A semi-analytical approach for analysis of the transient linear flow regime in tight reservoirs under three-phase flow conditions

Abstract

The majority of current techniques for production data analysis of hydraulically-fractured horizontal wells only consider single- or two-phase flow conditions. Three-phase flow is expected to occur in many practical scenarios of low-permeability reservoir production when water (frac fluid and/or formation) production is occurring along with hydrocarbon phases. In this paper, a new semi-analytical method is developed to allow production data analysis of wells producing from low-permeability (tight) reservoirs exhibiting three-phase during the transient linear flow period.The proposed technique is based on the modified black oil formulation and is developed by employing the Boltzmann transformation. The modified black oil formulation comprises three highly nonlinear Partial Differential Equations (PDE's) that describe the flow of water, oil and gas in the reservoir. The use of the Boltzmann transformation allows conversion of this set of governing fluid flow PDE's to a set of Ordinary Differential Equations (ODE's) that can be easily solved using numerical techniques such as the Runge-Kutta method. In this paper, oil relative permeability under three-phase flow conditions is modelled using Stone's well-known first model.A theoretical basis for the occurrence of straight lines on a plot of 1/q vs √t for any producing phase is provided herein. The slope of the line is a function of the flowing fluid properties, and the reservoir and fracture characteristics. One important advantage of using the Boltzmann transformation is that the linear flow parameter, xf√k, can be directly estimated as a part of the solution process without the need for defining any complex pseudo-variable transformations. The results demonstrate that the relative error in xf√k estimation is generally less than 10% for different fluid models and relative permeability curves. Moreover, the simultaneous solutions of the ODE's provide the pressure and phase saturations as unique functions of the Boltzmann variable; these can be used to find relationships between pressure and phase saturations that in turn can be utilized for other rate transient analysis applications. The derived pressure-saturation relationships are in excellent agreement with numerical simulation for different three-phase relative permeability models.The significance of this paper is that a new theoretical and practical framework is introduced for the first time to analyze three-phase production data for transient linear flow under constant flowing bottomhole pressure conditions. These conditions are of practical importance for many tight production scenarios where water, in addition to hydrocarbon phases, are mobile in the reservoir. The results are validated using numerical simulation.