Effect of Hysterisis on Flow in Porous Media
Authors: Cunha L.B. , Bonet E.J. 
Published: December 2007

Abstract
The hysterisis phenomenon affects the flow behaviour in various occasions. In particular in the WAG process the cyclic injections of wetting and non-wetting (water and gas) fluids causes the appearance of the hysterisis phenomena. To access its magnitude we performed experimental tests and analytical studies. We measured in different formations the relative permeabilities in the imbibition and drainage cycle, using oil and water. We monitored the rates and pressures at the extremes of the rock sample and also the internal saturation evolution with electrodes. These data permitted to obtain the fractional flow curve for the imbibition and drainage process. We then performed a flow test, which result, could be compared with different mathematical approaches proposed in the literature to describe this behaviour. There are in the literature different proposals to calculate the flow in porous media honouring the hysteretic flow. We used three semi-analytical and one numerical method to describe the experimental test. Finally we compared the results.

Introduction
The hysterisis phenomena in flow in porous media manifest itself in different occasions. In the oil exploitation we could mention the cyclic injection of gas and water for the WAG process, in the ascending and descending of the water cone, in the gas injection and production of storage aquifers, in the miscellar flooding process where an oil bank is displaced by water(1)(2)(3)(4)(5)(6). In hydrology the ascending and descending of the water table in the vadoze zone presents a hysterisis phenomena(7).
Different rocks and fluids present different degrees of hysterisis. It is believed that this phenomena depends on, besides fluid properties, morphological and topological aspects of porous media(8).
Normally the hysterisis manifests itself in different petrophysical properties. The more frequently cited in the literature are the capillary pressure, fractional flow curves and relative permeabilities.
The first drainage presents a very pronounced hysterisis. In oil fields, normally the first drainage occurs in the secondary oil migration and is not present in the exploitation stage. For this reason we will not consider it in this article and only periods after the first imbibition and secondary drainage will be considered.
In our paper we will focus mainly on the two-phase relative permeabilities, and its derived fractional flow curve. The main reason for not measuring capillary hysterisis is the fact that semi-analytical models based on hyperbolic equation based on mass conservation do not consider the capillary pressure. Also, the literature shows that this is an acceptable simplification.
For our two-phase system there is some controversy about the behaviour of different curves.
Our experimental results confirm, what is usually acceptable, that the hysterisis effect is important only on the non-wetting phase permeability and the imbibition permeability is greater than the drainage(3)(9)(10)(11).
We measured the relative permeability in a steady-state process and calculated and compared experimental tests with one numerical and three semi-analytical methods(12)(13)(14)(15)(10)(11)(16)(17(18).

Experimental Setup and Tests
The experimental setup is presented in figure 1. Searching for different degrees of rock hysterisis we performed tests in two formation rock-types: Botucatu and Berea sandstones. These sandstones have its petrographycal and petrophysical characteristics described elsewhere(19).
Figure 2 presents two test examples: one in the Botucatu formation where, we considered did not present significant hysterisis behaviour, and another in the Berea formation where the hysterisis effect was present although not very intense.
In this article we will present results obtained from this Berea sample. Its main geometric and petrophysical characteristics are given in the table 1. In the same table are presented the characteristics of the fluids used in the tests.
The relative permeabilities obtained by a steady state method, as well as the fractional flow results, for the drainage and imbibition cycle are presented in figure 3.
As already mentioned the wetting phase relative permeabilities are coincident while the non-wetting phase relative permeabilities present the hysterisis effect with the imbibition oil relative permeability presenting higher values for the same saturation.