An Extended Musket Differential Mass Balance Equation Applied to Declined Curve Analysis For Wells Undergoing Combined Drived Mechanisms
Open Access
- Author:
- Li, Yijie
- Graduate Program:
- Petroleum and Mineral Engineering
- Degree:
- Master of Science
- Document Type:
- Master Thesis
- Date of Defense:
- July 07, 2020
- Committee Members:
- Gregory R King, Thesis Advisor/Co-Advisor
Arash Dahi Taleghani, Committee Member
John Wang, Committee Member
Mort D Webster, Program Head/Chair - Keywords:
- Material Balance
Decline Curve Analysis
Combined Drive
boundary-dominated flow
Segereated flow
Aquifer Modeling
Dispersed flow
Forchheimer
gas injection
water injection
primary gas cap - Abstract:
- The objective of this thesis is to study the performance and decline behavior of wells and reservoirs undergoing combined drive mechanisms. The methodology used for these analyses is a novel, semi-analytical method valid from the reservoir to the wellbore during boundary-dominated flow regime. Fetkovich (1973) provided the theoretical basis for Arps’ decline curve analysis [Arps, 1945]. This work formed the foundation of Rate Transient Analysis (RTA). In his seminal work, Fetkovich (1987) assumed single-phase flow of oil wells producing from reservoirs undergoing rock-fluid expansion drive. Camacho-Velazquez (1987), Camacho-Velazquez and Raghavan (1989), and Camacho-Velazquez (1991) extended this work by considering two-phase, gas-oil pseudo-functions and allowing for oil production from reservoirs undergoing rock-fluid expansion and solution gas drive mechanisms. Camacho-Velazquez (1987) assumed both transient and boundary-dominated flow, Camacho-Velazquez and Raghavan (1989) assumed boundary-dominated flow only, while Camacho-Velazquez (1991) assumed transient flow only. Inherent to the work of these authors was the assumption of dispersed flow; however, although not discussed in these papers, segregated flow can be approximated with the use of straight-line relative permeability curves. In this work, I extend decline curve analysis for boundary-dominated flow further by allowing for three-phase flow from reservoirs undergoing simultaneous drive mechanisms which consider (1) natural drive mechanisms of rock-fluid expansion, solution gas drive, gas cap expansion (from a primary gas cap), and active water drive (from either bottom-water or edge water aquifers) which may act alone or concurrently and (2) engineered drive mechanisms, such as, partial or full pressure maintenance from gas and/or water injection (note, the proposed methodology is only valid for pressure maintenance and is not valid for oil displacement by either gas or water injection). Historically, non-Darcy flow has been considered in high flow rate gas wells, in which, it has been treated as a rate dependent skin factor, SND = DS q, in the Productivity Index defined by Darcy’s Law and has been assumed to act only in the near-wellbore vicinity. Alternatively, Fetkovich (1987) and Fetkovich (1996) considered non-Darcy flow of both gas and oil wells by use of the empirical Rawlins and Schellhardt (1936) Equation. The methodology proposed in this thesis considers oil and gas production from either Darcy flow or non-Darcy flow using the more theoretically grounded quadratic Forchheimer Equation [Forchheimer (1901)]. The proposed generalized differential material balance equations are derived for three-phase, boundary dominated flows with water encroachment from either a bottom-water or edge-water aquifer. The water transport from the aquifer to the reservoir is assumed to honor the Fetkovich Pseudo-Steady State Aquifer Model; however, other aquifer models may also be implemented. In this thesis, four coupled, nonlinear ordinary differential equations (ODEs) are derived which describe the gas saturation, water saturation (and, by extension, the oil saturation with use of the algebraic saturation constraint equation, So = 1 – Sw - Sg), reservoir pressure, and aquifer pressure behavior with time during the production process. This system of ODEs is formulated using an extended Muskat-Like Method [Lu (2019)] and further refined using a symbolic Gaussian Elimination procedure applying a variant of Gaussian Elimination which avoids division (the back substitution step still requires division however). The proposed ODEs for (dP_a)/dt , (dP_R)/dt, (dS_g)/dt, and (dS_w)/dt are all coupled, first-order, non-linear ODEs which are pressure, saturation, and production rate dependent. The final system of ODEs is then solved using the Fourth Order Runge-Kutta method for systems of equations. While the Runge-Kutta method is solved in the time-domain, the solutions for the saturation and pressure changes are presented in the time domain, the equations are also formulated in the average reservoir pressure and cumulative produced oil domains. The application of the proposed semi-analytical methodology is then used to compare the impact of the more generalized physics and flexible reservoir management strategies on the behavior of a single production well and the behavior of the Arps’ [Arps’ (1945)] decline constants D and b. In addition, the proposed methodology can be used as an alternative to decline curve analysis. While this thesis considers single wells, the methodology can be used to consider multiple production wells. In addition, this study investigates several alternates to dispersed flow, allowing for the development of a secondary gas cap.