In this paper, a semi-analytical solution of the pseudosteadystate (PSS) productivity index under non-Darcy flow condition is proposed. Based on the model, a new method of optimization of the fracture conductivity has been developed for non-Darcy flow in the fracture. Meanwhile, the effects of the Reynolds number, the proppant number and the fracture conductivity on the dimensionless productivity index have been discussed in detail. Results show that: (1) the classic UFD (Unified Fracture Design) curves of the Darcy-flow model underestimate the effect of the proppant number and are unsuitable for the non-Darcy-flow fracture optimization. Our discretized model provides a new tool to obtain the optimal fracture parameters for the case of non-Darcy flow condition. (2) for a given penetration ratio, the non-Darcy flow behavior exerts a strong influence on the productivity index with the dimensionless fracture conductivity CfD= 0.1 -1000 and a considerable productivity index drop can be observed. For the extremely low (CfD<0.1) or high (CfD>1000) dimensionless fracture conductivity, the effect of non-Darcy flow becomes negligible. (3) the effect of the non-Darcy flow on the productivity index becomes pronounced at large value of proppant number, Np. For a given proppant number, the optimal fracture conductivity is slowly increasing as the Reynolds number increases. However, the magnitude of the effect on the productivity index is gradually declining. At the Reynolds number less than 5, the non-Darcy flow has a relatively strong impact on the productivity index and an apparent fall of the maximum productivity index can be noticed, especially for a large proppant number. Beyond the value of 5, the declining trend of the maximum productivity index gradually slows down as the Reynolds number increases.
Bibliographical noteThis work was supported by the National Natural Science
Foundation of China No. 51204148 and the Fundamental Research Funds for the
- productivity index
- non-Darcy flow
- vertically fractured well
- optimization of the fracture parameters
- rectangular reservoir