The partial differential equation describing the flow of fluid in a porous media was derived by combining the continuity equation (material balance), Darcy's equation of flow and the equation of state. The partial differential equation was linearized by neglecting high order gradient terms such as The resulting partial differential equation is the well known parabolic diffusivity equation. This diffusivity equation which incorporates the effect of wellbore storage and skin factor on the inner boundary condition has been solved and the solution is used to analyze the pressure drawdown. The pressure buildup testing theory was derived from the pressure drawdown theory described above.
By definition, a pressure buildup test consists of two parts; the drawdown (flow) and shut-in periods. Traditionally, parts; the drawdown (flow) and shut-in periods. Traditionally, the equations defining a buildup test were derived using the diffusivity equation utilizing the principle of superposition. Thus it was implicitly assumed that the equations governing fluid flow were linear and that constancy of inner boundary conditions prevail throughout the drawdown and shut-in periods. Although these conditions are usually met, certain periods. Although these conditions are usually met, certain conditions may exist that prevent application of superposition. For example, requirement of linear partial differential equation which is essential for the application of superposition, is not fulfilled when the turbulent condition prevails during drawdown period. During the shut-in period however, the fluid velocity period. During the shut-in period however, the fluid velocity inside the reservoir diminishes quickly and turbulent effect ceases to exist. This turbulent flow is known to occur in high permeability gas reservoirs producing at high rates. permeability gas reservoirs producing at high rates. The turbulent flow is usually accounted for by modifying the fluid flow equation for laminar flow by adding turbulent flow term.
Where the term B is the turbulence factor. If B is, equal to zero, the second term of the right hand side of Eq. 1 disappears. Pressure gradient thus becomes linearly dependent on velocity and final partial differential equation describing laminar fluid flow in porous medium will be linear as described earlier. If B is larger than zero, then the governing equation will be non-linear leading to a non-linear partial differential equation. partial differential equation. Wattenberger and Ramey solved the non-linear equation representing the turbulent flow during the flow, period. They have developed the following equation to) period. They have developed the following equation to) describe pressure behavior under turbulent flow during drawdown period for hydrocarbon at pressures above 3000 psi. psi.(2)
Similarly an equation for pressures below 2000 psi has been developed.