DERIVATION OF THE PARTIAL DIFFERENTIAL EQUATION GOVERNING FLUID FLOW IN PETROLEUM RESERVOIRS (original) (raw)
2022
Having understanding about equations of fluid flow and transport through porous media is very important for various applications such as in oil and gas production and petroleum reservoirs simulation. But modeling fluid flow and transport in porous media, on the other hand is still an enormous technical challenge. To capture the best model of fluid flow, true description of fluid interaction such as capillary pressure and relative permeability is inevitable. Considering these parameters, the complexity of numerical calculation will increase. The modeling of such physical flow process mainly requires solving the mass and momentum conservation equations associated with equations of capillary pressure, saturation and relative permeability. Due to that, solution of the governing equations for fluid flow and transport requires knowledge of functional relationships between fluid pressures, saturations, and permeabilities which has formulated on the basis of conceptual models of fluid-porous media interactions. Therefore, in this work, the basic fluid flow and transport equations have been developed for a hierarchy of models: single phase, two-phase, black oil, volatile oil, compositional, thermal, and chemical. This hierarchy of models correspond to different oil production stages. Their governing differential equations consist of the mass and energy conservation equations and Darcy's law. I have chosen to start with the simplest model for single phase flow and to end with the most complex model for chemical flooding. This approach can be reversed; that is, I can start with the chemical model, and in turn derive the thermal, compositional, volatile oil, black oil, two-phase, and single-phase models.
Mathematical models of petroleum reservoirs have been utilized since the late 1800s. A mathematical model consists of a set of equations that describe the flow of fluids in a petroleum reservoir, together with an appropriate set of boundary and/or initial conditions. This chapter is devoted to the development of such a model.
Parametric analysis of diffusivity equation in oil reservoirs
Journal of Petroleum Exploration and Production Technology, 2016
The governing equation of fluid flow in an oil reservoir is generally non-linear PDE which is simplified as linear for engineering proposes. In this work, a comprehensive numerical model is employed to study the role of non-linear term in reservoir engineering problems. The pervasive sensitivity analysis is performed on rock and fluid properties, and it is shown that rock permeability and fluid viscosity are the most significant parameters which influence the pressure profile. Moreover, the critical reservoir radius was determined in four different cases including heavy and light oil along with high and low permeable rocks, in which the pressure difference of linear and non-linear diffusivity equation is significant. Results of this study give an insight into proper simplification of diffusivity equation and provide an engineering-based decision strategy for different reservoir properties having certain rock and fluid properties in oil reservoirs. The results show the significance o...
Oil displacement by water and gas in a porous medium: the Riemann problem
Bulletin of the Brazilian Mathematical Society, New Series, 2016
In this work we present the construction of the Riemann solution for a system of two conservation laws representing displacement in immiscible three-phase flow. The porous medium is initially filled with oil and small amounts of water and gas; then a fixed proportion of water and gas is injected. We use the wave curve method to determine the wave sequences in the Riemann solution for arbitrary initial and injection data in the above mentioned class. We show the L 1 Loc-stability of the Riemann solution with variation of data. We do not verify uniqueness of the Riemann solution, but we believe that it is valid.
Darcy and non-Darcy Flows in Fractured Gas Reservoirs
SPE Reservoir Characterisation and Simulation Conference and Exhibition, 2015
In this paper, we discuss the application of Forchheimer's model in flow problems where high sensitivity to capillary pressure is involved. The Forchheimer equation is basically composed of a Darcy term and a non-Darcy term to account for the inertial effects involved by high flow rates inside porous media. However, the Forchheimer equation does not offer an easy and systematic approach for estimating the flow, given that the  parameter, a non-Darcy flow coefficient, also known as the Forchheimer coefficient, is specific to different reservoir characteristics and must be determined accurately through experiments. It is, therefore, very important, before applying the Forchheimer model, to have an idea of its domain of applicability.
The Fluid Flow Motion through Porous Media for Slightly Compressible Materials
The multiphase flow in porous media is a topic of various big complexities for a long time in the field of fluid mechanics. This is a subject of important technical applications, most probably in oil recovery from petroleum reservoirs and also in others. The single phase fluid flow through a porous medium is generally defined by Darcy's law. In the petroleum industry and in other technical applications, the transport phenomenon is modeled by postulating a multiphase analysis of the Darcy's law. In this analysis, the distinct pressures are defined for each phase with the difference and well known as capillary pressure. That is determined by the interfacial tension, geometry of micro pore and the chemistry of the surface related to the solid medium. In regarding flow rates, the relative permeability is defined that gives the relationship between the volume flow rate of each fluid and the pressure gradient. In the present paper, there is an analysis about the mathematical laws and equations for the slightly compressible flow and rock and the analysis and important results have been founded. The analysis show that velocity of fluid related to any phase is inversely proportional to the viscosity of the fluid. The capillary pressure of the capillary tube is inversely proportional to the radius of tube and increases with increasing values of the surface tension of the fluid. It also varies inversely with the radii of curvature for the interface of the fluid. The pressure exerted by the fluid varies positively with its velocity and varies inversely with the absolute permeability of the porous medium.
Multiphase Linear Flow in Tight Oil Reservoirs
SPE Reservoir Evaluation & Engineering, 2017
Summary The main objective of this work is to gain a general understanding of the performance of tight oil reservoirs during transient linear two-phase flow producing at constant flowing pressure. To achieve this, we provide a theoretical basis to explain the effect of different parameters on the behavior of solution-gas-drive unconventional reservoirs. It is shown that, with the Boltzmann transformation, the highly nonlinear partial-differential equations (PDEs) governing two-phase flow through porous media can be converted to two nonlinear ordinary-differential equations (ODEs). The resulting ODEs simplify the calculation of the reservoir performance and avoid the tedious calculation inherent in solving the original PDEs. Thus, the proposed model facilitates sensitivity studies and rapid evaluation of different hypotheses. Moreover, successful conversion of the highly nonlinear PDEs (in terms of distance and time) to the ODEs (in terms of the Boltzmann variable) implies that the s...
New equations for binary gas transport in porous media, Part 2: experimental validation
2003
A rigorous understanding of the mass and momentum conservation equations for gas transport in porous media is vital for many environmental and industrial applications. We utilize the method of volume averaging to derive Darcy-scale, closure-level coupled equations for mass and momentum conservation. The up-scaled expressions for both the gas-phase advective velocity and the mass transport contain novel terms which may be significant under flow regimes of environmental significance. New terms in the velocity expression arise from the inclusion of a slip boundary condition and closure-level coupling to the mass transport equation. A new term in the mass conservation equation, due to the closure-level coupling, may significantly affect advective transport. Order of magnitude estimates based on the closure equations indicate that one or more of these new terms will be significant in many cases of gas flow in porous media.