Growth rate of a penny-shaped crack in hydraulic fracturing of rocks, 2

The deformation and growth of a vertical penny-shaped crack, fractured hydraulically, is investigated when fluid is injected from an inlet at the center of the crack at a constant flow rate. The total flow rate at the inlet is divided into three parts: flow rate extracted from an outlet hole at an arbitrary distance above the center, fluid loss rate from the crack surface, and total fluid mass change in the crack. Two cases are considered: in case 1, inlet flow rate is initially greater than the sum of the outlet flow and fluid loss rates, and in case 2 the reverse holds true. Two subcases a and b are also considered, depending on the values of outlet pressure. Ranges of the inlet flow rate and the outlet pressure are discussed for which the crack attains stationary states and the fluid can be extracted continuously. Subcase b, where the outlet pressure is less than or equal to the difference between the tectonic stress and the fluid head at the inlet, is found to be more practical, and reasonable outlet flow rates are obtained in this case. It is also found that case 2b is preferable to case 1b to obtain the fluid with higher temperature. Results are expected to be of use in considerations of heat extraction from hot dry rock.