Based on Huang’s accurate tri-sectional nonlinear kinematic equation(1997),a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary.This model contains double moving boundaries,including an internal moving boundary and an external moving boundary,which are diferent from the classical Stefan problem in heat conduction:The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary.Through a similarity transformation,the nonlinear partial diferential equation(PDE)system is transformed into a linear PDE system.Then an analytical solution is obtained for the dimensionless simplified mathematical model.This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications.Finally,through plotted comparison curves from the exact analytical solution,the sensitive efects of three characteristic parameters are discussed.It is concluded that with a decrease in the dimensionless critical pressure gradient,the sensitive efects of the dimensionless variable on the dimensionless pressure distribution and dimensionless pressure gradient distribution become more serious;with an increase in the dimensionless pseudo threshold pressure gradient,the sensitive efects of the dimensionless variable become more serious;the dimensionless threshold pressure gradient(TPG)has a great efect on the external moving boundary but has little efect on the internal moving boundary.
Numerical simulation of two-phase flow in fractured karst reservoirs is still a challenging issue.The triple-porosity model is the major approach up to now.However,the triple-continuum assumption in this model is unacceptable for many cases.In the present work,an efficient numerical model has been developed for immiscible two-phase flowin fractured karst reservoirs based on the idea of equivalent continuum representation.First,based on the discrete fracture-vug model and homogenization theory,the effective absolute permeability tensors for each grid blocks are calculated.And then an analytical procedure to obtain a pseudo relative permeability curves for a grid block containing fractures and cavities has been successfully implemented.Next,a full-tensor simulator has been designed based on a hybrid numerical method(combining mixed finite element method and finite volume method).A simple fracture system has been used to demonstrate the validity of our method.At last,we have used the fracture and cavity statistics data fromTAHE outcrops in west China,effective permeability values and other parameters from our code,and an equivalent continuum simulator to calculate the water flooding profiles for more realistic systems.
