THUS

DIFFUSIVITY EQUATION

Developing flow equations for Radial Flow system– Continuity equation– Equation of fluid motion (Darcy’s Equation)– Substitute Darcy’s Equation into continuity

equation

VEH model required DIFFUSIVITY EQUATION FOR RADIAL FLOW

Van Everdingen & Hurst unsteady state model

Van Everdingen & Hurst unsteady state model

The diffusivity equation in radial form expresses the relation between pressure and radius and time for a radial system such as drainage from an aquifer, where the driving potential of the system is the water expandability and the rock compressibility:

2p 1 p c p k ------ + --- ------ = ------- ----- where = ---------- r2 r r k t c

An exact analytical solution of the diffusivity equation for specified boundary and initial conditions define this pressure time profile and therefore will allow the calculation of the rate of water influx into the reservoir, provided the proper data are available.

The VEH analysis was for two cases (a)The pressure case: the pressure at the inside boundary is

known and the outside boundary is closed or the reservoir is infinite

(b)The rate case: the rate is known at the inside boundary. At the outside boundary there is no flow or the pressure is constant or the reservoir is infinite

To enable VEH analysis to be applicable for different reservoirs, they produced a more general equation by generating dimensionless functions

Dimensionless time tD, in place of real time, tDimensionless radius, rD, which is re/ro where re is the radius of aquifer and ro is the radius of oil reservoir

The dimensionless form of the diffusivity equation is, 1 pD pD ---- ----- rD ------ = ----- rD rD rD tD

kt re 2kh∆pWhere: tD= ---------- , rD = ------ , pD = -------------- μøcro

2 ro qμ

tD = time, dimensionlesst = time, secondsk = permeability, darcyμ = viscosity, centipoiseØ = porosity, fractionc = effective aquifer compressibility vol/vol/atmro = reservoir radius, centimeters

Converting the above equation to more commonly used units of t= days; k = millidarcies; μ = centipoises, ø = fraction; c = vol/vol/psi and r = feet then kt

tD= 6.323 x10-3 ---------- μøcro

2

THIS EQUATION IS USED IN WELL TESTING

DAKE reproduced VEH data solution in graphical form as below for infinite and finite reservoir

DAKE reproduced VEH data solution in graphical form as below for infinite and finite reservoir

Diffusivity equation

Dimensionless form of Diffusivity equation

Several slightly different solutions of dimensional diffusivity equation are presented in the petroleum literature

This solution, which describes a classical flow problem, is referred to by many names: • Lord Kelvin’s point source• Theis solution and• Continuous line source solution

THE CONTINUOUS LINE SOURCE SOLUTION

Fig. Line source solution versus finite wellbore radius

The number of terms between parentheses in above equation depends upon the magnitude of X and the desired accuracy

Following Table

Can be written as an approximation

The line source solution is itself an approximation for the pressure behavior due to a finite radius wellbore in an infinite reservoir

Combining all these equations and solving explicitly for real pressure any point in the reservoir located at a distance r (ft) from a flowing well at a constant rate q (STB/D) for a period of time t (hours):

Applying the log approximation to the Ei function, the combined equation becomes

The above two equations are typically used to find the pressure drop P = Pi – P( r, t) that will have occurred at any radius r from a well producing at a rate q after the well has flowed for a time period t.

At the well, the above equation becomes

The line corresponds to the time the pressure transient behaves as if the reservoir were infinite. The slope is used to calculate the reservoir permeability.

If we plot P = Pi – Pwf instead of Pwf, as shown in Figure, then the slope m is positive and becomes