Based on the KGD scheme, this paper investigates, with both analytical and numerical approaches, the propagation of a hydraulic fracture with a fluid lag in permeable rock. On the analytical aspect, the general form of normalized governing equations is first formulated to take into account both fluid lag and leak-off during the process of hydraulic fracturing. Then a new self-similar solution corresponding to the limiting case of zero dimensionless confining stress ($T=0$) and infinite dimensionless leak-off coefficient ($L=∞$) is obtained. A dimensionless parameter $R$ is proposed to indicate the propagation regimes of hydraulic fracture in more general cases, where $R$ is defined as the ratio of the two time-scales related to the dimensionless confining stress $T$ and the dimensionless leak-off coefficient $L$. In addition, a robust finite element-based KGD model has been developed to simulate the transient process from $L=0$ to $L=∞$ under $T=0$, and the numerical solutions converge and agree well with the self-similar solution at $T=0$ and $L=∞$. More general processes from $T=0$ and $L=0$ to $T=∞$ and $L=∞$ for three different values of $R$ are also simulated, which proves the effectiveness of the proposed dimensionless parameter $R$ for indicating fracture regimes.

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