modelControlVolumeL4

Control volume of an aquifer with 2D mass flow

Extends from TransiEnt.Basics.Icons.Fluid_Volume.

Information

1. Purpose of model

Discretized aquifer control volume model for 2D (radial-vertical) flow. Implements the coupled energy balance, mass continuity, and Darcy pressure equations for an [N_r × N_z] grid of aquifer cells. Includes optional buoyancy-driven vertical flow and temperature-dependent fluid properties (density, dynamic viscosity).

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

L4: Fully discretized control volume array. Each cell is ideally mixed (lumped). Physical effects considered:

Radial and vertical mass transport (Darcy flow, buoyancy)
Enthalpy transport by mass flow
Heat conduction via thermal resistors (coupled through port[N_r, N_z])
Temperature-dependent dynamic viscosity and density

3. Limits of validity

- No chemical reactions; single-component fluid.

- Density differences at radial cell interfaces are interpolated linearly between adjacent cells.

- Buoyancy switched on/off via Setting record boolean.

4. Interfaces

Interface diagram: fluid ports on all four boundaries, internal heat port array, signal outputs q_long and q_trans

fluidPortEast[N_z]: fluid inlet at inner radial boundary (Fluid, inlet)

fluidPortWest[N_z]: fluid outlet at outer radial boundary (Fluid, outlet)

fluidPortUp[N_r]: fluid outlet at top boundary (Fluid, outlet)

fluidPortDown[N_r]: fluid inlet at bottom boundary (Fluid, inlet)

port[N_r, N_z]: internal heat port array, one per cell — connected externally to the thermal resistor network (Heat)

q_long[N_r, N_z]: radial Darcy velocity output for dispersion calculation (Signal, outlet)

q_trans[N_r, N_z+1]: vertical Darcy velocity output for dispersion calculation (Signal, outlet)

5. Nomenclature

$T_{i,j}$	T[N_r, N_z]	cell temperature [K]
$p_{i,j}$	p[N_r, N_z]	cell pressure [Pa]
$U_{i,j}$	U[N_r, N_z]	internal energy of cell [J]
$\dot{H}_{i,j}$	H_flow[N_r+1, N_z]	enthalpy flow at radial interfaces [W]
$\dot{m}_{i,j}$	m_flow[N_r+1, N_z]	radial mass flow rate at interfaces [kg/s]
$\dot{Q}_{i,j}$	port[i,j].Q_flow	heat flow from thermal resistors [W]
$q_\mathrm{long}$	q_long[N_r, N_z]	radial Darcy velocity at cell centre (for dispersion) [m/s]
$q_\mathrm{trans}$	q_trans[N_r, N_z+1]	vertical Darcy velocity at horizontal cell interface (for dispersion) [m/s]
	n	porosity of aquifer [-]
$\rho_s$	rho_s	density of aquifer solid [kg/m³]
$c_{p,s}$	Cp_s	specific heat capacity of aquifer solid [J/(kg·K)]
	geo.V[i,j]	cell volume [m³]
$\rho_l$	rho_l	fluid density taken from TILMedia [kg/m³]
$c_{p,l}$	Cp_l	specific heat capacity of fluid taken from TILMedia [J/(kg·K)]
$\dot{H}_{\mathrm{buoy}}$	H_flow_buoyancy[N_r, N_z+1]	buoyancy-driven enthalpy flow rate [W]
$\dot{m}_{\mathrm{buoy}}$	m_flow_buoyancy[N_r, N_z+1]	buoyancy-driven mass flow rate [kg/s]
$\Delta p$	delta_p_darcy[N_r+1, N_z]	Darcy pressure loss at each radial cell interface [Pa]

6. Governing Equations

Energy balance for each aquifer cell (i,j):

(n*cp*rho_l + (1-n)*rho_s*cp_s)*V*dT/dt = sum H_flow + sum H_buoy + Q_dot

Mass continuity for each aquifer cell:

V * d_rho/dt = sum m_flow + sum m_flow_buoy

Static pressure balance (Darcy pressure loss calculated in DarcyFlow_horizontal):

$0 = p_{i-1,j} - p_{i,j} - Delta_p_Darcy_{i,j}$

7. Remarks for Usage

Requires an enclosing model to declare inner Modelica.SimCenter (for fluid type) and inner variables delta_p_darcy, rho_FM, f_eta, delta_p_v, rho_FM_v, f_eta_v (accessed via inner/outer from DarcyFlow sub-models). Thermal resistors are connected externally via port[N_r, N_z].

8. Validation

Validated in Validation.Buoyancy.Val_homogenLayer.mo

9. References

(no remarks)

10. Version History

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

Model revised by Markus Gillner (markus.gillner@tuhh.de), June 2026

- Corrected the interface list (removed non-existent port_east/west/up/down; the model has a single internal heat port array port[N_r, N_z]), fixed a character-encoding artifact, and corrected the delta_p_darcy dimension.

Parameters

Type	Name	Default	Description
SI.Temperature	T_start	Parameters.T_initial	Start value for temperature

Connectors

Type	Name	Default	Description
TransiEnt.Basics.Interfaces.Thermal.FluidPortIn[N_z]	fluidPortEast
TransiEnt.Basics.Interfaces.Thermal.FluidPortOut[N_z]	fluidPortWest
TransiEnt.Basics.Interfaces.Thermal.FluidPortIn[N_r]	fluidPortDown
TransiEnt.Basics.Interfaces.Thermal.FluidPortOut[N_r]	fluidPortUp
Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a[N_r,N_z]	port
Modelica.Blocks.Interfaces.RealOutput[N_r,N_z + 1]	q_trans
Modelica.Blocks.Interfaces.RealOutput[N_r,N_z]	q_long

Components

Type	Name	Description
SI.Mass[N_r,N_z]	m
SI.SpecificEnthalpy[N_r,N_z]	h
SI.Temperature[N_r,N_z]	T
SI.Pressure[N_r,N_z]	p
Real[N_r,N_z]	drhodt
Real[N_r,N_z]	der_T
SI.Energy[N_r,N_z]	U
SI.SpecificHeatCapacity[N_r,N_z]	c_w
SI.EnthalpyFlowRate[N_r + 1,N_z]	H_flow
SI.MassFlowRate[N_r + 1,N_z]	m_flow
SI.MassFlowRate[N_r,N_z + 1]	m_flow_buoyancy	mass flow induced by temperature driven density differences
SI.EnthalpyFlowRate[N_r,N_z + 1]	H_flow_buoyancy
SI.Density[N_r + 1,N_z]	rho_FM
SI.Density[N_r,N_z + 1]	rho_FM_v	Mean Density at cross surface area of control volumes
SI.DynamicViscosity[N_r + 2,N_z]	f_eta	temperature and density dependent dynamic viscosity
SI.PressureDifference[N_r + 1,N_z]	delta_p_darcy
SI.PressureDifference[N_r,N_z + 1]	delta_p_v	pressure difference between two control volumes in vertical neighbourhood
SI.DynamicViscosity[N_r,N_z + 2]	f_eta_v
TILMedia.VLEFluid_ph[N_z]	fluidEast
TILMedia.VLEFluid_ph[N_z]	fluidWest
TILMedia.VLEFluid_pT[N_r,N_z]	bulk
TILMedia.VLEFluid_ph[N_r]	fluidUp
TILMedia.VLEFluid_ph[N_r]	fluidDown
ClaRa.SimCenter	simCenter
PressureLoss.DarcyFlow_horizontal	darcyFlow_horizontal
Records.Subsurface_Basic	Parameters
Geometry	geo
PressureLoss.DarcyFlow_vertical	darcyFlow_vertical
Records.Setting	setting