In reality fluid flow occurs into a wellbore with a finite radius. To simplify the calculation, Horner assumed the wellbore could be replaced by a line source and he approximated the resultant solution to the Radial Cylindrical Flow equation by a logrithmic function. In tight reservoirs these approximations are not valid and considerable error will occur. The line source solution and the finite radius wellbore solution are markedly different before tD/rD=25. Hence, superposition of solutions when tD/rD < 25 will give erroneous results. The error is dependent On reservoir parameters and is not significantly reduced by extending the flow time. Synthetic pressure buildup data was generated using "PDCI" approximation to the finite radius wellbore solution. The error of the Horner Plot was determined from the slope of the best fit regression line.
In conventional OST analysis the Horner Plot is normally used to obtain reservoir transmissibility. In tight reservoirs, the requirements for a valid Horner Plot are not attained due to limited time in the hole. The error is dependent upon the reservoir parameters and the dimensionless time of the shut-in. In reality, radial cylindrical flow occurs into a wellbore with a finite radius. Horner assumed the well could be approximated by a line sink and simplified the resultant solution using a logarithmic approximation. In tight reservoirs, these modifications result in considerable error.
To make well test analysis amenable to hand calculations, Horner simplified the solution to the radial cylindrical flow equation in two ways. First, he assumed the wellbore could be approximated by a line sink. The Exponential Integral solution or Line Source solution which results, is far simpler than the unwieldy integral Bessel functions arising from the finite radius wellbore assumption. Second, Horner note that for values of tD/rD > 25 the Exponential Integral could be approximated by a logrithmic function within 25t.1 Consequently, it has been generally assumed that the Horner Plot is valid for tD/rD2 < 25.2. In tight gas reservoir tests, tD/rD = 25 may not be attained or a substantial portion of the build-up occurs before tD/rD = 25. The Horner Plot results from superposition of a constant flow rate +q for time T + δT against a flowrate -g for shut-in time δT; hence the familiar plot of pressure versus reveals that the finite wellbore solution, log10 [ΔT + ΔT)/ ΔT]. Examination of Fig. 1 Reveals that the finite wellbore situation, the Exponential Integral and the log approximation to the Exponential Integral are markedly different in their behavoir before tD = radius. Horner assumed the well could be 25. Superposition of the logrithmic approximation when a substantial portion of the build-up or the entire build-up is less than tD = 25 will lead to erroneous results. For mathematical treatment of the preceeding refer to the Appendix.
The "PDCI" approximations to the finite radius well bore solution proposed by Edwardson, were used to generate synthetic Pressure Data3. A dimensionless time factor "TDF" was defined as
Equation (Available in full paper)