inflo

PETE 4052 Well Testing Spring 2002Lecture 7 Inflow Equations March 4

Inflow EquationsOutline0. Synopsis1. Special Cases of the Diffusivity Equation2. Steady-State Inflow in a Circle3. Pseudosteady-State Inflow in a Circle4. Dietz Shape Factors5. Effects of Shape, Skin, and Area6. Productivity Index

Related readingCraft et al., Sections 7.7-7.8, p. 242-248.Dake: Sections Chapter 6, p. 141-152.

7.0 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. If the pressure does not change with time, or if the compressibility is small and constant and the rate is constant, then certain approximations can be made and we can obtain easy-to-use and very useful inflow equations.

Key Concepts Definition of steady state Definition of pseudosteady state Integration to obtain inflow equations in terms of external pressure and average pressure Use of Dietz shape factors Productivity Index

7.1 Special Cases of the Diffusivity EquationIn Lecture 6 we derived the diffusivity equation for single-phase flow of an incompressible fluid,

.............................................(6.12b)

In this lecture we will consider two particular case:

Steady state: .........................................................(7.1a)

and

Pseudosteady state: ...................................................(7.1b)

These terms are widely used in reservoir engineering and you should be familiar with them. The

third case (in which varies with r and t) is usually called transient.

7.2 Steady-State Inflow in a CircleRewriting Eqn. (6.12b) using the steady-state assumption [Eqn. (7.1a)],

.........................................................(7.2)

We will solve this subject to the boundary conditions

of 10


...........................................................(7.3a)and

...........................................................(7.3b)We will also use the relation, valid for steady-state flow, of

..........................................................(7.4)

Note that this is now an ordinary differential equation, because the time derivative has been eliminated. We can manipulate and then integrate Eqn. (7.2) once to get

Integrating again,

Applying the boundary conditions at rw and re,

Thus, the equation for pressure is

.......................................(7.5)

This form makes the linearity of p in ln(r) for steady-state radial flow very clear. The radial gradient of p is

Using Darcy’s Law [Eqn. (7.4)],

of 10


......................................(7.6)

which is just our familiar radial flow formula for incompressible fluids. Note that q is not a function of r.

We can also express this in terms of the average pressure. For a continuous variable,

We can use Eqn. (7.5) for p to get

...........................(7.7)

The integral can be evaluated by parts (similar to Dake, p. 144; here ),

Finally, we have the average pressure equation in terms of and :

................................(7.8a)

For a well in a circular drainage area, and

..............(7.8b)

We can solve Eqn. (7.8b) for

of 10


.............................................(7.9a)

and use it with Eqn. (7.6) to get an inflow equation in terms of (assuming, again, that ):

........................(7.9b)

DiscussionThis development follows Dake’s for pseudo- or semisteady state, p. 141-144, except for steady state. The final result [Eqn. (7.9b)] is the same as his steady state inflow equation in terms of , given in Table 6.1. Note that the average pressure [Eqn. (7.8b)] is not the mean of the external and wellbore pressures. We can find the radius at which this pressure occurs by defining [using Eqn. (7.5)]

....................................(7.10a)

But is also given by Eqn. (7.8b) so that

..........................(7.10b)

Because and , the average pressure occurs at a radius greater than the mean

radius ( ). The manipulation is simpler if you recognize that .

The average pressure occurs at because so much of the volume is at high pressure … a property of being in a radial rather than a linear geometry. For a linear flow geometry, the average pressure is the mean pressure (and is equal to the pressure halfway through the domain in linear space). At what point would you expect the pressure to be equal to the average pressure in spherical coordinates?

7.3 Pseudosteady-State Inflow in a Circle

Definition of Pseudosteady State

of 10


Pseudosteady (or, in Dake’s terms, semisteady) state is defined as a regime in which the rate is constant and the pressure is changing at the same rate everwhere. The rate of pressure change can be determined by material balance. For a fluid of small and constant compressibility, and assuming pure depletion drive,

or

because rate was assumed to be constant. The minus sign is needed because for positvie q, the pressure change must be negative (and all other terms are positive). This is equivalent to writing

............................................(7.11a)

Dividing by and taking the limit as it goes to zero, we get

....................................................(7.11b)

We use partial derivatives because p is still a function of r as well as t.

DiscussionNote that our expression [Eqn. (7.11b)] differs from Dake’s Eqn. (5.10) by a factor B. We have converted the volume back to a reservoir basis, which is appropriate (a) for a material balance calculation and (b) because theh compressibilities would be expressed on a reservoir volume basis. Dake is a bit careless with his B’s and q’s.

We can rewrite Eqn. (7.11b) in terms of dimensionless pressure and time using

......................................(7.12a)

and

..................................................(7.12b)

Combining Eqns. (7.12) and (7.11b),

..................................(7.13)

Dimensionless variables are so cool!

DerivationThe derivation of the pseudosteady inflow equations are given in Dake, p. 141-144. The derivation is very similar to the steady state derivation given in the previous section. I urge you to review this derivation. The results we will use further are listed below:

Pressure at r: .....................................(7.14a)

of 10


q in terms of pe: ...............................................(7.14b)

Average pressure : ........................................(7.14c)

q in terms of : ..............................................(7.14d)

and

: ....................................................(7.14e)

Several of these formulae are given in Dake’s excellent Table 6.1 (attached at end of these lecture notes). Note that we have included skin in these expression, following Dake. You should be able to modify the steady-state equation to include skin, and understand why this works.

DiscussionIf we compare the steady-state and pseudosteady-state inflow expressions in terms of average pressure or in terms of external boundary pressure, we can see the denominator is smaller for the pseudosteady-state expressions (see Dake, Table 6.1). That is, the same pressure drop moves more fluid in the pseudosteady state system. We can understand this physically. In steady-state, all fluid must move all the way from re to rw. On the other hand, in pseudosteady-state, expansion supplies fluid at all r < re. Thus, less work has to be done to move the fluid and a smaller pressure drop is required for a given flow rate.

We can rewrite the equations for pressure in dimensionless terms asSteady-state: ....................................................(7.15a)

Pseudosteady-state ............................................(7.15b)

of 10


where we have used and . Notice that there is a quadratic term in

the pseudosteady-state expression in addition to the logarithmic term. This additional term is insignificant when rD/reD, or near the well. This implies that the gradients near the well must be

similar. On the other hand, far from the well the gradients are different (Figure 7.1).Note the following features of Figure 7.1: The gradient decreases with r for the pseudosteady case. This is true because the gradient at

the outer boundary is zero – it is depletion drive. Thus, the gradient must go from the

at rD = 1 to 0 at reD (in Figure 7.1, reD = 1000).

The decrease in gradient (in log space) for the pseuosteady state implies that the flow rate decreases. At reD the flow rate is zero. The flow rate rapidly increases as we move inward from the outer boundary, because a large volume is filled with expanding fluid for each small change in r at the outer boundary. On the other hand, near the well, only a small volume of fluid expansion occurs for comparable changes in r, and the flow rate (and thus the gradient) is nearly constant.

The rate is of course proportional to the gradient of ln(r). For steady state, it is constant. The

flow rate for pseudosteady state is given in Figure 7.2. The change flow rate . It is

parabolic in r, because the volume contributing to the flow is parabolic in . The gradient is zero at the outer boundary (Fig. 7.1), so the flow rate is zero here also (Fig. 7.2)

of 10

Comparison of Pressure Profiles

0

1

2

3

4

5

6

7

8

1 10 100 1000

Dimensionless Radius

Dim

ensi

on

less

Pre

ssu

re

Steady State Pseudosteady State

Figure 7.1 – Pressure profiles for steady- and pseudosteady-state flow.


of 10

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000

Dimensionless Radius

Dim

ensi

on

less

Rat

e, q

(rD)/

q(1

)

Figure 7.2 – Variation of flow rate with radius for pseudosteady state flow


7.4 Dietz Shape FactorsThe denominator of the pseudosteady inflow equation [Eqn. (7.14b)] can be rearranged to get

........................................(7.16)

where is the exponential of Euler’s constant, and is equal to 1.781 (Euler’s constant is 0.5772).

Why did we do this? This form (the last expression in Eqn. 7.16) is in the Dietz standard form. In this form, the inflow equation is written as

..............................................(7.17)

where A is the drainage area and CA is the Dietz shape factor (see Figure 6.4 in Dake). The amazing thing about the Dietz shape factors is that they can be used for any shape or area of reservoir. For the most part, these expressions were derived using the method of images, which we will discuss later in the course. For now, it is sufficient to appreciate that this work has been done for us, and the Dietz shape factors, in conjunction with Eqn. (7.17), allow as to compute the stabilized inflow rate for practically any geometry of interest.

Application

The drainage area can often be computed from material balance [Eqn. (7.11b)],

.........................................................(7.18)

In other cases, the area can be computed from maps or pattern spacing. The shape of the well drainage area can be estimated from maps, using the additional information that drainage volume is proportional to well rates [as can be deduced from Eqn. (7.11b)]. We will soon see how to

estimate and s from well tests. The average pressure can also be estimated from well tests.

These are all the parameters we need to estimate the stabilized inflow rate. This relationship is very useful for understanding depletion behavior or reservoirs.

of 10


7.5 Effects of Shape, Skin, and AreaThe relative importance of several parameters can be deduced from Eqn (7.17), if we use our knowledge of skin and the properties of logarithms.

First, the basic effects: Flow rate decreases as area increases for a given bottomhole pressure and average pressure.

The change in the denominator is logarithmic; increasing A by a factor of e is like adding 1 to the skin factor.

Flow rate increases as the shape factor increases; again, the dependence is logarithmic. Increasing CA by a factor of e is like decreasing the skin factor by 1.

In terms of relative importance of the effects, it would be rare for well drainage areas within a given reservoir to vary by factors larger than e2, which is like changing s by 2. Examining Figure 6.4, CA varies from about 0.1 for very skewed drainage areas to over 30 for very symmetric areas; the corresponding change in logarithms is about 5.5 – which is like changing s by 5.5. Thus, it appears that the effect of area will generally be smaller than the effect of shape. Furthermore, the effect of shape can (in extreme cases) be as large as the effect of reasonable well-to-well differences in skin factor. In reservoirs with high skin, the variability of skin will control the variability in inflow.

Finally, observing flow rates and the rate of change of average pressure gives us no information about individual parameters such as CA, s, and k. We can only deduce the constant in front of the pressure difference . This is known as the productivity index.

7.6 Productivity IndexThe pseudosteady inflow equation in Dietz form is

.............................................(7.17)

If we define the productivity index, or PI, as the ratio of the flow rate to the pressure drop, then it is

.............................................(7.19)

This leads to a definition of flow efficiency analogous to the previous steady-state radial case, but now we have

..................................................(7.20)

Equation (7.20) shows that the effect of skin on rate depends upon drainage area and shape. Thus, one cannot determine any of these parameters from pseudosteady- or steady-state data. We will use pressure transient tests to isolate these parameters.

We will discuss a succession of pseudosteady states and decline curves in Lecture 7A.

of 10

inflo

Documents