Theoretical and Experimental Bases for the Dual-Water Model for Interpretation of Shaly Sands

Abstract

Abstract A simple petrophysical model proposed by Waxman and Smits (WS)1 in 1968 and Waxman and Thomas (WT)2 in 1972 accounts for the results of an extensive experimental study on the effects of clays on the resistivity of shaly sands. This model has been well accepted by the industry despite a few inconsistencies with experimental results. It is proposed that these inconsistencies resulted from the unaccounted presence of salt-free water at the clay/water interface. Electrochemistry indicates that this water should exist, but is there enough to influence the results? Both a theoretical study and reinterpretation of Waxman-Smits-Thomas data show that there is. The corresponding new model starts from the Waxman and Smits concept of supplementing the water conductivity with a conductivity from the clay counterions. The crucial step, however, is equating each of these conductivity terms to a particular type of water, each occupying a representative volume of the total porosity. This approach has been named the "dual-water" (DW) model because of these two water types - the conductivity and volume fraction of each being predicted by the model. The DW model has been tested on most of the core data reported in Refs. 1 and 2. The DW concept is also supported by log data3 and has been successfully applied to the interpretation of thousands of wells. However, the scope of this paper remains limited to the theoretical and experimental bases of the DW model. The Petrophysical DW Model The purpose of this model is to account for the resistivity behavior of clayey sands. For petrophysical considerations, a clayey formation is characterized by its total porosity, ft; its formation factor, F0; its water saturation, SwT; its bulk conductivity, Ct; and its concentration per unit PV of clay counterions, Qv. The formation behaves like a clean formation with identical parameters ft, F0, and Swt but containing a water whose conductivity, Cwe, differs from the bulk formation water. Neither the type of clays nor their distribution influences the results. Since the formation obeys Archie's laws,Equation 1 The clayey sand equivalent water conductivity, Cwe, can be considered a mixture of two waters. 1. A clay water surrounds the clay particles but has a conductivity independent of the type and amount of clay. Its conductivity, Ccw, comes exclusively from the clay counterions. The volume fraction of clay water, Vcw, is directly proportional to the counterion concentration, QvEquation 2 where vQ is the amount of clay water associated with 1 unit (meq) of clay counterions. 2. The water further away from the clay is called far water. Its conductivity, Cw, and ionic concentration correspond to the salinity of bulk-formation water. The volume fraction of this water, Vfw, is the balance between the total water content and the clay water.Equation 3 The implicit assumption is that the far water is displaced preferentially by hydrocarbons.