Steinar Evje

Monotone Difference Approximations Of BV Solutions To Degenerate Convection-Diffusion Equations

We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and, in general, are not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin [ Northeastern Math J., 5 (1989), pp. 395--422] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase flow equation, and hyperbolic conservation laws. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present context is to show that the (strongly degenerate) discrete diffusion term is sufficiently smooth. We provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes. Finally, we make some concluding remarks about monotone difference schemes for multidimensional equations.

On a rough AUSM scheme for a one-dimensional two-phase model

Abstract We are interested in exploring Advection Upstream Splitting Method (AUSM) schemes for hyperbolic systems of conservation laws which do not allow any analytical calculation of the Jacobian. For this purpose, we consider a two-phase model which has been used for modeling of unsteady compressible flow of oil and gas in pipes. The model consists of two mass conservation equations, one for each phase, and a common momentum equation. Since no analytical Jacobian can be obtained it is more difficult to use classical schemes such as Roe- and Godunov-type schemes. We propose an AUSM scheme for the current two-phase model obtained through natural generalizations of ideas described in M.-S. Liou [J. Comput Phys 129(2) (1996) 364]. A main feature of AUSM is simplicity and efficiency since it does not require the Jacobian. In particular, we prove that the proposed AUSM type scheme preserves the positivity of scalar quantities such as pressure, fluid densities and volume fractions. This guarantees that the scheme can handle the important and delicate case of transition from two-phase to single-phase flow without introducing negative masses. Many numerical results are included to confirm the accuracy and robustness of the proposed AUSM scheme. In particular, it is demonstrated that the AUSM scheme gives low numerical dissipation at volume fraction contact discontinuities and is able to produce stable and non-oscillatory solutions, also when more complex slip relations are used, that is, when the relative motion of one phase with respect to the other is more or less complicated. This makes the scheme suitable for simulations of many important two-phase flow processes.

Weak Solutions for a Gas-Liquid Model Relevant for Describing Gas-Kick in Oil Wells

The article was originally published at; http://dx.doi.org/10.1137/100813932; it is made available here with permission.

On the Wave Structure of Two‐Phase Flow Models

We explore the relationship between two common two‐phase flow models, usually denoted as the two‐fluid and drift‐flux models. They differ in their mathematical description of momentum transfer between the phases. In this paper we provide a framework in which these two model formulations are unified. The drift‐flux model employs a mixture momentum equation and treats interphasic momentum exchange indirectly through the slip relation, which gives the relative velocity between the phases as a function of the flow parameters. This closure law is in general highly complex, which makes it difficult to analyze the model algebraically. To facilitate the analysis, we express the quasi‐linear formulation of the drift‐flux model as a function of three parameters: the derivatives of the slip with respect to the vector of unknown variables. The wave structure of this model is investigated using a perturbation technique. Then we rewrite the drift‐flux model with a general slip relation such that it is expressed in term...

Weakly Implicit Numerical Schemes for a Two-Fluid Model

The aim of this paper is to construct semi-implicit numerical schemes for a two-phase (two-fluid) flow model, allowing for violation of the CFL criterion for sonic waves while maintaining a high level of accuracy and stability on volume fraction waves. By using an appropriate hybridization of a robust implicit flux and an upwind explicit flux, we obtain a class of first-order schemes, which we refer to as weakly implicit mixture flux (WIMF) methods. In particular, by using an advection upstream splitting method (AUSMD) type of upwind flux [S. Evje and T. Flatten, J. Comput. Phys., 192 (2003), pp. 175--210], we obtain a scheme denoted as WIMF-AUSMD. We present several numerical simulations, all of them indicating that the CFL-stability of the WIMF-AUSMD scheme is governed by the velocity of the volume fraction waves and not the rapid sonic waves. Comparisons with an explicit Roe scheme indicate that the scheme presented in this paper is highly efficient, robust, and accurate on slow transients. By exploiting the possibility to take much larger time steps, it outperforms the Roe scheme in the resolution of the volume fraction wave for the classical water faucet problem. On the other hand, it is more diffusive on pressure waves. Although conservation of positivity for the masses is not proved, we demonstrate that a fix may be applied, making the scheme able to handle the transition to one-phase flow while maintaining a high level of accuracy on volume fraction fronts.

A Continuous Dependence Result for Nonlinear Degenerate Parabolic Equations with Spatially Dependent Flux Function

We study entropy solutions of nonlinear degenerate parabolic equations of form,where k(x) is a vector-valued function and f(u),A(u) are scalar functions. We prove a result concerning the continuous dependence on the initial data,the flux function,and the diffusion function A(u). This paper complements previous workk(x)f(u) [7] by two of the authors, which contained a continuous dependence result concerning the initial data and the flux function k(x)f(u).

Relaxation schemes for the calculation of two-phase flow in pipes

In this paper, we are interested in some basic investigations of properties of the relaxation schemes first introduced by Jin and Xin [1]. The main advantages of these schemes are that they neither require the use of Riemann solvers nor the computation of nonlinear flux Jacobians. This can be an important advantage when more complex models are considered where it is not possible to perform analytical calculations of Jacobians and/or when considering fluids with nonstandard equation of state. We apply the schemes (relaxing and relaxed) to a certain two-phase model where Jacobians cannot in general be calculated analytically. We first demonstrate that the original relaxation schemes of Jin and Xin produce a poor approximation for a typical mass transport example which involves transition from two-phase flow to single-phase flow. However, by introducing a slight modification of the original relaxation model by splitting the momentum flux into a mass and pressure part, we obtain some flux splitting relaxation schemes which for typical two-phase flow cases yield a more accurate and robust approximation.

Control-Oriented Drift-Flux Modeling of Single and Two-Phase Flow for Drilling

We present a simplified drift-flux model for gas-liquid flow in pipes. The model is able to handle single and two-phase flow thanks to a particular choice of empirical slip law. A presented implicit numerical scheme can be used to rapidly solve the equations with good accuracy. Besides, it remains simple enough to be amenable to mathematical and control-oriented analysis. In particular, we present an analysis of the steady-states of the model that yields important considerations for drilling practitioners. This includes the identification of 4 distinct operating regimes of the system, and a discussion on the occurrence of slugging in underbalanced drilling.Copyright

Global weak solutions for a gas liquid model with external forces and general pressure law

In this work we show existence of global weak solutions for a two-phase gas-liquid model where the gas phase is represented by a general isothermal pressure law, whereas the liquid is assumed to be incompressible. To make the model relevant for pipe and well-flow applications we have included external forces in the momentum equation representing, respectively, wall friction forces and gravity forces. The analysis relies on a proper combination of the methods introduced in [S. Evje and K. H. Karlsen, Commun. Pure Appl. Anal., 8 (2009), pp. 1867–1894], [S. Evje, T. Flatten, and H. A. Friis, Nonlinear Anal., 70 (2009), pp. 3864–3886], where a two-phase gas-liquid model without external forces was studied for the first time, and on techniques that have been developed for the single-phase gas model. As a motivation for further research, some numerical examples are also included demonstrating the ability of the model to describe the ascent of a gas slug due to buoyancy forces in a vertical well. Characteristic ...

A Model for Spontaneous Imbibition as a Mechanism for Oil Recovery in Fractured Reservoirs

The flow of oil and water in naturally fractured reservoirs (NFR) can be highly complex and a simplified model is presented to illustrate some main features of this flow system. NFRs typically consist of low-permeable matrix rock containing a high-permeable fracture network. The effect of this network is that the advective flow bypasses the main portions of the reservoir where the oil is contained. Instead capillary forces and gravity forces are important for recovering the oil from these sections. We consider a linear fracture which is symmetrically surrounded by porous matrix. Advective flow occurs only along the fracture, while capillary driven flow occurs only along the axis of the matrix normal to the fracture. For a given set of relative permeability and capillary pressure curves, the behavior of the system is completely determined by the choice of two dimensionless parameters: (i) the ratio of time scales for advective flow in fracture to capillary flow in matrix