Multiphase Non-Darcy Flow in Proppant Packs

Abstract

Abstract This paper builds on a model for single-phase non-Darcy flow in porous media presented in SPE 89325.1 In this work the generalized equation for single-phase non-Darcy flow is extended to multiphase flow conditions. Laboratory measurements of gas-water flow were conducted in several flow cells to determine relative permeability curves and the non-linear relationship between individual phase flow-rate and induced potential gradient over a wide range of input fractional flow conditions and over a range of Reynolds Numbers (Re). A model is presented to predict multiphase non-Darcy flow in porous media that is applicable to high-deliverability reservoirs and hydraulically fractured wells. Introduction The importance of multiphase non-Darcy flow on well productivity and impairment has been widely recognized.2–9 The combination of high-velocity and multiphase flow has huge effects on productivity and stimulation effectiveness in hydraulically fractured wells. It is also important in near-well pressure drop in high permeability retrograde condensate reservoirs and frac-pack or gravel pack completions. Many papers have addressed methods to predict the combined effects of non-Darcy and multiphase flow.10–17 Several of these are based on work previously conducted at Stim-Lab.11–13 All these previous works have assumed either a constant value of the Forchheimer inertial coefficient (ß) or have relied on a combined average Reynolds Number to define the non-Darcy flow state, or both. In most cases the combined effects of inertial losses and multiphase flow have been expressed using a modified value of ß. The current work presents an alternate method of predicting the non-Darcy multiphase relationship between potential gradient and phase flow rate. The model is based on extensive laboratory measurements of many proppants at elevated confining stress, pore pressure, and temperature. The results can be applied to flow in proppant packs for hydraulically fractured wells, and to high-velocity flow in reservoir rocks, typically in the near-well region. Review of Single-Phase Non-Darcy Flow Model The general form for the non-Darcy flow equation described in SPE 89325 can be written as in Equation 1. If the exponent (E) is set to 1.0 and the value of kmr is set to zero, the general form of Equation 1 reduces identically to the Forchheimer Equation. Equation (1) To adequately model non-linear flow behavior in porous media the value of Re can be expressed in terms of the ratio of ???/µ to some transition constant, T, as defined by Equation 2. The advantage of this approach is that the value of T is a constant for any system, and is not dependent upon the experimental conditions under which it is determined, unlike ß, which is dependent upon the conditions of measurement. Equation (2) The transition constant, T, can be determined from the median particle diameter of the porous medium using T=1/(2Dm), with Dm in centimeters. Assuming the value of b used in the calculation represents a range of flow rates in which deviation from Forchheimer's equation is small, the value of T can be determined from T=1/(kdß) where ß must be in units of atm-sec2/g and k in darcies. The value of T can also be determined directly from the Forchheimer plot by taking T equal to the value of ???/µ when the value of the linear Forchheimer curve is exactly twice the y-axis intercept value. This corresponds to a value of Re=1.0 in Equation 2. The general model given by Equation 1 has been verified by comparison to laboratory flow data at very high Reynolds Numbers.18 Lopez-Hernandez found general values of kmr=0.002 and E=0.89 for sands and man-made proppants from 100-mesh to 12/18 mesh size.