In this paper the authors discuss the numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model is defined by an initial-boundary nonlinear convection-diffusion system of four partial differential equations: a parabolic equation for the pressure, two convection-diffusion equations for the concentrations of brine and radionuclide and a heat conduction equation for the temperature. The pressure appears within the concentration equations and heat conduction equation, and the Darcy velocity controls the concentrations and the temperature. The pressure is solved by the conservative mixed volume element method, and the order of the accuracy is improved by the Darcy velocity. The concentration of brine and temperature are computed by the upwind mixed volume element method on a changing mesh, where the diffusion is discretized by a mixed volume element and the convection is treated by an upwind scheme. The composite method can solve the convection-dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy. The mixed volume element has the local conservation of mass and energy, and it can obtain the brine and temperature and their adjoint vector functions simultaneously. The conservation nature plays an important role in numerical simulation of underground fluid. The concentrations of radionuclide factors are solved by the method of upwind fractional step difference and the computational work is decreased by decomposing a three-dimensional problem into three successive one-dimensional problems and using the method of speedup. By the theory and technique of a priori estimates of differential equations, we derive an optimal order result in $L^2$ norm. Numerical examples are given to show the effectiveness and practicability and the composite method is testified as a powerful tool to solve the well-known actual problem.