Fractional Flow, Relative Permeability & Capillarity: real example of basic concepts
This brief post illustrates a common example of how basic concepts like capillary pressure, relative permeability and fractional flow, were used to explain the production behaviour in a well drilled and completed in a Mesozoic carbonates field.
In my opinion, a very important part of the petrophysical learning process consists in going back, after a well has been completed and tested, and reviewing the analysis and the consistency of the expected results with the actual outcome.
Well-1, displayed in the picture below, was completed in two different intervals. The first interval (deeper) produced water with some oil traces (2%); the second interval (shallower) produced 100% oil.
Prior to the completion, a detailed petrophysical analysis was performed in the well (integrating all the available data, including well logs, core measurements, drilling cuttings descriptions, etc.). Interestingly, in the deeper interval that tested water, the computed water saturation was between 40 and 45%. In addition, the core and drilling cuttings presented significant oil staining and fluorescence.
In the fist track of the picture above are presented the completed intervals, the second track displays the mineralogy (blue is limestone, magenta is dolomite). Track 6 displays resistivity curves, track 7 permeability and track 8 shows water saturation (Sw) computed from well logs. The last track (9) displays again the Sw from log analysis (blue) and the Sw computed from a saturation-height model (red). The dashed blue line that crosses all the tracks is the estimated Free Water Level (FWL) required in the saturation-height model to match the Sw from logs.
There are several saturation-height methods available in the literature, some of them are capillary pressure-based and others wireline-log based. The SPE-71326 paper from B. Harrison and X.D. Jing (2001) describe some of them:
Capillary pressure-based method from Leverett (1941)
Capillary pressure-based method from Johnson (1987)
Log-based method from Cuddy et al. (1993)
Capillary pressure and Log-based method from Skelt-Harrison & Skelt (1995, 1996)
Other interesting and very comprehensive source of saturation-height functions can be found in the document written by Worthington, P.F., 2002 (Application of saturation-height functions in integrated reservoir description, in M. Lovell and N. Parkinson, eds., Geological applications of well logs: AAPG Methods in Exploration No. 13, p. 75–89).
In the specific example presented in this post, no capillary pressure data was available and a log-based method was used. The non-linear Porosity versus Height function, relating log water saturation to height-above-FWL (applied to each different rock type) was:
Sw = [a + b.Log(h) + c.Log(h)^2 + d. Log(h)^3 ] / [ϕ^f]
h = Height above FWL
ϕ = Porosity
Based on the water saturation-height functions a conceptual model of fluid distribution was built (picture below). Well-1 is displayed with the computed water saturation curve and the FWL is showed as a dashed blue line. The reservoir was divided in 4 main zones:
The upper one has the best rock properties: good porosity, good permeability, and good capillary pressure profile. Transition zone is short and the Oil-Water contact (OWC) is close to the FWL.
The second zone is tight with high capillary forces resulting in a long transition zone.
Third zone has moderate porosity and permeability values. Capillary forces produce a transition zone longer than the one in the first zone with better rock properties.
Deeper zone is similar to the second zone.
Relative Permeability and Fractional Flow
The picture below illustrates the correlation between capillary pressure, relative permeability, fractional flow, the conceptual fluid distribution within the reservoir, and the expected initial production behaviour.
In the upper part of the picture, the relative permeability of Oil (Kro) is presented as a green curve, relative permeability of water (Krw) is the blue curve, and the fractional flow (Fw) is the magenta curve. The lower part of the picture shows the capillary pressure curve in red.
At water saturations lower than the Critical (Swc) or equal to the Irreducible (Swirr) the relative permeability of the water is zero (there is no free or mobile water) and a clean oil production is expected.
As the water saturation increases, the relative permeability of oil gradually decreases and becomes zero at the residual oil saturation (Sor). At this point, only water is expected to be produced. In the reservoir, this corresponds to the water-oil contact (WOC).
At water saturations higher than Swc and lower than 1-Sor, oil and water production is expected and in the reservoir this corresponds to the transition zone. The fraction of water and oil flowing is given by the fractional flow curve, and not only depends on the porous media properties but also on the fluids properties. The main fluid property affecting the flow is the viscosity. For example, if gas and oil have the same relative permeability, more gas than oil will flow because of the significant difference in viscosity.
The fractional flow equation for the simplest case of horizontal flow with negligible capillary pressure gradient is:
Fw = 1 / [ 1 + (Kro* μo) / (Krw* μw) ]
The picture below shows the relative permeability and fractional flow curves modeling for the reservoir presented in this example. It uses the correlation presented by Mohamad Ibraim-Koederitz (2000), and takes into account the wettability, lithology, porosity, permeability, critical water saturation, residual oil saturation, oil and water viscosity, and oil and water formation volume factors.
In this picture are presented two fractional flow curves in magenta. One is computed using an oil viscosity of 1.5 cp and the other using 42 cp. The purpose of displaying the two cases is to highlight the impact of the oil viscosity in the production behaviour.
The deeper interval that tested water in the Well-1 has a water saturation of about 45%. In addition, the actual oil viscosity of the reservoir is 42 cp. Using the proper fractional flow curve in the picture above, the expected water cut (water fraction) is about 97%, which is very close to test results.
Other relative permeability correlations were used to compare the results. Some of them are presented in the picture below:
As a final check, the analysis of the water salinity and its chemical composition confirmed that the water produced in the test was formation water.