The Effect of Displacement History on Three-Phase Capillary Pressure and Relative Permeability Derived From Triangular Pore Cross-Sections With Contact-Angle Hysteresis

We present a model of mixed-wet triangular tubes that calculates three-phase capillary pressure and relative permeability curves. Several fluid configurations may occur in triangular pore cross-sections, and capillary displacements may either occur as piston-like displacements of the fluids occupied in the bulk, or as piston-like displacements of the fluids in layers. To our knowledge, this latter type of displacement has not been analyzed before in mixed-wet pores. Using minimization of Helmholtz free energy, we derive accurate three-phase capillary entry pressures for such layer displacements, accounting for contact-angle hysteresis. Numerical examples are presented to illustrate how the entry pressures for the different possible displacements relate to each other during gas and water invasion into pores with a specific fluid configuration. It turns out that the entry pressures for related displacements are consistent. This implies that pores occupied by the same fluid in the bulk portion must have the same fluid configuration for a constant value of capillary pressure. With this model we calculate three-phase capillary pressure and relative permeability, and explore how the saturation-dependencies of these quantities change according to saturation-reversal points. We simulate the sequence of processes primary drainage, imbibition and gas invasion, for different maximum capillary pressures P max ow after primary drainage. In the simulation results presented here, we find that the oil and gas relative permeability, and their saturation-dependencies, are sensitive to variations of P max ow , while the water relative permeability is less sensitive. Such effects are absent in cylindrical tubes. This is caused by the capillary entry pressures, which are strongly affected by hinging interfaces in the corners of angular pores when contact-angle hysteresis is assumed. Thus the choice of pore geometry is important if hysteretic capillary pressure and relative permeability relationships are simulated using network models. With respect to these findings, relative permeability and capillary pressure correlations should be formulated with parameters that strongly depend on saturationreversal points such that different saturation-dependencies can be accounted for in subsequent invasion processes. Introduction Relative permeability and capillary pressure are required as functions of the saturations to solve the equations for threephase flow in reservoir simulation. These relationships are normally formulated as simple correlations with adjustable parameters. In the reservoir, situations may occur where one of the phases appears or disappears, e.g., during phase transitions between gas and oil, or when a zero residual oil saturation is approached by drainage through continuous spreading layers in the crevices of the pore space. To implement these scenarios in a numerical reservoir simulator without creating convergence problems, the correlations must account for a smooth transition between twoand three-phase flow. In the oil industry three-phase capillary pressure and relative permeability curves have traditionally been predicted from corresponding two-phase measurements. However, both experimental and numerical work have shown that this practice may not be valid. Moreover, micromodel studies of threephase flow have revealed that the fluid distribution and the displacement mechanisms at the pore scale may be more complex than for two phases.1, 2 These findings emphasize the need for direct measurements of three-phase capillary pressure and relative permeability curves for various conditions. However, in three-phase flow there is an infinite number of possible displacement paths because of two independent saturations. Hence, it is impractical to perform time-consuming measurements of a vast amount of different processes for several rock and fluid properties. This points out the importance of developing physically-based pore-scale network models 3–6 to compute the relative permeability and capillary pressure curves. A pore-scale model, tuned to reproduce the measured data, may be employed to predict these quantities for displacement paths not covered by the measurements. 2 J.O. HELLAND AND S.M. SKJAEVELAND SPE 99743 In a recent paper7 we presented a simple bundle-of-triangulartubes model where the tubes have triangular cross-sections. This is a very simplistic approach for modelling of realistic reservoir rocks. Such a simple model does not incorporate the effect of interconnected pore networks, and hence phase entrapment is absent. However, the angular pore shapes allow for other important physical processes, such as the development of mixed wettability at the pore scale,8, 9 and drainage through oil layers along the corners.10 Several fluid configurations can occur in a triangular cross-section, and this requires careful analysis of all possible displacements. Based on the method proposed by van Dijke and Sorbie,11 we derived accurate expressions for the three-phase capillary entry pressures that account for contact-angle hysteresis and the possibility of simultaneous displacement of the fluids occupying the cross-sections. However, piston-like invasion was only assumed to occur as a displacement of fluids occupied in the bulk portion of the pore. More recently, van Dijke et al.12 employed their method to also account for piston-like displacements of fluid layers in the corners of the pores. In the present paper we explain how to caculate entry pressures for such displacements in a model of mixed-wet pores with contact-angle hysteresis. The developed pore model was employed to simulate threephase capillary pressure curves for various conditions and to analyze the corresponding saturation-dependencies. 7 It was found that the reversal point after primary drainage may strongly affect the saturation-dependencies of three-phase capillary pressure in subsequent displacement processes. Experimental measurements have shown that reversal point after primary drainage is related to wettability.13 This in turn affects the saturationdependencies of three-phase capillary pressure and relative permeability.14–18 In triangular tubes, the maximum capillary pressure after primary drainage, and thereby the reversal point, is related to wettability.19 Thus, there exists a relationship between the saturation-dependencies and the maximum capillary pressure for triangular tubes. The present paper represents extensions of the work by Helland and Skjaeveland7 in two directions: First, we extend the simple pore model to account for layer displacements for mixedwet conditions and contact-angle hysteresis, following the method by van Dijke et al.12 Second, we extend the model to also account for calculations of three-phase relative permeability. The paper is organized as follows: First we describe a method to calculate advancing and receding contact angles for three phases. Then we briefly describe the main features of the model, before derivations of three-phase entry pressures for bulk and layer displacements are described in detail for cases of gas invasion. Then we present numerical examples to examine how the different entry pressures relate to each other. With the present model we calculate relative permeability and explore if the saturationdependencies are sensitive to variations of reversal points in the displacement history.