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This paper explores the extent of an analogy postulated earlier between the usual energy-based statistical mechanics and mass-dispersion phenomena. It is shown that in the equilibrium case the interaction function as used in the energy-based statistical mechanics of solids or weakly coupled gases entails a corresponding result in mass-dispersion systems. Examples of specific transport equations are calculated. The analogy can also be extended to irreversible thermodynamics. It is shown that Ziegler's generalized Onsager relations entail corresponding results in the case of mass dispersion.It is shown that an approach based on the theory of Markov processes can also be used for the description of mass-based statistical mechanics. The conditions necessary for the maintenance of canonical invariance are indicated. Applications of the above theory to hydrological problems are indicated.

Abstract Difference analog equations of high-order accuracy describing stable miscible displacement are presented. The high-order-difference scheme eliminates almost all the numerical smearing, which is the result of the normally used approximations, and leaves only the effect of the physical dispersion in the solution. In a one-dimensional system, the technique involves an addition of a negative dispersion term to the continuity equation. The negative dispersion term, which is called the numerical dispersion coefficient, depends upon the flow velocity, the time-step size and the block size. The procedure is extended to multidimensional systems. As a check, comparisons of the computed results with analytical solutions in one- and two-dimensional systems are made. The error-function solution in a one-dimensional miscible system and a five-spot fractional flow curve computed from the potentiometric model are considered as the analytical potentiometric model are considered as the analytical solutions in the two cases. Based on the computed results, it has been concluded that the effect of the term with the negative numerical dispersion coefficient accounts for almost all the truncation error within the practical limits of oilfield problems. Introduction Miscible flooding is one of the most efficient hydrocarbon recovery processes known at the present time. Its application in the field, however, present time. Its application in the field, however, remains limited because of economic requirements. Recently, the development of vertical miscible floods has made the process economically attractive. The result is that many oil companies have proposed and started such floods in suitable reservoirs. The amount of solvent required to flood an entire reservoir miscibly can be roughly calculated by analytical methods. Since these methods are derived for a one-dimensional system, some important questions remain unanswered. Some of these questions are:How will the solvent bank develop under the proposed injection scheme?What will be the condition of the solvent bank at breakthrough? These questions can only be answered by two- or three-dimensional studies. Known numerical techniques simulating miscible displacement give considerable smearing of concentration in the liquid-liquid transition zone when applied to a Held case. The effect of numerical smearing in this case is significantly larger than the effect of physical dispersion. Therefore, in the past, immiscible simulators with controlled numerical past, immiscible simulators with controlled numerical smearing were used to simulate miscible floods. For example, in one case the two-phase saturation profile was displaced as near piston-like as possible, profile was displaced as near piston-like as possible, neglecting mixing due to molecular diffusion and convective dispersion. In another case, capillary pressure and relative permeability functions were pressure and relative permeability functions were modified to simulate the dispersion effect. It was stated that the second method is applicable to a system where numerical smearing is considerably less than the effect of physical dispersion. This limits the use of the second method to a system such as gas-gas displacement. In this article a high-order difference scheme for miscible equations is given. This scheme eliminates most of the numerical smearing, leaving only the effect of physical dispersion in the transition zone. The procedure is developed for one- and two-dimensional systems. Calculated results are compared with analytical solutions. It is shown that the procedure can be extended to a three-dimensional system. FORMULATION OF THE PROBLEM The differential equations governing a miscible displacement process for an incompressible system are the continuity equations and the flow equations. Numerical solutions of this set of equations have been described by Peaceman and Rachford and Warren et al. Their methods are adequate to simulate miscible displacement in laboratory systems. If one applies them to a field-size system, significant truncation errors are introduced in the results. Usually these errors are large compared with the effect of physical dispersion on the results. SPEJ P. 277

A previously noted analogy between certain transport equations and the heat conductivity equation is further analyzed. It is shown thatPrigogine's nonequilibrium formalism can be directly applied to statistical processes where there is a non-negative constant of the motion. If the total mass (of contaminant, tracer etc.) contained in a fluctuating system is taken as such a constant of the motion, a statistical theory of the corresponding transport process is obtained.

Abstract This paper presents an extension of an earlier one, which presented an improved numerical technique for solving multicomponent stable miscible flow equations. Diffusion-convection equations for each component together with a pressure equation are solved. Second-order correct approximations for individual component flows, resulting in numerical dispersion correction terms, are employed in solving the diffusion-convection equations. Backward-in-time implicit and Crank-Nicolson approximation techniques are used to solve the pressure equation. Pressure and saturation distributions using first- and second-order correct finite-difference approximations for individual component flows are compared. One- and two-dimensional systems are considered for these comparisons. The results show that in a compressible miscible displacement process, first-order correct finite-difference scheme process, first-order correct finite-difference scheme for individual component flows results in considerable numerical smearing in the saturation distribution. The pressure distribution, which depends upon the saturation distribution, is very sensitive to numerical smearing. Therefore, high-order correct approximations for individual component flours are necessary. The results also demonstrate That the earlier approach employed to reduce numerical smearing in the numerical solution of incompressible miscible flow processes can be extended to compressible miscible flow systems. Introduction Many investigators have tried to solve in some fashion the system of equations governing the miscible displacement process. In most cases, even though the system was compressible, an incompressible system was considered. One may state several reasons for this. Since the system of equations governing compressible miscible flow is nonlinear, analytical solution of such a system, even in a one-dimensional case, is not known In a multidimensional case it may be a practical impossibility. The only methods known to solve the above problem are the numerical methods. One of such methods is presented here. Basic flow equations for the individual components present in the compressible miscible flow system present in the compressible miscible flow system are assumed. These flow equations together with the individual component continuity equations are used to arrive at the pressure equation. Crank-Nicolson and backward-in-time approximation schemes are used in an iterative procedure to solve the pressure equation. Flows of individual components are calculated with first- and second-order correct finite-difference approximations. It is also shown that a semirigorous technique, which eliminates most of the numerical smearing from the numerical solution of an incompressible miscible system, can also be used for a compressible miscible flow system. It has been observed that pressure distributions in a one-dimensional case are pressure distributions in a one-dimensional case are significantly different when the individual component flow terms are approximated with first- and second-order correct finite-difference approximations. Also, the order of approximation of the flow equations has more effect on the pressure distribution than the order of approximation (time approximation) used to solve the pressure equation.