Adaptive Mesh Refinement (AMR) Technique For Steam Thermal Recovery In Fractured Petroleum Reservoirs

Steam injection is a widely used efficient recovery technique for fractured reservoirs containing viscous oil.The difficulty for the numerical simulations of this problem is the existence of very sharp temperature and saturation fronts around the phase change regions.Due to the rapid variations of physical quantities across the fronts,very fine grids are required to satisfy the simulation accuracy.If applying uniformly fine grids to the whole domain,very huge CPU time is needed.Therefore, it's quite reasonable to consider the use of adaptive mesh refinement(AMR) technique,which is capable of using fine grids in the area with steep gradients but coarser grids where the variations of temperature and saturations are slower,to track the moving fronts inside the calculation domain.The object of this paper is to apply the AMR technique to the numerical simulations of steam thermal recovery processes in fractured petroleum reservoirs. For homogeneous fractured reservoirs,we suggest using the same AMR grid structures for fracture and matrix,for it is not only convenient but also effective.The numerical examples indicate the AMR technique results are both accurate and fast compared with the solutions under referenced uniformly fine grids.The numerical example for SAGD process also shows that the steam chamber could easily reach the top of the reservoir,which is harmful to the thermal efficiency of steam energy for the SAGD process in fractured reservoirs.However,there are several mathematical difficulties to face with under heterogeneous circumstances.Firstly,it's found that since the matrix-fracture mass exchange is relevant to the matrix permeability and the communications between the adjacent matrix blocks are very tiny,the spatial variations of matrix permeability will cause saturation discontinuities between the neighboring matrix blocks.Directly performing refining (downscaling) operations or coarsening(upscaling) operations upon these matrix quantities may lead to difficulties tied to the upscaling or downscaling of the exchange term.In order to avoid this problem,we suggest using two separated grid systems for the fracture and matrix equations,respectively.The AMR grid structure is applied to the fracture equation,while uniform fine-grids are maintained for the matrix equation.Under the consideration that there is no direct communication between the matrix blocks under DP model,the fracture and matrix equations can be decoupled by a special decomposition approach during the Newton-Raphson procedure for solving the nonlinear resulting equations.After that,only fracture varibles under AMR grid structures are left to participate in the large spare matrix computations.Moreover,this decomposition approach is also validated under DK model,as long as we use the variables of the neighboring matrix blocks at the last Newton-Raphson iteration step.The numerical examples show that the proposed AMR technique is fast and can give good accuracy compared to the reference fine-grid results.The improvement of the computing efficiency obtained with the proposed AMR technique comes from the use of the adaptive fracture grid as well as the special decomposition approach.Second,how to calculate the effective permeability of coarse grids is a difficult problem that has attracted much attention.For the equivalent permeability under multi-level grid,the situation becomes even more complicated.As most upscaling techniques have their origins in statistical physics or require the use of periodic boundary conditions,these cells in such levels may include insufficient information to satisfy the precondition of the upscaling methods.To deal with the problem,we suggest not treating the coarse cells isolately but considering the influence of the other region of the domain.We start from the large scale averaging theory whose closure variables are able to provide the description of the deviation map of the pressure and velocity of the domain.With the help of the information,we are able to obtain the equivalent of an arbitrary region inside the domain.This technique provides an appropriate way to calculate the equivalent permeability for multi-level grids,without introducing any boundary condition upon them.The numerical examples presented in this paper indicate that the proposed model may provide relatively good approximations for the equivalent permeabilities of the coarse cells of multiple nested gridsAMR simulations are much faster than the uniformly fine-grid simulations. Therefore,it enables us to implement large amount of simulations rapidly with different parameters.It is of help to find out problems and make better decisions for pertroleum recovery.After having implemented large amount of numerical experiments upon the steam drive process for a one-dimensional fractured reservoir, we find that oil displacement in matrix is dominated by oil-water capillary pressure only under certain conditions.When conditions are changed to decrease the amount of water entering into fractured media from the boundary of the flow field,water in fracture may be vaporized to superheated steam.In these cases,the appearance of superheated steam in fracture rather than in matrix will decrease the fracture pressure and generate the pressure difference between matrix and fracture,which results in oil flowing from matrix to fracture.In summary,several problems relevant to the AMR numerical simulations of steam thermal recovery for fractured petroleum reservoirs are carefully stutied in this work.The numerical results indicate the considerable adavantage of AMR technique in computational efficiency.And we also observe some interesting behaviors for oil displacement in fractured reserviors.We expect the AMR technique to have a practical help to the fractured reservoir simulations in the future.