On the periodic injection of fluid into, and its extraction from, a confined aquifer

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Abstract
We consider the periodic injection and extraction of fluid from a line well in a horizontal saturated aquifer of finite thickness as part of an aquifer thermal energy storage system. We focus on the case in which the injected fluid is dense relative to the original fluid in the aquifer and we explore the competition between the driving pressure and buoyancy force in controlling the dispersal of the injected fluid through the aquifer. We show that, after each cycle, a progressively larger fraction of the injected fluid is extracted, while the remainder of the injected fluid gradually migrates away from the well such that, after time $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}t$ , the position of the leading edge of the injected fluid, $x_{nose}(t)$ , scales as $ x_{nose}(t)\sim x_{nose}(\tau )\sqrt{t/\tau }$ , where $\tau $ is the period of injection. If the fluid is extracted from the base of the layer, then, near the well, the thickness of the injected fluid at the end of the extraction cycle tends to a constant value, which decreases with injection rate. We also show that there is a class of self-similar exchange-flow solutions that develop when a saturated porous layer of thickness $H$ is in contact with a stratified fluid reservoir, filled to thickness $F_0H