Transient Pressure Behaviour Under Non- Darcy Flow, Formation Damage and Their Combined Effect for Dual Porosity Reservoirs
Authors: Zeng F. , Zhao G. , Xu X. 
Published: March 2010

Abstract
A semi-analytical model is presented to study transient pressure behaviour of vertical wells in a dual porosity reservoir under the influence of non-Darcy flow inside the reservoir (Forchheimer number), formation damage around the wellbore (mechanical skin factor), and their combined effect (near-wellbore non-Darcy skin factor). Unique transient pressure behaviour is presented in both semi-log pressure and log-log pressure derivative plots. Semi-log pressure plots suggest that for a vertical well producing with a constant sandface flow rate, its pressure responses exhibit two linear portions for Darcy flow, even with non-Darcy skin factor (i.e. the near-wellbore non-Darcy skin factor is larger than zero). However, the first linear portion will dismiss if non-Darcy flow presents inside the reservoir (i.e. the Forchheimer number is larger than zero). Further pressure derivative analyses in a log-log plot suggest that:both Forchheimer number and nearwellbore non-Darcy skin factor lead to a narrower and steeper transition region between pure wellbore storage and radial flow region for build up tests and a wider and gentler one for drawdown tests; andthe derivative curves under the effect of the Forchheimer number always exhibit a slight convex shape in the radial flow region, rather than a horizontal line with a constant value of 0.5 for Darcy flow.
The type curves presented may be applied to estimate mechanical skin factor, Forchheimer number and near-wellbore non- Darcy skin factor from a single rate test. Synthetic cases show that ignoring the globally distributed non-Darcy flow causes errors of up to 300% in interpreting dual porosity characteristic parameters. Both synthetic and field case studies suggest that the non-Darcy flow has a significant effect on the interpretation of the inter-porosity flow coefficient, ?, and less effect on the interpretation of the storativity ratio, ?.

Introduction
The Forchheimer equation suggests that high velocity gas flow in porous media obeys the following quadratic equation:
Equation (available in full paper)
It has been recognized that Forchheimer non-Darcy flow is introduced by gas inertial effects; thus, ß?is called the inertial factor. Forchheimer non-Darcy flow is a universal phenomenon in gas reservoir development. It could affect reservoir parameter estimations, well productivity evaluation and production performance analysis. Observations in the field(1) and the laboratory(2, 3) have suggested that, in the fracture system of a naturally fractured reservoir, the pressure loss is dominated by the square term in the Forchheimer equation.
Studies on modelling non-Darcy flow in naturally fractured reservoirs are limited. The first attempt was reported by Villalobos-L. et al.(4). Based on numerical results of a radial liquid flow model with the Forchheimer non-Darcy flow, these authors derived an expression of a rate-dependent pseudo-skin term similar to that in homogeneous single porosity reservoirs.
Wu(5) modelled non-Darcy flow in a naturally fractured system using the Multiple Interacting Continua (MINC) method by applying the results of Darcy flow to approximate the characteristic length of non-Darcy flow distance between fractures and the matrix crossing the interface for the dual porosity formation. He observed that semi-log plots of pressure drop versus time are extremely sensitive to the values of non-Darcy coefficient.