Abstract
A multiple interacting continua (MINC) method is presented, which is applicable for numerical simulation presented, which is applicable for numerical simulation of heat and multiphase fluid flow in multidimensional, fractured porous media. This method is a generalization of the double-porosity concept. The partitioning of the flow domain into computational volume elements is based on the criterion of approximate thermodynamic equilibrium at all times within each element. The thermodynamic conditions in the rock matrix are assumed to be controlled primarily by the distance from the fractures, which leads to the use of nested gridblocks. The MINC concept is implemented through the integral finite difference (IFD) method. No analytical approximations are made for coupling between the fracture and matrix continua. Instead, the transient flow of fluid and heat between matrix and fractures is treated by a numerical method. The geometric parameters needed in simulation are preprocessed from a specification of fracture spacings and apertures and geometry of the matrix blocks.
The numerical implementation of the MINC method is verified by comparison with the analytical solution of Warren and Root. Illustrative applications are given for several geothermal reservoir engineering problems.
Introduction
In this paper, we present a numerical method for simulating transient nonisothermal, two-phase flow of water in fractured porous medium. The method is base on a generalization of a concept originally proposed by Barenblatt et al. and introduced into the petroleum literature by Warren and Root, Odeh, and others in the form of what has been termed the "double-porosity" model. The essence of this approach is that in a fractured porous medium, fractures are characterized by much porous medium, fractures are characterized by much larger diffusivities (and hence, much smaller response times) than the rock matrix. Therefore, the early system response is influenced by the matrix. In seeking to analytically solve such a system, all fractures were grouped into one continuum and all the matrix blocks into another, resulting in two interacting continua coupled through a mass transfer function determined by the size and shape of the blocks, as well as the local difference in potentials between the two continua. Later, Kazemi and Duguid and Lee incorporated the double-porosity concept into a numerical model. For a more detailed description of the concept and its application, see Refs. 6 through 8.
Very little work has been done in investigating nonisothermal, two-phase fluid flow in fractured porous media. Moench and coworkers used the discrete fracture approach to study the behavior of fissured, vapor-dominated geothermal reservoirs. The purpose of our work is first to generalize the double-porosity concept into one of many interacting continua. We then incorporate the MINC model into a simulator for nonisothermal transport of a homogeneous two-phase fluid (water and steam) in multidimensional systems. Our approach is considerably broader in scope and more general than any previous models discussed in the literature. The MINC previous models discussed in the literature. The MINC method permits treatment of multiphase fluids with large and variable compressibility and allows for phase transitions with latent heat effects, as well as for coupling between fluid and heat flow. The transient interaction between matrix and fractures is treated in a realistic way. Although the model can permit alternative formulations for the equation of motion, we shall assume that, macroscopically, each continuum obeys Darcy's law; in particular, we shall use the "cubic law" for the flow of particular, we shall use the "cubic law" for the flow of fluids in fracture. While the methodology presented in this paper is generally applicable to multiphase compositional thermal systems, our illustrative calculations were restricted to geothermal reservoir problems.
The numerical method chosen to implement the MINC concept is the IFD method. In this method, all thermophysical and thermodynamic properties are represented by averages over explicitly defined finite subdomains, while fluxes of mass or energy across surface segments are evaluated through finite difference approximations. An important aspect of this method is that the geometric quantities required to evaluate the conductance between two communicating volume elements are provided directly as input data rather than having them generated from data on nodal arrangements and nodal coordinates. Thus, a remarkable flexibility is attained by which one can allow a volume element in any one continuum to communicate with another element in its own or any other continuum. Inasmuch as the interaction between volume elements of different continua is handled as a geometric feature, the IFD methodology does not distinguish between the MINC method and the conventional porous-medium type approaches to modeling. porous-medium type approaches to modeling. SPEJ
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A Newton-Modified Weighted Arithmetic Mean Solution of Nonlinear Porous Medium Type Equations