Abstract
Reactive transport in porous media with dissolution and precipitation has important applications in oil and gas industry and groundwater remediation. In this work, we present a simulation method for reactive flow in porous media of two salts that share an ion. The method consists of a front-tracking solver that uses the Riemann solutions of the underlying set of hyperbolic partial differential equations. In addition to the discontinuities stemming from the nonlinearities of the flux function, the flux function for the corresponding advection reaction equation also admits discontinuities for a heterogeneous medium. Here, we solve the Riemann problem for the governing nonlinear hyperbolic system with a discontinuous flux function. We use mass balance across the interface and the non-decreasing sequence of velocity of waves to seek the unique solution for this problem. Furthermore, a model is provided for mixing of streamlines at the well to estimate the amount of precipitate. In the use of streamline methods, we have modified the definition of time-of-flight to allow the model to be easily utilised for the heteregeneous case. The simulation method is applied to model dissolution through injection of an unsaturated fluid. It is shown that the first dissolution shock, which causes induced precipitation due to the co-ion effect, results in accumulation of precipitate at the well.
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Acknowledgements
We would like to thank our colleagues Dr. Marc Hesse and Prof. Eric Mackay for the fruitful discussions. We would also like to show our gratitude for the travel fund provided through Adrian Todd Golden Key Student Fund of Institute of Petroleum Engineering at Heriot-Watt University and an Energy Technology Partnership travel grant.
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This research is financially supported by the Energy Industry Doctorate Program Grant ETP136 of the Energy Technology Partnership, Scotland and Energi Simulation.
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Appendix: Secondary variables
Appendix: Secondary variables
In this section, we derive the expressions for the secondary variables in terms of the primary variables NA and NB. The secondary variables nα are used for the analysis of the corresponding eigen-problem and to identify wave types.
For the secondary variables nA and nB, we note that they are continuous but not necessarily differentiable across the borders. In general
A fluid in region I is unsaturated with respect to both ions A and B and hence
In region II, we have saturation with respect to both ions and therefore
To find concentrations for region III, we use Eq. (5b) to obtain nB and then replace it in Eq. (4a) and solve for nA,
Similar to region III, the concentrations for region IV can be obtained
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Ghaderi Zefreh, M., Nilsen, H.M., Lie, K.A. et al. Streamline simulation of a reactive advective flow with discontinuous flux function. Comput Geosci 23, 255–271 (2019). https://doi.org/10.1007/s10596-018-9771-3
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DOI: https://doi.org/10.1007/s10596-018-9771-3