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Linear Flow Models

Introduction to Linear Flow

The Predico AFA linear flow model follows the work of T. A. Blasingame [2009]. According to Blasingame, Linear Flow Analysis is NOT technically a traditional RTA.

This model forms the basis for the Stimulated Rock Volume (SRV) model in the AFA autoRTA modules (both oil and gas)

The physical system given by T. A. Blasingame is shown below:


image-20230508-024325.png
Linear Flow System (T.

For the no-flow Boundary Condition , the solution given in Laplace space is:

image-20230508-023836.png

For the constant Boundary Condition , condition, the solution given in Laplace space is:

image-20230508-023946.png

where:

xD = Length of reservoir

s = Laplace parameter, or Laplace space equivalent of time.

and:

The log-log plot for constant rate solution in an finite linear reservoir, for “No Flow“ outer boundary (observation point = xD = 0)

image-20241031-052630.png

The log-log plot for constant well rate solution in a finite linear reservoir, with constant pressure outer boundary (observation point = xD = 0)

image-20241031-052954.png

Types of Linear Flow

Linear Channel Flow

Linear channel flow occurs when there is effectively 1 direction flow in an apparent channel deposit towards a vertical well. For examole, refer to Linear Channel Flow example for a demonstration).

When using the linear flow equations presented in Solutions for the Diffusivity Equation under Different Flow Regimes , the variable “w” term represents channel width.

image-20241111-030630.png


Linear Fracture Flow: Vertical Well

In this example using the work of Wattenbarger et al [1998], linear flow occurs as fluid moves in 1D towards a vertical well with a large stimulation.

When using the linear flow equations presented in Solutions for the Diffusivity Equation under Different Flow Regimes , the “w” term 2 * xf, or xf = w/2.

image-20241111-010339.png

Using the more detailed transient models (wellbore storage through to pseudo-steady state), linear flow is demonstrated by Clarkson [2021] below for both infinite and finite conductivity fractures using the Bourdet (Well Testing) derivative

image-20241111-053413.png

Horizontal Well

Although other flow regimes can occur with a horizontal well, the linear flow regime can be observed if the majority of flow is perpendicular the horizontal well (prior to PSS):


image-20250123-054408.png


MFHW Linear Flow (Independent Fracture)

In this example, the fracture half length calculated represents the total calculated from all the independent fractures (refer to Multi-Fractured Horizontal Well). That is:

Where nf = the total number of fractures contributing to production.

A schematic by Clarskon [2021], is provided below:

image-20241111-052618.png

MFHW Compound Linear Flow (Independent Fracture)

According to Clarkson [2021], this scenario represents the effective producing WELL half-length (which may be representative of the total contributing fractures to well production).

A schematic by Clarkson is provided below

image-20241111-052705.png


References:

  • T. A. Blasingame. “Petroleum Engineering 620 - Fluid Flow in Petroleum Reservoirs: Reservoir Flow Solutions Lecture 3 - Linear Flow Solutions: Infinite and Finite-Acting Reservoirs Cases”, 2009. Texas A&M University Course Notes.

  • T. A. Blasingame “Analysis of Well Performance v20230723“. Texas A&M,

  • Wattenbarger et al, “Production Analysis in Linear Flow into Fractured Tight Gas Wells“, SPE 39931, Rocky Mountain Regional/Low Permeability Reservoirs Symposium & Exhibition, Denver Colorado, 5-8

  • Clarkson, C.R., Unconventional Reservoir Rate-Transient Analysis, 2021

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