A new analytically derived shape factor for gas–oil gravity drainage mechanism

Matrix–fracture exchange term is an important parameter in simulation of naturally fractured reservoirs using double porosity concept. A matrix block which is located in gas-cap is going to be produced under gravity drainage retarded by capillary force. Hence, oil can be produced as long as gravitational force is greater than the capillary force. The interaction of these forces which controls the oil production from a matrix block may be expressed by a factor which is matrix–fracture transfer shape factor. Customarily, this factor has been used as a history matching parameter as it is still not well understood and the formulations used in commercial simulators are not precise. Therefore, a mathematically derived shape factor which reflects the gravity drainage behavior into an exchange term using shape factor concept is essential. Traditionally, a constant shape factor has been used in most of the commercial reservoir simulators. In this paper, we use analytical modeling to develop a time dependent shape factor for gas–oil gravity drainage mechanism for a single matrix block in a gas-cap. The obtained model is verified against several fine-grid numerical simulations as well as two major existing models being used in a commercial simulator. The obtained shape factor starts from a dimensionless shape factor of 2 (σLz2 = 2) and demonstrates a transient behavior at early times and then converges to π2/4 at late times. The capillary force distribution throughout the matrix block is also obtained during the process showing a nonlinear distribution at early times which turns into a linear one at late times. The linear behavior of capillary force distribution corresponds to late time stabilization of shape factor. Moreover, the mathematical derivation predicts very early time constant maximum drainage rate after which drainage rate starts to decline. The results obtained in this paper will improve our understanding of the gas–oil gravity drainage in fractured reservoirs and will find application in more accurate representation of this process in dual-porosity simulators.