Resultados por GIXRD obtenidos para el acero 1.4970

CAPITULO V. RESULTADOS, ANÁLISIS Y DISCUSIÓN

V.2 GIXRD y cambios morfológicos en la superficie

V.3.2 Resultados por GIXRD obtenidos para el acero 1.4970

conducted by producing a well at constant rate for some time, shutting the well in (usually at the surface), allowing the pressure to build up in the wellbore, and recording the BHP as a function of time. In other words, we can model buildup tests by use of superposition in time as a “two-rate” problem. This is illustrated in Fig. 1.17 as two wells, one producing at flow rate q for time (tp)Dt) and the other producing at

a rate (0*q) for time (tp)Dt*tp).

. . . (1.198)

A buildup test is generally run for a short time, so transient flow is expected. Therefore, we use the transient-radial-flow (line-source) solution for the required constant-rate solutions.

pD(tD)+ * 1₂Ei

ǒ

* r2

ƫ

(1.199) . . . and pD(DtD)+ * 1₂Ei

ƪ

₄₍* 1_Dt D)

ƫ

. . . (1.200) We substitute Eqs. 1.199 and 1.200 into Eq. 1.198 to obtain

pwD+ qD

Ǌ

* 1₂Ei

ƪ

* 1 4

ǒ

tp) Dt

Ǔ

ƫ

ƪ

* 1₂Ei

ǒ

* 1 4(DtD)

Ǔƫǋ

+qD 2

Ǌ

ƪ

4(* 1DtD)

ƫ

* Ei

ƪ

* 1 4

ǒ

tp) Dt

Ǔ

ƫǋ

. . . . (1.201)

D.p 70.6 q B _kh J.l

[I (,1688

_n <I> J.l _kt _

2]

_/

Constant Rate Solution _

log (D.p)

Observed Variable Rate Pressure Data (Distorted by Wellbore Storage)

log(t)

Fig. 1.18-Effect of well bore storage (variable rate) on pressure profile.

Recall the log approximation for the Ei function, Ei( - x) = In(1.781x). Substituting this into Eq. 1 .201 gives

We know that In(a) - In(b) = In(a/b); therefore, Eq. 1.202 becomes

We recall the definitions of the following dimensionless variables, kh

(P

i - Pws)

Pwo _{= 1 41.2qB,u '}

where _Pws= shut-in BHP, and

qo = = 1

(1.204)

(1.205) because qO = flow rate just before shut-in = q.

0.0002 63 71'1.t

f').fo = 2 .. . . • . . . • . . . .. (1.20 6)

¢,uctrw

. . . .. (1.20 7)

Substitute Eqs. 1.204 through 1 .20 7 into Eq. 1 .203 to obtain

(

¢,uctr�v

)}

x 0.000 2 63 7kf').f _ _ 70.6qB,u

[(

fp + f').t

)]

Pi Pws - _{kh In} f').t _ _ 70.6qB,u + Pws - Pi _kh In f').t . . . . .. (1.208) We know that In (x) = 2.303 log(x), therefore Eq. 1.208 becomes

_ _ 70.6(2.303)qB,u

[(

t" + f').t

)]

Pws - Pi _kh log f').t

Constant Rate Solution

log (D.p)

Observed Variable Rate Pressure Data (Distorted by Wellbore Storage)

log(t)

Fig. 1.19-Effect of deconvolution on variable-rate pressure pro· file.

_ _ 1 62.6qB,u

[(

tl' + f').t

)]

- Pi _kh log f').t . . . . .. (1.209)

This equation models a pressure-buildup test and forms the basis for some test-analysis techniques. An important characteristic of this relation is that a plot of _Pws(pressure recorded during the build up test) vs. the logarithm of the function [(�) + f').t)/f').t] should be a straight line with a slope inversely proportional to the formation permeability. We discuss pressure-buildup analysis in more detail in Chap. 2.

1.8 Deconvolution

With superposition (discussed in Sec. 1.7), we can calculate the vari able-rate solution from a known constant-rate solution. There are times when we know the variable-rate pressure response and wish to calculate a constant-rate pressure profile. This is especially useful when wellbore storage distorts pressure data; wellbore storage causes variable sandface rates, and thus a variable-rate pressure profile. For example, if wellbore storage distorts flow data from an infinite-acting well, the pressures measured at the sandface would not match the infi nite-acting solution derived for constant rate. Fig. 1.18 shows this. To analyze pressure data distorted by well bore storage, we would have to use solutions that incorporate well bore-storage effects. There have been solutions developed with wellbore storage as an in ner-boundary condition for many reservoir models; however, we must know what the reservoir model is to use these solutions. If we could calculate a constant-rate pressure profile from the well bore storage distorted data, we could eliminate the well bore-storage ef fects and have a solution that would indicate the reservoir model, rather than requiring a priori knowledge of the model.

Deconvolution is a technique that can be used to remove the ef fects of wellbore storage from the measured pressure profile. In this section, we discuss some simple methods of deconvolution in rate normalization 12-14 and Laplace transform deconvolution. 15-18 Fig.

1.19 illustrates the effects of deconvolution.

1.8.1 Rate-Normalization Methods. We can describe discrete changes in rate with superposition in time; however, if the rate is changing smoothly (as a function of time), we can use rate-normal ization methods. 12-14 _{These methods group together the variables} dependent on time; for example, the constant-rate, infinite-acting reservoir solution is given by

( ) = 70.6qB,u

[

(

- 948¢,uCtr�v

)

]

t kh EI kt 2s .

For variable rates, this equation becomes

( ) = 70.6q(t)B,u

[ . (

- 948¢,uCtr�

)

- 2

]

kh El kt s .

. . .. (1.210)

... (1 .211 )

Grouping together functions of time on the left side of the equation, we have a rate-normalized pressure term that can be used in pressure transient analysis (as described in Chap. 2).

I1p(t)

= 70.68#

[

(

- 94S1>Wlr�

)

- 2

]

q(t) kh EI kt S . . . . .. (1.21 2) Ref. 1 4 gives a more complete discussion of this equation, includ ing its limitations.

1.8.2 Laplace Transform Deconvolution. Kuchuk and Ayestaran 15 presented the idea of deconvolution by use of Laplace transforma tions to convert the convolution integral into a form that could be solved algebraically for the constant-rate pressure profile. To use this method, they needed to express the sandface flow rate and/or vari able-rate pressure profile as approximation functions. On the basis of ideas of van Everdingenl6 and Hurst, 17 the authors developed an ex ponential series model to fit the flow rate data. Use of this method is limited when the rate profile is not represented accurately by the ex ponential series model. The use of a numerical-inversion routine is another disadvantage because of inherent instability. Other authors, including Blasingame et al., 18 have introduced more stable Laplace transform methods by use of different approximations to fit the mea sured data functions. These methods are again limited by the choice of the functions that fit the measured pressure data and sandface rates. In this section, we present a general development of Laplace transform deconvolution 15 and show how to represent the rate func tion so that a direct inversion from Laplace space exists.

In Sec. 1. 7.1 , we developed the convolution integral

f dqo(r)

Pwo(to) = ----cJT[Pso(to - r) + sldr. o

. . . .. (1.19 6)

With variables with dimensions, Eq. 1.19 6 becomes

I1Pw(t) =

f

I1ps/t - r)dr , ... (1.213) o

where I1pw = measured pressure drop during test, qO = reference rate, q(r) = sandface flow rate, and I1psf = constant-rate pressure be havior of the reservoir at the sandface (i.e. , the pressure data that would have been obtained from a constant-rate flow test if wellbore storage had not distorted the test data).

Looking at Eq. 1.213, we have a problem with direct calculation of I1psf because it is "locked" inside the integral. We can use a tech nique that we used to develop many of the solutions to the diffusivity equation presented in this chapter-Laplace transforms.

We need to use a theorem that will allow us to take Laplace trans forms of Eg. 1.213. This is Duhamel's theorem,2 which states

{j

!(')g« - ')<I'

] �

ljt(t)jl,[g(t)], " " " " '" (1.21 4)

Our equation, however, is in the form

f

!'(r)g(t - r)dr. . ... (1.21 5)

We need to use the property of Laplace transforms, which relates the transform of the derivative of a function,

£[f'

(t)

]

, to the transform of the function,]l:u):

£[f'(t)]

= uj(u) - f(t = 0) ... (1.21 6)

FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA

Therefore, taking Laplace transforms of Eq. 1.213 by use of Duha mel's theorem gives

(1.21 7) Because flow rate is zero at initial time, we have

I1p,/U) = u{jo(u)l1ps/u)' ... (1.21 S)

which can be rearranged algebraically as follows to solve for I1ps/

11-

11-_{Pif u{jo'}= PIV ... (1.219) We can calculate I1Psf and then invert to "real space" to obtain I1p sf' This can be done by use of a numerical-inversion routine, such as the Stehfesr7 algorithm, or by choosing functions for I1Psf, I1pw, and/or qD so that direct inversion is possible. Many different meth ods have been presented in the literature; we present a simple direct inversion method in this chapter known as the beta deconvolu tionlS-17 method.

Beta Deconvolution. van Everdingen 16 and Hurst 17 suggest that the sandface rate for wellbore-storage-dominated data may be approximated by an exponential in the form

(1.220) Taking the Laplace transform of Eg. 1.220 gives

- I 1 (3

qo = Ii - _u₊_{(3 =}_u(u₊_(3f . ... (1.221 ) We can now substitute Eq. 1.221 into Eq. 1.219 and rearrange the equation to obtain a Laplace transform relationship that has a direct inversion.

_ 1 u(u + (3) _

I1ps = Ii (3 I1p,v

�

I1Pw + I1p

lV

' ... (1.222)

Inverting this equation gives 1 d

I1ps = {idi(l1pw) + I1pw . ... (1.223)

Eq. 1.223 implies that, if we have measured pressure-drop data, I1pw, that are distorted by wellbore storage and if we can fit the f1ow rate vs. time data with an exponential function to find (3, we can cal culate the constant-rate pressure drop, I1Psf by use of this relation ship. Note that we need the derivative of the pressure drop in this equation; this can be calculated numerically by use of finite-differ ence methods.

1.9 Chapter Summary

In Sec. 1.2, we presented the partial-differential equations describ ing the flow of fluids in porous media for several systems of interest. These systems include radial flow of a single-phase, slightly com pressible liquid (Eq. 1.39), radial flow of a single-phase gas (Eq. 1.51 ), and multiphase flow (Eq. 1.65). We presented approximate linearizations of the single-phase gas-flow equation in terms of pressure (Eq. 1.53), pressure squared (Eq. 1.5S), and pseudopres sure (Eq. 1.64). For each of these systems, we arrived at the ap propriate partial-differential equation by combining the continuity equation, an equation of motion, and an EOS. Throughout the sec tion, we made the following assumptions.

1. Flow is radial.

2. Darcy's law describes the relationship between flow velocity and pressure gradient.

3. The porous medium is uniform and isotropic. 4. Gravity effects are negligible.

5. Conditions are isothermal.

6. Effective penneability is independent of pressure.

To develop Eq. 1.39 for slightly compressible liquids, we further assumed Points 7 and 8.

7. The fluid has a small, constant compressibility; i.e., the EOS is given by P = PbexP[c(P - Pb)].

8. The compressibility/pressure-gradient squared product,

c(ap/ar)2, is negligible.

For gases, Eq. 1.51, we made Assumptions 1 through 6 and added Assumption 9.

9. The density is given by the real gas law, P = pM/zRT.

Although Eq. 1.51 is rigorous, it is also nonlinear. To take advan tage of the solution methods available for linear PDE's, we would like to linearize Eq. 1.51, even at the expense of making more re strictive assumptions.

10. The term p/fAz is constant with respect to pressure. Using Assumption 10, we obtain Eq. 1.53, identical to the expres sion that describes flow of a slightly compressible liquid (Eq. 1.39).

This equation is linear for gases, provided that Assumption 11 fol lows.

11. The term fACt is constant with pressure.

If instead we assume the following, we obtain Eq. 1.58.

1 2. The term fAz is constant.

Eq. 1.58 has the same fonn as Eq. 1.39, with pressure replaced by pressure squared. As with Eq. 1.39, this equation is linear provided that Assumption 11 is true.

By defining the real gas pseudopressure, we obtain Eq. 1.64. This equation depends on Assumptions 1 through 6 and 9, but not 10 or

1 2. It is linear if Assumption 11 holds.

Finally, for multiphase flow, Assumptions 1 through 6, along with the following additional assumptions, result in Eq. 1.65.

13. Pressure and saturation gradients are small.

14. Capillary pressure is negligible.

In Sec. 1.3, we turned our attention to a description of the more common initial and boundary conditions. A single initial condition, that of uniform pressure at time t = 0, is used throughout this text. The outer-boundary conditions of interest include the following.

1. Infinite-acting reservoir, where the pressure approaches the initial pressure at distances far from the well bore for all times.

2. No-flow outer boundary, where the partial derivative of the pressure with respect to r is zero for r= reo

3. Constant-pressure outer boundary, where the pressure is constant and equal to the initial pressure for all times at some dis tance re from the well.

Inner boundary conditions of interest include (1) constant sand face production rate, (2) constant-pressure production, (3) wellbore storage, and (4) skin factor.

In Sec. 1.4, we developed appropriate dimensionless variables for constant-rate production, radial flow. We stated expressions for di mensionless pressure, dimensionless radius, dimensionless time, di mensionless well bore-storage coefficient, and skin factor. Using these dimensionless variables, we wrote the diffusivity equation and the associated initial and boundary conditions in dimensionless fonn. We also stated dimensionless variables for radial flow, production at constant pressure. We stated expressions for dimensionless pres sure, dimensionless rate, and dimensionless cumulative production. As with the constant-rate case, we wrote the diffusivity equation and the associated initial and boundary conditions in dimensionless form by use of these dimensionless variables.

Finally, we stated expressions for dimensionless pressure, dimen sionless length, and dimensionless time for constant-rate produc tion from a linear system. We stated these expressions first for the general case, then for a well with a vertical hydraulically in duced fracture.

In Sec. 1.5, we presented solutions to the diffusivity equation for several reservoir models, including models (1) with transient radial flow and constant-rate production from a line-source well, (2) with out skin factor, (3) with both skin factor and wellbore storage, (4)

with pseudosteady-state radial flow and constant-rate production from a cylindrical-source well in a closed reservoir, (5) steady-state radial flow and constant-rate production from a cylindrical-source well in a reservoir with a constant-pressure outer boundary, and (6)

transient linear flow and constant-rate production from a hydrauli cally fractured well. We tied these solutions to field applications through several examples.

In the next three sections, we explored three extremely useful techniques arising from the linearity of the diffusivity equation: su perposition in space in Sec. 1.6; superposition in time, or convolu tion, in Sec. 1.7; and deconvolution in Sec. 1.8.

In Sec. 1.6, we discussed the use of superposition in space. Super position in space may be used to detennine the pressure drop at any given point in a multi well reservoir as a function of time simply by adding the pressure drops that would result from each well consid ered independently. In many cases of interest, superposition in space may be used to develop solutions for a single well in a reser voir with one or more no-flow or constant-pressure boundaries by use of the method of images.

Superposition in time, discussed in Sec. 1.7, may be used to calcu late the pressure drop for a well where the production rate is a piece wise constant function of time. By approximating an arbitrary rate history as a sequence of constant rates and letting the width of each rate interval go to zero, we obtained an expression for the convolu tion integral (Eq. 1.196). The convolution integral allows us to cal culate the pressure as a function of time for an arbitary rate history, for any reservoir geometry, from the given rate history and the well bore pressure solution for constant-rate production from the same reservoir geometry. Superposition in time may also be used to model pressure-buildup tests by considering the test as a two-rate problem. Finally, in Sec. 1.8, we introduced deconvolution, the inverse problem to that of convolution. In convolution, we use a known variable rate and a known pressure response to constant-rate pro duction to calculate the (unknown) pressure response to the known variable rate; in deconvolution, we use a known variable rate and a known pressure response to that variable rate to infer the (unknown) pressure response to constant-rate production. If this deconvolution can be perfonned without having to make any assumptions as to the reservoir model, the resulting constant-rate pressure response can then be used to infer the reservoir model. It can also be analyzed by use of any pressure transient test analysis methods developed for constant-rate production. This process can be especially useful for analyzing tests distorted by wellbore storage if the sandface flow rates can be measured or calculated.

The simplest deconvolution method we discussed was that of rate normalization, where the pressure change as a function of time is di vided by the rate, also as a function of time. This method is applica ble whenever the rate is changing slowly and smoothly.

We also introduced a general class of convolution methods re ferred to Laplace transfonn deconvolution, along with beta decon volution, a special case of Laplace transform deconvolution.

1. 10 Discussion Questions

For each of the following variables, state the field units, define the variable, and discuss how it might be measured or estimated. What are likely sources of uncertainty in the measurement or estimate of each quantity?

a. porosity, <p b. viscosity, fA

c. fonnation net pay thickness, h d. well radius, rw e. water saturation, Sw f. oil saturation, So g. gas saturation, Sg h. formation compressibility, q i. oil compressibility, Co j. gas compressibility, cg k. flow rate, q

I. oil formation volume factor, Bo

m. gas formation volume factor, Bg

n. water formation volume factor, s'v o. time, t

p. pressure, P

q. permeability, k

FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA 25 2. Explain Fig. 1.1. In which direction do we assume fluid flows?

What is the flow rate in the q and z directions?

3. On what assumptions is the continuity equation (Eq. 1.18) based?

4. Why is the differential form of Darcy’s law (Eq. 1.19) written in terms of potential F instead of pressure? What is the physical significance of F?

5. On what assumptions is the equation of state (Eq. 1.28) based? 6. What is the major assumption used in writing the diffusivity equation for gas in terms of pressure (Eq. 1.53)? In what field situations is this assumption reasonable? In what situations is this assumption likely to be violated?

7. What is the major assumption used in writing the diffusivity equation for gas in terms of pressure-squared (Eq. 1.58)? In what field situations is this assumption reasonable? In what situations is this assumption likely to be violated?

8. What is the major assumption used in writing the diffusivity equation for gas in terms of pseudopressure (Eq. 1.64)? In what field situations is this assumption reasonable? In what situations is this assumption likely to be violated?

9. What is the physical significance of the total mobility for a three-phase system (Eq. 1.66)?

10. Explain the significance of each of the terms in the equation for the total compressibility (Eq. 1.67).

11. Which term in the total compressibility equation (Eq. 1.67) is most important for an oil reservoir just above the bubble point pressure of 1500 psi, with a water saturation of 15%? For an oil reservoir just below the bubble point pressure of 1500 psi, with a gas saturation of 5% and a water saturation of 15%? For a low pressure gas reservoir at 150 psi with a water saturation of 25%? For a geopres- sured gas reservoir at 12000 psi with a water saturation of 20% and a formation compressibility of 30 10–6_{? Make any assumptions}

about fluid properties as reasonable as possible.

12. What are the assumptions used in developing the diffusivity equation for multiphase flow (Eq. 1.65)? In what field situations are these assumptions reasonable? In what situations are these assumptions likely to be violated?

13. The diffusivity equation for radial fluid flow in porous media (Eq. 1.68) is first order in time and second order in space. How many boundary conditions must we specify in order to solve this equation?

14. We have assumed permeability is isotropic in this chapter. How would the presence of an anisotropic permeability affect the pressure response for a vertical well producing at constant rate?

15. Under what field conditions would the assumption of constant pressure at the gas-liquid interface (Eq. 1.91) be valid? Under what conditions would it not be valid?

16. Under what field conditions would the wellbore storage coefficient for a single-phase fluid (Eq. 1.95) not be constant?

17. Does the presence of a zone of altered permeability affect the pressure response in the reservoir at some distance from the well- bore for a fixed flow rate q? If the pressure response away from the wellbore is not affected by the presence of a zone of altered permeability, why do we ever need to remove skin damage or stimulate a well?

18. Which of the following values appear in the definition of dimensionless pressure for constant rate production (Eq. 1.104)—permeability, porosity, net pay thickness, viscosity, formation volume factor, compressibility, flow rate, wellbore radius? Which appear in the definition of dimensionless time (Eq. 1.107)?

In document INSTITUTO POLITECNICO NACIONAL ESCUELA SUPERIOR DE FISICA Y MATEMATICAS (página 117-127)