We describe the reaction which occurs when formation fluid containing Ba in solution is displaced by and mixes with injected water, containing sulphate, to form BaSO4. The model is based on the conservation of mass of both the barium and sulphate, assuming that each species is advected with the flow, and that at the interface between the two fluids, dispersive and diffusive mixing of the fluids leads to precipitation. The model equations are solved numerically, and a simple asymptotic model for the reaction, valid at long time, is developed. This identifies that in a linear geometry, the mass of solid which forms when the front has distance x from the source, decays at a rate proportional to 1/x1/2. The modelling approach may also be used to describe injection into a radial geometry, leading to an analogous asymptotic model for the decay of solid precipitate with distance from the injector well.
The formation of precipitate within permeable rock is a well-known process, and there are several different styles of reaction. These include
frontal reactions such as occur during acidising when the rock matrix interacts with the injected fluid (1 Schecter, 1992; 2: Economides et al. 1993),
thermal gradient reactions (3:Phillips, 1991) when injected liquid migrates through a temperature gradient, and
pressure induced reactions, such as occur near production wells, as the fluid decompresses, releases CO2, thereby reducing the solubility of Ca and leading to calcite precipitate.
However, a further important reaction is the BaSO4 precipitation reaction in which formation and injected (sea) water mix, and precipitate scale. BaSO4 formation is especially important in some North Sea wells and also in new provinces, including those offshore Angola. In some north sea fields, formation and injected water appear to mix in the producing wells. This leads to such massive scale deposits that some wells require to be shut-in every week or so, during which time chemical inhibitor may be injected into the reservoir.
In this work, we develop a model to describe the how the mixing of the formation and injected seawater in a porous rock, including an effective dispersion/diffusion term to model the mixing of the two fluids, leads to scale formation in situ. Since the reaction is driven by the mixing of the two fluids, we are able to identify the dominant terms in the reaction-diffusion-advection system, and thereby develop a simplified asymptotic model for the rate of precipitation. We compare this with a numerical model to validate the approach, and draw some conclusions in terms of the implications for precipitation within the reservoir rather than in the well.