Non-Destructive Techniques to Determine the Effective Stress Coefficient of Sandstone Formations
Authors: Frempong P.K. , Butt S.D. 
Published: March 2010

Abstract
The concept of effective stress coefficient, "n," is critical for the study of stress and pressure-dependent behaviour of rocks. This parameter is used to scale the pore fluid pressure for the computation of effective stress and impacts on the accuracy of pore pressure and saturation predictions in reservoirs. However, "n" is relatively unknown and difficult to estimate and, in many cases, is often incorrectly assumed to be equal to one. We present a general review of "n" and give a theoretical proof that "n" is less than one for porous rocks. We also introduce a laboratory methodology to measure the value of "n" employing ultrasonic P-wave velocity (Vp) and attenuation quality factor (Qp). We then confirm the value of "n' experimentally by using this laboratory method to estimate "n" of quartz sandstone. We compare our experimental value of "n" to both theoretical and practical results obtained from other literature that used different estimation methodology. We argue from our experimental results that:"n" is not one but varies from zero to one;there is no single value of "n" for a particular reservoir rock;"n" depends on internal factors (porosity and pore geometry) and external factors (pore pressure and confining pressure) of the rock;velocity derived "n" and quality factor derived "n" are slightly different for the same rock; and,for the quartz sandstone used in this experiment, the velocity derived "n" is smaller and also less sensitive to changes in pressure than quality factor derived "n."

Introduction
Most of the pioneering work on what is now known as effective stress coefficient was done by Terzaghi(1), who showed that most rocks are porous to some degree due to the pore spaces and micro cracks within them. The effect of these pores on the strength of rocks has widely been documented(1–5). A rock with a few or without pores is considerably stronger than porous ones since pore spaces are unsupported within the rock. With the presence of unsupported spaces, the intact rock between the pores has to take larger loads to sustain an overall applied stress. This important phenomenon has a direct bearing on the effective stress of the rock (Figure 1). Thus, the effective stress (Pe) is not directly equal to the differential pressure (Pd). Many writers(4–12) have highlighted the problem caused by this phenomenon and the resulting uncertainties associated with the estimated pressures and saturation in reservoirs. This uncertainty is increasingly significant for reservoir engineering due to the expanding use of seismic techniques for remote estimation of abnormal pore pressures for drill planning and for monitoring reservoir pore pressure changes during production. For example, Carcione and Helle(13) applied a new technique for estimating reservoir pressures from seismic data to an overpressured gas field in the Norwegian North Sea and concluded that once a reliable velocity field was determined for the reservoir formation, the most important part of the prediction process was the determination of the effective stress coefficients and related dry-rock moduli.