darcy Namespace Reference

Pressure-based Darcy flow in porous media. More...

Functions
void	validate_material_properties (const dmnType &domain)
void	validate_element_support (const mshType &mesh)
void	b_darcy (ComMod &com_mod, const int eNoN, const double w, const Vector< double > &N, const double h, Array< double > &lR)
void	construct_darcy (ComMod &com_mod, const mshType &lM, const SolutionStates &solutions)
void	darcy_2d (ComMod &com_mod, const int eNoN, const double w, const Vector< double > &N, const Array< double > &Nx, const Array< double > &al, const Array< double > &yl, Array< double > &lR, Array3< double > &lK)
void	darcy_3d (ComMod &com_mod, const int eNoN, const double w, const Vector< double > &N, const Array< double > &Nx, const Array< double > &al, const Array< double > &yl, Array< double > &lR, Array3< double > &lK)

Detailed Description

Pressure-based Darcy flow in porous media.

This namespace implements the Darcy equation for perfusion of porous media with intrinsic dimensions 2 and 3. Material coefficients are homogeneous within each solver domain, permeability is isotropic, and Stokes-flow assumptions apply. The assembled pressure strong form is

\[ \rho \beta \frac{\partial p}{\partial t} - \nabla \cdot \left(\frac{\rho K}{\mu}\nabla p\right) = \rho s. \]

This discretizes the pressure-only strong form. Velocity is not an independent unknown. After pressure is solved, Darcy velocity is evaluated as the derived field

\[ \boldsymbol{q} = -\frac{K}{\mu}\nabla p. \]

The model quantities and their admissible ranges are:

• \(p\): pressure unknown [M/L/T^2].
• \(\boldsymbol{q}\): derived Darcy velocity [L/T].
• \(K\): configured intrinsic scalar permeability [L^2]. Darcy_permeability defaults to \(10^{-15}\) and must satisfy \(K > 0\).
• \(\mu\): configured dynamic viscosity [M/L/T]. Darcy_fluid_viscosity defaults to 1 and must satisfy \(\mu > 0\).
• \(\rho\): configured reference fluid density [M/L^3]. Fluid_density defaults to 0.5 and must satisfy \(\rho > 0\).
• \(\beta\): configured storage/compressibility [L*T^2/M]. Darcy_compressibility defaults to 0 and must satisfy \(\beta \ge 0\).
• \(s\): configured volumetric source provided by Source_term [1/T]; it defaults to 0 and is constant within each configured domain.

Darcy flux output

On supported two- and three-dimensional meshes, the pressure gradient is reconstructed in the mesh coordinates and the derived Darcy flux is

\[ \boldsymbol{q} = -\frac{K}{\mu}\nabla p. \]

Cardiovascular porous-flow context

The following works describe future multi-compartment and microcirculation model extensions than the single-compartment formulation implemented here:

• C. Michler et al., "A computationally efficient framework for the simulation of cardiac perfusion using a multi-compartment Darcy porous-media flow model," DOI 10.1002/cnm.2520.
• G. Montino Pelagi et al., "Modeling cardiac microcirculation for the simulation of coronary flow and 3D myocardial perfusion," DOI 10.1007/s10237-024-01873-z.

Function Documentation

b_darcy()

void darcy::b_darcy	(	ComMod &	com_mod,
		const int	eNoN,
		const double	w,
		const Vector< double > &	N,
		const double	h,
		Array< double > &	lR
	)

Assemble a Darcy boundary contribution into the element residual.

Parameters

[in]	com_mod	Common solver state retained for the common assembly interface.
[in]	eNoN	Number of element nodes.
[in]	w	Weighted boundary quadrature measure.
[in]	N	Shape-function values at the quadrature point.
[in]	h	Prescribed boundary flux contribution.
[in,out]	lR	Element residual.

construct_darcy()

void darcy::construct_darcy	(	ComMod &	com_mod,
		const mshType &	lM,
		const SolutionStates &	solutions
	)

Assemble Darcy volume contributions for all supported elements in a mesh.

Parameters

[in,out]	com_mod	Common solver state and assembly interface.
[in]	lM	Mesh whose Darcy elements are assembled.
[in]	solutions	Solution states used for element-local fields.

darcy_2d()

void darcy::darcy_2d	(	ComMod &	com_mod,
		const int	eNoN,
		const double	w,
		const Vector< double > &	N,
		const Array< double > &	Nx,
		const Array< double > &	al,
		const Array< double > &	yl,
		Array< double > &	lR,
		Array3< double > &	lK
	)

Assemble the residual and tangent for an intrinsic two-dimensional element.

Parameters

[in]	com_mod	Common solver state.
[in]	eNoN	Number of element nodes.
[in]	w	Weighted volume quadrature measure.
[in]	N	Shape-function values at the quadrature point.
[in]	Nx	Mapped spatial shape-function derivatives.
[in]	al	Element-local pressure rates.
[in]	yl	Element-local pressure state.
[in,out]	lR	Element residual.
[in,out]	lK	Element tangent matrix.

darcy_3d()

void darcy::darcy_3d	(	ComMod &	com_mod,
		const int	eNoN,
		const double	w,
		const Vector< double > &	N,
		const Array< double > &	Nx,
		const Array< double > &	al,
		const Array< double > &	yl,
		Array< double > &	lR,
		Array3< double > &	lK
	)

Assemble the residual and tangent for an intrinsic three-dimensional element.

Parameters

[in]	com_mod	Common solver state.
[in]	eNoN	Number of element nodes.
[in]	w	Weighted volume quadrature measure.
[in]	N	Shape-function values at the quadrature point.
[in]	Nx	Mapped spatial shape-function derivatives.
[in]	al	Element-local pressure rates.
[in]	yl	Element-local pressure state.
[in,out]	lR	Element residual.
[in,out]	lK	Element tangent matrix.

validate_element_support()

void darcy::validate_element_support

(

const mshType &

mesh

)

Reject mesh types that Darcy assembly and flux output do not implement.

Parameters

[in]

mesh

Mesh whose element type is checked.

Note

mshType::lFib denotes a one-dimensional mesh embedded in the ambient geometry, not a myocardial material fiber direction. Darcy is currently limited to intrinsic dimensions 2 and 3.

validate_material_properties()

void darcy::validate_material_properties

(

const dmnType &

domain

)

Validate the configured Darcy material coefficients for a domain.

Parameters

[in]

domain

Solver domain whose material properties are checked.