darcyInterTransportFoam models the surface flow as a transient, immiscible, two-phase flow using the VOF interface capturing approach. This method separates the two phases (i.e., air and water) by a continuous, free interface, whose location and shape are determined as a property of the phase fraction field, αi (–) (Schulze and Thorenz, 2015). Such deformation of the interface allows us to account for the actual flow dynamics in the surface domain, which in turn leads to a correct estimate of the flow, solute and heat exchanges across the sediment-water interface.

As commonly assumed for free-surface fluids, the surface flow is additionally considered to be incompressible (Young et al., 2010). Consequently, there is no need for equations of state to relate the thermodynamic variables (Versteeg and Malalasekera, 2007) and the two-phase flow is adequately governed by the continuity equation (Eq. 1), the Navier-Stokes equations (Eq. 2) and the αi transport equation (Eq. 6).

The VOF method solves the surface flow dynamics considering the two immiscible phases as one fluid. The resulting single-phase is assumed to be Newtonian and incompressible, according to the physical properties of both surface phases. As a result, the continuity equation can be simplified to: 1∇⋅Usur,i=0, and the momentum equation to: 2∂ρsur,iUsur,i∂t+∇⋅ρsur,iUsur,iUsur,i-∇⋅μsur,i∇Usur,i+∇Usur,jT︸τij=-∇prghsur,i-g⋅xi∇ρsur,i+ρsur,iSUsur,i, where Usur,i and Usur,j respectively denote the single-phase velocity field in the x and y directions (m s⁻¹), t is the time (s), ρsur,i and μsur,i are respectively the single-phase density (kg m-3) and dynamic viscosity fields (N m⁻²), τij is the deviatoric viscous stress tensor field (N m⁻²), prghsur,i is the modified pressure field (Pa) obtained by subtracting the hydrostatic pressure to the static pressure, g is the gravitational acceleration vector (m s-2), xi is the cell position vector field (m) and SUsur,i accounts for any source/sink term field (m s⁻¹). prghsur,i enables to avoid potential steep pressure gradients caused by hydrostatic effects as well as simplifying the definition of pressure BCs (Rusche, 2003).

The single-phase parameters ρsur,i and μsur,i are calculated by weighted averaging of the physical properties of both surface fluids based on αi: 3ρsur,i=αiρwsur+1-αiρasur,4μsur,i=αiμwsur,i+1-αiμasur,i, where ρwsur and ρasur are respectively the densities of the surface water and the air (kg m⁻³) and μwsur,i and μasur,i represent the surface water and air dynamic viscosity fields (N m⁻²). The latter properties result from: 5μpsur,i=μphyspsur+μturb,i, where p is the subscript for the respective surface phase, w (water) or a (air), and μphyspsur and μturb,i are respectively the physical and turbulent dynamic viscosities (N m⁻²). The μturb,i field is computed by the selected turbulence model. As most OpenFOAM solvers, darcyInterTransportFoam allows to choose among different Reynolds-Averaged Navier-Stokes (RANS), Large Eddy Simulation (LES) and hybrid RANS-LES Detached Eddy Simulation (DES) models to simulate the turbulent hydrodynamics. For the test case in this study, the widely applied k-ω SST (Shear Stress Transport) RANS model (Menter, 1994) was chosen to resolve the different scales of turbulence.

An advanced transport equation is used to accurately determine the distribution of αi through the surface domain and, therefore, the position of the free surface. This is achieved by introducing an additional convective term into the standard αi transport equation that “compresses” the free surface (Berberović et al., 2009), resulting in a sharper interface resolution which ultimately enables proper conservation of αi throughout the simulation domain (Weller, 2002): 6∂αi∂t+∇⋅αiUsur,i+∇⋅[(1-αi)αiUr,i]︸Compression term=0, where Ur,i=Uwsur,i-Uasur,i is the relative velocity field (m s⁻¹) between the two surface phases, known as the compression velocity (Berberović et al., 2009).

The resultant αi is strictly bounded between 0 (air) and 1 (water) in each computational cell of the surface domain. Any value in between will imply a mixture of both phases. As for the interface, the VOF method estimates its location as a property of αi rather than explicitly computing it, being this normally considered to be around αi=0.5.

Following the Usur,i, psur,i and αi solutions, the newly implemented fields of surface water depth, dsur,i (m), piezometric level, piezosur,i (m) and surface hydraulic head, hsur,i (m), are subsequently calculated according to Bernoulli's equation.

The new FCM provides two simulation options for modelling the transport of a conservative solute in the surface domain. The first option solves the one-fluid advection-diffusion equation previously implemented in hyporheicScalarInterFoam, based on the formulation introduced by Teuber (2020). Further details on this approach can be found in Haroun et al. (2010), Nieves-Remacha et al. (2015) and Severin (2017).

However, in the new solver the computation of the surface diffusion coefficient field, Dsur,i (m² s⁻¹), differs from that in hyporheicScalarInterFoam in that it also accounts for the contribution of the physical (or molecular) diffusivity, Dphys (m² s⁻¹), to the total diffusivity across the domain, a factor particularly relevant in low-turbulence zones or for fine-scale processes: 7Dsur,i=Dphys+νturb,iScturbDturb,i, where Scturb is the turbulent Schmidt number (–) and Dturb,i is the turbulent diffusivity, (m² s⁻¹). Both Dphys and Scturb have to be defined by the user. Alternatively, Dsur,i can be specified via a constant value for the whole surface domain, offering a simplified yet practical setup not available in the original solver.

The second simulation option is newly implemented and designed to reduce computational costs when the focus is limited to a single simulated phase. Based on the two-phase advection-diffusion equation contained in the standard scalarTransport function object (OpenCFD Ltd., 2023), this approach uses the computed αi to delimit the extension of the solute transport simulation within the system. Accordingly, the passive transport of any species is only simulated in those cells whose αi is equal to 1, skipping the rest in the computation. The described governing equation has the subsequent form: 8∂Csur,i∂t+∇⋅αiUsur,iCsur,i-∇⋅αiDsur,i∇Csur,i=αiSCsur,i, where Csur,i is the surface concentration field (kg m⁻³) and SCsur,i accounts for any concentration source/sink field (kg m⁻³).

The latter approach was the one selected to model the conservative solute transport in the test case presented in this paper.

Analogously to solute transport, darcyInterTransportFoam enables to solve the surface heat transfer through two new simulation approaches. One of these is based on Aryal (2019) and simulates the distribution of temperature in both surface phases using the following energy equation: 9ρCpsur,i∂Tsur,i∂t+∇⋅ρCpsur,iUsur,iTsur,i-∇⋅Ksur,i∇Tsur,i=ρCpsur,iSTsur,i, where ρ is the density (kg m⁻³), Cpsur,i is the surface specific heat capacity field (m² s⁻² °C⁻¹), Tsur,i is the surface temperature field (°C), Ksur,i is the surface thermal conductivity field (kg m s⁻³ °C⁻¹) and STsur,i represents any heat source/sink field (°C). As for ρsur,i and μsur,i, the ρCpsur,i field is computed as a single-phase property by weighted averaging of ρ and Cp of both phases based on αi. Ksur,i is likewise evaluated as a function of αi, with the thermal conductivity of each surface phase, Kpsur (kg m s⁻³ °C⁻¹), defined as: 10Kpsur=ρpsurCppsurvphyspsurPrpsur, where vphyspsur, Cppsur, Prpsur denote, respectively, the physical kinematic viscosity (m² s⁻¹), the specific heat capacity (m² s⁻² °C⁻¹) and the Prandtl number (–) of either the air or water phase.

The second simulation approach restricts the computation of heat transfer to a single surface phase, thereby reducing computational demands. Similarly to solute transport, this is done based on the αi solution. As such, the transfer of heat is skipped in all those computational cells whose αi differs from 1. The resulting energy equation has the form of: 11ρwsurCpwsur∂Tsur,i∂t+∇⋅ρwsurCpwsurαiUsur,iTsur,i-∇⋅Ksur,i∇Tsur,i=ρwsurCpwsurαiSTsur,i, As occurred with solute transport, this latter simulation option was chosen to solve the temperature dynamics and exchanges produced in the implemented test case.

The introduced solver version assumes the subsurface domain to remain fully saturated, regardless of temporal variations in hydraulic head. Accordingly, groundwater flow is simulated under steady-state conditions, with the solution depending on the transient surface hydrodynamics transferred to the subsurface across the domain interface and on the selected BCs. Subsurface flowpaths, as well as solute and heat processes, are driven by the resulting saturated hydraulic head gradients within the porous medium.

On the other hand, in contrast to the prior solver version, darcyInterTransportFoam defines the subsurface geological parameters, namely, porosity, θi (–), hydraulic conductivity, Kij (m s⁻¹), and longitudinal and transverse dispersivities, αL,i and αT,i (m), as 3D scalar and tensor fields. This novel feature allows to extend the applicability of the code to actual alluvial aquifers commonly characterized by heterogeneous θi, αL,i and αT,i scalar fields as well as heterogeneous and anisotropic Kij tensor fields.

Considering the fully saturated assumption, the 3D groundwater flow is described by the following continuity equation for a heterogeneous, anisotropic aquifer: 12∇⋅-Kij∇hsub,i=0, where hsub,i is the subsurface hydraulic head field (m). Mass conservation is thus solved based on hsub,i, in contrast to the modified pressure field, prghsub,i (Pa), used in hyporheicScalarInterFoam. While both variables are closely related, they are not identical, as prghsub,i does not include the velocity head: 13prghsub,iρwsubg=hsub,i-qi22gθi2, where ρwsub is the groundwater density (kg m⁻³) and qi is the specific discharge field (m s⁻¹). Consequently, prghsub,i approximates hsub,i in the subsurface, as kinetic energy effects are typically negligible due to the low flow velocities in porous media. However, solving the groundwater flow based on hsub,i allows for the incorporation of the surface water velocity effects at the SW-GW interface, thereby accounting for their potential influence on the overall subsurface dynamics. This contribution was omitted in the original version of the solver.

Following the head solution, qi (Haitjema and Anderson, 2016) is sequentially computed using Darcy's law: 14qi=-Kij∇hsub,i, Darcy's law is generally applicable to slow-moving groundwater flows characterized by Reynolds numbers smaller than 10 (Bear, 1972), a condition that is satisfied in most natural groundwater environments. The application of Darcy's law to non-Darcy-flow in porous media may lead to an overestimation of the groundwater flow rates.

The conservative solute transport in the porous medium is modelled using the same advection-diffusion-dispersion equation as in hyporheicScalarInterFoam. Nevertheless, in darcyInterTransportFoam the hydrodynamic dispersion coefficient tensor field, Dsub,ij (m² s⁻¹), can either be specified as spacially uniform and constant through a single value or calculated as the sum of the effective molecular diffusion coefficient field, Ddiff,ij (m² s⁻¹), and the mechanical dispersion coefficient field, Ddisp,ij (m² s⁻¹). This choice between two alternatives represents an improvement over its predecessor, where only the second option was available, and aims to provide greater flexibility to the user in solute transport simulations. Moreover, unlike in hyporheicScalarInterFoam, the Ddiff,ij tensor corrects for tortuosity the user-defined molecular diffusion coefficient, Dm, based on Boudreau (1996), ensuring a more accurate representation of solute transport in complex subsurface environments: 15Ddiff,ij=Dm1-2ln⁡θiJij, where Jij is the tensor of ones. The Ddisp,ij tensor follows the same formulation as in the original solver, with the default setting αT,i=0.1⋅αL,i (Cardenas et al., 2008) applied when no field is specified for αT,i at the start of the simulation.

The new FCM solves the transport of heat at the subsurface domain assuming local thermodynamic equilibrium in the ground. Thus, both the groundwater and the porous medium (Twsub and Tpm, respectively) are considered to be at the same temperature Twsub=Tpm=Tsub. This assumption simplifies the definition of the heat transfer model to a single energy equation (Rubin, 1974), derived by combining the energy balance equations of both subsurface phases (i.e., water and porous medium): 16ρCpsub,i∂Tsub,i∂t+∇⋅ρwsubCpwsubUsub,iTsub,i-∇⋅(Ksub,i∇Tsub,i)=ρCpsub,iSTsub,i, where Tsub,i is the subsurface temperature field (°C), Cpwsub is the groundwater specific heat capacity (m² s⁻² °C⁻¹), Ksub,i is the subsurface thermal conductivity field (kg m s⁻³ °C⁻¹) and STsub,i represents any heat source/sink field (°C). The overall specific heat capacity field, ρCpsub,i, is computed by weighted averaging of ρ and Cp of the groundwater and the porous medium based on θi. Ksub,i is also averaged in the same way, with the thermal conductivity of groundwater, Kwsub (kg m s⁻³ °C⁻¹), calculated analogously to Kpsur and the thermal conductivity of the porous medium, Kpm (kg m s⁻³ °C⁻¹), prescribed by the user as a single constant value. By default, the variables used to compute Kwsub are assigned the values of their surface counterparts, although user-defined values can also be specified.

The updates implemented in the solver comprise modifications to the structure and organization of the code (Sect. S1), along with improvements to existing functionalities and the addition of new ones (Sect. S2).

The solver updates described below include the names of all the cited parameters in the code (i.e., files, variables, fields, dictionaries, etc.), along with the corresponding point numbers (in brackets) referring to their extended descriptions in Sect. S2:

Computation of additional hydrology-related fields, such as the specific discharge (q), the water depth (dSur, dSub) or hydraulic head (hSur, hSub). Accordingly, the groundwater flow is solved based on hSub, in contrast to the modified pressure field (pTildeSub) used in the original model (1).

Definition of a reference for the free-surface (hRef) in the surface domain. This optional feature allows to specify a water height at the surface outlet patch that, combined with the appropriate BCs for pressure and velocity, enables the flow to leave the simulation domain freely (3).

Possibility to set the porosity (theta) and the longitudinal and transverse dispersivities (alpha_L and alpha_T) as heterogeneous via scalar fields, as well as the hydraulic conductivity (K) as heterogeneous and anisotropic through a tensor field. This non-uniform field definition provides full flexibility in assigning the previous parameters throughout the subsurface domain, enabling, e.g., the setup of layers with diverse geologic characteristics, a heterogeneous parameter distribution or the definition of impermeable zones (5).

Simplification of the scheme used to select the processes to be solved by the model. As a result, the choice of the flow, the solute transport as well as the heat transfer simulation can be made jointly for both surface and subsurface domains (flow, soluteTransport and heatTransfer options in the simulationProperties dictionary, respectively). Furthermore, as an alternative to this joint selection, the simulation of each process can be also separately specified for each domain. The different process selection options as well as their corresponding implications can be checked in Fig. 1 (7).

Full reformulation of the passive solute transport module, including both the code organization and the modelling approach. Regarding the later, both the surface diffusion and the subsurface hydrodynamic dispersion fields (DSur and DSub, respectively) can be either computed through new or partially modified equations or specified as uniform throughout the domain via a single value (constDSur or constDSub). Moreover, the transport of conservative solutes (CSur) can be optionally solved in one or two phases at the surface domain (solutePhases). All the solute transport related expressions are gathered within the corresponding header files CSurEqn.H and CSubEqn.H in the surface and subsurface domains. A thorough description of the novel transport equations is provided in Sect. 2.1.1 and 2.1.2 (8).

Addition of a heat transfer module for the simulation of the temperature distribution in both surface and subsurface domains. Both the code organization and the modelling scheme of this new module are based on those implemented in its solute transport counterpart. Consequently, two phase-based simulation options (heatPhases) are included in the TSurEqn.H file to simulate the transfer of heat in the surface domain (TSur), whereas the energy equation required to resolve the temperature distribution in the ground (TSub) is contained in TSubEqn.H. Further details about the implemented heat transfer equations can be found in Sect. 2.1.1 and 2.1.2 (9).

Modification of two-way data mapping (Sect. 2.2) for a more accurate conveyance of the information between the domains. In this way, the data is no longer interpolated but directly assigned to the matching nodes of both meshes at their respective interface patch. This new method ensures the exact transmission of the solution from one domain to another (hSur, q, CSur/CSub and TSur/TSub) (10).

Reformulation of the convergence check at the SW-GW interface. Rather than relying on a fixed maximum number of correction loops, water and solute flux consistencies are evaluated at each time step by comparing the exchange fluxes predicted from the surface domain with those computed from the groundwater solution. Convergence is considered achieved when both water and solute flux mismatches (couplePhiMaxMismatch and coupleSolutePhiMaxMismatch) fall below the specified tolerances (couplePhiTol and couplePhiSoluteTol) (Fig. 1) (11).

Four new utilities were developed to be used with darcyInterTransportFoam (Sect. S3). Additionally, some of them are also compatible with other OpenFOAM solvers. The complete new set was used to perform pre- and post-processing actions in the implemented test case:

checkMeshesMatch: checks whether the number of faces of both meshes and their coordinates match at the interface patch. The fulfilment of both conditions ensures the appropriate fitting of both surface and subsurface meshes. Only applicable to OpenFOAM models consisting of two adjacent simulation meshes.

changeZRefMesh: performs two optional actions: (1) translation of the mesh points according to a user-specified reference mesh elevation; and (2) adjustment of hRef to the actual mesh location. Both actions are independent of each other. Applicable to further OpenFOAM models.

restoreZRefMesh: initially designed to be used in combination with changeZRefMesh but also applicable on its own. This utility conducts three optional and independent actions: (1) translation of the mesh points according to the meshShiftZ value; (2) reestablishment of the original, user-defined hRef (if previously applied changeZRefMesh); and (3) adjustment of the computed hydrological fields (piezoSur, hSur and hSub) based on meshShiftZ. meshShiftZ can be either previously determined by changeZRefMesh or provided by the user in the constant directory. Applicable beyond darcyInterTransportFoam.

transferMeshDecomp: transfers the domain decomposition from one mesh to another through the interface patch of both meshes. A prior decomposition of both domains (decomposePar utility) is required to generate the necessary cellDist and cellDecomposition files for its execution. Furthermore, it optionally overwrites the cellDecomposition file of the reference and/or target mesh once the transfer is done. Only applicable to OpenFOAM models consisting of two adjacent modelling meshes.

One new boundary condition for Usur,i and another two for hsub,i were implemented along with the new solver (Sect. S4). Analogously to the novel utilities, some of the new BCs are darcyInterTransportFoam-specific while others can also be applied to other OpenFOAM solvers. The two hsub,i boundary conditions were used in the test case:

darcyGradHead: sets at the patch the ∇hsub,i resulting from the defined qi BC. It is based on the previously developed darcyGradPressure BC (Horgue et al., 2015). Applicable to OpenFOAM models other than darcyInterTransportFoam.

surfaceHydrostaticHead: specifies at the inlet, outlet or aquifer (i.e., bottom) patch of the subsurface domain the hydrostatic hsub,i computed at the equivalent surface boundary. Accordingly, single values will be assigned to the subsurface inlet and outlet patches (max⁡piezosur,i), whereas an entire head field will be prescribed to the bottom boundary of the subsurface domain (piezosur,i). In this early version, its use is restricted to the above-mentioned boundaries. Only applicable to darcyInterTransportFoam.

The surface and subsurface meshes for the test case were generated using blockMesh, one of the mesh generation utilities of OpenFOAM. Accordingly, the resulting meshes consist of 3D, structured, hexahedral blocks defined by either straight or curved edges. To enable full control over the mesh generation process, the blockMeshDict file required for running blockMesh was generated for each mesh using a set of user-defined MATLAB scripts (Pardo-Álvarez, 2026b). These scripts were also used to generate other mesh-related dictionaries (e.g., topoSetDict) and input data files (e.g., setFieldsDict) associated with the resulting structured meshes.

A high-resolution (0.25 m) Digital Elevation Model (DEM), acquired on 20 March 2015 using drone-based photogrammetry (Schilling et al., 2017), was used to define the interface boundary of both domains (i.e., the riverbed patch) (Fig. A3). The surface and subsurface meshes are approximately 2.5 and 10 m high, respectively, and both are about 235 m long and 50 m wide. However, rectangular inflow and outflow stretches of around 6.25 and 12.5 m long, respectively, were added to each mesh to prevent potential perturbations caused by the geometric features of the inlet and outlet boundaries. Moreover, the smooth transition between these regular sections and the actual riverbed topography was achieved through approximately 12.5 m long segments. Consequently, the total length of each mesh was increased by around 43.75 m (Fig. A3). The other two dimensions remained unaltered.

As for the mesh discretization, a mesh sensitivity study was conducted to determine the optimal resolution for each mesh, leading to the selection of a surface mesh with approximately 2×106 cells and a subsurface mesh with about 8×106 elements. Identical cell counts in the longitudinal and transversal directions are imposed for both domains to ensure effective interface node matching and, thus, successful coupling of their solutions.

Moreover, the meshes were gradually refined vertically towards the SW-GW interface to capture the complex dynamics across this boundary and satisfy the y+ (dimensionless wall distance) requirement near the riverbed in the surface domain (Fig. 3). In both meshes, the smallest elements are located close to the interface (0.001 m² face area), while the largest cells are situated within the air-phase in the surface domain (0.24 m²) and at the bottom boundary in the subsurface domain (0.1 m²). Especially large element sizes were defined for the surface air-phase, since detailed processes in this region are not of interest for the present study. This phase is only simulated to account for the water level fluctuations.

All fluid parameters were assumed to stay constant over time in agreement with the simulated steady-state flow and transport dynamics. Common values of νphysp and ρp at ambient temperature were defined for the air and water (Table A1 in the Appendix). Regarding the solute transport parameters, a Dphys of 4.2×10-10 m² s⁻¹, corresponding to fluorescein sodium (NaFl), and a Scturb of 1 were applied to both surface fluids. Dm was assumed to be equal to Dphys. Moreover, heat parameters, such as Prp, were inferred from the fluid temperatures specified for each domain. Further details on the fluid properties are provided in Table A1.

Regarding the geological properties, two layers of different homogeneous and isotropic Kij and θi were defined in the subsurface domain (Fig. 4). The upper layer, located at the first 2 m beneath the riverbed, has a Kij of 2.8×10-3 m s⁻¹ and a θi of 0.41. These values were assigned via a setFieldsDict using a triangulated surface geometry. The rest of the aquifer corresponds to the second layer, whose Kij and θi are 6.4×10-3 m s⁻¹ and 0.1, respectively. Moreover, eight impermeable regions of different depths (from 0.25 to 1 m) were defined below the surface dry areas (i.e. the riverbanks and several gravel bars) by means of a triangulated surface geometry using the water table height as reference (Fig. 4). Within these regions, the values of Kij and θi are small enough to ensure the flow cannot pass through them. Thereby, Kij is set to 10⁻²⁰ m s⁻¹ and θi to 10⁻³. Additional geological parameters, such as αL,i and αT,i, were also prescribed at the aquifer. In this way, a different homogeneous αL,i was assigned to each subsurface region. Particularly, αL,i of 3 and 5 m were set, respectively, to the upper and lower layers, while a field of 10⁻⁷ m was prescribed to the impervious zones. αT,i was approximated by 1/10 of αL,i. The rest of the ground parameters can be found in Table A1.

All surface and subsurface fields were initialized with values as close as possible to the expected solution to help the model reach the steady state faster. Some values were obtained from a spin-up run, and thus directly specified in the field data files for the entire simulation domain (Tables A2, A3 and A4), while others were assigned to specific mesh locations through different setFields actions. The latter was the case of the surface fields alpha.water (αi), U (Usur,i), C (Csur,i) and T (Tsur,i).

On the other hand, as it happened with the fluid and geologic properties, all the prescribed boundary conditions remained constant during the simulation to represent a steady-state scenario. The inlet of the surface domain is split into two patches, with αi prescribed accordingly on each one: water enters the domain through the lower patch (inletWater) via a specified constant discharge (10 m³ s⁻¹), whereas the upper half (inletAir) is reserved for the air flow by means of a total pressure condition. The latter condition is also applied to the top boundary (atmosphere). Additionally, water also flows into the domain through the waterJet patch (0.42 m³ s⁻¹). The water level at the surface outlet (0.25 m) is fixed by combining a hydrostatic pressure condition with a reference free-surface elevation (hRef). At the riverbed, Usur,i is initially set to 0 m s⁻¹, switching to a heterogeneous field after the first coupling iteration. The impact of the riverbed roughness in the flow dynamics is also considered through the application of a rough wall function to νturb,i at this patch. As for the subsurface domain, the inlet and outlet boundaries are prescribed with an upstream qi of 2×10-5 m s⁻¹ and a downstream hydrostatic hsub,i condition, respectively. Furthermore, hsub,i is originally fixed to zero at the riverbed, pending the first iteration of the coupling scheme to change its value. Regarding the solute transport and heat transfer simulation, constant input values of 5 kg m⁻³ and 2 °C are set at the inletWater patch and 10 kg m⁻³ and 8 °C at the waterJet and subsurface inlet boundaries, whereas both inletAir and atmosphere patches are prescribed with a zero-gradient outflow condition with specified input for the case of reversing flow. Similar mixed BCs are defined at the riverbed in the two domains, mapping the values from the coupled domain in case of inflow. Both surface and subsurface outlet patches are set to a zero-gradient condition. The remaining boundaries, namely banks and aquifer, are specified as impermeable using a slip condition. As such, the flow can move along the patch but not across it.

The full suite of initial and boundary conditions prescribed for the test case is detailed in Appendix A. For more detailed information on their use and implementation, refer to the online OpenFOAM User's Guide (OpenCFD Ltd., 2023).

The terms of the partial differential equations governing the flow, solute transport and heat transfer are transformed into algebraic equations through different spatial and temporal discretization schemes. The complete suite of numerical schemes used in the present test case can be checked in Table A5.

The discretized equations are then solved using a range of iterative linear-solvers. The selected linear-solver, along with the preconditioner and tolerance needed to achieve an accurate solution, is defined for each dynamic variable in Table A6.

In the surface domain, the 3D velocities and pressure are solved iteratively using the PIMPLE pressure-velocity coupling algorithm (OpenCFD Ltd., 2023). This iterative procedure is necessary for the VOF method to effectively solve the pressure field while satisfying the continuity equation. The pressure correction is achieved via the Poisson equation, with convergence attained through 3 inner-correctors and 1 outer-corrector. As for the subsurface solution, the hydraulic heads are corrected for mesh non-orthogonality using 1 non-orthogonal corrector.

The model setup is completed with the definition of the simulation control options. The initial timestep was set to 0.001 s, allowing a maximum timestep of 1 s. Moreover, the timestep size was dynamically adjusted according to a specified maximum Courant number of 0.5 and a maximum alpha Courant number of 1.