Transport Properties¶

Transport properties are evaluated for a specified homogeneous phase or an already resolved pair of coexisting phases. They are not equilibrium conditions: a viscosity or interfacial-tension correlation does not establish phase stability, choose an EoS root, or perform a flash. The caller must supply the phase composition, density, pressure, and root required by the selected model.

torch-flash uses SI at the public boundary:

dynamic viscosity \(\eta\) in Pa s;
kinematic viscosity \(\nu\) in m²/s;
thermal conductivity \(\lambda\) in W/(m K);
surface or interfacial tension \(\sigma\) in N/m; and
diffusion coefficient \(D\) in m²/s.

The packaged coefficients are identified by transport.pedersen-2024. The common chapter-level source is Pedersen, Christensen, and Shaikh, Phase Behavior of Petroleum Reservoir Fluids, 3rd ed. (2024), chapter 10, doi:10.1201/9780429457418. Primary equation sources are cited with each model.

Viscosity¶

Dynamic and kinematic viscosity¶

Dynamic viscosity relates shear stress to velocity gradient. Kinematic viscosity divides out the homogeneous phase mass density:

\[ \nu = \frac{\eta}{\rho_m}. \]

kinematic_viscosity implements this SI definition and requires \(\eta\geq0\) and \(\rho_m>0\).

Pedersen corresponding states¶

corresponding_states_viscosity maps a petroleum mixture to a methane reference state. In compact form, its reference coordinates are

\[ T_0 = T\frac{T_{c,0}}{T_{c,m}}\frac{\alpha_0}{\alpha_m}, \qquad P_0 = P\frac{P_{c,0}}{P_{c,m}}\frac{\alpha_0}{\alpha_m}, \]

and the mixture viscosity is

\[ \eta_m = \left(\frac{T_{c,m}}{T_{c,0}}\right)^{-1/6} \left(\frac{P_{c,m}}{P_{c,0}}\right)^{2/3} \left(\frac{M_m}{M_0}\right)^{1/2} \frac{\alpha_m}{\alpha_0}\, \eta_0(T_0,\rho_0). \]

Here \(T_{c,m}\) and \(P_{c,m}\) are the Pedersen mixture reducing properties, \(M_m\) is the correlation's mixture molecular-weight parameter, and \(\alpha_m/\alpha_0\) is the dense-fluid shape correction. The methane reference density \(\rho_0\) is solved with the published BWR reference equation before the Hanley methane viscosity is evaluated. The formulation originates with Pedersen et al., doi:10.1016/0009-2509(84)87009-8; the methane reference equation is from Hanley, McCarty, and Haynes, doi:10.1016/0011-2275(75)90010-7.

The phase argument selects the methane-reference density branch. It does not identify the physical phase of the supplied mixture.

heavy_oil_corresponding_states_viscosity retains the conventional corresponding-states result above a 75 K methane-reference temperature, uses the Lindeloff heavy-oil branch below 65 K, and linearly blends the two in between. Its dimensionless third and fourth CSP factors are explicit tensors. Changing them is calibration and requires a separately recorded dataset, objective, bounds, and validation result.

For the low-reference-temperature branch, the representative molecular mass is

\[ M = M_n \left[ \frac{M_w}{\mathrm{Visfac}_3(3\mathrm{rd\ CSP})M_n} \right]^{\mathrm{Visfac}_4(4\mathrm{th\ CSP})} \]

when \(M_w/M_n>1.5\), with the corresponding capped-ratio expression from Eq. 10.34 below that threshold. Here

\[ \mathrm{Visfac}_3=0.2252\,T/M_n+0.9738, \qquad \mathrm{Visfac}_4=0.5354\,\mathrm{Visfac}_3-0.1170. \]

evaluate_heavy_oil_corresponding_states_profile is the high-level phase-aware pressure-profile operation. It solves one bubble point per unique temperature, retains the feed above the bubble point, batches all sub-bubble flashes at that temperature, and passes their equilibrium liquid compositions to the viscosity model. Boundary and flash convergence, vapor fraction, and log-fugacity residual remain explicit in HeavyOilCSPProfile.

fit_heavy_oil_csp_factors calibrates both factors against every supplied state simultaneously. Its full-batch objective is the mean squared logarithmic viscosity ratio, and smooth bounds keep both physical factors positive. Full-batch PyTorch LBFGS with a strong-Wolfe line search is the default for this two-parameter problem; the result also reports the log-viscosity sensitivity matrix, singular values, numerical rank, and condition number. The phase model is prepared separately and is not silently refitted.

The Heavy Oil 5 study uses the bundled characterization.pedersen-heavy-aromatic-2004 C200 correlations and PR78. The heavy-aromatic characterization is defined by Pedersen, Milter, and Sørensen, doi:10.2118/88364-PA.

Lohrenz-Bray-Clark¶

lbc_viscosity combines a dilute-gas mixture contribution with a reduced density polynomial:

\[ \eta = \eta^\star + \frac{ \left(a_1+a_2\rho_r+a_3\rho_r^2+a_4\rho_r^3+a_5\rho_r^4\right)^4 -10^{-4} }{\xi_m}, \qquad \rho_r=\rho_n\sum_i x_iV_{c,i}. \]

The numerical polynomial uses the conventional LBC source units internally; the API accepts molar density in mol/m³ and critical volume in m³/mol and returns Pa s. Critical volume is therefore a scientific input, not a generic EoS placeholder. Characterized C7+ cuts can use lbc_pseudocomponent_critical_volume, while measured or fitted critical volumes can be supplied explicitly.

The model is from Lohrenz, Bray, and Clark, doi:10.2118/915-PA. Its five coefficients can be passed as a differentiable tensor, but an untuned LBC calculation is not automatically predictive for heavy oils.

Lee gas correlation¶

lee_gas_viscosity implements the Lee-Gonzalez-Eakin natural-gas correlation:

\[ \eta = 10^{-4}K\exp\left[ X\left(\frac{\rho_m}{62.4}\right)^Y \right] \quad\text{cP}, \]

with \(K\), \(X\), and \(Y\) defined from temperature in degrees Rankine and mixture molar mass in g/mol. The public function converts SI temperature, mass density, and molar mass into the published units and converts the result back to Pa s. It is a gas correlation and must not be used for a condensed liquid merely because the inputs are finite. The source is Lee, Gonzalez, and Eakin, J. Petroleum Technology 18 (1966), 997-1000, doi:10.2118/1340-PA.

One-parameter friction theory¶

friction_theory_viscosity decomposes the cubic-EoS pressure into repulsive and attractive contributions:

\[ \eta = \eta_0 + \kappa_r P_r + \kappa_a P_a + \kappa_{rr}P_r^2. \]

The temperature-dependent friction coefficients are mixed from the published component reducing relations. The same cubic EoS, mixing rule, interactions, and phase root determine \(P_r\) and \(P_a\). The bundled parameterization is defined only for SRK and PR76; PR78 is not silently treated as PR76.

The implementation follows the one-parameter model of Quiñones-Cisneros, Zéberg-Mikkelsen, and Stenby, doi:10.1016/S0378-3812(00)00474-X. Its reported scope is nonpolar fluids and n-alkane mixtures.

Thermal conductivity¶

Methane reference correlation¶

methane_thermal_conductivity evaluates the Hanley methane correlation as a sum of dilute, first-density, dense-fluid, low-temperature, and optional critical-enhancement terms:

\[ \lambda(T,\rho) = \lambda^{(0)}(T) +\lambda^{(1)}(T)\rho +\Delta\lambda_{\mathrm{dense}}(T,\rho) +\Delta\lambda_{\mathrm{low}\ T}(T,\rho) +\Delta\lambda_c(T,\rho). \]

The public density is mol/L, matching the numerical form of the reference correlation. methane_critical_thermal_conductivity_enhancement exposes the Vicentini-Missoni enhancement separately and rejects states whose BWR isothermal compressibility is nonpositive. The defining methane equations are from Hanley, McCarty, and Haynes, doi:10.1016/0011-2275(75)90010-7.

Christensen-Fredenslund mixture model¶

corresponding_states_thermal_conductivity maps a homogeneous mixture to the methane reference and applies a conductivity scale:

\[ \lambda_m = F_\lambda \left(\lambda_0-\lambda_{\mathrm{int},0}\right) +\lambda_{\mathrm{int},m}. \]

\(F_\lambda\) contains the mixture critical-property, effective-mass, and shape-factor ratios. The two internal-energy terms use ideal-gas heat capacities supplied by the caller. Following the published mixture model, the pure-methane critical enhancement is omitted from the mapped reference term. The model source is Christensen and Fredenslund, Chemical Engineering Science 35 (1980), 871-875; the validation measurements are separate and originate in doi:10.1021/je60083a034.

Surface and interfacial tension¶

Brock-Bird pure-fluid surface tension¶

For a nonpolar pure fluid, brock_bird_surface_tension evaluates

\[ \sigma = P_c^{2/3}T_c^{1/3}(0.133\alpha-0.281) (1-T_r)^{11/9}, \]

with the Riedel parameter

\[ \alpha = 0.9076\left[ 1+\frac{T_b}{T_c-T_b}\ln\left(\frac{P_c}{1\ {\rm atm}}\right) \right]. \]

The source equation produces dyn/cm from K and atm; the API converts to N/m. It requires \(0<T<T_c\) and \(0<T_b<T_c\). The defining paper is Brock and Bird, doi:10.1002/aic.690010208.

Weinaug-Katz and Danesh¶

For already coexisting liquid and vapor phases, weinaug_katz_interfacial_tension forms the phase parachor contrast

\[ \mathcal{P} = \sum_i P_i\left(\rho_Lx_i-\rho_Vy_i\right), \qquad \sigma=\lvert\mathcal{P}\rvert^4 \]

in the conventional mol/cm³ and dyn/cm units before returning SI. Published pure-component parachors are available through published_parachors; petroleum pseudo-components can use parachor_from_molar_mass.

With danesh_exponent=True, the fixed fourth power is replaced by the Danesh gas-condensate density-dependent exponent. Both liquid and vapor mass densities must then be supplied explicitly. Neither form performs a flash or infers phase identity.

Lee-Chien¶

lee_chien_interfacial_tension constructs separate liquid and vapor phase parachors from critical properties, Riedel parameters, and the Lee-Chien \(B_i\) coefficients before applying the phase density contrast. The bundled published_lee_chien_b table contains only the eight components reproduced in Pedersen Table 10.20. Additional compounds require explicit coefficients; they are never inferred from another component family.

The implementation follows Lee and Chien, SPE-12643-MS (1984), doi:10.2118/12643-MS, through Pedersen chapter 10.

Liquid n-paraffin diffusion¶

hayduk_minhas_n_paraffin_diffusion_coefficient evaluates the infinite-dilution normal-paraffin correlation

\[ D_{AB}^{\infty} = 13.3\times10^{-12} \frac{ T^{1.47} \eta_B^{\,10.2/V_A-0.791} }{ V_A^{0.71} }. \]

In the defining numerical equation, \(T\) is in K, solvent or bulk-phase viscosity \(\eta_B\) is in cP, and the diffusing solute's normal-boiling molar volume \(V_A\) is in cm³/mol. The public API accepts Pa s and m³/mol and returns m²/s. \(V_A\) is a solute descriptor: it is not the current bulk specific volume or EoS phase volume.

The source is Hayduk and Minhas, doi:10.1002/cjce.5450600213. The correlation is not a concentrated-mixture Maxwell-Stefan model.

PyTorch behavior and failure modes¶

All public transport functions preserve dtype, device, leading batch dimensions, and gradients through tensor inputs and explicit fitted parameters. Float64 is the reference precision for dense and near-critical states. A transport calculation can still be scientifically invalid even when its tensor result is finite:

the supplied EoS root may not represent the intended homogeneous phase;
a methane reference mapping can approach a branch or critical ill-conditioning;
a required critical volume, parachor, or Lee-Chien coefficient may be unavailable;
heavy-oil factors or LBC coefficients may be unidentified for the fluid; and
an empirical correlation may be outside its fluid-family or state range.

Nonphysical inputs and nonpositive results raise explicit errors. The transport validation report shows the experimental agreement and model-form limitations for the selected reservoir-gas, heavy-oil, and CO2/methane cases.