A Critical Review For Proper Use Of Water-Oil-Gas Transfer Functions In Dual-Porosity Naturally Fractured Reservoirs-Part II
Authors: Al-Kobaisi Mohammed , Kazemi Hossein , Ramirez Benjamin Alejandro , Ozkan Erdal , Atan Safian 
Published: February 2016

Abstract
This paper is Part II of SPE 109821. In Part I, we discussed the viability of the use of simple transfer functions to accurately account for fluid exchange resulting from capillary, gravity and diffusion mass transfer for immiscible flow between fracture and matrix in dual-porosity numerical models. Here we will show additional information on several relevant topics, which include (1) flow of a low concentration, water-soluble surfactant in the fracture and the extent to which the surfactant is transported into the matrix, (2) an adjustment to the transfer function to account for the early slow mass transfer into matrix before the invading fluid establishes full connectivity with the matrix, and (3) an analytical approximation to the differential equation of mass transfer from a fracture to the matrix and a method of solution to predict oil drainage performance.
Numerical experiments involving single-porosity, fine-grid simulation of immiscible oil recovery from a typical matrix block by water, gas, or surfactant-augmented water in an adjacent fracture were performed. Results emphasize the viability of the transfer function formulations and their accuracy in quantifying the interaction of capillary and gravity forces to produce oil depending on the wettability of the matrix. For miscible flow the fracture-matrix mass transfer is less complicated because the interfacial tension between solvent and oil is zero; nevertheless, gravity contrast between solvent in the fracture and oil in the matrix creates convective mass transfer and drainage of the oil.

Introduction
Naturally fractured reservoirs contain a great amount of the known petroleum hydrocarbons worldwide, but the oil recovery from these reservoirs has been rather low. Characterization and quantification of fractures in these reservoirs is a very difficult task; nonetheless, when natural fractures significantly contribute to fluid movement and hydrocarbon drainage in the reservoir, a dual-porosity approach is adopted to quantify reservoir performance. The dual-porosity concept can be perceived and quantified in several ways as shown in Fig. 1.
The dual-porosity concept was conceived based on the premise that a very high conductive fracture medium was formed as an interconnected network of secondary porosity within a pre-existing porous rock of primary porosity. A third medium of lower conductivity fractures (that is, microfractures) can be added to the flow system in some important applications. Regardless of the formulation, the flow in the high conductivity fracture network takes place at high velocities from one gird cell to another irrespective of the flowing phase. In two- or three-phase flow, there is usually a local exchange of fluids between the fractures and the adjacent matrix at comparatively low velocities. Contrast in fluid velocities in the two flow systems is a very important issue in naturally fractured reservoirs because, in multi-phase flow, typically water or gas can move rapidly in the fractures and surround the matrix blocks partially or totally. Once a matrix block is surrounded partially or totally by a particular fluid, then transfer of fluid phases and components takes place between the fracture and matrix. Deciphering the recovery mechanisms and describing the pertinent equations of mass transfer constitute the heart of this paper-both Part I and II. Similar issues extend to any variants of the dual-porosity concept, such as the triple-porosity, irrespective of the idealization concept.

Physical Perspective
First, let us consider a naturally fractured reservoir containing a single-phase fluid, such as gas. For this case, the reservoir is produced by fluid expansion via production wells. The production mechanism is rather simple in that the producing well creates a pressure gradient in the fractures connected to the well, which, in turn, fractures create a pressure drawdown on the adjacent matrix to create matrix flow. In this scenario all connected fractures play a role in bringing gas to the wells. In fact, the early models of Barenblatt, et al (1960) and Warren and Root (1962) pertain to this mechanism. We should note that, in these publications, the driving force for matrix depletion is the pressure differential between the fracture and matrix only.