Solubility

Methods

Coupled H2O–CO2 volatile solubility (dissolved-content) closures can be set with:

GeoParams.MaterialParameters.Solubility.Liu2005_Solubility Type

Liu2005_Solubility(; coeffs=(b1..b6, c1..c4), Pref=1MPa, Tref=1K)

Coupled H2O–CO2 solubility for silicic (rhyolite) melt after Liu et al. (2005), as used by Degruyter & Huber (2014). compute_dissolved returns the dissolved H2O and CO2 mass fractions of the melt as a function of pressure, temperature, and the CO2 mole fraction of the gas X_co2:

$$m_{H_{2}O}=(b_1{P}_w^{1/2}+b_2{P}_w+b_3{P}_w^{3/2})\frac{T_{ref}}{T}+b_4{P}_w^{3/2}+P_c(b_5{P}_w^{1/2}+b_6{P}_w)$$$$m_{C{O}_2}=P_c(c_1+c_2{P}_w)\frac{T_{ref}}{T}+P_c(c_3{P}_w^{1/2}+c_4{P}_w^{3/2})$$

with the dimensionless partial pressures $P_w=P(1-X_{co2})/P_{ref}$ and $P_c=P{X}_{co2}/P_{ref}$. With Pref=1MPa, Tref=1K these equal the reference partial pressures in MPa and reproduce the Liu (2005) numbers; because they are ratios of like-dimensioned quantities, the closure is dimensionally homogeneous and nondimensionalizes cleanly. The fitted coeffs are dimensionless; isdimensional tracks the reference GeoUnits.

References

• Liu, Y., Zhang, Y., Behrens, H. (2005), Solubility of H2O in rhyolitic melts at low pressures and a new empirical model for mixed H2O-CO2 solubility, JVGR 143, 219-235, https://doi.org/10.1016/j.jvolgeores.2004.09.019

• Degruyter, W., Huber, C. (2014), A model for the eruption frequency of upper crustal silicic magma chambers, EPSL 403, 117-130, https://doi.org/10.1016/j.epsl.2014.06.047

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GeoParams.MaterialParameters.Solubility.Mafic_Solubility Type

Mafic_Solubility(; coeffs=(b1..b10, c1..c4), T0=273.15K, Tref=1K, Pref=1MPa)

Coupled H2O–CO2 solubility for mafic (basalt) melt. Dissolved H2O follows the mafic polynomial of the Scholz/Degruyter–Huber reference in the dimensionless groups $T_C=(T-T_0)/T_{ref}$ (numerically °C) and $P_m=P/P_{ref}$ (numerically MPa); dissolved CO2 reuses the Liu (2005) rhyolite CO2 block. Nondimensionalizes like Liu2005_Solubility. The H2O polynomial has no floor at zero and goes negative outside its calibration (high X_co2, low P); the dissolved H2O output is floored at zero rather than returned as a negative mass fraction.

References

• Degruyter, W., Huber, C. (2014), A model for the eruption frequency of upper crustal silicic magma chambers, EPSL 403, 117-130, https://doi.org/10.1016/j.epsl.2014.06.047

• Liu, Y., Zhang, Y., Behrens, H. (2005), Solubility of H2O in rhyolitic melts at low pressures and a new empirical model for mixed H2O-CO2 solubility, JVGR 143, 219-235, https://doi.org/10.1016/j.jvolgeores.2004.09.019

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GeoParams.MaterialParameters.Solubility.GasMixture Type

GasMixture(; Cp_h2o=3880J/kg/K, Cp_co2=1200J/kg/K, M_h2o=18.02e-3kg/mol, M_co2=44.01e-3kg/mol)

H2O–CO2 gas-mixture properties keyed on the CO2 mole fraction of the gas X_co2 (Degruyter & Huber 2014). The effective molar mass is

$$m_g=M_{H_{2}O}(1-X_{co2})+M_{C{O}_2}{X}_{co2}$$

and the mass-weighted specific heat is

$$c_g=\frac{M_{H_{2}O}{c}_{H_{2}O}(1-X_{co2})+M_{C{O}_2}{c}_{C{O}_2}{X}_{co2}}{m_g}$$

with the reference convention $c_g=0$ at $X_{co2}=0$ — a discontinuity, since the formula's own limit as $X_{co2}\to 0$ is $c_{H_{2}O}$.

All fields are GeoUnits, so the struct nondimensionalizes, but both accessors read values rather than Quantitys: neither return carries units.

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Computational routines

To evaluate dissolved H2O and CO2 (mass fractions) within a user routine, use this:

GeoParams.MaterialParameters.Solubility.compute_dissolved Function

compute_dissolved(s::AbstractSolubility, P, T, X_co2) -> (m_h2o, m_co2)

Dissolved H2O and CO2 mass fractions of the melt (P in Pa, T in K, X_co2 the CO2 mole fraction of the gas). Also callable as compute_dissolved(s; P, T, X_co2) and compute_dissolved(s, args::NamedTuple).

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GeoParams.MaterialParameters.Solubility.compute_dissolved! Function

compute_dissolved!(m_h2o, m_co2, MatParam, Phases, args)

In-place dissolved H2O and CO2 over a domain. args is a NamedTuple of P, T, X_co2 index-matched to the output arrays. Also accepts a single solubility or phase struct in place of (MatParam, Phases).

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Partial derivatives (via ForwardDiff, useful for implicit/Newton solves) are available for pressure, temperature, and gas composition:

GeoParams.MaterialParameters.Solubility.∂dissolved_∂P Function

∂dissolved_∂P(s, P, T, X_co2) -> (∂m_h2o/∂P, ∂m_co2/∂P)

ForwardDiff partial derivatives of compute_dissolved with respect to pressure. Companions ∂dissolved_∂T, ∂dissolved_∂Xco2.

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GeoParams.MaterialParameters.Solubility.∂dissolved_∂T Function

∂dissolved_∂T(s, P, T, X_co2) -> (∂m_h2o/∂T, ∂m_co2/∂T)

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GeoParams.MaterialParameters.Solubility.∂dissolved_∂Xco2 Function

∂dissolved_∂Xco2(s, P, T, X_co2) -> (∂m_h2o/∂X_co2, ∂m_co2/∂X_co2)

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Given a target dissolved H2O content, the gas composition X_co2 that produces it can be found with:

GeoParams.MaterialParameters.Solubility.find_Xco2 Function

find_Xco2(s::AbstractSolubility, P, T, m_h2o_target; X0=0.5, tol=1e-8, max_iter=50) -> X_co2

Invert compute_dissolved for the gas composition: solve for X_co2 ∈ [0,1] such that compute_dissolved(s, P, T, X_co2)[1] == m_h2o_target, at fixed pressure P and temperature T. Uses a safeguarded Newton iteration (via ∂dissolved_∂Xco2) bracketed by bisection, so an out-of-bounds Newton step always falls back to a bisection halving instead of leaving [0,1].

Throws an ErrorException if m_h2o_target is infeasible at this P,T (outside the achievable range between X_co2=0 and X_co2=1) or if the iteration fails to converge within max_iter steps — this never returns a clamped or out-of-tolerance answer.

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Gas-mixture helpers for the energy equation:

GeoParams.MaterialParameters.Solubility.compute_gas_heatcapacity Function

compute_gas_heatcapacity(s::GasMixture, X_co2)

Mass-weighted specific heat of the H2O–CO2 gas mixture; zero at X_co2 == 0 (reference convention).

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GeoParams.MaterialParameters.Solubility.effective_molar_mass Function

effective_molar_mass(s::GasMixture, X_co2)

Effective molar mass of the H2O–CO2 gas mixture at CO2 mole fraction X_co2.

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Note that ε_g (exsolved-gas volume fraction) and X_co2 (CO2 mole fraction of the gas) are typically ODE state variables owned by the solver, not closed algebraically at a point — the closures above are the entries to that mass-balance, not a replacement for it.