Chapter 5 The Basic Differential Equation for Radial Flow in a Porous Medium

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Chapter 5The Basic Differential Equation for
Radial Flow in a Porous Medium

• 5.1 Introduction
• To derive and to solve the radial fluid flow in
  porous medium

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5.2 Derivation of the Basic radial differential
equation

• Assumptions
• -- The reservoir is homogenous in all rock
• properties and isotropic with respect to
  permeability
• -- hconst. and hperfh
• -- Single phase fluid
• Why not Cartesian geometry?

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• Conservation of mass
• Mass flow rate (in) - mass flow rate (out)
• Rate of change of mass in the volume element
• Using Darcys law for a radial flow

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• Since

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5.3 Conditions of Solution

• Radial flow equation
• The most common and useful analytical solution is
  for the

?? constant terminal rate solution (Chapter
78)
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• Radial flow equation
• The three most common conditions
• (1) Transient --- Early time no boundary effect
• (Infinite acting
  reservoir)
• (2) Semi steady state --- The effect of the outer
  boundary has been felt.
• 
  where

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• (3)Steady state
• due to natural water influx or the injection of
  some fluid and

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5.4 The Linearization of Equation 5.1 for
Fluids of small and constant compressibility

From Eq.(5.4)
Note
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It is for the flow of liquids or for cpltlt1 c in
equation(5.20) is the total ,or saturation
weighted compressibility
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Chapter 6 Well Inflow Equations for Stabilized
Flow Conditions

• 6.1 Introduction
• --- Solutions of the radial diffusivity equation
  for liquid flow
• -- Semi-steady state
• -- Steady state

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6.2 Semi-Steady state solution
From Eq.(5.8), Such as

From Eq(5.10), such as
From Eq(5.20), such as
Eq(5.10)(5.20)
Integration
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6.4 Example of the Application of the Stabilized
Inflow Equations

• Steam injection

Ts Ts
Temperature
Tr Tr
rh rw rh
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Exercise 6.1 Wellbore Damage

• (1) show the skin factor maybe expressed as

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Solution
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(2) Before stimulation

• After stimulation

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6.5 Generalized Form of Inflow Equation Under
Semi-steady State Conditions

• Semi-steady state equation in terms of the avg.
  pressure Eq.(6.12)

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