modelDarcyFlow_horizontal

Momentum balance (with mass continous equation) of Darcy

Extends from ClaRa.Basics.Icons.Delta_p.

Information

1. Purpose of model

Implements Darcy's law for radial (horizontal) groundwater flow through a porous medium in cylindrical geometry. Computes mass flow rates and Darcy velocities between all N_r+1 radial interfaces of the aquifer grid, including the well-screen interface.

2. Level of detail, physical effects considered, and physical insight

L4: Resolved mass flow field for each radial cell interface across the full [N_r+1, N_z] grid. Viscosity and density are provided by the enclosing ControlVolumeL4 via Modelica's inner/outer mechanism. Inertial effects are neglected (creeping flow assumption).

3. Limits of validity

- Valid for low Reynolds number flow (Darcy regime).

- Isothermal permeability: k is constant, temperature-dependent viscosity is injected via inner/outer.

- No well-bore storage or skin effects modeled.

4. Interfaces

No connectable ports. Variables are accessed via Modelica inner/outer mechanism from the enclosing ControlVolumeL4:

delta_p_darcy[N_r+1, N_z] — radial pressure differences (outer, set by ControlVolumeL4)

rho_FM[N_r+1, N_z] — fluid density at cell interfaces (outer)

f_eta[N_r+2, N_z] — dynamic viscosity at cell interfaces (outer)

5. Nomenclature

$\dot{m}$	`m_flow[N_r+1, N_z]`	radial mass flow rate at each interface [kg/s]
	`v_darcy[N_r+1, N_z]`	radial Darcy velocity at each interface [m/s]
$\Delta p$	`delta_p_darcy`	pressure difference driving radial flow [Pa]
	`k`	radial permeability of aquifer [m²]
$\eta$	`f_eta`	dynamic viscosity of fluid [Pa·s]
$\rho$	`rho_FM`	fluid density at interface [kg/m³]
	`z_A[N_r, N_z]`	height of control volume [m]
	`r_r[N_r, N_z]`	centre radius of control volume [m]
	`A_s[N_r, N_z]`	outer lateral surface area of control volume [m²]

6. Governing Equations

Radial Darcy mass flow at inner cell interfaces (i = 1 ... N_r) and outer boundary (i = N_r+1):

m_flow = 2*pi*z*rho*k*Delta_p / (eta*(ln(r_outer)-ln(r_r)))

Darcy velocity from mass flow rate and interface area:

m_flow = v_D * A_s * rho

7. Remarks for Usage

This model must be instantiated inside ControlVolumeL4 so that the inner/outer variable declarations are satisfied. It cannot be used as a standalone component.

8. Validation

Tested in check model "ATES.Validierung.Buoyancy.Val_homogenLayer"

9. References

(no remarks)

10. Version History

Model created by Markus Gillner (markus.gillner@tuhh.de), May 2026

Parameters

Type	Name	Description
Real	k	permeability of aquifer
Real	eta	viscosity of aquifer
SI.Density	rho_l	Density of undisturbed or intial fluid
Integer	N_r
Integer	N_z
SI.Area[N_r,N_z]	A_s	outer surface of volume
SI.Length[N_r,N_z]	z_A	heights of control volumes in aquifer
SI.Length[N_r,N_z]	r_r	radius of the center of the control volumes to well
SI.Length[N_r]	r	radius of the outer edge of the control volumes to well
SI.Length	r_0	Outer diameter of well screen

Components

Type	Name	Description
SI.Pressure[N_r + 1,N_z]	delta_p_darcy	pressure loss over a representative elementary volume (REV), to be calculated in control volume
SI.Density[N_r + 1,N_z]	rho_FM	density of fluid in REV, to be calculated in control volume
SI.DynamicViscosity[N_r + 2,N_z]	f_eta	temperature and density dependent dynamic viscosity
SI.MassFlowRate[N_r + 1,N_z]	m_flow	mass flow rate of fluid into a REV
SI.Velocity[N_r + 1,N_z]	v_darcy	Darcy velocity m^2/m^3