Use of the Method of Characteristics to Study Three-Phase Flow

Gonzalez, Marta (PDVSA Intevep) | Araujo, Mariela (PDVSA Intevep)



Numerical simulation of multiphase flow is the most acceptable method for the prediction of oil recovery. Recently, the modeling of the three-phase flow has become critical since it may occur in a variety of conditions such as when gas is injected in a reservoir, or when gas or condensate appears as the reservoir pressure drops. Gas injection includes hydrocarbon gas, CO2, air, nitrogen or steam injection as well as WAG or gravity drainage. For such processes, the modeling of fluid displacement must include hysteresis in the relative permeability curves.

In this work, the method of characteristics is used to study the effect of rock geometry and wettability condition on the saturation paths for a three-phase system. With this method the partial differential equations used to describe fluid transport are converted into a set of ordinary differential equations which are integrated simultaneously. The method gives information about the flow behavior (fractional flow curves) and the saturation trajectories represented on a ternary diagram.

Imbibition and drainage processes are modeled using relative permeability curves calculated with a fractal pore model, on samples of strong wettability contrast: water wet, intermediate and oil wet respectively. The effect of the rock geometry and wettability condition on the saturation trajectories is analyzed.

Herein, the most relevant contribution of this work is a fast and accurate method to study the gemoetry and wettability effects on three-phase flow properties, such as relative permeability. These properties are relevant for the understanding of oil production and the design and selection of the more appropriate oil recovery strategies.


The prediction of the recovery efficiency of a reservoir under many tertiary recovery processes such as gas injection or WAG requieres an adequate use of three-phase flow properties. The numerical simulation of multiphase flow under three-phase conditions requires the use of transport properties which are dependent on the particular process the fluids are undergoing. Among these properties relative permeabilities and capillary pressures are known to be dependent on the saturation and the saturation history, effect known as hysteresis1-3.

Hysteresis effects must be considered when the displacement mechanism changes between drainage and imbibition in two-phase flow2. The saturation oscillations associated to WAG processes lead to hysteresis in the fluid properties, making necessary the consideration of these effects in the reservoir simulation if a good description of the fluid behavior is expected. In addition, several tertiary recovery processes have shown the hysteresis effects related to the displacement cycle being considered4. A better understanding of the behavior of three phase flow is essential to incorporate hysteresis effects on flow models.

The number of phases present in the porous medium is relevant when studying three-phase flow are modeled. For two-phase systems there are only two relevant saturation sequences known as imbibition and drainage, since the saturation of the two fluids satisfy S1+S2=1. The problem gets more complicated under three-phase conditions since the fluid saturations satisfy the relation S1+S2+S3= 1. Terms such as drainage and imbibition are now meaningless. For any two fluids it is required to indicate if the saturation increases or decreases, as for a DDI case where the water saturation (first letter) decreases, the oil saturation (denoted by the second letter) decreases and the gas saturation (third letter) increases. For three-phase flow there are six different saturation sequences, fact that adds complexity when three-phase relative permeability are generated from two-phase data as done conventionally5.