OPM - Reference Documentation for ewoms

Immiscible Two-Phase Model Using the Richards Equation. More...

Collaboration diagram for Richards:

Files
file	richardsproperties.hh
	Contains the property declarations for the Richards model.

Classes
class	Ewoms::RichardsBoundaryRateVector< TypeTag >
	Implements a boundary vector for the fully implicit Richards model. More...

class	Ewoms::RichardsExtensiveQuantities< TypeTag >
	Calculates and stores the data which is required to calculate the flux of fluid over a face of a finite volume. More...

struct	Ewoms::RichardsIndices
	Indices for the primary variables/conservation equations of the Richards model. More...

class	Ewoms::RichardsIntensiveQuantities< TypeTag >
	Intensive quantities required by the Richards model. More...

class	Ewoms::RichardsLocalResidual< TypeTag >
	Element-wise calculation of the residual for the Richards model. More...

class	Ewoms::RichardsModel< TypeTag >
	This model implements a variant of the Richards equation for quasi-twophase flow. More...

class	Ewoms::RichardsNewtonMethod< TypeTag >
	A Richards model specific Newton method. More...

class	Ewoms::RichardsPrimaryVariables< TypeTag >
	Represents the primary variables used in the Richards model. More...

class	Ewoms::RichardsRateVector< TypeTag >
	Implements a vector representing mass, molar or volumetric rates. More...

Detailed Description

Immiscible Two-Phase Model Using the Richards Equation.

In the unsaturated zone, Richards' equation is frequently used to approximate the water distribution above the groundwater level. It can be derived from the two-phase equations, i.e.

$\frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t} - \mathrm{div} \left\{ \rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K}\; \mathbf{grad}\left[ p_\alpha - g\rho_\alpha \right] \right\} = q_\alpha,$

where $\alpha \in \{w, n\}$ is the index of the fluid phase, $\rho_\alpha$ is the fluid density, $S_\alpha$ is the fluid saturation, $\phi$ is the porosity of the soil, $k_{r\alpha}$ is the relative permeability for the fluid, $\mu_\alpha$ is the fluid's dynamic viscosity, $\mathbf{K}$ is the intrinsic permeability tensor, $p_\alpha$ is the fluid phase pressure and $g$ is the potential of the gravity field.

In contrast to the "full" two-phase model, the Richards model assumes that the non-wetting fluid is gas and that it thus exhibits a much lower viscosity than the (liquid) wetting phase. (This assumption is quite realistic in many applications: For example, at atmospheric pressure and at room temperature, the viscosity of air is only about $1\%$ of the viscosity of liquid water.) As a consequence, the $\frac{k_{r\alpha}}{\mu_\alpha}$ term typically is much larger for the gas phase than for the wetting phase. Using this reasoning, the Richards model assumes that $\frac{k_{rn}}{\mu_n}$ is infinitely large compared to the same term of the liquid phase. This implies that the pressure of the gas phase is equivalent to the static pressure distribution and that therefore, mass conservation only needs to be considered for the liquid phase.

The model thus choses the absolute pressure of the wetting phase $p_w$ as its only primary variable. The wetting phase saturation is calculated using the inverse of the capillary pressure, i.e.

$S_w = p_c^{-1}(p_n - p_w)$

holds, where $p_n$ is a reference pressure given by the problem's referencePressure() method. Nota bene, that the last step assumes that the capillary pressure-saturation curve can be uniquely inverted, i.e. it is not possible to set the capillary pressure to zero if the Richards model ought to be used!

Files

Classes

Detailed Description