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# Methods and Applications of Analysis

## Volume 12 (2005)

### Number 3

### On the Riemann Solutions of the Balance Equations for Steam and Water Flow in a Porous Medium

Pages: 325 – 348

DOI: http://dx.doi.org/10.4310/MAA.2005.v12.n3.a7

#### Authors

#### Abstract

Conservation laws have been used to model a variety of physical phenomena and therefore the theory for this class of equations is well developed. However, in many problems, such as transport of hot fluids and gases undergoing mass transfer, balance laws are required to describe the flow.

As an example, in this work we obtain the solutions for the basic one-dimensional profiles that appear in the clean up problem or in recovery of geothermal energy. We consider the injection of a mixture of steam and water in several proportions in a porous rock filled with a different mixture of water and steam. We neglect compressibility, heat conductivity and capillarity and present a physical model for steam injection based on the mass balance and energy conservation equations.

We describe completely all possible solutions of the Riemann problem. We find several types of shock between regions and develop a scheme to find the solution from these shocks. A new type of shock, the evaporation shock, is identified in the Riemann solution. This work generalizes the work of Bruining et. al., where the condensation shock appears. It is a step towards obtaining a general method for solving Riemann problems for a wide class of balance equations with phase changes.

#### Keywords

Porous medium; steamdrive; Riemann solution; balance equations; multiphase flow

#### 2010 Mathematics Subject Classification

35L60, 35L67, 76S05