The Riemann Solution for Carbonated Waterflooding

We develop a Riemann solver for transport problems related to oil recovery. We consider one dimensional incompressible flow in porous media involving several chemical components, namely $H_2O$, $H^+$, $OH^-$, $CO_2$, $CO_3^{2-}$, $HCO_3^-$ and decane, which are in chemical equilibrium in aqueous and oleic phases. As there is mass transfer between phases and the partial molar volume differs between aqueous and oleic phases leading to a variable total Darcy velocity, fractional flow theory does not easily apply. Recall that for upscaled equations the convection terms completely dominate the diffusion terms; this is why our basic model considers the limit of zero diffusion coefficients. The Riemann solution for this model can therefore be applied for upscaled transport processes in enhanced oil recovery involving geochemical aspects. We formulate three conservation equations, in which we substitute regression expressions that are obtained by geochemical software (PHREEQC). Gibbs phase rule together with charge balance shows that compositions can be rewritten in terms of the pH only. We use the initial and boundary conditions for carbonated aqueous phase injection in an oil reservoir containing connate water with some carbon dioxide. We compare the Riemann solution with a numerical solution, which includes capillary and diffusion effects. The structure of the Riemann solution for constant oil viscosity, from left (upstream) to right (downstream), consists of two rarefaction waves connected by a chemical shock; the latter is continued with a constant state and finally a fast Buckley-Leverett saturation shock. In the first rarefaction wave only the saturation changes, while in the second one both saturation and composition change. The connection point between the rarefaction waves can be constructed from a curve of states where the two characteristic velocities coincide. The significant new contribution is the effective Riemann solver we developed to obtain solutions for oil recovery problems including geochemistry and a space dependent total Darcy velocity.