Productivity of a Horizontal Well Appendices A and B

Abstract

APPENDIX A Two appendices describe the mathematical and algebraic details that led to the formulas provided in the text of this paper. Appendix A presents the general solution, and briefly describes the techniques for deriving simple, closed form expressions for single, double, and triple sums of infinite series. Appendix B describes these procedures in a little more detail, and also indicates certain methods for averaging the variable wellbore pressures in the general anisotropic cases. Uniform Flux Boundary Condition For well problems similar to the one treated in this paper, a uniform flux, or a uniform pressure is commonly imposed as a boundary condition at the well surface. Recognizing that neither is entirely correct, the question that has been debated is whether one is preferable to the other, or whether both give satisfactorily accurate solutions. Muskat showed that the uniform flux boundary condition gives values accurate to 0.5 percent. We also investigated the implication of the uniform flux assumption. We used our exact solution (Equations Al-A3) and computed the wellbore pressure, pwf at various locations y along the well length L. We did this for isotropic and anisotropic systems where the wells were located at the center, or away from the center. For the anisotropic runs the value of kx was equal to, or twice as large as, that of ky, and was ten times that of kz. Also L/b-0.5. We found that the maximum variation in pwf values was 7 psi for the worst case. This was the anisotropic case with psi for the worst case. This was the anisotropic case with kx=2ky, kx=l0kz, and the well was located away from the center. For the other cases, the variation was less than 3 psi. Since one expects pwf to be of the order of several hundred, if not thousand pounds, we concluded that the uniform flux assumption resulted pounds, we concluded that the uniform flux assumption resulted inappropriately uniform pressure. Therefore, we expect a uniform pressure boundary condition to result in approximately uniform flux. pressure boundary condition to result in approximately uniform flux. Thus we concluded that these two boundary conditions lead to approximately the same solution.