11In this paper we discuss one-dimensional models for two-phase Enhanced Oil Recovery (EOR) floods (oil displacement by 12 gases, polymers, carbonized water, hot water, etc.). The main result presented here is the splitting of the EOR mathematical model 13 into thermodynamical and hydrodynamical parts. The introduction of a potential associated with one of the conservation laws and 14 its use as a new independent coordinate reduces the number of equations by one. The (n) Â (n) conservation law model for two-15 phase n-component EOR flows in new coordinates is transformed into a reduced (n À 1) Â (n À 1) auxiliary system containing just 16 thermodynamical variables (equilibrium fractions of components, sorption isotherms) and one lifting equation containing just 17 hydrodynamical parameters (phase relative permeabilities and viscosities). The algorithm to solve analytically the problem includes 18 solution of the reduced auxiliary problem, solution of one lifting hyperbolic equation and inversion of the coordinate transfor-19 mation. The splitting allows proving the independence of phase transitions occurring during displacement of phase relative 20 permeabilities and viscosities. For example, the minimum miscibility pressure (MMP) and transitional tie lines are independent of 21 relative permeabilities and phases viscosities. Relative motion of polymer, surfactant and fresh water slugs depends on sorption 22 isotherms only. Therefore, MMP for gasflood or minimum fresh water slug size providing isolation of polymer/surfactant from 23 incompatible formation water for chemical flooding can be calculated from the reduced auxiliary system. Reduction of the number 24 of equations allows the generation of new analytical models for EOR. The analytical model for displacement of oil by a polymer 25 slug with water drive is presented.

Completion of water injection wells directly into production oil zone has become a common practice on offshore development projects. Hence, injectivity tests started to play an important role for reservoir management of these fields. In an injectivity test a single phase (water) is injected continuously for a certain period of time in an oil saturated zone followed by a falloff period. Though single-phase flow of a slightly compressible fluid analysis methods are well documented in literature, transient two-phase solutions and analysis techniques need further development. Analytical solutions that model pressure behavior during a single layer injectivity test were recently presented in literature. This paper presents a new analytical solution for a vertical water injection well completed in a layered oil reservoir of infinite extent. To the best authors’ knowledge, the proposed solution was not presented before. Two phase immiscible displacement of oil by water in a multilayer reservoir is modeled by a system of partial differential equations. This system is weakly non-linear and can be decoupled by assuming steady-state conditions in the two-phase region. The solution is then obtained by the method of characteristics on the two-phase region coupled with the infinite acting single phase radial flow solution valid for the oil region. The solution derived in this work can be used to compute the wellbore pressure, injection rates and the water front for each individual layer without resorting of a numerical flow simulator. Parameter estimation from field data can be obtained from the proposed solution if a production logging rate profile is available. Two synthetic examples are used to verify the solution derived in this work. The first example considers the behavior of a two-layer reservoir whereas in the second case a reservoir with seven layers is analyzed. The wellbore pressure is compared to a finite difference simulator. An excellent agreement is obtained for both examples.

The hydraulic diffusivity equation that governs the flow of compressible fluids in porous media is nonlinear. Although the gaswell test analysis by means of the pseudopressure function has become a standard field practice, the effect of viscosity and gascompressibility variation with pressure is often neglected. Moreover, in field operations, the gas well is submitted to a variable rate production to determine well/reservoir properties and an estimation of the absolute open flow (AOF). For slightly compressible fluids, variable rate can be properly handled by superposition in time. Unfortunately, superposition cannot be casually justified for gas reservoirs because of its nonlinear behavior.In this paper, a general solution that properly accounts for both fluid property behavior and variable rate is presented. The proposed solution, which is derived from the Green's-function method by recasting the effect of the viscosity-compressibility product variation as a nonlinear source term, can handle variable gas rate for several well/reservoir geometries of practical interest. From the general solution, an analytical expression for variable-rate tests of a fully penetrating vertical well in an infinite gas reservoir is derived. This expression is applied to a synthetic data set to calculate the pressure response for a buildup test in an infinite homogeneous reservoir. The results compared with a commercial finitedifference numerical simulator show close agreement for both drawdown and buildup periods. It is also shown that the dimensionless pseudopressure converges to the slightly compressible fluid solution for long shut-in times. Thus, during those long times, Horner analysis and log-log derivative plot can be applied to obtain good estimation of reservoir parameters, as discussed previously in literature.

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The hydraulic diffusivity equation that governs the flow of compressible fluids in porous media is nonlinear. Although gas well test analysis by means of the pseudo pressure function has become a standard field practice, the effect of viscosity and gas compressibility variation with pressure is often neglected. Moreover, in field operation, the gas well is submitted to a variable rate production in order to determine well/reservoir properties and an estimation of the absolute open flow (AOF). For slightly compressible fluids, variable rate can be properly handled by superposition in time. Unfortunately superposition cannot be offhand justified to gas reservoir due to its non-linear behavior. In this work a general solution which properly accounts for both fluid property behavior and variable rate is presented. The proposed solution, which is based on Green's Functions method by considering the viscosity-compressibility product change as a non-linear source term, can handle variable gas rate for several well and reservoir geometries of practical interest. From the general solution, an analytical expression for variable rate tests of a fully penetrating vertical well in an infinite gas reservoir is derived. This expression is applied to a synthetic data set to calculate the pressure response for a buildup test in an infinite homogeneous reservoir. The results compared to a finite difference numerical simulator shows close agreement. It is also shown that the dimensionless pseudo pressure converges to the slightly compressible fluid solution for long shut-in times. Thus, at such long times, Horner analysis and log-log derivative plot can be applied to obtain good estimative of reservoir parameters as already discussed previously in literature.