Numerical Modeling of Multiphase Flow Toward Fractured Horizontal Wells In Heterogeneous Nanoporous Formations

Abstract

This paper presents an anomalous diffusion based approach for modeling two-phase oil/water flow in fractured nanoporous media, such as that encountered in unconventional oil and gas reservoirs. Production is assumed to be from a multiply fractured horizontal well and the focus of the discussions is 1D linear flow in the region between hydraulic fractures. To account for the complexity and heterogeneity of the fractured medium, anomalous diffusion is considered in the flow domain between hydraulic fractures. To do so, the conventional Darcy phase fluxes are replaced by more general fractional flux laws which incorporate the non-local and hereditary nature of flow by means of fractional space-time phase and capillary pressure gradients. A modified IMPES scheme is derived for linear oil/water flow under specified terminal pressure production, and the impact of sub- and super-diffusion on phase rates, as well as pressure and saturation distributions across the system is studied.

Methods of Decline Curve Analysis for Shale Gas Reservoirs

“…There have been many studies on anomalous diffusions and the corresponding fractional diffusion equations. For example, Zhou and Selim [46] showed that subdiffusion in porous media can be characterized by a long-tailed profile in the spatial distribution of densities; Metzler and Klafter [47] illustrated a fractional diffusion equation with respect to a non-Markovian diffusion process; Chaves [42] derived the same equation to describe Levy flights; Nigmatullin [48] first considered the fractional initial boundary value problem and derived several theoretical and numerical results; Luchko et al [49,50] obtained the maximum principle and the unique existence of the generalized solution; Sakamoto and Yamamoto [44] studied the uniqueness of weak solutions and the asymptotic behavior; Luchko and Zuo [51] studied the explicit form of the forward solutions for subdiffusion equations with Dirichlet or Neumann boundary conditions; Chen and Raghavan [52] constructed a one-dimensional, fractional-order transient diffusion equation to model diffusion in complex geological media; The pressure distribution was derived in terms of the Laplace transformation and the Mittag-Leffler function; Holy and Ozkan [53] presented a fractional diffusion-based approach for modeling two-phase flow in fractured porous media, such as that encountered in unconventional reservoirs. These fractional diffusion models have shown better accuracy in fluid flow mechanisms in a complex porous medium, such as groundwater contamination and shale gas production.…”

Novel Mathematical Model for Transient Pressure Analysis of Multifractured Horizontal Wells in Naturally Fractured Oil Reservoirs

“…The theory of anomalous diffusion has been regarded as a practical alternative to conventional models for the non-Darcy flow effect in the classic trilinear flow model. , After that, the theory of anomalous diffusion has been updated for nanoporous media , and applied to more complex cases . This work adopts the theory of anomalous diffusion for non-Darcy flow in natural fractures.…”