Hyperbolic-elliptic models for well-reservoir flow

Steinar Evje; Kenneth H. Karlsen; Steinar Evje; Kenneth H. Karlsen

doi:10.3934/nhm.2006.1.639

Networks and Heterogeneous Media

2006, Volume 1, Issue 4: 639-673. doi: 10.3934/nhm.2006.1.639

Previous Article Next Article

Hyperbolic-elliptic models for well-reservoir flow

Steinar Evje ^{1
,},
Kenneth H. Karlsen ^{2
,}

1.
International Research Institute of Stavanger, University of Stavanger, P.O. Box 8046, N--4068 Stavanger
2.
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo

Received: 01 September 2006
Primary: 35L65, 35L60; Secondary: 35K40.

We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starting point is an integral equation representing unsteady single-phase 3-D porous media flow and the 1-D isothermal Euler equations representing unsteady well flow. This $2 \times 2$ system of conservation laws is coupled to the integral equation through natural coupling conditions accounting for the flow between well and surrounding reservoir. By imposing simplifying assumptions we obtain various hyperbolic-parabolic and hyperbolic-elliptic systems. In particular, by assuming that the fluid is incompressible we obtain a hyperbolic-elliptic system for which we present existence and uniqueness results. Numerical examples demonstrate formation of steep gradients resulting from a balance between a local nonlinear convective term and a non-local diffusive term. This balance is governed by various well, reservoir, and fluid parameters involved in the non-local diffusion term, and reflects the interaction between well and reservoir.
- Non-local conservation law,
- coupled well-reservoir flow,
- advanced well,
- hyperbolic-elliptic model,
- entropy weak solution,
- existence,
- uniqueness
Citation: Steinar Evje, Kenneth H. Karlsen. Hyperbolic-elliptic models for well-reservoir flow[J]. Networks and Heterogeneous Media, 2006, 1(4): 639-673. doi: 10.3934/nhm.2006.1.639

Related Papers:
Abstract

We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starting point is an integral equation representing unsteady single-phase 3-D porous media flow and the 1-D isothermal Euler equations representing unsteady well flow. This $2 \times 2$ system of conservation laws is coupled to the integral equation through natural coupling conditions accounting for the flow between well and surrounding reservoir. By imposing simplifying assumptions we obtain various hyperbolic-parabolic and hyperbolic-elliptic systems. In particular, by assuming that the fluid is incompressible we obtain a hyperbolic-elliptic system for which we present existence and uniqueness results. Numerical examples demonstrate formation of steep gradients resulting from a balance between a local nonlinear convective term and a non-local diffusive term. This balance is governed by various well, reservoir, and fluid parameters involved in the non-local diffusion term, and reflects the interaction between well and reservoir.
Reader Comments

Your name:*

Email:*
© 2006 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)