This paper concerns the use of network principles to study displacement phenomena in porous media. The information presented is for equal-viscosity, equal-density miscible displacements. The paper explains the reasons for using an interconnected network of capillary tubes to model the interconnected network of pores in a reservoir rock. A method is presented for defining the heterogeneity of a presented for defining the heterogeneity of a network of tubes based on tube-size and tube-location distribution functions. A technique is described for constructing a network whose heterogeneity models the heterogeneity of pores in a reservoir rock. The use of networks to provide information which can be used in the solution of reservoir engineering problems is illustrated with example calculations of the effect of heterogeneity on fingering, breakthrough, and selective plugging in linear systems, and the effect of heterogeneity on areal sweep efficiency in a five-spot pattern. Introduction Oil in a reservoir is contained in an interconnected three-dimensional network of pores. Direct evidence of the nature of this network of pores comes from examination of petrographic thin sections and three dimensional Scanning Electron Microscope (SEM) pictures of the pores. The SEM pictures show that the pores in a reservoir rock are channels through which flow can occur. These channels have highly irregular configurations so irregular that it is not practical at this time to calculate flow behavior through individual channels or through the interconnected network of the channels. It is practical, however, to use a computer to calculate flow behavior in an interconnected network of capillary tubes and several investigators have studied the problem of using a network of tubes to model a network of pores. pores. Fatt pioneered the idea of using a network of cubes model for reservoir engineering studies. He demonstrated that capillary pressures, relative permeabilities, and electrical resistivities permeabilities, and electrical resistivities calculated for a network model have the same characteristics as those measured for real pores in reservoir rocks. From this, Fatt concluded that the network of tubes is a valid model of real porous media. Rose reinforced Fatt's conclusion and showed that computers can be used to study the displacement characteristics of networks and to obtain results "…which can be supposed to have a direct bearing on the mechanics of petroleum recovery…" This paper takes two steps beyond the work of Fatt and Rose. First, it describes a technique for constructing a network whose heterogeneity models the heterogeneity of natural pores. This is done by matching calculated equal-viscosity miscible displacement behavior in the network with measured behavior in a laboratory core. Second, it illustrates the use of the network model for calculating the effects of heterogeneity on fingering, breakthrough, and plugging in linear systems and areal sweep efficiency in a five-spot pattern. The networks used in the studies in this paper consist of several hundred interconnected capillary tubes of different sizes. Four different types of connections or configurations were investigated and are shown below. These configurations are discussed in detail later in the paper. SPEJ P. 99

Fluid crossflow can significantly affect sweep efficiency in heterogeneous reservoirs. The importance of fluid crossflow relative to purely longitudinal convective transport in a twodimensional setting depends on several factors. Rock properties such as porosity, permeability, the ratio of vertical to horizontal permeability, and length scale of correlation are important factors. Fluid properties such as phase densities, phase viscosities, and interfacial tension are also factors. Coupled rock-fluid properties, for example, wettability, relative permeabilities, and capillary pressure are also factors. In addition to these factors, flow velocity, system dimensions, and boundary conditions also affect sweep efficiency. In this paper we examine the role of capillary forces, in addition to viscous and gravity forces, on sweep efficiency of immiscible displacements in a heterogeneous porous medium. A fully implicit, black oil, reservoir simulation model was used to compute displacement performance in two-dimensional, fine-grid (x-z) cross-sections. The results presented in this paper clearly show the importance of capillary crossflow as a recovery mechanism, in addition to viscous and gravity crossflow, in displacements in heterogeneous reservoirs.

Part II of this series extends the network technique to all miscible displacement mobility ratios and introduces a heterogeneity factor called H. As a result, the network model can be used to study the general miscible displacement case, i.e., miscible displacements with all mobility ratios in linear or areal flow systems having a range of heterogeneities. Engineering charts are presented which show the relationship between recovery, mobility ratio, heterogeneity and pore volumes injected. Introduction Part I of this series dealt with equal viscosity, and equal-density miscible displacements. Part II extends the network technique to miscible displacements with all mobility ratios. Part II also introduces a heterogeneity factor, H, used in designing network models. The equal-density limitation is retained. Plug flow is assumed in the capillary tubes. The Plug flow is assumed in the capillary tubes. The original fluid and injected fluid are considered ideal so there are no heat or volumetric effects from mixing. Viscosity of the mixtures is discussed in the section on Calculations. Part II is divided into three main sections. The first discusses the calculation methods. The second shows comparisons of data calculated by the network technique and data measured for real porous media. These comparisons demonstrate that porous media. These comparisons demonstrate that network models can indeed be used to predict the performance of displacements in real porous media. performance of displacements in real porous media. The third section illustrates the application of network methods to reservoir engineering problems. The illustration is accomplished with a series of charts that relate oil recovery to heterogeneity, mobility ratio, and pore volumes injected for the special cases of a linear system (length/width = 3/1) and five-spots. CALCULATIONS Two primary steps are required to calculate the effect of heterogeneity and mobility on miscible displacement efficiency. The first is to design a network with the desired heterogeneity. The second is to calculate the displacement phenomena that occur as the injection fluid advances through the network. A network is designed by specifying the following:Type linear or areal flow models.Tube configuration diamond, hexagonal, etc. In this paper all networks are the diamond configurations shown in Fig. 1.Size number of tubes in the network.Heterogeneity by using the heterogeneity factor (see Fig 2a).Tube radius distribution function all tube radius distribution functions are the single modal type shown in Fig. 2a.Tube location distribution all location distributions are random. SPEJ P. 345

The Inexact Adaptive Newton method (IAN) is a modification of the Adaptive Implicit Method! (AIM) with improved Newton convergence. Both methods simplify the Jacobian at each time step by zeroing coefficients in regions where saturations are changing slowly. The methods differ in how the diagonal block terms are treated.On test problems with up to 3,000 cells, IAN consistently saves approximately 30% of the CPU time when compared to the fully implicit method. AIM shows similar savings on some problems, but takes as much CPU time as fully implicit on other test problems due to poor Newton convergence.