3-D Analytical Solution to the Diffusivity Equation for Finite Sources With Application to Horizontal Wells

Abstract

Abstract Presenting Discrete Flux Element (DFE) Method this paper provides the solution to the diffusivity equation for horizontal wells with regular and irregular geometries. DFE method is used to derive the equation of potential and its derivative with uniform potential as well as uniform flux Inner Boundary Conditions (IBC). The results showed that the equivalent pressure point moves in time and is not the same as the equivalent derivative point. Pressure derivative with respect to ln(tD) reflects the wellbore length and wellbore distance to the no-flow boundary. Introduction Potential distribution around a partially penetrating well, either horizontal or vertical, is to be obtained through solving the diffusivity equation in 3-D. Solution to the diffusivity equation for the sources that are fully penetrated, can be found directly. However in the cases where the sources are partially penetrated and/or they have irregular geometry, direct solution is impractical. This paper presents a new method as Discrete Flux Element (DFE) that permits calculating potential distribution inside the reservoir for partially penetrating wells with irregular geometry. Gravity is not neglected therefore this solution can be used to study special flow problems such as: coning, where gravity plays an important role in modeling the physics of the problem (Azar-Nejad, Tortike and Farouq Ali and Azar-Nejad and Tortike). Therefore the term potential is used throughout the paper. However, if one neglects the gravity effect one can use pressure instead. The reservoir under study is a rectilinear reservoir i.e. an infinite horizontal slab. The other type of boundary conditions can be constructed by the Method of Images. The reservoir is assumed isotropic, however, anisotropy can be introduced through well-known transformation rules. All dimensions are made dimensionless with respect to 2ht, where ht is the reservoir height. Therefore radial flow is represented by a unit slope line in a plot of potential (pressure) against ln(tD). Previous Studies Potential distribution due to a vertical partially penetrating well was first studied by Muskat under steady state conditions. The method was based on superposition of different line sources with uniform flux IBC and different lengths. To model the spherical flow at the bottom of the well Muskat introduced a point source there. The strength of each line source was found in such a way that the potential along the well surface was almost the same (within 2%). Muskat showed that the solution of uniform flux IBC at point z = 0.75lp, on the well is identical to the potential due to uniform potential IBC. Gringarten and Ramey and Raghavan, modeled a fracture with infinite conductivity, under unsteady state conditions, by dividing the fracture length into different segments each having uniform flux IBC. Gringarten et al., suggested that a uniform flux solution calculated at point x = 0.732xf is identical to the solution of an infinite conductivity IBC. This point was called equivalent pressure point. Clonts and Ramey, modeled a horizontal well with uniform flux IBC, applying Green's function and Newman's product rule. Comparing a horizontal well with a fracture Clonts et al., postulated that pressure calculated at point x = 0.732xf would be identical to an infinite conductivity (uniform pressure IBC) solution. P. 247