Simulation of Relative Permeability Hysteresis to the Nonwetting Phase

Abstract

ABSTRACT In reservoir fluid flow, the situation described by an increase in nonwetting phase saturation followed by an increase in wetting phase saturation causes a relative permeability hysteresis effect that is well known. The imbibition relative permeability is a function of the historical maximum nonwetting phase saturation, hence there is no single imbibition curve applicable for use in reservoir modeling. This paper presents a method which allows the calculation of imbibition relative permeability starting at any saturation. The data required are the drainage curve, the historical maximum nonwetting phase saturation, and a minimum of one additional point on some corresponding experimental imbibition curve. In this development, all imbibition curves are shown to be parallel. Also significant is the fact that the residual nonwetting phase saturation can be calculated without a complete experimental specification of the imbibition curve. INTRODUCTION Whenever reservoir rocks exhibiting a strong wettability preference for a specific phase experience a change in saturation history from an initial drainage to an imbibition process, the nonwetting phase is subject to entrapment by the wetting phase. This entrapment of the nonwetting phase results in hysteresis of the relative permeability to this phase. Such occurrences were initially characterized by Geffen, et al.¹ and are well documented in the literature. Gross errors can result in reservoir modeling if the phenomenon is ignored. For example, drainage data used instead of imbibition data in a gas reservoir with a strong water drive could result in predicted recoveries twice the amount actually observed. Hysteresis must be considered to facilitate modeling the variety of saturations which are encountered in certain types of EOR schemes involving alternate injection of wetting and nonwetting phases and also in irregularly developed oil wet formations under waterflood where oil is swept back into regions previously waterflooded. Theoretical and empirical models have been developed that attempt to describe the phenomenon. One such model which has gained considerable popularity was published by Land. 2 He subsequently experimentally verified the model. 3 One of Land's significant contributions was that he empirically demonstrated a relationship between the value of the historical maximum nonwetting phase saturation and the value of nonwetting phase saturation ultimately trapped upon complete resaturation by the wetting phase. He then showed how this relationship could be used to calculate imbibition relative permeabilities to the nonwetting phase. It is not obvious from Land's development that such calculations can be done without the knowledge of the pore size distribution parameter, ε. Killough4 made use of Land's historical maximum-residual nonwetting phase saturation relationship in his treatment of relative permeability hysteresis to establish the endpoint of the imbibition curve. For intermediate values of the curve, the use of Killough's parametric interpolation method requires that a parameter λ be specified. The main purpose of this paper is to establish a procedure which requires only: a drainage curve, the historical maximum nonwetting phase saturation, a minimum of one point on the imbibition curve, and Land's relationship between the historical maximum and final trapped values of nonwetting phase saturation. Neither the pore size distribution factor, ε, in Land's equation nor the value of λ in Killough's approach is necessary to calculate imbibition relative permeability to the nonwetting phase. Subordinate aims of this paper are:With one qualification, to show the imbibition curve used in the procedure in conjunction with the drainage curve need not envelop all anticipated imbibition curves (i.e., the procedure is extrapolative as well as interpolative).To demonstrate that estimates of the endpoint values of the imbibition curves can be determined using the procedure. These values of end points may be practically difficult to obtain experimentally and when measured are subject to uncertainty.To show that any two calculated imbibition curves are parallel for a specific set of data.