Linear Horizontal Flow Analysis

Subtopics:

Constant Rate Solution

Summary of Equations for Linear Fracture Flow

Derivative

Summary of Equations for Linear Fracture Flow Derivative Analysis

Linear horizontal flow is a flow regime that exists only in horizontal wells. It occurs in the transition between the vertical radial flow, and elliptical or horizontal radial flow regimes. The purpose of analyzing linear horizontal flow data is to determine the permeability in the y-direction (ky).

Constant Rate Solution

For the constant rate solution, the equations for linear channel flow apply with the channel width (w) replaced by effective wellbore length (Le) and permeability (k) replaced by permeability in the y-direction (ky) as follows:

For oil and water:

For gas:

Linear horizontal flow data forms a straight line when placed on a semi-log plot of pressure versus a square root time function (p vs. sqrt (time function)) for a buildup as shown below.

The slope (m) of this line is used to calculate the square root of permeability in the y-direction (sqrt(ky)) as follows:

For oil and water:

For gas:

Permeability in the y-direction (ky) can then be directly calculated by taking the square of the result shown above.

Summary of Equations for Linear Channel Flow (Field Units)

	Flow Period	Square Root of Permeability in the Y-Direction (md0.5)
Oil	Drawdown
	Buildup
Gas	Drawdown
	Buildup

Derivative

The signature of linear horizontal flow data on a derivative plot is a straight line with a slope of 1/2, between the vertical radial and elliptical, or horizontal radial flow regions. The position of this line is used to calculate the square root of permeability in the y-direction (sqrt(ky)).

Note:

The square root of permeability in the y-direction increases as the position of this line moves to the right.

Starting with the constant rate solution (see above) the derivative with respect to the logarithm of square-root time is:

For oil and water:

For gas:

Taking the logarithm of both sides:

For oil and water:

For gas:

This result is linear with time and, as a result, the derivative of linear horizontal flow data falls on a straight line with a slope equal to 1/2 on a log Dp versus log Dt plot as shown below.

Using any derivative point on this line, the square root of permeability in the y-direction (sqrt(ky)) can be determined as follows:

For oil and water:

 

For gas:

Summary of Equations for Linear Channel Flow Derivative Analysis (Field Units)

	Flow Period	Square Root of Permeability in the Y-Direction (md0.5)
Oil	Drawdown
	Buildup
Gas	Drawdown
	Buildup