Research Spotlight Collection and Retrospective Examination of Relative Permeability Data on Steady-State 2-Phase Flow in Porous Media

Collection and Retrospective Examination of Relative Permeability Data on Steady-State 2-Phase Flow in Porous Media

M.S. Valavanides

TEI Athens, Civil Infrastructure Eng. Dept, Laboratory of Hydraulics, marval@teiath.gr

Abstract and scope of work

The scope of this note is to highlight current research efforts towards elucidating and understanding the principal mechanism in two-phase flow in porous media, namely, the interplay /dialect between viscosity, capillarity and porous medium structure, and its effect on the operational performance of such flows/processes – expressed by the interstitial arrangement(s) of the flow and its stability, the efficiency of the process (flowrate/power), etc.

Experimental evidence on the phenomenology of steady-state two-phase flow in porous media is recorded in the well-known relative permeability diagrams published in the literature [2, 3]. An extensive retrospective examination of such data is being conducted in order to reveal and identify important process characteristics, specifically:

1. to provide experimental evidence on the existence of optimum operating conditions, i.e. conditions whereby process efficiency (here considered in terms of oil produced per kW dissipated by the process) attains maximum values, a flow characteristic already predicted by the DeProF theory [1],
2. to delineate a universal operational efficiency map for steady-state two-phase flow in porous media; this map is based not only on appropriate modelling considerations but also on extensive experimental evidence,
3. to examine the combined effects of the capillary number, the viscosity ratio and the porous medium structure on the operational performance of the process,
4. to examine if (the effect of) the porous medium structure (geometry and topology) can be described (evaluated) with only a few parameters in an indirect fashion, i.e. not from geometrical and topological considerations but considering its effect on the operational performance of the flow.

Overview

Optimum operating conditions for steady-state two-phase flow in pore networks were first predicted by the DeProF theory [1]. The operational efficiency of the sought process, considered in terms of oil produced per kW dissipated by the process, is measured by the energy utilization index,

(1)

where, r is the oil/water flow rate ratio and is the reduced mechanical power dissipation (including the effect of bulk viscosities and interfacial hysteresis on strain rates). Ca, the capillary number, and r, the oil-water flow-rate ratio, are the process operational parameters;  is the specific rate of mechanical energy dissipation of the two-phase flow, and  is the rate of mechanical energy dissipation for equivalent one-phase flow of water. Extensive simulations using the DeProF mechanistic model revealed the existence of optimum operating conditions in the form of a smooth and continuous locus [r*(Ca)] in the domain of the process operational parameters (Fig. 6 in [1]).

The transformation, originally introduced in [1],

and

(2)

where  is the oil/water viscosity ratio, valid for steady-state flow conditions, was implemented to map laboratory data sets of relative permeabilities k_ri(S_w), i=o,w into corresponding energy utilization values f_EU(r). Such an indicative transformation is presented in Figure 1.

(a)
(b)
Figure 1 Steady-state relative permeability for oil (red squares) and water (light blue diamonds) and energy utilization index, fEU, (filled dots in red & green), eqn (1), corresponding to flow-rate ratio values r for two-phase co-current flow in a fine sand pack. The values of r and fEU are computed by transforming – via eqs (2) – the original measurements published by Bentsen [2], see diagrams on the left. The data sets pertain to (a) favourable and (b) unfavourable viscosity ratios κ. The values rx and r* mark the flow-rate ratio values for which equal relative permeabilities and local maximum operational efficiency are attained. The effect of the system and operational parameter values on rx & r* are described in [4, 5 & 6].

Current Status

So far, reconstructions similar to those presented in Figure 1 were delivered from 19 published works and for as many as 83 relative permeability diagrams [3], pertaining to a variety of conditions of steady-state two-phase flow in sand packs, plug cores, glass or virtual micro-models, etc. (Table I).

Observations show a universal trend of the process operational efficiency in terms of Ca, r and κ. Such a trend can be cast into an operational efficiency map [4]. Referring to this map, one may consistently and rationally resolve the extent to which disconnected oil flow and associated capillarity effects regulate the flow, and one could use it as a normative tool for flow characterization (capillary dominated / viscous dominated) or as a guiding tool for designing more efficient processes.

At present, a complementary laboratory study on steady-state 2-phase flow in sand packs for fluid systems of equal viscosity (κ=1) is in progress in the “Laboratory of Transport Phenomena & Porous Materials” at the Institute of Chemical Eng. Sciences, Foundation of Research & Technology Hellas (ICE-HT/FORTH).

Updates with new data will be provided in due time at the ImproDeProF project website, http://users.teiath.gr/marval/ArchIII/ImproDeProF.html.

Many exploratory actions have already been taken with respect to tasks (a)-(d) already described in the introduction. The preliminary results that have been delivered and presented in a series of works [3, 5, 6] are promising. Thus, there is significant evidence in justifying additional efforts in this area and the success of the outlined research [tasks (a)-(d)] is likely to be reached.

Invitation to incorporate additional relative permeability data / laboratory studies

An open invitation is addressed to colleagues and to the InterPore society members for exchanging points of view and for discussing the possibility of collaboration in this particular area of research.

If you have performed a similar study, delivered laboratory or numerical data, or if you are aware of such data that were not included in the referenced work [3], please contact Dr. Marios Valavanides (marval@teiath.gr) to provide your input, citation, reference, and other relevant information.

If you plan to perform a similar laboratory or numerical study on steady-state 2-phase flow in porous media, Dr. Marios Valavanides would be eager and willing to exchange ideas and/or points of view.

You may also want to check how your relative permeability data would look and what latent information could be revealed under transformation (2). To do so, you may download the handy excel file http://users.teiath.gr/marval/ArchIII/relpermtrans.xls to your PC, input your steady-state relative permeability data and discover what are the local conditions of optimum operation, what is the character of flow (capillary/viscous) for each relative permeability measurement (run) and what is the maximum attainable process operational efficiency in your system.

Proposal to create a relative permeability databank

Apart from collaboration and exchanging of information, ideas etc., the creation of a databank containing relative permeability diagrams /studies resulting from laboratory experiments / measurements and/or numerical simulations (pre-existing or new) may provide the necessary evidence and would further leverage following attempts to model the process and deal with tasks (a) to (d). In that context, it is suggested /proposed:

1. to build a databank containing a broad collection of relative permeability diagrams (immiscible steady-state in the beginning , unsteady-state eventually), and
2. to provide a normative methodology for comparing all available studies under the same -universal- measure and in terms of the pertinent process operational parameters /independent variables.

Such a databank will be available to researchers in the particular field or to the scientific community as a whole, free-of-charge, provided the original data providers consent on such a possibility, the appropriate acknowledgements are given and pertinent copyright material restrictions are not in effect (if the latter is not the case, even the citation information would be valuable for a researcher). With direct access to such a databank a researcher may invest more of his time in the retrospective examination of relative permeabilites over a broad spectrum of system/flow conditions instead of delving into libraries to locate such data. Any suggestions on the particular task are more that welcome (and should be fwd to marval@teiath.gr).

Acknowledgements

The preliminary research work presented above has been co-funded by the European Union (European Social Fund) and Greek national resources in the frameworks of the “Archimedes III: Funding of Research Groups in TEI of Athens” (MlS 379389) a RTD project (ImproDeProF), of the “Education & Lifelong Learning” Operational Program.

References

1. Valavanides, M.S. (2012) “Steady-State Two-Phase Flow in Porous Media: Review of Progress in the Development of the DeProF Theory Bridging Pore- to Statistical Thermodynamics- Scales” Oil & Gas Science and Technology, 67(5) pp. 787-804 [http://dx.doi.org/10.2516/ogst/2012056]
2. Bentsen, R.G. (2005) “Interfacial Coupling in Vertical, Two-Phase Flow Through Porous Media” Journal of Petroleum Science & Technology 23, pp. 1341-1380
3. Valavanides, M.S., Totaj, E. (2013) “Retrospective Examination of Relative Permeability Data on Steady-State 2-Ph Flow in Porous Media Transformation of Rel-Perm Data (k_ro, k_rw) into Operational Efficiency Data (fEU)” ImproDeProF /Archimedes III, project internal report [http://users.teiath.gr/marval/ArchIII/retrorelperm.pdf]
4. Valavanides, M.S., Kamvyssas, G. (2013) “Operational Efficiency Map of Steady-State Two-Phase Flow in Porous Media Processes” InterPore2013, 5th International Conference on Porous Media Prague, 21-24 May [http://users.teiath.gr/marval/publ/Valavanides_Kamvyssas_InterPore5_2013.pdf]
5. Valavanides, M.S., Kamvyssas, G., Totaj, E. (2013) “Retrospective Examination of Relative Permeability Data and Operational Efficiency Aspects for Steady-State 2-Phase Fow in Porous Media” 6^th Panhellenic Symposium on Porous Media, Kavala, September 9-10 [http://users.teiath.gr/marval/publ/Valavanides_etal_PSPM6_2013.pdf]
6. Valavanides, M.S. (2013) “Capillary vs Viscous Flow: Introduction of a Normative Methodology for the Characterization of 2-Ph Flow in P.M.” 6^th Panhellenic Symposium on Porous Media, Kavala, September 9-10 [http://users.teiath.gr/marval/publ/Valavanides_PSPM6_2013.pdf]