A General Differential Mass Balance Equation and its Applications

Open Access
Lu, Chengcheng
Graduate Program:
Petroleum and Mineral Engineering
Master of Science
Document Type:
Master Thesis
Date of Defense:
March 20, 2019
Committee Members:
  • Gregory R King, Thesis Advisor
  • Turgay Ertekin, Committee Member
  • EUGENE MORGAN, Committee Member
  • Muskat
  • Forchheimer
  • Gas-Oil Ratio
  • Solution gas drive
This study derives equations for a generalized differential material balance equation for two-phase, boundary dominated solution-gas drive reservoirs during primary recovery of oil. It studies how oil saturation changes with pressure,(dS_o)/dP, during oil and gas production. The derived equation is a generalization of the Muskat Equation, and the resulting ordinary differential equation, ODE, or System of ODEs is solved using the Fourth Order Runge-Kutta method. The Runge-Kutta method is used to solve the developed ODE in pressure domain for oil saturation or the system of ODEs in time domain for pressure and oil saturation. This thesis also outlines the solution of the resulting equations in other domains, such as, the Cumulative Produced Oil, N_p domain ((dS_o)/(dN_p ) and dp/(dN_p ) ), and the Cumulative Produced Gas, G_p domain ((dS_o)/(dG_p ) and dp/(dG_p ) ). This thesis illustrates how the generalized Muskat Equation can be derived based on the pseudo steady-state flow concept, full transient flow (based on well test assumptions), or from the divergence theorem. These derivations show that (dS_o)/dP is a function of gas-oil ratio, GOR, and this thesis considers several different methods to analyze the gas-oil ratio. The basic analysis uses Darcy’s law to describe mass transport for both oil and gas phases. Then, it compares the results with non-Darcy flow described by the Forchheimer Equation and allows for a GOR based on combined non-Darcy and Darcy flow (gas transport by Forchheimer Flow with oil transport by Darcy flow or transport of both phases by Forchheimer flow). When considering non-Darcy’s flow, the non-Darcy coefficient, β_ND can be either constant or a function of prevailing pressure and saturation conditions. In addition, this study investigates several variants of segregated flow, allowing for the development of a secondary gas cap (the original Muskat Method only allows for dispersed flow and no secondary gas cap). These segregated flow models do not allow gas coning from the secondary gas cap. This study expands on the segregated flow by (1) suggesting a new variant of segregated flow combining segregated and dispersed flow and (2) allowing for gas coning using a published coning correlation. The full suite of GOR models governing (dS_o)/dP then becomes: dispersed flow (Darcy flow in terms of pressure, Darcy flow in terms of two-phase pseudo pressure, and non-Darcy flow), segregated flow in non-coning reservoirs (standard segregated flow model, published segregated flow correlation model assuming hydrostatic pressure, and a novel mixed segregated-dispersed flow correlation), and segregated flow with coning based on published coning correlations. The study then further develops the analysis from saturation changes in the pressure domain to saturation and pressure change in time domain. The resulting model is then used to investigate Arps decline analysis to study how flow rate changes over time impact the behavior of D and b parameters in Arps decline curve. The models developed in this study using MATLAB software are compared with a suite of numerical models developed with CMG specifically for this project.