capillary number

The capillary number is a dimensionless ratio that expresses the competition between viscous forces and capillary forces acting at fluid interfaces within a porous medium, defined as Nc equals viscosity times velocity divided by interfacial tension (Nc = µv/sigma), where µ is the displacing fluid viscosity in Pascal-seconds, v is the interstitial flow velocity in metres per second, and sigma is the interfacial tension between the displacing and displaced fluids in millinewtons per metre. This ratio governs whether a flowing fluid can mobilize oil ganglia trapped in pore throats after a displacement front has passed, with low capillary numbers indicating that capillary retention forces dominate and oil remains trapped at residual saturation, while elevated capillary numbers indicate that viscous drag forces overcome the capillary pressure holding oil in place. In conventional Western Canada Sedimentary Basin waterfloods, the capillary number typically ranges from 10 to the minus 7 to 10 to the minus 6 because formation brine viscosity is 0.3 to 0.8 mPa-s, interstitial velocities are 0.1 to 1.0 metres per day, and crude-brine interfacial tension is 15 to 30 mN/m at reservoir temperature; at these values the capillary retaining forces far exceed viscous mobilizing forces, leaving 20 to 50% of original oil in place trapped as residual oil saturation in the swept zone after waterflooding. Recovering this trapped oil requires raising the capillary number by one to three orders of magnitude, achieved either by reducing interfacial tension through surfactant flooding (capable of reducing sigma from 20 mN/m to 0.001 to 0.1 mN/m, raising Nc by two to four orders of magnitude), increasing displacing fluid viscosity through polymer flooding (raising µ by a factor of 10 to 100 compared to injection brine), or combining both effects in alkaline-surfactant-polymer floods that simultaneously reduce interfacial tension and improve sweep efficiency. In the Cardium Formation at Pembina and Lochend, reservoir engineers calculate capillary numbers using core flood measurements of residual oil saturation at various flow rates and interfacial tensions to establish the capillary desaturation curve, which plots residual saturation against capillary number and identifies the critical Nc above which significant oil mobilization begins and the plateau Nc at which residual saturation reaches its minimum achievable value. Viking Formation waterflood pools at Provost and Redwater show residual oil saturations of 20 to 30% in swept zones, and capillary number analysis guides surfactant slug volume and concentration design to shift the pool average Nc from its current waterflood value into the desaturation curve transition zone, targeting a reduction of 8 to 15 percentage points in residual saturation within economic chemical flood design constraints. The capillary number concept also applies to gas injection enhanced recovery, where viscous fingering and capillary trapping at the gas-oil interface influence the efficiency of miscible and immiscible carbon dioxide floods in WCSB Devonian carbonate reservoirs; raising the capillary number through injection pressure optimization above minimum miscibility pressure effectively eliminates the gas-oil interfacial tension term, producing first-contact miscibility conditions where sigma approaches zero and trapped oil ganglia dissolve into the gas phase without requiring mechanical mobilization. Reservoir simulation models for chemical EOR incorporate capillary desaturation curves as input functions that relate local capillary number to residual phase saturation at every grid cell, allowing the simulator to track how a surfactant or polymer flood front gradually reduces trapped oil saturation as the chemical slug advances through the swept zone. Laboratory determination of capillary desaturation curves requires steady-state core flood experiments at reservoir temperature and net overburden stress, with measurements at a minimum of five capillary number values spanning the range from background waterflood conditions to the maximum achievable Nc under the proposed EOR design, ensuring that the model accurately captures the transition zone slope and the minimum residual saturation plateau. Understanding capillary number magnitude, the parameters that control it, and the shape of the capillary desaturation curve enables reservoir engineers, EOR specialists, and chemical flood designers to assess whether a given pool is an economic candidate for chemical flooding and to optimize the surfactant or polymer formulation that will deliver the required Nc increase at affordable chemical cost per incremental barrel.

• Definition and formula: Capillary number Nc equals µv/sigma, where µ is displacing fluid viscosity (Pa-s), v is interstitial velocity (m/s), and sigma is interfacial tension (mN/m). The ratio compares viscous drag mobilizing trapped oil ganglia against capillary pressure retaining them in pore throats. Higher Nc favors oil mobilization; waterflood Nc values of 10 to the minus 7 to 10 to the minus 6 in WCSB reservoirs leave 20 to 50% residual oil in swept zones.
• Capillary desaturation curve: A plot of residual oil saturation versus capillary number, measured in core flood experiments at reservoir conditions, showing a flat plateau at waterflood Sor for low Nc, a steep transition zone where oil begins to mobilize as Nc increases, and a lower plateau at minimum achievable residual saturation. The critical Nc marks the onset of mobilization; EOR designs target Nc one to two orders of magnitude above this threshold for economic residual oil recovery.
• Surfactant flooding and interfacial tension reduction: Surfactant EOR raises the capillary number primarily by reducing interfacial tension from 15 to 30 mN/m under waterflood conditions to 0.001 to 0.1 mN/m with optimized surfactant formulations, increasing Nc by two to four orders of magnitude. In WCSB Cardium and Viking pilots, target ultra-low interfacial tensions of 0.01 mN/m require surfactant selection matched to the specific crude oil composition, formation brine salinity (typically 20,000 to 100,000 mg/L in central Alberta), and reservoir temperature of 45 to 65 degrees Celsius.
• Polymer flooding and viscosity increase: Polymer flooding raises Nc through the µ term by increasing displacing fluid viscosity 10 to 100 times above formation brine viscosity, improving both capillary number and macroscopic sweep efficiency by reducing mobility ratio. WCSB polymer floods in Cardium and Lloydminster heavy oil pools use partially hydrolyzed polyacrylamide at 300 to 2,000 mg/L concentration to achieve design viscosities, with capillary number calculations confirming the increment of residual oil mobilization expected beyond the sweep efficiency improvement alone.
• Miscible CO2 flooding: Carbon dioxide injection above minimum miscibility pressure (MMP) effectively drives sigma to zero, creating first-contact or multi-contact miscibility that eliminates the capillary trapping mechanism entirely. In WCSB Devonian carbonate reservoirs targeted for CO2-EOR or acid gas storage-with-EOR, maintaining reservoir pressure above MMP throughout the flood pattern ensures that Nc remains in the mobilization regime and gravity-stable injection geometry minimizes viscous fingering losses.

Capillary Number Analysis for a Cardium EOR Pilot at Pembina

A reservoir engineering team evaluating a surfactant-polymer EOR pilot in a mature Cardium Formation waterflood unit at Pembina field calculated the background capillary number using core analysis data: formation brine viscosity of 0.55 mPa-s, average interstitial velocity of 0.35 m/day (4.1 times 10 to the minus 6 m/s), and crude-brine interfacial tension of 22 mN/m, yielding Nc of 1.0 times 10 to the minus 7. The capillary desaturation curve from restored-state core floods at 50 degrees Celsius showed a critical Nc onset at 5 times 10 to the minus 5 and a minimum Sor of 8% (down from 26% at waterflood Sor) at Nc above 10 to the minus 3. The team designed an alkyl ether sulfate surfactant blend at 0.3% active concentration that achieved interfacial tension of 0.008 mN/m in the pilot brine at 50 degrees Celsius, raising the capillary number to 2.8 times 10 to the minus 3 and placing the design firmly on the minimum-Sor plateau of the desaturation curve. Projected incremental recovery factor was 11 percentage points of original oil in place over the 5-year pilot, justifying the pilot capital of $4.2 million.

Related Terms

Residual oil saturation is the trapped oil fraction that the capillary number must overcome; it is the primary target of chemical EOR programs designed using capillary desaturation curve analysis. Interfacial tension is the denominator in the capillary number formula and the primary lever in surfactant EOR, with ultra-low interfacial tension formulations achieving the two to four order-of-magnitude Nc increase needed to mobilize residual oil. Capillary pressure curve provides the pore-scale data, including entry pressure and threshold pore throat radius, needed to interpret why the capillary desaturation curve has its specific shape for a given reservoir rock. Polymer flooding raises the capillary number through the viscosity term while simultaneously improving macroscopic sweep efficiency, and combined surfactant-polymer (SP) or alkaline-surfactant-polymer (ASP) floods attack both the µ and sigma components simultaneously. Minimum miscibility pressure is the CO2-EOR analog of capillary number optimization: above MMP the interfacial tension between injected CO2 and crude oil approaches zero, effectively achieving infinite capillary number and eliminating capillary trapping as a recovery limitation.