Johan Olav Helland

Physically Based Capillary Pressure Correlation for Mixed-Wet Reservoirs From a Bundle-of-Tubes Model

It is shown that the main characteristics of mixed-wet capillary pressure curves with hysteretic scanning loops can be reproduced by a bundle-of-triangular-tubes model. Accurate expressions for the entry pressures are employed, truly accounting for the mixed wettability and the diverse fluid configurations that arise from contact-angle hysteresis and pore shape. The simulated curves are compared with published correlations that have been suggested by inspection of laboratory data from core plug experiments.

Three-Phase Capillary Pressure Correlation For Mixed-Wet Reservoirs

We present new three-phase capillary pressure correlations that could be employed to model the dynamics of three-phase transition zones in mixed-wet reservoirs. The capillary pressures are expressed as a sum of two terms. One term is a function of a decreasing saturation and the other term a function of an increasing saturation. Thus the correlations depend on the type of displacement process, i.e., the direction of saturation change. The two-saturation dependency, together with the inclusion of adjustable parameters, ensures that the correlations account for different wettability conditions, saturation histories, and different relationships between the three capillary pressures. The correlations are compatible with a smooth transition between twoand three-phase flow if one of the phases appears or disappears. In particular, if the gas saturation becomes zero, it is shown that the correlations are reduced to a previously published two-phase correlation validated for oil/water systems in mixed-wet rock. Capillary pressure curves for various conditions, computed using a previously developed bundle-of-triangular-tubes model, are compared with the correlations, and the match is excellent in all cases. Finally, the correlations are validated experimentally by centrifuge measurements performed on water-wet cores.

Footprints of spontaneous fluid redistribution on capillary pressure in porous rock

Pore-scale imaging of two-phase flow in porous media shows that pore filling occurs as cooperative events with accompanying spontaneous fluid redistribution in other parts of the pore space. We present a level set method that controls saturation quasi-statically to model experiments controlled by low, constant flow rates and demonstrate that our method can describe the observed displacement mechanisms. The level set approach determines states of capillary equilibrium, which generally are different for displacement protocols constrained by saturation and pressure. Saturation-controlled simulations of drainage in sandstone show spontaneous fluid redistributions with abrupt pressure jumps and cooperative behavior, including snap-off and interface retraction events, consistent with experimental observations. Drainage capillary pressure curves are lower when saturation, rather than pressure, controls displacement. Remarkably, these effects are less significant for imbibition processes where the development of hydraulically connected wetting phase moderates the cooperative behavior and associated pressure jumps.

Pore-Scale Study of the Impact of Fracture and Wettability on Two-Phase Flow Properties of Rock

PORE-SCALE STUDY OF THE IMPACT OF FRACTURE AND WETTABILITY ON T H E TWO-PHASE FLOW PROPERTIES OF ROCK DMITRIY SILIN, JONATHAN AJO-FRANKLIN, JOHAN OLAV HELLAND, ESPEN JETTESTUEN, AND DIMITRIOS G. HATZIGNATIOU Abstract. Fractures and wettability are among other factors that can strongly affect the two- phase flow properties of porous media. Maximal-inscribed spheres (MIS) and finite-difference flow simulations on computer-generated structures mimicking micro-CT images of fractured rock suggest the character of the capillary pressure and relative permeability curves modification by natural or induced fracture and wettability alteration. 1. Introduction The presence of two or more different immiscible fluids (e.g., water, oil, gas) makes the pore space saturation and fluid flow multiphase. Any oil or gas recovery operation has to deal with two or three-phase flow. The same is true for subsurface injection, for example, for the purpose of CO 2 geologic sequestration. Capillary pressure and relative permeability curves are conventional concepts characterizing the capability of porous materials to store the fluids and the possibility of fluid flow and migration [16]. These curves can be determined from core laboratory experiments. Recently, this traditional approach has been complemented by digital rock methods [18, 21, 22] including pore-network modeling [2, 4, 6–8, 14, 19] and simulations on 2D and 3D micro-tomography data [9, 12, 15, 17]. The method of Maximal Inscribed Spheres (MIS) [23, 24] uses digitized micro-CT data as input information for evaluation of fluid distribution in capillary equilibrium. This method estimates two-phase fluid occupation of the pores, computes capillary pressure and relative permeability curves. The results are in agreement with experimental data [25, 27, 28]. In this study, the MIS method is employed to estimate the impact of natural fractures and wettability on capillary pressure and relative permeability curves. Micro-CT imaging of fractured rock can be difficult. In this study, we generate (via computer) a synthetic material with angular grains. We use micro CT data for samples of Bentheimer sandstone to validate the procedure. After model validation, a fracture is simulated by parting sand grains whose centers are on difference sides of the fracture plane. 2. The method of maximal inscribed spheres The idea of the MIS method of is simple [23, 27, 28]. In capillary equilibrium, the curvature of the fluid-fluid interfaces is determined by the capillary pressure, which is the pressure difference between wetting and nonwetting fluids. For a given radius of curvature, R, the nonwetting fluid saturation is estimated by evaluating the relative pore volume that can be covered by spheres that can be inscribed in the pores, and whose radii are greater than or equal to R. Such a calculation assumes a zero contact angle at the interface between the solid and the two fluids. In order to account for a non-zero contact angle, the MIS computation is modified. Namely, each inscribed Date : August 17, 2012. Key words and phrases. Pore-scale flow modeling, maximal inscribed spheres, wettability, fracture.

The relationship between capillary pressure, saturation and interfacial area from a model of mixed-wet triangular tubes

Fluid-fluid interfacial area is recognized as an important parameter in understanding various multiphase flow processes in porous media. Mass transfer processes such as dissolution, adsorption and volatilization occur across interfaces and are strongly related to interfacial area. Moreover, the coefficient for interfacial mass transfer is assumed to be proportional to the interfacial area. It has also been observed experimentally that surfactants and bacteria may preferentially accumulate at the interfaces and affect the subsequent fluid transport. To quantify the efficiency and consequences of these processes, the magnitude of the interfacial area is required. In this paper we use a simple, yet physically-based, bundle-of-triangular-tubes model to calculate interfacial area for mixed-wet conditions and contact angle hysteresis. In such a representation of the pore network, capillary displacements may either occur as piston-like displacements of the fluids occupied in the bulk, or as piston-like displacements of fluids in layers. Accurate expressions for the associated capillary entry pressures are implemented. The simulated interfacial area vs. saturation data displays the same general trends as experimental measurements. We also derive analytical expressions for the relationship between specific interfacial area, capillary pressure and saturation for primary drainage. Based on these expressions, we formulate flexible correlations for subsequent invasion processes. The correlations are compared with the simulated data, and good agreement is obtained. The proposed correlations are consistent with the well-known Brooks- Corey correlation and may also be implemented in reservoir simulators. Finally, we use our model to explore the conjecture by Hassanizadeh and Gray who suggested that hysteresis can be eliminated in the relationship between capillary pressure, saturation and specific interfacial area. We find that hysteresis may be significant for both water-wet and mixed-wet conditions as long as contact angle hysteresis is assumed.

Simulation of Three‑Phase Capillary Pressure Curves Directly in 2D Rock Images

Pore-scale modeling of three-phase capillary pressure in realistic pore geometries could contribute to an increased knowledge of three-phase displacement mechanisms and also provide support to time-consuming and challenging core-scale laboratory measurements. In this work we have developed a semi-analytical model for computing three-phase capillary pressure curves and the corresponding three-phase fluid configurations in uniformly-wet rock images encountered during tertiary gas invasion. The fluid configurations and favorable entry pressure are determined based on free energy minimization by combining all physically allowed gas-oil, gas-water, and oil-water arc menisci in various ways. The model is shown to reproduce all threephase displacements and capillary entry pressures that previously have been derived in idealized angular tubes for gas invasion at uniform water-wet conditions. These single-pore displacement mechanisms include (i) gas invasion into pores occupied by oil and water leading to simultaneous displacement of the three fluids, (ii) simultaneous invasion of bulk gas and surrounding oil into water filled pores, and finally (iii) the pure two-phase fluid displacements in which gas invades pores occupied by either water or oil. The proposed novel semi-analytical model is validated against existing analytical solutions developed in a star-shape pore space, and subsequently employed on an SEM image of Bentheim sandstone to simulate three-phase fluid configurations and capillary pressure curves at uniform water-wet conditions and different spreading coefficents. The simulated fluid configurations for the different spreading coefficients show similar oil layer behaviour as previously published experimental three-phase fluid configurations obtained by computed microtomography in Bentheim sandstone. The computed saturation paths indicate that three-phase oil-water capillary pressure is a function of the water saturation only, whereas the three-phase gas-oil capillary pressure appears to be a function of two saturations. This is explained by the three-phase displacements occurring in the majority of the simulations, in which gas-water interfaces form immediately during gas invasion into oil- and water-saturated pore shapes.

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.