2022 11 03 CHOA eJournal

Synopsis of Recent Papers (SPE205883, SPE210445) on Relative Permeability Expressions Better Suited for Simulation of Gravity Drainage Processes Subodh Gupta When my esteemed colleagues at the editorial committee of CHOA Journal asked me to provide a synopsis of my recent papers on relative permeability, developed in the quest of proper simulation of SAGD, and other gravity drainage processes, I gladly accepted. While the actual papers are publicly available (see references), this brief writeup should serve to tell what these papers are about.

“If proper rel. perms are not used … a history match of the actual slower rise of steam chamber will falsely indicate existence of impermeable barriers even … where none really exist.”

Motivation for these papers has been to address issues with the simulation of gravity drainage processes such as SAGD or SAP. (Gupta S., 2021) deals with and addresses a problem in much used Stone-II model for the simulation of SAGD. In reality, the residual oil saturation in SAGD chamber from retrieved cores decreases with time and is found to be as low as 1-2 %. But application of Stone-II predicts Sor settling down at around 20%.

Figure 1. Simulation with Stone-II vs. reality

While use of the new fundamentals-based rel. perms presented last year fixed that problem (Figure 1), it did not address another issue related to the gravity drainage processes - the counter current movement of gas and liquids and the resultant slow-down of each on account of mutually acting interfacial shear stress. Counter current movement essentially comes into the picture on account of gravity in the potential gradient term (p0g sin0). In absence of gravity, both the gas and the liquids have the same driving force (dP⁄dx).

If proper rel. perms are not used in simulation of SAGD or SAP, a history match of the actual slower rise of steam chamber will falsely indicate existence of impermeable barriers even in perfectly homogeneous reservoirs, where none really exist (Figure 2).

Figure 2. Simulation with gravity-free vs. gravity incorporated rel. perms

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This issue is taken up and resolved with the development of more comprehensive rel. perms incorporating gravity in (Gupta S., 2022). As before the porous medium is approximated by a bundle of tubes having a size distribution. This is the same approach taken by the classical researchers. The difference here comes from the fact as to how the fluid saturations are distributed in the pores of difference size and based on recent insights from X-ray tomography imaging published by Alhosani (2021). In Figure 3, the Figure 3. Modern insight into pore occupancy order left diagram depicts the classical view, and the right one incorporating the new insight. In the classical approach it was assumed the wetting phase occupies the smallest pores, gas phase the largest pores and the intermediate phase the intermediate sized pores.

The X-ray imaging from Alhouseni et al. clearly shows that all pores are accessible to all phases, with saturation in the smallest pores dominated by wetting phase, in the largest pores by gas, and in the intermediate pores by the intermediate phase. Thus, in most pores all three phases co-exist and co-flow. This necessitates development of multi-phase flow equations in tubular channels as opposed to the use of Poeseuille’s equation classically, justified as in any given pore size only the flow of a single phase occurred.

This difference also naturally explains why relative permeability expressions should be temperature dependent. When these expressions are used in conjunction with a poresize distribution and tortuosity factor (to account for the difference in linear tubular channels vs. tortuous flow in porous media) the resultant rel. perm equations are given as shown here (Figure 4).

Figure 4. New gravity-incorporated rel. perms

“… X-ray imaging … clearly shows that all pores are accessible to all phases, with saturation in the smallest pores dominated by wetting phase, in the largest pores by gas, and in the intermediate pores by the intermediate phase.

Thus, in most pores all three phases co-exist and co-flow.”

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They are evidently more involved compared to gravity-free equations presented in the 2021 paper as they also have the ratios of various potential gradient terms (with specific definition of each in the paper), in addition to the expected functions of various phase saturations.

When angle of tilt in flow is set to zero or when value of gravity is set to zero, these equations revert back to the equations presented earlier. In this way the model presented here is internally consistent, and more comprehensive than the earlier model.

To graphically visualize the kind of difference gravity and the resultant counter current flow can introduce in rel. perms, some reasonable values are fed in an example with just two phases – gas and water. For illustrative purposes the expressions for two phases, gas - water system and the three-phase system are delivered separately in the paper an as a consistency check when saturation for one of the phases is set to zero, the 3-phase expressions collapse to the 2-phase one.

The gravity-incorporated vs gravity-neutral relative permeability curves – two phase system:

Figure 5. Graphical visualization of changes: gravity-free vs. gravity incorporated rel. perms CHOA eJournal – 3 NOVEMBER 2022

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In the presented graph (Figure 5), with the incorporation of gravity, the water rel. perm shifts downward.

But the gas rel. perm does not change perceptibly. This is reasonable to expect as the gravity dependent factor in krg becomes negligible due to viscosity of gases being 2 orders of magnitude smaller than that of liquid.

“… gravity-incorporated relative permeabilities [equations are presented] … that show a slowdown of draining liquids in a counter current flow situation … which otherwise might give a false impression of being on account of permeability barriers.”

In conclusion, (Gupta S., 2022) extends the derivation contained in (Gupta S., 2021) and starting with proper force balance in a tilted or vertical flow scenario, presents gravity-incorporated relative permeabilities. These rel. perm expressions are more comprehensive and revert to the earlier expressions when gravity dependent term vanishes. This also implies that they are anisotropic, being different for horizontal flow vs. the vertical flow. When used, these rel. perms show a slow-down of draining liquids in a counter current flow situation. In simulation of SAGD, they show the expected slow-down of chamber rise even in a homogeneous reservoir which otherwise might give a false impression of being on account of permeability barriers.

References Abdulla Alhosani, Branko Bijeljic, M. J. (2021). Pore-Scale Imaging and Analysis of Wettability Order, Trapping and Displacement in Three-Phase Flow in Porous Media with Various Wettabilities.

Transport in Porous Media. doi:10.1007/s11242-021-01595-1 Gupta, Subodh. (2021). Issue with Stone-II Three Phase Permeability Model, and A Novel Robust Fundamentals-Based Alternative to It. Paper presented at the SPE Annual Technical Conference and Exhibition. Dubai, UAE: SPE. (SPE205883) Gupta, Subodh. (2022) "Physics-Based 3-Phase Relative Permeability for Gas-Liquid Counter-Current Flow." Paper presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, USA: SPE. (SPE210445)

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