Equilibrium Component Mass Conservation Equations

STOMP components comprise the subsurface systems of interest. Available components and component phases vary based on the operational mode (Table 1). The component mass conservation equation equates the time rate of change for each STOMP component mass within a control volume with the flux of that component mass crossing the control volume surface. Phase partitioning of components are computed assuming equilibrium conditions for all modes of STOMP (except HYDT-KE). This implies that in geologic media the time scale for thermodynamic equilibrium is significantly shorter for component transport.

Equilibrium ComponentsOperational ModesPhases














n-Petroleum components




Salt/Hydrate Inhibitor




Table 1. Equilibrium components and component phases for STOMP modes in liquid (l), gas (g), NAPL (n), precipitated salt (p), hydrate (h), and ice (i) phases.

Mass Conservation Equation

The left-hand-side of the equation represents the volumetric integral of the mass accumulation term for component (j) in all phases (γ) over the node volume. The right-hand-side of the equation represents the surface integral of flux terms over the node volume as well as the volumetric integral of source/sink terms.

Where the mass accumulation term is

and component (j) can exist in the aqueous (l), nonaqueous liquid (n), gas (g), hydrate (h), ice (i), and precipitated (p) phases under equilibrium conditions, depending on the operational mode. 

Mass Flux

The flux is a combination of advective and diffusive components:

Advective Term

Advective fluxes are represented by the Darcy velocity.


Diffusive Term

Diffusive fluxes are computed from gradients in molar concentration, considering molecular diffusion, but ignoring hydraulic dispersion.

where a combined diffusion-dispersion coefficient, D, replaces the classical Fickian diffusion coefficient.

Sorbed oil

For the oil component,o, the mass conservation equation is modified to include mass accumulation of sorbed oil to the rock/soil phase.

Sorbed Oil

Osmotic Flux Term

In STOMP-W, -WAE, -WS, -WASE, and -WAS, there is an osmotic flux term, which accounts for the flow of aqueous fluid by osmotic pressure for simulations with coupled salt transport.

Osmotic Flux Term


Discretized Mass Conservation Equation

The mass conservation equations, shown above, are discretized by assuming a piecewise profile to express the variation in primary variables between node points and integrating over the node volume. The mass accumulation terms (i.e., left-hand-side terms) are integrated over the node volume according to

Discretized Mass Flux

Defining flux directions parallel to the surface normal allows the surface integrals to be converted to summations over all node surfaces.

where west (W), east (E), south (S), top(T), and bottom (B) node surfaces are considered.


This transformation strictly requires an orthogonal grid system for the flux directions to be aligned with the surface normals.  Nonorthogonal systems will yield mass balance errors.

Discretized Advective Term

Darcy fluxes are discretized, in the six coordinated directions, using upwind interfacial averaging (uw) for the component mass fraction, phase density, and relative permeability and harmonic averaging (h) for the intrinsic permeability and phase viscosity.

These default interfacial averaging schemes can be altered through user input.

Discretized Diffusive Term

Diffusive fluxes are discretized, in the six coordinate directions, using harmonic (h) averaging for the combination of terms which comprise an effective diffusion coefficient.

The default interfacial averaging scheme can be altered for each parameter through user input.

Time Discretization

The mass conservation equations are discretized in time using a fully implicit scheme, where the time levels are indicated with superscripts. The primary unknowns for the mass conservation equations are intrinsic properties at node volume centroids (node grid point) for time level t+δt.

The residual equation for each component is then the difference between left-hand-side and right-hand-side.



  Dissolution of water in the NAPL phase is neglected in all operational modes except STOMP-HYDT-KE.

  Dissolution of air in the NAPL is neglected.

  In Water-Oil (WO) and Water-Oil-Air (WOA) modes, oil exists in the diffusive pore space as liquid oil in the NAPL phase, dissolved oil in the aqueous phase, and as oil vapor in the gas phase.

  Salt transport occurs by advection and diffusion-dispersion through only the aqueous phase.

  Following the low solubility assumption for dissolved air and oil in the aqueous phase, water diffusion-dispersion through the aqueous phase is neglected.




(In order of appearance)

time, s

volume of element n, m3

mass accumulation term for component j , kg/m3
surface of element n, m2
advective flux of component j, kg/m2s
unit surface normal vector
mass fraction of component j in phase γ
specific mass source of phase γ, kg/m3 s

diffusive porosity

density of phase γ, kg/m3

saturation of phase γ
Darcy velocity vector of phase γ, m/s
diffusive-dispersive flux of component j for the phase γ, kg/m2 s
relative permeability of phase γ
intrinsic permeability, m2
kinematic viscosity of phase γ, Pa s
pressure of phase γ, Pa
acceleration of gravity, m/s2
unit gravitational direction vector
molecular weight of component j, kg/kg mol
molecular weight of phase γ, kg/kg mol
phase tortuosity for phase γ
diffusion coefficient of component j for phase, m2/s
mole fraction of component j in phase γ
total porosity
area of surface, m2
phase index
aqueous liquid phase
nonaqueous liquid phase

gaseous phase
hydrate phase
ice phase
precipitated salt phase
solid phase

node surface index 

negative surface (west, south, bottom)
positive surface (west, south, bottom)
component index
oil component
water component
salt component
upwind weighting scheme
harmonic weighting scheme

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