Analytical method for transient flow rate with the effect of the quadratic gradient term

Owing to neglecting the quadratic gradient term (QGT) in the governing equation, the conventional method proposed by van Everdingen and Hurst [Trans. AIME, 1949, 186, 305-324] for obtaining the transient flow rate is inconsistent with material balance and may lead to errors in the prediction of the flow rate for the wells producing at a constant bottomhole pressure. This paper extends the conventional method to consider the effect of the QGT. If the pressure solution in Laplace space under the constant-rate-production condition is known, the flow-rate solution in Laplace space under the constant-pressure-production condition including the effect of the QGT can be determined by the proposed method. Flow-rate solutions for horizontal wells in reservoirs with different lateral outer boundaries under the constant-pressure-production condition are obtained by the proposed method and the conventional method, respectively. The difference between the flow-rate solutions with and without the effect of the QGT is qualitatively and quantitatively analyzed. The proposed method for obtaining the transient flow rate is consistent with material balance, and thus it can be used to obtain more accurate transient flow rate.