抄録
The purpose of this study is to develop practical and versatile analytical solutions of well testing for the reservoir with radial property variation. Here, “practical” means that the calculation cost is not so expensive and “versatile” means that the algorisms are numerically stable”. Two analytical solutions for this problem are presented and improved. The first is the perturbative solution which was introduced by Oliver. Although this solution could treat the continuous function of permeability variation, the first order perturbation remains some errors when the center of the perturbation is set to constant. Thus, we develop a new function to estimate an adequate average permeability which should be applied in order to obtain the good approximation of pressure calculation. The result is excellent; however, this solution is a little tedious to apply for the porosity variation problems. In addition, it is difficult to expand this solution for the problem that the reservoir property changes with time. The second solution is multi-composite solution which is appeared in several papers. “Multi-composite model” treats of permeability and porosity variation as stair-step functions, however, the excellent approximation will be obtained on the condition that the “steps” are enough fine. The fine steps require high calculation cost. This paper presents several mathematical techniques in order to save this cost as well as to obtain numerical stability. Moreover, “Stepwise approach” enables us to treat the time dependent reservoir parameter, such as permeability change with time. Because “Stepwise approach” does not require any superposition of rate, calculation speed is very fast even in the case that production rate is frequently changed.
