Universality of Riemann solutions in porous media (original) (raw)
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The convex permeability three-phase flow in reservoirs
We focus on a system of two conservation laws representing a large class of models of immiscible flow in porous media relevant for petroleum engineering. The Riemann solutions are found for a range of initial and injection conditions important in applications, including the injection of two fluids (water, gas) into a horizontal reservoir containing a third fluid (oil) to be displaced. Despite loss of hyperbolicity, there is evidence that the solution for each such data exists and is unique. Also, the solution depends L1 continuously on the Riemann data. Such solutions always present a lead shock involving the fluid already present and one of the other injected fluids. There is a special solution separating solutions according to which of the injected fluids is present in the lead shock, the separatrix solution. This class of solutions was discovered recently for a particular model with quadratic permeabilities, see Azevedo et al. (2010). We reveal the nature of the separatrix soluti...
On a universal structure for immiscible three-phase flow in virgin reservoirs
We discuss the solution for commonly used models of the flow resulting from the injection of any proportion of three immiscible fluids such as water, oil, and gas in a reservoir initially containing oil and residual water. The solutions supported in the universal structure generically belong to two classes, characterized by the location of the injection state in the saturation triangle. Each class of solutions occurs for injection states in one of the two regions, separated by a curve of states for most of which the inter-stitial speeds of water and gas are equal. This is a separatrix curve because on one side water appears at breakthrough, while gas appears for injection states on the other side. In other words, the behavior near breakthrough is flow of oil and of the dominant phase, either water or gas; the non-dominant phase is left behind. Our arguments are rigorous for the class of Corey models with convex relative perme-ability functions. They also hold for Stone's interpolation I model [5]. This description of the universal structure of solutions for the injection problems is valid for any values of phase viscosities. The inevitable presence of an umbilic point (or of an elliptic region for the Stone model) seems to be the cause of this universal solution structure. This universal structure was perceived recently in the particular case of quadratic Corey relative permeability models and with the injected state consisting of a mixture of water and gas but no oil [5]. However, the results of the present paper are more general in two ways. First, they are valid for a set of perme-ability functions that is stable under perturbations, the set of convex permeabilities. Second, they are valid for the injection of any proportion of three rather than only two phases that were the scope of [5].
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.
Classification of the umbilic point in immiscible three-phase flow in porous media
2012
We consider the flow in a porous medium of three fluids that do not mix nor interchange mass. Under simplifying assumptions this is the case of oil, water and gas in a petroleum reservoir. For a simple geometry, the horizontal displacement of a pre-existent uniform mixture by another injected mixture gives rise to a Riemann problem for a system of two conservation laws. Such a system depends on laboratory-measured relative permeability functions for each of the three fluids. For Corey models each permeability depends solely on the saturation of the respective fluid, giving rise to systems containing an umbilic point in the interior of the saturation triangle. It has been conjectured that the structure of the Riemann solution in the saturation triangle is strongly influenced by the nature of the umbilic point, which is determined by the quadratic expansion of the flux function nearby. In 1987 it was proved that, for very general Corey permeabilities, umbilic points have types I or II of Schaeffer&Shearer's classification.
Loss of real characteristics for models of three-phase flow in a porous medium
Transport in Porous Media, 1989
In this paper we examine the generalized Buckley-Leverett equations governing threephase immiscible, incompressible flow in a porous medium, in the absence of gravitational and diffusive/dispersive effects. We consider the effect of the relative permeability models on the characteristic speeds in the flow. Using a simple idea from projective geometry, we show that under reasonable assumptions on the relative permeabilities there must be at least one point in the saturation triangle at which the characteristic speeds are equal. In general, there is a small region in the saturation triangle where the characteristic speeds are complex. This is demonstrated with the numerical results at the end of the paper. Key words. Oil reservoir models, three-phase flow. O. Symbols and Notation a, b, c, d A,B,C,D det J dev J J L m P r• R entries of Jacobian matrix coefficients in Taylor expansion of At, A~, A. determinant of matrix J deviator of matrix J Jacobian matrix linear term in Taylor expansion for J near (sv, Sa) = (0, 1) slope of r § pressure eigenvectors of Jacobian matrix real line * -Eng-48.
A new set of equations describing immiscible two-phase flow in isotropic porous media
arXiv: Fluid Dynamics, 2016
Based on non-equilibrium thermodynamics we derive a set of general equations relating the partial volumetric flow rates to each other and to the total volumetric flow rate in immiscible two-phase flow in porous media. These equations together with the conservation of saturation reduces the immiscible two-phase flow problem to a single-phase flow problem of a complex fluid. We discuss the new equation in terms of the relative permeability equations. We test the equations on model systems, both analytically and numerically.
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.
Three-phase immiscible displacement in heterogeneous petroleum reservoirs
Mathematics and Computers in Simulation, 2006
We describe a fractional-step numerical procedure for the simulation of immiscible three-phase flow in heterogeneous porous media that takes into account capillary pressure and apply it to indicate the existence of a so-called "transitional" wave in at least some multi-dimensional flows, thereby extending theoretical results for one-dimensional flows. The step procedure combines a second-order, conservative central difference scheme for a pertinent system of conservation laws modeling the convective transport of the fluid phases with locally conservative mixed finite elements for the associated parabolic and elliptic problems.
Advances in Water Resources, 2020
In this article we consider a two-phase flow model in a highly heterogeneous porous column. The porous column consists of homogeneous blocks, where the porosity and permeability vary from one block to the other. The flow direction is perpendicular to the layering of the porous column, and hence can be approximated by one-dimensional model equations. The periodic change in porosity and absolute permeability enforce the fluid to be trapped at the interface between the blocks, leading to a highly varying saturation. In order to capture the effective behavior, upscaled equations for the average saturation are derived via homogenization. This technique relies on a notion of periodicity and allows averaging over any number of blocks that may have any internal distributions of the rock parameters. Moreover, the present article also derives effective equations for randomly distributed layers of different porosity and absolute permeability. Numerical experiments are performed which show good...