Analytical model of cold water front movement in a geothermal reservoir

Injection of cooled geothermal water back into the producing formation is a procedure that maintains reservoir pressure and increases energy extraction efficiency, but, because reinjected fluids tend to be much colder than the reservoir rock, injection can also cause cooling of the fluid produced from nearby wells. It is therefore essential to determine the cold front velocity in a geothermal reservoir. Constant thermal properties of both rock and fluid are generally assumed in order to solve this problem. In this paper, the rock density and heat capacity of the water–rock system as functions of temperature are assumed. Using the method of characteristics, an analytical solution is obtained. It is shown that the variable heat capacity of rock leads to a temperature-dependent speed of propagation of the thermal front. This results in a steepening of the front when cold water is injected into a hot zone, and eventually the formation of a discontinuous solution, or shock. A method is proposed for finding such discontinuous solutions and an equation for the velocity of the thermal front is presented. The difference between the front velocity obtained by means of the weak solution presented here and the classical model with constant thermal properties varies between about 1 and 14%.