Polymer flooding is one of the most used chemical enhanced oil recovery (CEOR) technologies worldwide. Because of its commercial success at the field scale, there has been an increasing interest to expand its applicability to more unfavorable mobility ratio conditions, such as more viscous oil. Therefore, an important requirement of success is to find a set of design parameters that balance material requirements and petroleum recovery benefits in a cost-effective manner. Then, prediction of oil recovery turns out to handle more detailed information and time-consuming field reservoir simulation. Thus, for an effective enhanced oil recovery project management, a quick and feasible tool is needed to identify projects for polymer flooding applications, without giving up key physical and chemical phenomena related to the recovery process and avoiding activities or projects that have no hope of achieving adequate profitability. A detailed one-dimensional mathematical model for multiphase compositional polymer flooding is presented. The mathematical formulation is based on fractional flow theory, and as a function of fluid saturation and chemical compositions, it considers phenomena such as rheology behavior (shear thinning and shear thickening), salinity variations, permeability reduction, and polymer adsorption. Moreover, by setting proper boundary and initial conditions, the formulation can model different polymer injection strategies such as slug or continuous injection. A numerical model based on finite-difference formulation with a fully implicit scheme was derived to solve the system of nonlinear equations. The validation of the numerical algorithm is verified through analytical solutions, coreflood laboratory experiments, and a CMG-STARS numerical model for waterflooding and polymer flooding. In this work, key aspects to be considered for optimum strategies that would help increase polymer flooding effectiveness are also investigated. For that purpose, the simulation tool developed is used to analyze the effects of polymer and salinity concentrations, the dependence of apparent aqueous viscosity on the shear rate, permeability reduction, reversible–irreversible polymer adsorption, polymer injection strategies on petroleum recovery, and the flow dynamics along porous media. The practical tool and analysis help connect math with physics, facilitating the upscaling from laboratory observations to field application with a better-fitted numerical simulation model, that contributes to determine favorable scenarios, and thus, it could assist engineers to understand how key parameters affect oil recovery without performing time-consuming CEOR simulations.
A mathematical model for water injection in vertical porous media initially saturated with oil and water is presented. The mathematical formulation takes the form of a nonlinear convection-diffusion equation. Its contribution comes from consideration of the three chief forces (viscous, capillary and gravity) in oil recovery processes. The model is general in that it can use any shape for relative permeability and capillary pressure functions, and it is developed to allow analysis of these forces individually or in a combined manner. By using Corey-type functions for relative permeability, logarithmic functions for capillary pressure, and Peclet and Gravity dimensionless numbers, the flow equation is written in a dimensionless form. In order to represent the physics of the oil-water displacement more accurately, variable saturation-dependent coefficients for the diffusive (capillary) and convection (viscous and gravity) terms were used. Thus, a nonlinear equation is obtained. A numerical model based on the finite-difference formulation with a fully implicit scheme was implemented to obtain the solution to this equation. The analytic solution for the diffusion-convection equation for the semi-infinite problem published elsewhere and the well-known Buckley-Leverett solution were used to validate the numerical algorithm. The numerical model allows evaluation of the influence of each of these three forces on the magnitude and direction of the dimensionless water velocity. Water velocity is defined as the sum of the velocity contribution of each force (viscous, capillary and gravity). This model also helps determine favorable scenarios for each force. For instance, the analytic equation and the numerical results show the cases in which one force dominates the others, under given petrophysical and fluid properties and oil or water injection velocity. Finally, by setting the proper boundary and initial conditions, this model can be used to simulate any displacement in which these three forces interact. INTRODUCTION At the end of primary recovery processes, when pressure in the reservoir has depleted, it is necessary to provide extra energy to the reservoir; one way to achieve this is by water injection. Waterflooding requires previous studies to forecast the performance and pattern of the displacement and to select the best recovery conditions.
Reservoir performance during waterflooding is important to reservoir engineers. Analytical and semi-analytical flow models with different assumptions have been presented and used widely to describe the flow dynamics of such a process. Most assume that one of the terms, typically capillary forces, can be neglected or considered constant diffusion coefficients, so the simplified diffusive-convective type of flow equations can be solved analytically or by numerical methods. Obtaining analytical or semi-analytical solutions to non-linear diffusive-convective flow equations, including capillary, gravity and viscous forces simultaneously, has been a challenge. This paper presents a theoretical study of the effects that controlling flow parameters have on saturation profiles and breakthrough time during oil recovery by waterflooding. A mathematical non-linear diffusive-convective type model for immiscible oil-water displacement in one-dimensional vertical homogeneous porous media considering the three chief forces (capillary, gravity and viscous) is derived and solved numerically by using a finite-difference formulation with fully implicit scheme in time and central differences in space. Dimensionless equations are written so that any of the three forces can be investigated independently; capillary and gravity forces can be "turned on or off." The effects of varying fluid viscosity, injection flow rate, system length or wettability, for both displacing and displaced fluids, can be understood thoroughly. The flow model is versatile enough that it allows for variations of the shape of the relative permeability and capillary pressure functions. The impact of these functions in the driving forces and on oil recovery is analyzed. The contribution of each of the forces to the dimensionless water velocity and its impact on oil recovery was studied in four flow cases:viscous,viscous-gravity,viscous-capillary andviscous-gravity-capillary; all possible flow cases in water injection problems were considered. Graphical results are discussed. INTRODUCTION In primary recovery stage, a reservoir's energy is due to high pore pressure. Pressure drop between the reservoir and the wells allow fluids flow through the reservoir to eventually reach production wells, and, for this reason, when there is not sufficient energy to establish fluid movement, a secondary recovery method such as water injection is needed to add energy to the reservoir and thereby maintain its pressure or displace fluids. Before implementing a water injection process, like all oil recovery methods, various scenarios should be evaluated and main parameters identified with the purpose of increasing oil recovery after applying a water injection process.
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