Analysis of Buildup and Drawdown Tests With Variable Wellbore Storage and Skin

Abstract

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 (∂2p/∂r2). The resulting partial differential equation is the well known parabolic diffusivity equation. This diffusivity equation which incorporates the effect of welibore storage and skin factor on the inner boundary condition has been solved1 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, 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 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 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. The turbulent flow is usually accounted for by modifying the fluid flow equation for laminar flow by adding turbulent flow term.

Non-Darcy Skin Effect with a New Boundary Condition

“…Wattenbarger and Ramey (1968) (8) The authors approximated the effect of turbulence by an additional skin drop (Dq g ) in the equation. Soliman et al (1991) had claimed that this approximation is valid as long as turbulent flow regime does not prevail through a large portion of the reservoir. That is, approximating the turbulence effect for tests with a radius of investigation of more than 50ft will yield an acceptable accuracy from approximating the turbulence effect with an additional rate dependent skin term (D).…”

“…This study uses the novel framework suggested by Soliman et al (1991) to model the post-Darcy turbulence effect with different skin values during drawdown and buildup. The authors had used unit step functions to model the problem of changing skin as the well is shut-in.…”

Derivation and application of mathematical model for well test analysis with variable skin factor in hydrocarbon reservoirs

“…Some researchers believed that skin factor is related to turbulence effect of fluids due to the fact that high speed non-Darcy flow will produce additional pressure drop at the bottom of the well. [13][14][15][16][17] Moreover, the authors also proposed their own mathematical model of skin factor that changes with production rate. However, variable skin factor caused by sand migration lacked systematic study.…”