Analytical Solutions of a One-Dimensional Linear Model Describing Scale Inhibitor Precipitation Treatments

The deposition of mineral scales such as barium sulfate and calcium carbonate in producing oil wells is a well-known problem in the oil industry, costing many millions of dollars per year to solve. The main preventative measure for managing downhole scale is to inject chemical scale inhibitor (SI) back into the producing well and out into the near well reservoir formation in a so-called “squeeze” treatment. The scale inhibitor is retained in and subsequently released from the formation by the two main mechanisms of “adsorption” ((Formula presented.)) and “precipitation” ((Formula presented.)). As the well is brought back onto production, the scale inhibitor then desorbs (adsorption) or re-dissolves (precipitation) and the low SI concentration that is present in the produced water effectively retards the scale deposition process. A complete model of SI retention must have a full kinetic (Formula presented.) model embedded in a transport model for flow through porous media. In this paper, we present a subcase of this model involving only kinetic precipitation ((Formula presented.)). The simple quasi-linear problem with an infinite source of precipitate is straightforwardly soluble using conventional methods for a precipitate described by a solubility (Formula presented.) and a dissolution rate (Formula presented.). However, the problem with a finite amount of precipitate is more complex and novel analytical solutions are presented for the (transient) behavior for this case. The mathematical difficulty in this latter system arises because, when the precipitate is fully dissolved close to the system inlet, a moving internal boundary develops along with some related flow regions defined by the parameters of the problem. The problem is solved here by making certain assumptions about the internal moving point at position (Formula presented.), where (Formula presented.), and we derive an expression for the velocity of this point ((Formula presented.)). From this, we then build the solution for all possible regions which may develop (depending on the problem parameters). Understanding the behavior of this idealized system gives us some practical formulae for precipitation squeeze design purposes. It also serves as an important set of reference solutions in the search for analytical solutions of more complex cases of the (Formula presented.) model which we will investigate in forthcoming papers.