s function (Milly, 1982)

P_cap = +97.783 x 10^A

with

where S_lr = CP(1)

Restriction: CP(1) ≥ 0.

ICP = 6 Leverett’s function (Leverett, 1941; Udell and Fitch, 1985)

with

(T) = surface tension of water (supplied internally)

f(S_l) = 1.417 (1 – S*) – 2.120 (1 – S*)² + 1.263 (1 – S*)³

where

S* = (S_l - S_lr) / (1 – S_lr)

Parameters: P₀ = CP(1) S_lr = CP(2)

Restriction: 0 ≤ CP(2) < 1

ICP = 7 van Genuchten function (van Genuchten, 1980)

subject to the restriction

0 ≤ P_cap ≤ P_max

Here,

S* = (S_l - S_lr) / (S_ls – S_lr)

Parameters: CP(1) = = l – l/n

CP(2) = S_lr (should be chosen smaller than the corresponding

parameter in the relative permeability function; see

note below.)

CP(3) = P₀

CP(4) = P_max

CP(5) = S_ls

Notation: Parameter is m in van Genuchten’s notation, with m = l – l/n;

parameter n is often written as.

Note on parameter choices: In van Genuchten’s derivation (1980), the parameter S_lr for irreducible water saturation is the same in the relative permeability and capillary pressure functions. As a consequence, for S_l → S_lr we have k_rl → 0 and P_cap → , which is unphysical because it implies that the radii of capillary menisci go to zero as liquid phase is becoming immobile (discontinuous). Accordingly, we recommend to always choose a smaller S_lr for the capillary pressure as compared to the relative permeability function.

ICP = 8 no capillary pressure

P_cap = 0 for all saturations; no parameters

10.17.1.1 Fluid Phase Data

Note Variable Name Type Default Description

(1) Phase_number integer [1] Phase number; i ≤ Nphase

Phase_type list [liquid] Phase type

liquid

gas

Phase_name string [none] Phase name; name(s) must be enclosed

in quotation marks.

Mass_density real [0.0] Mass density

Compressibility real [0.0] Compressibility_ [LT² / M]

Viscosity real [0.0] Viscosity M / L T

Saturation real [0.0] Saturation S_i

Minimum_saturation real [0.0] Minimum saturation S_ir

Maximum_saturation real [0.0] Maximum saturation S_is

Mass_fraction_dissolved_solid real [0.0] mass fraction of total dissolved solid (TDS)

(2) formula_type integer [iphase] Relative permeability formula type

(Touma_Vauclin option)

(2) a_coefficient real [0.0] Coefficient A_i in curve fit formula

(2) b_coefficient real [0.0] Coefficient B_i in curve fit formula

(2) Normalizing_pressure real [0.0] Normalizing pressure p_o in curve fit

formula

• Diffusivity / Dispersivity

Diffusivity list [none] Diffusivity

Type list [*] Form of diffusivity matrix:

isotropic / anisotropic if isotropic only k₁₁ need be specified.

k_11 real [0.0] Diffusivity k₁₁

k_22 real [0.0] Diffusivity k₂₂

k_33 real [0.0] Diffusivity k₃₃

k_12 real [0.0] Diffusivity k₁₂

k_23 real [0.0] Diffusivity k₂₃

k_13 real [0.0] Diffusivity k₁₃

Notes/

(1) The wetting phase must be defined as phase_number = 1.

(2) The relative permeability is assumed in this case to be given by a curve fit to the experimental data as:

formula_type = 1

formula_type = 2

where p_c = p₂ –p₁ = capillary pressure; p₀ = normalizing pressure; and S_i = degree of saturation.

References / Bibliography

1. Touma, J. and M. Vauclin, “Experimental and Numerical Analysis of Two-Phase Infiltration in a Partially Saturated Soil,” Transport in Porous Media, Vol. 1, 1986, pp. 27-55.

10.17.1.2 Relative Permeability and Capillary Pressure Data

Note Variable Name Type Default Description

Material_set_number integer [1] Material set number Numat

Data_type list [none] Data type:

Relative_permeability Relative permeability

Capillary_pressure Capillary pressure

• Relative Permeability Case

Data must follow in the form:

(1) < S₁, k_r1(S₁) , k_r2 (S₁) >

< etc..., terminate with a blank record >

• Capillary Pressure Case

Data must follow in the form:

(2) < S₁, p_c1(S₁), p_c2(S₁) >

< etc..., terminate with a blank record >

Notes /

(1) S₁ = degree of saturation for phase 1; S₂ = 1 - S₁

k_r1 (S₁) = relative permeability for phase 1

k_r2 (S₁) = relative permeability for phase 2

(2) p_c1 (S₁) = capillary pressure for phase 1 invasion

p_c2 (S₁) = capillary pressure for phase 1 drainage

EXAMPLE

Scalar_diffusion_model /

material_type = linear /

material_name = scalar_diffusion /

number_of_phases = 2 /

max_number_of_data_points = 11

material_set_number = 1 /

porosity = 0.30 /

permeability /

type = isotropic /

name = intrinsic /

k_11 = 2.1248e-11

phase_number = 1 /

mass_density = 62.4 /

compressibility = 1.0e-6 /

viscosity = 2.088543e-5

phase_number = 2 /

mass_density = 49.92 /

compressibility = 1.0e-6 /

viscosity = 8.3541723e-5

material_set_number = 1 /

data_type = Relative_permeability

0.20 0.00 0.60

0.25 0.02 0.47

0.30 0.04 0.38

0.35 0.07 0.31

0.40 0.09 0.25

0.45 0.13 0.18

0.50 0.17 0.13

0.55 0.22 0.09

0.60 0.28 0.05

0.65 0.35 0.02

0.70 0.45 0.00

material_set_number = 1 /

data_type = Capillary_pressure

0.200 208.854 208.854

0.225 173.349 173.349

0.250 148.287 148.287

0.300 121.136 121.136

0.350 102.339 102.339

0.400 87.719 87.719

0.500 68.922 68.922

0.650 52.213 52.213

0.700 48.663 48.663

Notes . .