Advective Transport Modeling

Key Takeaways

• The Peclet number Pe = v × L / D is the dimensionless ratio of advective to diffusive transport at length scale L. At Pe greater than 10, advection dominates and solute moves primarily at the bulk fluid velocity; at Pe less than 0.1, diffusion dominates and concentration gradients are smoothed independent of flow. In typical WCSB waterflood operations, interstitial velocities of 0.1-5 m/d, dispersivities of 1-20 m at well spacing scales, and molecular diffusion coefficients of approximately 10^-9 m²/s give field-scale Pe values of 10^4 to 10^6, firmly in the advection-dominated regime. This means that the timing of tracer or waterfront breakthrough at producing wells is controlled almost entirely by the permeability distribution and the injector-producer geometry, not by diffusion. Dispersion modifies the sharpness of the breakthrough front but does not significantly change the timing of first arrival. Pe at the pore scale (L = pore throat diameter 0.01-0.5 mm, v = 10^-4 m/s) is typically 0.01-10, meaning diffusion contributes to pore-scale mixing but advection still dominates pore-to-pore transport.
• Dispersivity α_L is scale-dependent, increasing from approximately 0.001-0.01 m at the core plug scale to 1-10 m at the inter-well scale to 10-100 m at the field scale. This scale effect arises because heterogeneity in permeability, which causes velocity variations that spread solute, becomes more pronounced at larger observation scales: at the core scale, only pore-scale grain sorting controls velocity variation; at the inter-well scale, bedding, cementation, and facies heterogeneity all contribute. The practical implication is that dispersivity cannot be reliably upscaled from laboratory corefloods to field-scale models without correction: using core-plug α_L = 0.005 m in a field-scale tracer model with 200 m well spacing would severely underpredict dispersion and overpredict the sharpness of solute breakthrough fronts. Field-scale α_L values are best estimated from inter-well tracer test breakthrough curve analysis, typically yielding 3-15 m for Viking and Cardium sandstone waterfloods in Alberta.
• The retardation factor R quantifies the degree to which a solute is slowed relative to the carrier fluid by sorption onto the solid phase. For a linear sorption isotherm (q_s = K_d × C, where q_s is the mass sorbed per mass of rock), R = 1 + ρ_b × K_d / φ. Conservative tracers (NaBr, NaCl, tritiated water HTO, deuterium oxide D2O) have K_d approximately equal to zero and R approximately equal to 1, traveling at the mean fluid velocity; they are ideal for characterising fluid flow paths and swept pore volumes. Partitioning tracers (hexanol, isopropanol, fluorobenzoate esters) partition between the aqueous phase and residual oil with partition coefficient K_ow, giving R = 1 + ρ_b × S_o × K_ow / (φ × S_w), where S_o is residual oil saturation. By co-injecting a conservative and a partitioning tracer and comparing their breakthrough times, the residual oil saturation in the swept volume can be calculated directly: S_o = (R-1) × φ × S_w / (ρ_b × K_ow). This single-well or inter-well partitioning tracer test (SWCTT or IWPTT) provides a direct in-situ measurement of residual oil saturation for EOR potential assessment.
• Permeability heterogeneity is the dominant control on advective transport outcomes at the reservoir scale, overriding the effects of dispersion in most cases. In a heterogeneous formation with a Dykstra-Parsons coefficient V_DP greater than 0.6 (typical of many WCSB sandstone reservoirs), the fastest-flowing high-permeability streaks carry injected tracers and water to the producing wells at velocities 10-100 times faster than the bulk average, creating very early breakthrough in the highest-perm layers while significant swept-out volumes in low-permeability intervals lag far behind. This causes the characteristic early breakthrough and long tailing shape of tracer breakthrough curves from heterogeneous reservoirs: the early part of the BTC (concentration versus pore volumes injected) reflects the fastest flow paths, while the long tail at concentrations well below peak reflects late drainage of slow paths. A model that assumes homogeneous permeability will predict a symmetric, bell-shaped BTC and miss both the early breakthrough (with its implication for flood efficiency) and the tailing (with its implication for chemical retention times in EOR).
• Numerical methods for solving the ADE include finite-difference methods (accurate but can introduce numerical diffusion on coarse grids when the cell Peclet number exceeds 2; stabilised by upstream weighting but at the cost of added numerical dispersion), particle tracking (virtual particles advected by the velocity field plus random walk diffusion; avoids numerical diffusion but requires many particles for low-noise results), and streamline-based simulation (divides the 3D velocity field into a set of 1D streamlines connecting injectors to producers; solves the 1D ADE along each streamline analytically or semi-analytically; efficient for large waterflood and tracer problems). CMG-GEM and Schlumberger ECLIPSE handle reactive transport and compositional systems; MODFLOW/MT3DMS is the regulatory standard for groundwater applications; streamline tools (FRONTSIM, 3DSL) are standard for waterflood pattern optimisation and tracer matching in WCSB conventional oil fields.