Layered petroleum reservoirs with cross-flows

Studied is a cylindrical reservoir consisting of three layers: a water-containing bottom layer, and two oil-containing top layers from whose upper layer oil is produced. For its solution, a corrected version of the finite Hankel transform for Neumann-Neumann boundary conditions was used together with numerical inversion of the Laplace transform. The effects of the water zone on the unsteady state pressure in the reservoir were evaluated at distances away from the well and at the well-bore itself. We found that the vertical pressure drop increases gradually with time and is more significant in the vicinity of the well-bore. For constant production and at any time t, smaller reservoirs experience higher pressure drops than larger ones. For the reservoir investigated, we found that for nondimensional time t_Dw<10⁴ the presence of a second fluid (water) has no effect on the pressure drop. Of all the formation and fluid properties investigated, porosity has the largest effect on pressure.Nomenclature c₁, c₂ Oil and water compressibilities, vol/vol/atm, vol/vol/psi - h Height of water zone from bottom of reservoir, cm, ft - h_D h/r_w, non-dimensional - H Height of reservoir, cm, ft - H_D H/r_w, non-dimensional - J₀, J₁ Bessel functions of the first kind, zero and first-order - K_r2, K_r1 Oil and water zones, horizontal permeabilities, darcies, md - K_z2, K_z1 Oil and water zones, vertical permeabilities, darcies, md - k₁ n=1, 2, 3... - k₂ n=1, 2, 3... - k_1,0 - k_2,0 - p(r, z, t) P(r, z, 0)–P(r, z, t), atm, psi - P(r, z, t) Pressure at any layer in the reservoir, atm, psi - P(r, z, 0) Initial pressure at any layer in the reservoir, atm, psi - P_D , non-dimensional - q Constant production rate of well, cc/sec, barrels/day - r Radius of reservoir, cm, ft - r_D r/r_w, non-dimensional - r_e Drainage radius, cm, ft - r_eD re/r_w, non-dimensional - r_w Well-bore radius, cm, ft - t Time, sec, hr - _Dw (k_r2t)/(₂₂c₂r_w²), non-dimensional - Y₀, Y₁ Bessel functions of the second kind, zero and first-order - z Distance z measured vertically upward from bottom of reservoir, cm, ft - Z_D z/r_w, non-dimensional - z₁ Height of the bottom of the producing layer, cm, ft - z_1D z₁/r_w, non-dimensional - z₂ Height of the top of the producing layer, cm, ft - z_2,D z₂/r_w, non-dimensional - _n nth positive root of equation (18) - ₁ k_z1/k_r1, non-dimensional - ₂ k_z2/k_r2, non-dimensional - _{1 11}c₁/k_r1, hydraulic diffusivity of layer I - _{2 22}c₂/k_r2, hydraulic diffusivity of layers II and III - ₂, ₁ Viscosity of oil and water, cp, cp - _{n n}/r_w, l/cm, l/ft - ₂, ₁ Porosity of oil and water-filled zones, fraction - (₁/₂) (k_z2/k_z1), non-dimensional