Determining Relative Permeability from Unsteady-State Radial Fluid Displacements

Abstract

The practical interpretation formulae which can be used for determination of the relative permeability data from the production and/or injection performance data are derived based on the formulation of the near-well-bore immiscible fluid displacement process. The formulations for convenient calculation of the relative permeability data similar to the JBN method are provided and validated using the laboratory data obtained by the radial immiscible fluid displacement tests in horizontal porous disks. The applications demonstrate the capabilities of the present interpretation method. Introduction By-and-large, the present relative permeability determination methods have been developed for applications in the linear flow systems such as in the laboratory core plugs. However, determination of the relative permeability data from the injection and production well data requires a radial flow interpretation method. For this purpose, the convenient interpretation formulae elaborated and demonstrated by the authors (Tóth et al.[1–6]) previously for determination of relative permeability from unsteady-state two-phase linear fluid displacements in laboratory core tests under constant rate or pressure conditions are transformed for displacement in the radial distance around a well. Special formulae for laboratory simulation of such radial flow in circular disks of relatively small thickness are also derived for interpretation of laboratory data. The interpretation method developed in this paper is somewhat similar to the JBN method for displacement in core plugs but applicable for radial flow and also improved in accuracy by means of special straight-line plotting schemes. In contrast to the JBN method, however, the present method yields the relative permeability functions instead of calculating the point values of the relative permeability data. The new approach taken in this study is based on the theory of the displacement processes according to the Buckley-Leverett model and Darcy's law considering the variation in the cross-sectional area with the radial distance (Marle[7], Latil[8]) Parallel to the above-mentioned theoretical development, also the laboratory investigations of the radial flow systems have been carried out to derive and verify the relevant displacement relationships and to determine the relative permeability functions from the production data. The laboratory flow tests have been carried out using horizontal disks of typical sandstone samples in 4" outer diameter sizes containing 0.5" diameter holes in the center. The upper and lower flat surfaces of these disks have been coated with epoxy. The disk-shaped cores have been first saturated with oil. Then, the oil has been displaced by injecting or pumping water through the hole inside the disks to represent the flow condition around a well. The production data obtained under constant injection rate and constant pressure difference conditions have been measured accurately and used with the present interpretation formulae. The relative permeability curves have been constructed using such data and compared with each other. Fluid Displacement in Radial Systems Radial rock samples were prepared for investigations where the fluid is flowing from the mantle of the disk-shaped samples to a small diameter well drilled into the core center. The core is considered homogeneous. Its porosity is F and permeability is k. The height of the core sample is h, the radius of the axial well is rw, and the external radius is re. The core is saturated with a fluid denoted by a subscript k. Then, this pore fluid is displaced by another fluid denoted by a subscript d. Both fluids are assumed incompressible and immiscible with each other. The volumetric rate of the injected fluid is qi. The effect of the capillary forces is supposed to be negligible (Pc=0) during the displacement processes. The pressure at the mantle (fluid inlet face) is pe, the pressure inside the small diameter well (fluid outlet face) is pw. Thus, the pressure difference between the outer and inner faces of the disk is ?p = pe - pw (see Fig. 1). In this section, the theoretical foundations of the relative permeability determination from radial flow immiscible displacements are presented by extending the application of the Tóth et al.[1–6] approach.

Determining Relative Permeability From Unsteady-State Radial Fluid Displacements

“…1). In this section, the theoretical foundations of the relative permeability determination from radial flow immiscible displacements are presented by extending the application of the Tóth et al [1][2][3][4][5][6] approach.…”

“…These equations can be called as the general flow equations. After the breakthrough (t > t a ), the following relationship has been proven to be valid for the different volumes (Tóth et al [1][2][3][4][5][6] ) and this relationship was found also valid for the radial system:…”

“…The average saturation of the displacing fluid in the radial core sample after t time is expressed as (Tóth et al [1][2][3][4][5][6] …”

“…As a result, its second derivative fulfils the condition that d 2 V i (t)/dt 2 >0. The following expressions can be written after the breakthrough time (Tóth et al [1][2][3][4][5][6] …”

“…For this purpose, the convenient interpretation formulae elaborated and demonstrated by the authors (Tóth et al [1][2][3][4][5][6] previously for determination of relative permeability from unsteady-state two-phase linear fluid displacements in laboratory core tests under constant rate or pressure conditions are transformed for displacement in the radial distance around a well. Special formulae for laboratory simulation of such radial flow in circular disks of relatively small thickness are also derived for interpretation of laboratory data.…”