PETE 4052
Lecture 6
Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
Equations for Single-Phase Porous Media Flow
Outline
0.
1.
2.
3.
4.
5.
6.
7.
Synopsis
The Conservation Equation
Use of Darcy’s Law in the Conservation Equation
The Case of Small and Constant Compressibility
The Linear Diffusivity Equation
Dimensionless Variables
Other Coordinate Systems and Notation
Discussion
Related reading
Craft et al., Sections 7.5-7.6, p. 231-234.
Dake: Sections 5.1-5.4, p. 131-139.
Earlougher: Sections 2.1-2.3, p. 4-6.
Horne, p. 35.
Matthews and Russel: Sections 2.1-2.3, p. 4-7.
6.0 Synopsis
The flow of a single, compressible fluid through porous, permeable rock can be described using a
partial differential equation known as the diffusivity equation. Modified forms of the diffusivity
equation can be used to describe gas flow. A similar equation can be derived for multiphase flow
as well, and that equation is the basis for reservoir simulation. Clearly, the diffusivity equation is
at the very heart of reservoir engineering and an intuitive understanding of this equation is
essential to all who would do reservoir engineering.
Key Concepts
 Mass conservation
 Use of constitutive equations for velocity and density
 The form of the linearized diffusivity equation
 Assumptions in the linearized diffusivity equation
 Dimensionless variables
 Definitions of steady, pseudosteady, and transient flow
 Radius of investigation
 Diffusivity
6.1 The Conservation Equation
Many physical systems – ranging from solar collectors to river deltas to flow in reservoirs – can
be analyzed using the principle of conservation. This principle is closely related to the idea of a
control volume in thermodynamics; it is based on the idea that the amount of “stuff” (energy,
mass, whatever) entering, leaving, created, and destroyed in a given volume must be balanced.
We will derive the conservation equation for a radial flow geometry, because this geometry is
especially useful for well testing and inflow analysis. We could do it for any geometry we chose.
Page 1 of 9
PETE 4052
Lecture 6
Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
Consider a cylindrical shell of radius r and thickness r (Figure 6.1).
Figure 6.1 – Radial control volume. The area of the outer face is



2 r  r h . The volume of the shell
is exactly  r  r  r h . If r is small, this is approximately equal to 2rrh . The pore volume
can be computed simply by multiplying by the porosity.
2
2
The mass balance for this shell is
(Flow in at r) - (Flow out at r  r)  Accumulation between r and r  r
Let us consider each of the terms in Eqn. (6.1) in turn:
Flow in at r: Mass Inflow  Area  Superficia l Vel.  Density  Time  2rh u (r ) t .......... (6.2a)
2 r  r hu (r  r ) t .............................................. (6.2b)
Flow out at r+r:
2rrh  t  t  2rrh  t ............................................ (6.2c)
Accumulation:
Combining Eqns. (6.1) and (6.2),
2rh u (r )  o t  2 r  r hu (r  r )  o t  2rrh  t  t  2rrh  t ....... (6.3a)
Dividing by 2rht ,
ru(r )  o (r )  (r  r )u (r  r )  o (r  r ) rt  t  rt
.................. (6.3b)

r
t
If we take the limit as r and t go to zero, then we get,


...................................................... (6.4)
 ru   r
r
t
Equation (6.4) is the conservation equation in radial coordinates. It states that the sum of the
partial differential derivatives in r and t is zero. This is also known as a divergence equation; all
conservation equations (for any quantity, in any coordinate system) can be expressed in a form
very similar to Equation 6.4.
This equation must be manipulated further to be useful: it includes dependent variables , , and
u, whereas we really want an equation in p only. We will use constitutive equations for these
quantities to get the desired equation.
Page 2 of 9
PETE 4052
Lecture 6
Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
6.2 Use of Darcy’s Law in the Conservation Equation
We will begin by using the differential form of Darcy’s Law in the conservation equation. In
differential form, Darcy’s Law is
k p
............................................................ (6.5)
u
 r
Substituting Eqn. (6.5a) into the conservation equation [Eqn. (6.4)],

   k p 
  r
 r  
r    r 
t
.............................................. (6.6)
1    k p  
 
r 
r r    r 
t
This gets us closer to the desired form, but the density and porosity still must be handled.
6.3 The Case of Small and Constant Compressibility
We will assume that the density and porosity are exponential functions of pressure:
 ( p)   R exp c o ( p  p R  ................................................(6.7a
 ( p)   R exp c f ( p  p R .............................................. (6.7b)

where

co 
1 
.......................................................... (6.8a)
 p
cf 
1 
.......................................................... (6.8b)
 p
and
Substituting Eqns. (6.7) into Eqn. (6.6),
 k p  
1  
   R expc o ( p  p R  R exp c f ( p  p R  ..... (6.9a)
r R expc o ( p  p R 
r r 
  r  t


Expanding the left-hand side of Eqn (6.9a),
 k p 
 

r R expc o ( p  p R 
r 
  r 
 k p  
  k p 
  R expc o ( p  p R    R expc o ( p  p R   r

  r
r   r 
  r  r
........... (6.9b)
p  k p 
  k p 
   R expc o ( p  p R   r

  R c o expc o ( p  p R   r
r   r 
r   r 
  k  p  2   k p 

  c o  r     r
r   r 
    r 
Similarly, expanding the right-hand side of Eqn. (6.9a),
Page 3 of 9
PETE 4052
Lecture 6
Well Testing
Equations for Single-Phase Porous Media Flow

Spring 2002
February 22


 R expc o ( p  p R  R exp c f ( p  p R  
t
 R  R c o  c f  expc o ( p  p R  expc f ( p  p R 
 (c 0  c f )
p
........................... (6.9c)
t
p
t
Recombining Eqns. (6.9b) and (6.9c),
2
   k  p 
p
  k p 


  (c 0  c f )
c
r
    r
 o

r     r 
r   r 
t
 k  p  1   k p 
p
 r
   (c 0  c f )
c o    
r r   r 
t
   r 
2
......................... (6.10)
 k  p  1   k p 
p
 r
  c t
c o    
r r   r 
t
   r 
where ct  (c 0  c f ) . For systems with connate water and/or immobile gas, the definition of total
2
compressibility can be extended to be
c t  c f  S o c o  S w c w  S g c g ............................................ (6.11)
Notice that the only dependent variable is now pressure; however, the porosity is still a function
of pressure.
6.4 The Linearized Diffusivity Equation
Although Equation (6.10) is in pressure, it is nonlinear. It is very difficult to solve nonlinear
partial differential equations, and we therefore seek a simplified, linear form to work with.
The nonlinearity comes from two different sources.
 k  p 
The gradient-squared term: c o  r  
   r 
p
The porosity term: c t
t
In many systems, the porosity varies slowly and it can be replaced by it value at the average
pressure. This is usually adequate.
2
The gradient-squared term is often small because (1) the gradient is small and (2) the oil
compressibility was assumed to be small. If these assumptions are made (see discussion), then the
linearized diffusivity equation can be written as
p
1   k p 
 r
  ct
.............................................. (6.12a)
r r   r 
t
If k/ is constant,
1   p  c t p
............................................... (6.12b)
r  
r r  r 
k t
This equation is very important! It will form the basis of our study of inflow and pressure
transient analysis (or well testing).
Page 4 of 9
PETE 4052
Lecture 6
Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
6.5 Dimensionless Variables
We used the concept of a dimensionless variable when we discussed the skin factor. We will
extend that discussion now to better understand the linearized diffusivity equation. It seems
sensible to make radius dimensionless on the wellbore radius,
r
rD 
rw
r  rD rw
................................................ (6.13)
 drD 
1 


r
dr rD rw rD
With this substitution, the diffusivity equation is
p
1 1  
k 1 p 
 rw rD
  c t


rw rD rw rD 
 rw rD 
t
.................................... (6.14)
1   k p 
2 p
 rD
  c t rw
rD rD   rD 
t
This definition needs no unit conversions; it works in consistent and field units. If we assume k
and  are constant,
p  c t rw2 p
1  
 rD

........................................... (6.15)
rD rD  rD 
k
t
As for our case with skin, we will define the dimensionless pressure as
2kh
pD 
( p i  p) ................................................. (6.16a)
qB
in consistent units or
0.00708 kh
pD 
( p i  p) ............................................. (6.16b)
qB
in field units. Now the differential equation is
2
1   p D  c t rw p D
 rD
 
......................................... (6.17)
rD rD  rD 
k
t
(We can choose the definition of dimensionless pressure pretty freely because p appears in all
terms in the equation; the reasons for this choice should be clear from the form of Darcy’s Law
and we will discuss it more later in the course). Because all other terms are dimensionless, the
definition of dimensionless time can be deduced from Eqn. (6.17):
kt
....................................................... (6.18a)
tD 
c t rw2
in consistent units or
0.0002637 kt
................................................... (6.18b)
tD 
ct rw2
in field units, t is given in hours – if it were in days, the conversion factor is 24 times larger.
Our final, dimensionless form of the equation is
1   p D  p D
 rD

............................................... (6.19)
rD rD  rD  t D
This form of the equation is simpler than earlier forms, because there are only variables and no
parameters in it. The most important thing about dimensionless variables is the way they scale the
Page 5 of 9
PETE 4052
Lecture 6
Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
solution. Regardless of the values of parameters such as k, , , ct, q, and so on….the
dimensionless pressure at the same (rD,tD) is the same for all infinite-acting radial flow
problems.
More generally, if the appropriate scaling is used in bounded or non-radial problems, one can
define dimensionless pressures that have this same property (for example, for a fractured well or a
well in the center of a square). We need not compute solutions for all different combinations of
parameters – the dimensionless solution is the scaled solution that can be used for all of these
cases.
6.6 Other Coordinate Systems and Notation
In linear coordinates, the diffusivity equation is
 2 p c t p

...................................................... (6.20)
k t
x 2
For any coordinate system, one can write the diffusivity equation in terms of the divergence,
which some of you may be familiar with
c t p
...................................................... (6.21)
2 p 
k t
Discussion
Assumptions
The steps and assumptions used to derive the linearized diffusivity equation are summarized in
Table 6.1, below:
Table 6.1 Derivation of the Linearized Diffusivity Equation
Eqn. Step
Assumptions
6.4 Conservation Equation
h is constant; radial geometry
6.6 Constitutive Equation for velocity
Darcy's Law holds
Porosity and fluid density are exponential
Constitutive Equation for density; nonlinear
6.11
functions of pressure; the pore and fluid
diffusivity equation
compressibility are constant
6.12a Diffusivity Eqn with variable k, 
The product fluid compressibility times the
square of the gradient of pressure is small
6.12b Linearized diffusivity equation
k,  and  are constant
The assumptions are very important to be familiar with. Study this table! In particular, consider
the following:
 The compressibility of gas may not be small or constant. Thus, this form of the diffusivity
equation may not be accurate. We will consider an alternative form, in real gas potential (or
pseudopressure) later in the course.
 The assumption that the product of the compressibility times the square of the gradient term
is small may not be valid (1) for gas or (2) near the well, where the gradients are high. These
problems can also be addressed using real gas potential.
 Unconsolidated rocks may have very high pore compressibilities, cf. This can make the
p
equation nonlinear, because both  and
depend on pressure. This problem is not solved
t
using pseudopressure, but may be reduced (but NOT eliminated) using pseudotime.
 The mobility k/ may not be constant, especially for gas or highly compressible rocks. This
can introduce an additional nonlinearity. Pseudopressure corrects for the viscosity variation,
and can be generalized to correct for permeability varying with pressure.
Page 6 of 9
PETE 4052
Lecture 6
Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
The Steady-State Equation in Dimensionless Form
We previously wrote Darcy’s Law as
2kh
q
( p(r )  p(rw ))
r
B ln( )
rw
We can rearrange this to get
r
2kh
( pi  p(r ))  ( pi  p(rw ))
ln( ) 
rw
qB
and introduce the definitions of rD (Eqn. 6.13) and pD (Eqn. 6.16) to get
ln( rD )  p D (rD )  p D (1)
If we use a special definition of pD that is sensible for steady-state systems (where pw is constant),
2kh
pD 
( p  pw )
qB
Then the dimensionless steady-state equation is simply
p D (rD )  ln( rD ) ...................................................... (6.22)
The dimensionless form is simpler for steady state systems, as well. As we will see later, we can
get this equation by solving the linearized diffusivity equation.
Diffusivity
k
appears in the diffusivity equation in all coordinate systems. It is known as the
c t
diffusivity and is usually represented with the symbol or “eta”:
k
.......................................................... (6.23a)

c t
or, in field units,
k  ft 2 
  0.0002637
............................................ (6.23b)
c t  hr 
The diffusivity is important because it controls the speed with which pressure information travels
through the reservoir. In general, higher values of  mean more rapid transmission of
information; you can see this because the diffusivity is multiplied by t (and divided by the square
of the wellbore radius) to get the dimensionless time. Thus, at same dimensional time a reservoir
with higher diffusivity will have a greater dimensionless time – things happen faster. Increasing
permeability has the same effect as decreasing any of the other three parameter (, , ct).
Consider why
 Higher mobility, k/ causes more rapid pressure transmission
 Higher porosity and compressibility cause slower transmission (ct is also known as the
storativity).

Some Basic Cases
Several reservoir systems of interest can be classified based on the value of the right-hand side of
Eqn. 6.19:
p D

 0 : Steady state, the pressure is not changing with time
t D
The group
Page 7 of 9
PETE 4052
Lecture 6


Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
p D
 constant : Pseudosteady state, usually occurs when rate is changing slowly and
t D
compressibility is nearly constant.
p D
 f (rD , t D ) : Transient flow.
t D

Radius of Investigation
In transient flow, the radius of investigation is a useful concept (Horne, p. 35). As we will see
later, that the solution of the partial differential equation predicts that there is an infinitesimally
small pressure change everywhere, instantly. However, from a physical and practical point of
view, the pressure changes propagate more slowly. One definition of the radius of investigation is
that, at any time, it is the closest boundary that would have had a detectable effect at the well. For
an infinite-acting well, this gives
kt
rinv  0.03
ct ................................................... (6.24a)
 0.03 t
With our previous definition of tD, Eqn. (6.18b),
rinv  0.03
ct rw2 t D
k
ct 0.0002637 k ......................................... (6.24b)
rDinv  1.8 t D
Sometimes an alternative definition of dimensionless time is used, based on area rather than
wellbore radius
0.0002637 kt
.................................................. (6.25a)
t DA 
ct A
For a well in the center of a circle,
0.0002637 kt
kt
................................. (6.25b)
t DA 
 8.938 10 5
2
ctre
ct re2
As we will see later, boundaries affect this system when t DA  0.1 so
kt
............................................. (6.25c)
0.1  8.938 10 5
ct rinv2
Solving for rinv,
rinv 
8.938  10 5 kt
0.10
ct rinv2
............................................. (6.26)
kt
 0.02897
ct
Eqn. (6.26) is about the same as Eqn. (6.24a), which was obtained by a much more complicated
analysis using derivatives of Green’s functions.
A couple of things are worth noting:
 We see (Eqn 6.24b) that diffusivity controls the radius of investigation.
 Radius of investigation increases with the square-root of time; this behavior is characteristic
of solutions to the diffusivity equation.
Page 8 of 9
PETE 4052
Lecture 6

Well Testing
Equations for Single-Phase Porous Media Flow
Spring 2002
February 22
The time to investigate near-well regions is typically too small to be well-represented in data
(see below), whereas in low-permeability reservoirs it may take a long time to detect
boundaries (see below).
Example 1: Radius investigated in one minute in a good-quality reservoir.
Assuming k = 100 md,  = 0.25, ct = 1010-6/psi,  = 0.8 cp, and t = 1/60 hr, rinv is
100 (1 / 60)
rinv  0.03
 27 ft . Thus, even after 1 minute, the part of the reservoir
(0.25)(0.8)(10  10 6 )
being investigated is far beyond the skin region. Wellbore weirdness (storage and temperature
transients on the pressure gauges) will usually obscure these data.
Example 2: Radius investigated in 1 week in a low-permeability, low-pressure gas reservoir.
Assuming k = 0.010 md,  = 0.05, ct = 20010-6/psi,  = 0.05 cp, and t = 168 hr, rinv is
0.010 (168)
rinv  0.03
 55 ft . So after one week, only 55 feet have been
(0.05)(0.05)(200  10 6 )
investigated. Because such tight gas reservoirs are usually completed on large, it is difficult to
detect boundaries 100’s or 1000’s of feet away.
The difference in these two time scales is solely due to the differences in diffusivity, . Example
1 is high-diffusivity, example 2 is low-diffusivity.
Page 9 of 9