Diffusivity Equation for Spherical Flow

DIFFUSIVITY EQUATION FOR SPHERICAL FLOW Spherical Flow Spherical flow regime occurs when the predominant flow pattern in the reservoir is toward a point. Spherical flow occurs when a vertical well is partially penetrated or during RFT/MDT/WFT tests. This flow regime is recognized as a -1/2 slope in the pressure derivative on the log-log diagnostic plot. Its presence enables determination of the spherical permeability. When spherical flow is followed by radial flow, both horizontal and vertical permeability can be quantified.

Diffusivity Equation In 1966, Charas studied transient spherical flow of fluids through porous media that can analyze and calculate the data of bore-hole test. According to the principle of mass conservation, the continuity equation of slightly compressible fluids for spherical flow in porous media may be written as:

(r².υr) = øc.

1/ r².

→ (1)

Where r is the spherical distance; υr is superficial velocity in the spherical direction; ø is porosity; c is system compressibility; t is time; p is pressure. The equation of flow through porous media for fluids is given by:

υrn =

→ (2)

k/µe .

where k is permeability; µe is effective viscosity; n is flow behavior index. When n = 1, Eq. (1) reduces to Darcy's law for Newtonian fluids. If n< 1, the fluid is pseudo plastic. If n > 1, the fluid is dilatant. Substituting Eq. (2) into Eq. (1), differentially calculating and neglecting pressure gradient squared term yield nonlinear partial differential equation for transient spherical flow of power-law fluids through porous media.

1/r2n .

(r2n .

) = n.c.ø (µe/k)1/n. (

When n = 1, Eq. (3) reduces to spherical diffusivity equation.

1/ r².

(r2.

) = cø. µe/k.

(n-1)/n

.

→ (3)