Stability And Error Estimates Of Local Discontinuous Galerkin Method For Simulating Wormhole Propagation

In the actual production and development of oil and gas,people are committed to researching technology for enhancing oil and gas production.Recently,the matrix acidization technique has been attracted considerable attention.This technology is achieved by injecting acid to dissolve the material around the well.The transport and reaction of the acid lead to increasing the permeability of the damaged area,and a well-connected flow channel frame can be established between the reservoir and the well.These channels are usually called wormholes.Wormholes penetrate deep into the formation and help push the oil and gas in the reservoir to the surface.Therefore,the productivity will be enhanced successfully.It is well known that for a given amount of acid,the relative increase in permeability is a strong function of the injection conditions.And only at a proper flow rate,wormholes(long conductive channels)are formed.Therefore,to form a wormhole with the best density has become a key task.In this paper,we will focus on studying the local discontinuous Galerkin(LDG)method for simulating traditional Darcy wormhole propagation and wormhole propagation with Darcy-Forchheimer model.Firstly,we apply two fully-discrete methods to the wormhole propagation.We give a theoretical analysis of the stability and error estimate for this method.Traditional LDG methods usually use the diffusion terms to control the convection terms to obtain the stability of some linear equations.However,the variables in the wormhole model are all coupled together,and the entire system is highly nonlinear.Therefore,it is very difficult to follow the previous idea to obtain the stability for this model.To overcome this difficulty,we have introduced a new auxiliary variable,which includes both convection and diffusion terms.In addition,we also adopt a special way for the time evolution of porosity,so as to obtain a physically relevant numerical approximation and controllable porosity growth rate.Based on these techniques,the problems caused by different time steps in the first-order fully-discrete method can be dealt with and the stability of the method can be obtained.For the second-order time integration,we obtain the optimal error estimate of pressure,velocity,porosity and concentration under weak temporal-spatial conditions.Secondly,we conduct a semi-discrete LDG method for the Darcy-Forchheimer model.We introduce two auxiliary variables and analyze the error estimate by establishing four energy inequalities.Fortunately,optimal error estimates for pressure,velocity,concentration and porosity under different norms are established.Finally,the corresponding numerical experiments are provided to verify the accuracy and efficiency of the LDG methods.The results are consistent with the theoretical analysis.