Technical Notes - Frontal Drive Production Mechanisms - A New Method for Calculating the Displacing Fluid Saturation at Breakthrough

A new graphical method, which is a modification of that proposed by Buckley and Leverett', is presented for the determination of the displacing fluid saturation at breakthrough for frontal drive mechanisms. This method is applicable to all types of fluids, which may be either soluble or insoluble in the oil at reservoir and injection conditions. This adaptation has unique advantages in handling problems in which either liquids or gases are the displacing fluids. INTRODUCTION In the displacement of oil from a reservoir with water or gas injected into the formation or by the expansion of a gas cap, the determination of breakthrough or flood advance is important to define the initial and subordinate phases of the operation. Thus, the determination of the displacing fluid saturation at breakthrough is fundamental to the establishment of over-all reservoir production. Methods for analyzing this saturation and for calculating production rates have been previously reported1,'*".". All of these methods utilize the rate equation for linear flow developed by Buckley and Leverett1. xs = QT - Qn/A? (?fu/?Su) .....(1) Sp where f, represents the volumetric fraction of displacing fluid in the total flowing fluid within the reservoir as is defined by: fo = 1/1+ ku/kd µn/µn .......(2) Since the time, ?, necessary for the flood front to travel distance ?X is unknown, Eq. 1 cannot be solved directly for the determination of the breakthrough point. Therefore, it becomes necessary to solve Eq. 1 simultaneously with an equation representing a fluid bal-ance on the reservoir behind the flood front. To develop such a fluid balance equation, it has been assumed that the fluid composition is uniform in the reservoir behind the flood front. Based upon this assumption, the follow-ing equation can be derived from a material balance on the displacing fluid: x = (?X) s Dx ? x=0 [ Snx - Sn1 ] dx = QT - Qn/A? It is readily observed that the displacing fluid satura-tion behind the flood front, Snx, may be determined through the simultaneous application of Eqs. 1 and 3. Although the approach may vary, this objective re-mains the same for all of the previously mentioned graphical or analytical methods. After Sox is determined, it is then possible to calculate the gas-oil and/or water-oil ratios and the rates during the initial and subordinate phases of production, and thereby define the production history of the reservoir. In this presentation only the de-termination of Snx is considered, and the reader is re-ferred to the several texts in reservoir engineering2,4,5 for a complete discussion on the methods for establish-ing the production history of a reservoir. PROPOSED METHOD FOR DETERMINING Snx Since at any time the quantity, QT - Qn/A? is a constant, Eq. 1 may be expressed in differential form as follows: dxsn = QT - Qn/A? d (? fn/? Sn) sn .....(4) Eq. 4 may be substituted directly into Eq. 3 to obtain: ? fn/? Sn ? [ Snz - Sn] QT - Qn/A? d ( ? fn/? Sn) = QT - Qn/A? ...............(5) Eq. 5 simplifies to the expression: ? fn/? Sn ? [ Snz - Sn] QT - Qn/A? d ( ? fn/? Sn) = 1 ....(6)