Ground-Water Flow and Solute Transport for the PHAST Simulator Ken Kipp and David Parkhurst

Outline   Concepts   Assumptions   Governing Equations   Numerical Implementation

Basic Concepts   Conservation of an extensive variable in an open system – –Mass – –Solute mass – –Solute moles – –Electrons – –Redox state   Mechanisms of transport – –Flow in porous media – –Advection – –Dispersion – –Diffusion   Mechanism of chemical reaction – –Mass action equilibria – –Kinetics

Water Balance Rate of water accumulation = Rate of water addition by advection + Rate of water addition by point sources

Solute Component Balance Rate of solute accumulation = Rate of solute addition by advection + Rate of solute addition by dispersion Rate of solute addition by diffusion + Rate of solute addition by equilibrium reactions Rate of solute addition by kinetic reactions + Rate of solute addition by point sources + +

Simplifying Assumptions   Single phase   Liquid water   Saturated porous media   Darcy flow   Isothermal   Dilute solutions   Constant density and viscosity   Compressible porous media for confined flow   Isotropic dispersion with empirical modification   One set of diffusion and dispersion parameters   No pure diffusive solute sources

Ground-Water Flow Equation   Interstitial velocity (Darcy flow)  Conservation of water mass (volume)

Flow Equation Parameters   Hydraulic conductivity (L/T)  Specific storage (1/L), confined flow

Flow Equation Results  Head field h(x,t)  Velocity field v(x,t)  Boundaries –Flow rates –Cumulative amounts  Assumed independent of solute transport

Solute Transport Equation  Conservation of solute mass for each component i

Solute Transport Parameters  Porosity  Dispersion coefficient tensor –Dispersivities –Effective molecular diffusivity  Modified dispersion coefficients for restricted anisotropic dispersion –Split into and –Applies to horizontal flow in layered anisotropic aquifer Dispersion Models  Isotropic dispersion

Dispersive Processes

Solute Transport Equation Results  Component concentration fields c i (x,t)  Breakthrough curves  Peak concentrations  Boundaries –Solute flow rates –Cumulative solute amounts

Water and Porous-Matrix Properties   Intrinsic water properties – –Density, – –Compressibility – –Viscosity   Intrinsic porous matrix properties – –Permeability – –Porosity – –Compressibility – –Dispersivity   Combined properties for simulator – –Hydraulic conductivity – –Storage coefficient – –Effective molecular diffusivity

Boundary Conditions for Flow   No flux (default)   Specified head – –adjacent open body of water – –determined from field data or larger model   Specified flux – –precipitation – –determined from field data or larger model   Leakage flux –adjacent leaky aquifer –extension of simulation region   River leakage flux   Well –injection or production well –observation well   Free surface –water table; unconfined flow –atmospheric pressure

Boundary Conditions for Solute Transport   Specified concentrations – –Only with specified head boundary – –Adjacent open body of water   Associated advective component fluxes – –Can be used with all boundary conditions – –Open boundary with incoming ground-water flux – –Outgoing flux leaves at resident boundary concentration

Leakage Boundary Condition   Leaky boundary – –Fluid flux a function of head – –Associated concentration for advective solute flux   Assumptions and limitations – –One-dimensional Darcian flow in confining layer – –Neglect transient fluid storage in confining layer – –Neglect transient solute storage in confining layer – –Outer aquifer conditions not affected by leakage flux

Leakage Flux

River Leakage Boundary Condition   River leakage boundary – –Fluid flux a function of head – –Associated concentration for advective solute flux   Assumptions and limitations – –One-dimensional Darcian flow in river bed – –Neglect transient fluid storage in river bed – –Neglect transient solute storage in river bed – –River conditions not affected by leakage flux – –Limit on maximum recharge flux from river

River Geometry

River Leakage Flux

Free-Surface Boundary Conditions   Two b.c. since location is an unknown 1. Atmospheric pressure 2. Kinematic condition neglected  Linear extrapolation of pressure to locate free surface elevation  Free surface allowed in any cell  Only one free surface at any horizontal location  Cells resaturate initially from below

Well-Bore Model   Methods – –Local well model with steady-state head profile at each cell layer – –Concentration from production well is layer- flow-rate weighted average   Options – –Specified flow rate (injection, production) – –Observation well   Allocation of flow – –By mobility – –By product of mobility and head difference

Well Geometry

Wells   Assumptions and limitations – –Well bore is finite radius cylinder incorporated as a source term – –Local steady-state radial-flow well-bore equation – –Hydrostatic pressure distribution in well bore – –Net flow in well bore must not reverse direction – –No reactions are considered in well bore

Initial Condition Options   Head distribution – –Defined by zones   Uniform value   Piecewise linear – –Water-table condition – –Steady-state flow field   Concentration distribution – –Defined by zones   Uniform value   Piecewise linear

Solving the Flow and Reactive- Transport Equations   Finite difference approximations   Equation discretization – –Point-distributed mesh – –Zonation of spatial properties – –Backwards-in-time or centered-in-time – –Upstream-in-space or centered-in-space   Operator splitting the reactive-transport equations

Discretization

Point-Distributed Grid   Elements and nodes  Porous-media properties defined by element

Point-Distributed Grid   Cells and nodes  Boundary and initial conditions defined by cell (node)

Zones for Property Definition  Selection of elements

Zones for Property Definition   Selection of cells and nodes

Application of Boundary Conditions

Well Discretization

Temporal Discretization Using Finite Differences   Generic equation  Centered-in-time, Crank-Nicolson;  = 0.5  Backward-in-time, Fully implicit;  = 1  Intermediate weighting; 0.5 <=  <= 1

Operator Splitting   Separating transport from reaction calculations   Solute Transport Equations (simplified):  Finite Difference Equations:

Sequential Non-lterative Approach   Transport step:  Reaction step:

Summary   Sequential Solution of Coupled Equation System – –Discretize in space and time yielding finite difference equations – –Operator split the reactive-transport equations – –Solve sequentially for each time step   Flow   Solute transport for each component   Equilibrium and kinetic chemical reactions

PHAST Information Flow Chart

Types of PHAST Simulations Steady-state ground-water flow Transient ground-water flow Steady-state ground-water flow and transient reactive transport Transient ground-water flow and transient reactive transport