Low Penetration Of Non-darcy Flow Moving Boundary Model With Stefan Conditions

It is an important strategic direction of the industrial development of petroleum of our country to develop scientifically and rationally low-permeability reservoir. It can be known from the development and well testing practice of low-permeability reservoir that fluid flow through low-permeability media exist threshold pressure gradient and present the special phenomenon that the moving boundary expands outwards constantly. Namely, the pressure is not spreading to infinite instantly. In place that moving border does not reach, the reservoir keep still. In addition, a large number of tests verify low-permeability fluid will deviate from Darcy flow. All these make low-permeability reservoir, stratum pressure distribute, bottom-hole pressure distribute and the differential of pressure are different from that of medium-high permeability and have their own characteristics, Worthy of our discussion deeply, in order to guide practice of developing better. In view of this, this paper set up a neoteric moving boundary model with Stefan condition and adopt moving border analog-tracing methods-Volume of Fluids function methods solve the model.The paper discuss the following content mainly:1. Using the general method to set up fluid flow through low-permeability media mathematics model and considering the influence of threshold pressure gradient, we set up a moving boundary model with Stefan condition.2. After turning the existence of the moving boundary problem into fixed point problem of a integral transformations, we utilize Schauder fixed point theorems and extreme value principle to prove the existence and uniqueness of the model. This enables us to go further numerical solved.3. To resolve the moving boundary with Stefan condition, it demands that moving boundary as a part of solution with controlling amount of host field is expressed for together. Because of nonlinear of the moving boundary question, can it only generally get its numerical solution.We utilize Volume of Fluids function methods tracking the moving-boundary. After defining the function of volume of fluid, the resolve is divided into two steps, namely tracing the moving boundary and resolving pressure. VOF equation use Lax-Wendroffgeshi (LW) scheme with precision of two steps. Pressure of host field is turned into initial-boundary value problem of variable coefficient parabolic equation on regular border. We utilize implicit difference method to solve it. Through error analysis, we prove implicit difference scheme is steady and convergent. We write the corresponding procedure with MATLAB to the resolving process.4. In addition, the paper provide computational example. Utilize the procedure we resolve one production well, summarize the pressure characteristic with threshold pressure gradient,and then solve and compare different production wells, draw a series of relation pictures, have summarized and threshold pressure gradient differently to the bottom-hole pressure and influence of moving boundary.We using boundary analog-tracing methods to study low-permeability reservoir is only a beginning, so there are some weak points. At the end, while drawing the conclusion, we put forward some constructive suggestion for interested reader carrying on the discussion.