A new type curve for well test analysis for non-newtonian fluids in petroleum reservoir is developed. The general analytical solution in Laplace variable presented by Ikoku and Ramey Jr.2 forms the mathematical basis of the proposed type-curves. The equation for the type curve was developed, following similar procedure to that of Bourdet et al10. The Bessel functions involved in the final solution of the non-newtonian case cannot be approximated by logarithm function as in the Newtonian case. Hence, the dimensionless group of the skin factor and wellbore storage coefficient used in the Bourdet et al's case was not used in this study. Instead, only the skin factor is retained. The dimensionless wellbore storage coefficient is grouped with the dimensionless time as usual.
The log-log plot of the pressure derivatives at the infinite acting radial flow lie on a straight line for each flow behavior index. This straight line intersects the Newtonian infinite acting pressure derivative line at tD/CD = 1. In addition to the unit slope line of the wellbore storage region, this point of intersection provides a fulcrum point for proper curve matching. Thus the characteristic mobility can be computed using this intersection point.
Apart from the conventional type-curve-matching method of analysis, the Tiab's direct synthesis (TDS) technique is developed for the evaluation of well test data in non-newtonian fluid flow. This is based on the long time solution as in the conventional case and the characteristic line of the type curve. The process does not involve type curve matching, but provides a direct method of evaluating the well test data from the log-log plot of the pressure and pressure derivatives. Two examples from the references 3 and 6 were used to validate the type curves and satisfactory results were obtained.
Several efforts have been made to develop general type curves to analyze pressure transient data for non-newtonian fluid flow in porous media. The problem has been how to combine the skin factor and wellbore storage coefficient into one dimensionless group as in Bourdet et al's type curves. Poollen et al 1 developed a relationship between the flow rate and pressure differential for a steady state flow while they used numerical method to solve the unsteady state flow. Their method lacked analytical methodology for well test analysis. The work of Ikoku et al2, 3 & 5 presented the analytical solution that is useful for well test analysis. Odeh and Yang 4 presented similar solution to that of Ikoku et al, but their method of analysis involved trial and error. The flow behavior index cannot be determined directly from Odeh and Yang's method. Vongvuthipornchai and Raghavan 6 developed type curves that include skin factor and wellbore storage coefficient in a similar manner to that of Bourdet type curves for Newtonian fluid. The mathematical equation for the type curve was deduced from their early time solution, but not from the general solution. The straight line of the Non-Newtonian pressure derivatives does not intercept the 0.5-line of the Newtonian pressure derivatives at tD/CD = 1. This raised the question of the validity of such type curve. Olarewaju 7 presented a series of type curves on non-newtonian fluid flow in naturally fractured and homogeneous reservoirs. The pressure derivatives on the type curve for the Newtonian fluid in double porosity case fell below the 0.5-line during radial flow regime. No trough or transition flow was observed. In the non-newtonian case, the extrapolation of the straight line of the pressure derivatives during radial flow does not intercept the 0.5-line at tD/CD = 1. We believe that this should be a unique characteristic feature as shown by Katime and Tiab 8. Eventhough Olarewaju combined wellbore storage and skin factors together as in Bourdet's solution for Newtonian case, the validity of his method is questionable.
In this study, the only dimensionless group for characterizing the type curve is the skin factor and flow behavior index. The wellbore storage coefficient is grouped along with the time as in the other existing type curves. The Tiab direct synthesis technique (TDS) which provides a method of well test analysis without type curve matching is also developed for the analysis of pressure transient data for non-newtonian fluid.
The following assumptions are made in deriving the mathematical formulae.