Physical processes and governing equations - Revision history

Klmos: /* Governing equations */

2019-05-29T14:32:22Z

‎Governing equations

Klmos: /* Contaminant transport */

2019-04-10T17:04:56Z

‎Contaminant transport

Klmos: /* Contaminant transport */

2019-04-10T17:04:04Z

‎Contaminant transport

Klmos: /* Governing equations */ Kf and Df non-bold

2019-04-10T17:02:28Z

‎Governing equations: Kf and Df non-bold

Klmos: /* Governing equations */ fixed D and K (bold)

2019-04-10T17:00:36Z

‎Governing equations: fixed D and K (bold)

Klmos: /* Governing equations */ More work on formatting of variables and equations

2019-04-10T16:56:21Z

‎Governing equations: More work on formatting of variables and equations

Klmos: corrected italics in tables

2019-04-10T16:45:28Z

corrected italics in tables

Klmos: /* Governing equations */ Improved tables

2019-04-10T16:33:00Z

‎Governing equations: Improved tables

Klmos: /* Contaminant transport */

2018-02-19T13:17:55Z

‎Contaminant transport

Klmos: /* Contaminant transport */

2018-02-15T13:20:23Z

‎Contaminant transport

@@ Line 52: / Line 52: @@
 The cubic law considers the fracture flow as flow between parallel plates with no roughness.
-With the ''Cubic law'', the fracture conductivity can be estimated base on the hydraulic aperture: <br>
+With the ''Cubic law'', the fracture conductivity can be estimated as: <br>
 <math> K_\text{f} = \frac{(2b)^2 \rho_\text{w} g}{12 \mu_\text{w}} </math>
@@ Line 123: / Line 123: @@
 === Coupling of flow and transport between fractures and matrix ===
 The fracture and the matrix system are coupled by the continuity of water fluxes and solute fluxes at the fracture-matrix interface.
-Furthermore, the hydraulic head and the solute concentration are continuous at the interface.
+Furthermore, the hydraulic head and the solute concentration are continuous at the interface between fractures and matrix.
 {{Return}}
 [[Category:Introduction]]

@@ Line 106: / Line 106: @@
 | <math> \rho_\text{b} </math> || bulk density || $ \text{kg/m}^3 $
 |-
-| <math> k_\text{d} </math> || sorption coefficient || $ \text{m}^3/kg} $
+| <math> k_\text{d} </math> || sorption coefficient || $ \text{m}^3\text{/kg} $
 |-
 | <math> c </math> || concentration || $ \text{kg/m}^3 $

@@ Line 90: / Line 90: @@
 ''Transport in fractures'': <br>
-<math> 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b D_\text{f} \nabla_T c_\text{f}) = q_\text{f}</math>
+<math> 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b D_\text{f} \nabla_\text{T} c_\text{f}) = q_\text{f}</math>
 Variables are:
@@ Line 117: / Line 117: @@
 |-
 | <math> \mathbf{D}_\text{m}\text{,} D_\text{f} </math> || hydrodynamic dispersion tensor and coefficient|| $ \text{m}^2\text{/s} $
 |}

@@ Line 53: / Line 53: @@
 With the ''Cubic law'', the fracture conductivity can be estimated base on the hydraulic aperture: <br>
-<math> \mathbf{K}_\text{f} = \frac{(2b)^2 \rho_\text{w} g}{12 \mu_\text{w}} </math>
+<math> K_\text{f} = \frac{(2b)^2 \rho_\text{w} g}{12 \mu_\text{w}} </math>
 {| class="wikitable"
@@ Line 69: / Line 69: @@
 | <math> q_\text{darcy} </math> || Darcy flux || <math> \text{m}^3/\text{(m}^2 \text{s)} </math>
 |-
-| <math> \mathbf{K}_\text{f} </math> || fracture conductivity || <math> \text{m/s} </math>
+| <math> K_\text{f} </math> || fracture conductivity || <math> \text{m/s} </math>
 |-
 | <math> 2b </math> || hydraulic fracture aperture || <math> \text{m} </math>
@@ Line 90: / Line 90: @@
 ''Transport in fractures'': <br>
-<math> 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b \mathbf{D}_\text{f} \nabla_T c_\text{f}) = q_\text{f}</math>
+<math> 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b D_\text{f} \nabla_T c_\text{f}) = q_\text{f}</math>
 Variables are:
@@ Line 116: / Line 116: @@
 | <math> R_\text{f} </math> || retardation factor accounting for sorption to the fracture walls || $ 1 $
 |-
-| <math> \mathbf{D}_\text{m}\text{,} \mathbf{D}_\text{f} </math> || hydrodynamic dispersion tensors|| $ \text{m}^2\text{/s} $
+| <math> \mathbf{D}_\text{m}\text{,} D_\text{f} </math> || hydrodynamic dispersion tensor and coefficient|| $ \text{m}^2\text{/s} $
 |}

@@ Line 43: / Line 43: @@
 ''Mass balance equation'' for incompressible fluid flow: <br>
-<math> S_\text{s} \frac{\partial h}{\partial t} - \nabla \cdot (k \nabla  h) = q_\text{w} </math>
+<math> S_\text{s} \frac{\partial h}{\partial t} - \nabla \cdot (\mathbf{K} \nabla  h) = q_\text{w} </math>
 ''Darcy's law'' (also used in the fractures with the fracture hydraulic conductivity $k_f$) relates water fluxes to hydraulic head gradients: <br>
-<math> q_\text{darcy} = -k \nabla h </math>
+<math> q_\text{darcy} = -\mathbf{K} \nabla h </math>
 The hydraulic conductivities in the fractures can be several orders of magnitude higher than the conductivity of the limestone matrix.
@@ Line 53: / Line 53: @@
 With the ''Cubic law'', the fracture conductivity can be estimated base on the hydraulic aperture: <br>
-<math> k_\text{f} = \frac{(2b)^2 \rho_\text{w} g}{12 \mu_\text{w}} </math>
+<math> \mathbf{K}_\text{f} = \frac{(2b)^2 \rho_\text{w} g}{12 \mu_\text{w}} </math>
 {| class="wikitable"
@@ Line 63: / Line 63: @@
 | <math> t </math> || time || <math> \text{s} </math>
 |-
-| <math> k </math> || hydraulic conductivity || $\text{m/s}$
+| <math> \mathbf{K} </math> || hydraulic conductivity || $\text{m/s}$
 |-
 | <math> q_\text{w} </math> || water sources/sinks || <math> \text{m}^3\text{/m}^3 </math>
@@ Line 69: / Line 69: @@
 | <math> q_\text{darcy} </math> || Darcy flux || <math> \text{m}^3/\text{(m}^2 \text{s)} </math>
 |-
-| <math> k_\text{f} </math> || fracture conductivity || <math> \text{m/s} </math>
+| <math> \mathbf{K}_\text{f} </math> || fracture conductivity || <math> \text{m/s} </math>
 |-
 | <math> 2b </math> || hydraulic fracture aperture || <math> \text{m} </math>
@@ Line 84: / Line 84: @@
 ''Transport equation'' for a dissolved species: <br>
-<math>(n + \rho_\text{b} k_\text{d}) \frac{\partial c}{\partial t} + \nabla \cdot (n \mathbf{v} c) - \nabla \cdot (n D_\text{m} \nabla c) = q_\text{m}</math>
+<math>(n + \rho_\text{b} k_\text{d}) \frac{\partial c}{\partial t} + \nabla \cdot (n \mathbf{v} c) - \nabla \cdot (n \mathbf{D}_\text{m} \nabla c) = q_\text{m}</math>
 The first term describes the storage (here including retardation due to linear sorption), the second and third terms describe advective and dispersive transport of a solute.
@@ Line 90: / Line 90: @@
 ''Transport in fractures'': <br>
-<math> 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b D_\text{f} \nabla_T c_\text{f}) = q_\text{f}</math>
+<math> 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b \mathbf{D}_\text{f} \nabla_T c_\text{f}) = q_\text{f}</math>
 Variables are:
@@ Line 116: / Line 116: @@
 | <math> R_\text{f} </math> || retardation factor accounting for sorption to the fracture walls || $ 1 $
 |-
-| <math> D_\text{m}\text{,} D_\text{f} </math> || hydrodynamic dispersion coefficients || $ \text{m}^2\text{/s} $
+| <math> \mathbf{D}_\text{m}\text{,} \mathbf{D}_\text{f} </math> || hydrodynamic dispersion tensors|| $ \text{m}^2\text{/s} $
 |}

@@ Line 43: / Line 43: @@
 ''Mass balance equation'' for incompressible fluid flow: <br>
-<math> S_s \frac{\partial h}{\partial t} - \nabla \cdot (k \nabla  h) = q_w </math>
+<math> S_\text{s} \frac{\partial h}{\partial t} - \nabla \cdot (k \nabla  h) = q_\text{w} </math>
 ''Darcy's law'' (also used in the fractures with the fracture hydraulic conductivity $k_f$) relates water fluxes to hydraulic head gradients: <br>
-<math> q_{darcy} = -k \nabla h </math>
+<math> q_\text{darcy} = -k \nabla h </math>
 The hydraulic conductivities in the fractures can be several orders of magnitude higher than the conductivity of the limestone matrix.
@@ Line 53: / Line 53: @@
 With the ''Cubic law'', the fracture conductivity can be estimated base on the hydraulic aperture: <br>
-<math> k_f = \frac{(2b)^2 \rho_w g}{12 \mu} </math>
+<math> k_\text{f} = \frac{(2b)^2 \rho_\text{w} g}{12 \mu_\text{w}} </math>
 {| class="wikitable"
@@ Line 74: / Line 74: @@
 |-
 | <math> \rho_\text{w} </math> || density of water || <math> \text{kg/m}^3 </math>
 |-
 | <math> g </math> || gravity constant || <math> \text{m/s}^2 </math>
@@ Line 82: / Line 84: @@
 ''Transport equation'' for a dissolved species: <br>
-<math>(n + \rho_b k_d) \frac{\partial c}{\partial t} + \nabla \cdot (nvc) - \nabla \cdot (n D_m \nabla c) = q_m</math>
+<math>(n + \rho_\text{b} k_\text{d}) \frac{\partial c}{\partial t} + \nabla \cdot (n \mathbf{v} c) - \nabla \cdot (n D_\text{m} \nabla c) = q_\text{m}</math>
 The first term describes the storage (here including retardation due to linear sorption), the second and third terms describe advective and dispersive transport of a solute.
@@ Line 88: / Line 90: @@
 ''Transport in fractures'': <br>
-<math> 2b R_f \frac{\partial c_f}{\partial t} + \nabla \cdot (2b v_f c_f) - \nabla \cdot (2b D_f \nabla_T c_f) = q_f</math>
+<math> 2b R_\text{f} \frac{\partial c_\text{f}}{\partial t} + \nabla \cdot (2b \mathbf{v}_\text{f} c_\text{f}) - \nabla \cdot (2b D_\text{f} \nabla_T c_\text{f}) = q_\text{f}</math>
 Variables are:
@@ Line 98: / Line 100: @@
 | <math> n </math> || porosity || $ - $
 |-
-| <math> q_\text{m} </math> || mass sources and sinks, also degradation || $ \text{kg/(m}^3 \text{s)} $
+| <math> q_\text{m}\text{,} q_\text{f} </math> || mass sources and sinks, also degradation || $ \text{kg/(m}^3 \text{s)} $
 |-
 | <math> c_\text{f} </math> || concentration in the fractures || $ \text{kg/m}^3 $
@@ Line 104: / Line 106: @@
 | <math> \rho_\text{b} </math> || bulk density || $ \text{kg/m}^3 $
 |-
-| <math> k_\text{d} </math> || sorption coefficient || $ \text{L/kg} $
+| <math> k_\text{d} </math> || sorption coefficient || $ \text{m}^3/kg} $
 |-
 | <math> c </math> || concentration || $ \text{kg/m}^3 $
 |-
-| <math> v </math> || flow velocity in the matrix || $ \text{m/s} $
+| <math> \mathbf{v} </math> || flow velocity in the matrix || $ \text{m/s} $
 |-
-| <math> v_\text{f} </math> || flow velocity in the fractures || $ \text{m/s} $
+| <math> \mathbf{v}_\text{f} </math> || flow velocity in the fractures || $ \text{m/s} $
 |-
 | <math> R_\text{f} </math> || retardation factor accounting for sorption to the fracture walls || $ 1 $

@@ Line 59: / Line 59: @@
 ! Variable !! Name !! Unit
 |-
-| <math> S_s </math> || specific storage || <math> 1/m </math>
+| <math> S_\text{s} </math> || specific storage || <math> \text{1/m} </math>
 |-
-| <math> t </math> || time || <math> s </math>
+| <math> t </math> || time || <math> \text{s} </math>
 |-
-| <math> K </math> || hydraulic conductivity || <math> m/s </math>
+| <math> k </math> || hydraulic conductivity || $\text{m/s}$
 |-
-| <math> q_w </math> || water sources/sinks || <math> m^3/m^3 </math>
+| <math> q_\text{w} </math> || water sources/sinks || <math> \text{m}^3\text{/m}^3 </math>
 |-
-| <math> q_{darcy} </math> || Darcy flux || <math> m^3/(m^2 s) </math>
+| <math> q_\text{darcy} </math> || Darcy flux || <math> \text{m}^3/\text{(m}^2 \text{s)} </math>
 |-
-| <math> k_f </math> || fracture conductivity || <math> m/s </math>
+| <math> k_\text{f} </math> || fracture conductivity || <math> \text{m/s} </math>
 |-
-| <math> 2b </math> || hydraulic fracture aperture || <math> m </math>
+| <math> 2b </math> || hydraulic fracture aperture || <math> \text{m} </math>
 |-
-| <math> \rho_w </math> || density of water || <math> kg/m^3 </math>
+| <math> \rho_\text{w} </math> || density of water || <math> \text{kg/m}^3 </math>
 |-
-| <math> g </math> || gravity constant || <math> m/s^2 </math>
+| <math> g </math> || gravity constant || <math> \text{m/s}^2 </math>
 |}
@@ Line 98: / Line 98: @@
 | <math> n </math> || porosity || $ - $
 |-
-| <math> q_m </math> || mass sources and sinks, also degradation || $ kg/(m^3 s) $
+| <math> q_\text{m} </math> || mass sources and sinks, also degradation || $ \text{kg/(m}^3 \text{s)} $
 |-
-| <math> c_f </math> || concentration in the fractures || $ mg/L $
+| <math> c_\text{f} </math> || concentration in the fractures || $ \text{kg/m}^3 $
 |-
-| <math> \rho_b </math> || bulk density || $ kg/m^3 $
+| <math> \rho_\text{b} </math> || bulk density || $ \text{kg/m}^3 $
 |-
-| <math> k_d </math> || sorption coefficient || $ L/kg $
+| <math> k_\text{d} </math> || sorption coefficient || $ \text{L/kg} $
 |-
-| <math> c </math> || concentration || $ mg/L $
+| <math> c </math> || concentration || $ \text{kg/m}^3 $
 |-
-| <math> v </math> || flow velocity in the matrix || $ m/s $
+| <math> v </math> || flow velocity in the matrix || $ \text{m/s} $
 |-
-| <math> v_f </math> || flow velocity in the fractures || $ m/s $
+| <math> v_\text{f} </math> || flow velocity in the fractures || $ \text{m/s} $
 |-
-| <math> R_f </math> || retardation factor accounting for sorption to the fracture walls || $ 1 $
+| <math> R_\text{f} </math> || retardation factor accounting for sorption to the fracture walls || $ 1 $
 |-
-| <math> D_m, D_f </math> || hydrodynamic dispersion coefficients || $ m^2/s $
+| <math> D_\text{m}\text{,} D_\text{f} </math> || hydrodynamic dispersion coefficients || $ \text{m}^2\text{/s} $
 |}