Research Article
Exact Solution for Non-Self-Similar Wave-Interaction Problem
during Two-Phase Four-Component Flow in Porous Media
S. Borazjani,
1
P. Bedrikovetsky,
1
and R. Farajzadeh
2,3
1
Australian School of Petroleum, he University of Adelaide, SA 5005, Australia
2
Shell Global Solutions International, Rijswijk, he Netherlands
3
Delt University of Technology, he Netherlands
Correspondence should be addressed to S. Borazjani; sara.borazjani@adelaide.edu.au
Received 6 September 2013; Revised 27 December 2013; Accepted 29 December 2013; Published 12 March 2014
Academic Editor: Shuyu Sun
Copyright © 2014 S. Borazjani et al. his is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Analytical solutions for one-dimensional two-phase multicomponent lows in porous media describe processes of enhanced oil
recovery, environmental lows of waste disposal, and contaminant propagation in subterranean reservoirs and water management
in aquifers. We derive the exact solution for 3×3 hyperbolic system of conservation laws that corresponds to two-phase four-
component low in porous media where sorption of the third component depends on its own concentration in water and also on
the fourth component concentration. Using the potential function as an independent variable instead of time allows splitting the
initial system to 2×2 system for concentrations and one scalar hyperbolic equation for phase saturation, which allows for full
integration of non-self-similar problem with wave interactions.
1. Introduction
Exact self-similar solutions of Riemann problems for hyper-
bolic systems of conservation laws and non-self-similar
solutions of hyperbolic wave interactions have been derived
for various lows in gas dynamics, shallow waters, and
chromatography (see monographs [1–8]). For low in porous
media, hyperbolic systems of conservation laws describe two-
phase multicomponent displacement [9, 10]. Consider
+
(,)
=0 (1)
(+())
+
( (,))
=0,
(2)
where s is the saturation (volumetric fraction) of aqueous
phase and f is the water lux. Equation (1) is the mass
balance for water and (2) is the mass balance for each
component in the aqueous solution. Under the conditions
of thermodynamic equilibrium, the concentrations of the
components adsorbed on the solid phase (a
i
) and dissolved in
the aqueous phase (c
i
) are governed by adsorption isotherms:
=(), =(
1
,
2
,...,
), =(
1
,
2
,...,
).
(3)
Exact and semianalytical solutions of one-dimensional low
problems are widely used in stream-line simulation for low
prediction in three-dimensional natural reservoirs [10]. he
sequence of concentration shocks in the one-dimensional
analytical solution is important for interpretation of labo-
ratory tests in two-phase multicomponent low in natural
reservoir cores.
he scalar hyperbolic equations (1) and (2), =0,
correspond to displacement of oil by water [9, 10]. he
( + 1) × ( + 1) system (1) and (2) describes two-phase
low of oleic and aqueous phases with components (such
as polymer and diferent salts) that may adsorb and be
dissolved in both phases. hese lows are typical for so-called
chemical enhanced oil recovery displacements, like injections
of polymers or surfactants, and for numerous environmental
lows [9, 10]. For polymer injection in oil reservoirs, =1
corresponds to polymer and =2,3,..., to diferent ions.
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 731567, 13 pages
http://dx.doi.org/10.1155/2014/731567