We present a network model of three-phase flow in water-wet porous media. To explain the high oil recoveries in gas injection and gravity drainage experiments, we show that the mechanism for oil recovery is flow through connected oil layers in the pore space that are on the order of a micron thick. We then describe a simple model for the configuration of oil, water, and gas in a single pore present an approximate expression for the conductance of oil layers. We use this expression to derive the oil relative permeability when flow is dominated by layer drainage. We show that for low oil and water saturations kro~So2, consistent with the results of several experiments. To predict kro for the full range of oil saturation we use a capillary equilibrium-based network model that can simulate any sequence of oil, water, and gas injection. We introduce a self-consistency procedure to ensure that the correct sequence of saturation changes in used in the network model to compute relative permeability. We then present relative permeabilities and oil recoveries for gas injection into different initial oil saturations, and for waterflooding a reservoir containing gas and oil. We show that the relative permeabilities are strongly affected by the fluid properties and by the type of displacement process.
Network models simulate multiphase flow through an idealized representation of the pore space to calculate average properties, such as relative permeability, capillary pressure, and oil recovery. Network models can predict multiphase flow properties directly if both the geometry of the porous medium and the displacement process are known precisely.1-5 Where a complete description of the flow physics and the pore structure is unknown or difficult to obtain, conceptual models can be developed. These models make approximations about the structure of the pore space and the flow processes. Although they cannot make direct predictions of multiphase properties, they can be used to provide insight into flow in porous media. With suitable tuning of parameters, the models can match experimental data and can then be used to make predictions for situations outside the range of available measurements. Examples of this approach include studies of relative permeability hysteresis and the effects of wettability in two-phase flow.6-11 In this paper, we use a conceptual model to study three-phase flow.
The flow of three phases - oil, water, and gas - occurs in a variety of different displacement processes in oil reservoirs and during pollutant transport and cleanup. Although there is now a large body of literature on three-phase relative permeability (for example, Oak et al.12 provides a review of studies up to 1990 and Jerauld13 provides a recent discussion of measurements in Prudhoe Bay), three-phase flow is not well understood, and current empirical models for relative permeability do not adequately describe the full range of possible behavior. A three-phase network model requires knowledge of the pore-scale displacement mechanisms, which are studied using micromodels that reproduce the anticipated behavior in real rock. As a result of these experiments, the displacement processes for three-phase flow in water-wet media are now fairly well understood.14-19 Based on these observations, several three-phase network models have been constructed.20-28 These models have predicted successfully oil recovery in micromodel experiments,20-22 and have computed three-phase relative permeabilities and capillary pressures.25-28
In this paper, we present a network model for water-wet media and use it to address two unique aspects of three-phase flow. First, oil may form a layer in grooves, crevices, roughness or corners of the pore space, sandwiched between water close to the solid surfaces and gas in the center of the pores. Flow through such layers is the mechanism by which oil may drain to low saturation during gas injection. We will present an approximate analytical model of oil layer conductance and use it to predict oil relative permeability in the layer drainage regime. Second, a three-phase displacement involves changes in two independent saturations. This is in contrast to two-phase flow (say, oil and water), where the water saturation can only increase (imbibition for a water-wet system) or decrease (drainage). The direction of the saturation change affects relative permeability and capillary pressure. In three-phase flow there is an infinite nnumber of possible routes in saturation space, all with potentially different relative permeabilities and oil recoveries. In our model we specify a sequence of saturation changes. However, in a macroscopic displacement, normally the boundary conditions are known - the gas and water fractional flows at injection wells and the initial saturation of oil and water in the reservoir. This displacement results in a certain sequence of saturation changes at a fixed point in the reservoir, but the network model does not automatically know what the sequence is. In this paper, we present a self-consistency procedure that allows the network model to find relative permeabilities for the right saturation path.
First, we will discuss the significance of oil layers in three-phase flow and describe how the network model computes oil layer conductance. Before describing the network model itself, we show how we can make predictions of oil relative permeability directly from expressions for the oil layer conductance. Then we will introduce the network model. Further details are provided elsewhere25,26. We then describe the self-consistency procedure. Last, we will present self-consistent saturation paths and relative permeabilities for gas injection and for waterflooding a reservoir containing oil and gas.