**5. Hydraulically fractured wells**

[18] linearized the partial-differential equation for the problem of a well intercepted by a vertical fracture. Their dimensionless pressure solution is given below:

Transient Pressure and Pressure Derivative Analysis for Non-Newtonian Fluids 163

$$P\_D\left(t\_D\right) = \frac{\left(3-n\right)^{2v} t\_D^v}{\left(1-n\right)\Gamma(1-v)} - \frac{1}{1-n} \tag{23}$$

Where = (1-*n*)/(3-*n*)

162 New Technologies in the Oil and Gas Industry

*tMAX* = 1.3 *tirN\_NN* = 0.0008 hr

is found with Equation 16 to be 179.7.

10

100

*t*\*

resulted to be 120 ft.

*P*' and

*P*, psi

1000

10000

From Figure 6 the following information was read:

<sup>1</sup> 541 69 psi *<sup>r</sup> P .*

<sup>1</sup> 105 45 psi *<sup>r</sup> t\* P .*

**Figure 6.** Pressure and pressure derivative for example 1

zone. They resulted to be 100 md and 4.5.

**5. Hydraulically fractured wells** 

*tr*1 = 0.3 *Pr*1 = 541.54 psi (*t*\**P*')*<sup>r</sup>*<sup>1</sup> = 105.45 psi *tr*2 = 120 *Pr*2 = 991.5 psi (*t*\**P*')*<sup>r</sup>*<sup>2</sup> = 39.02 psi

**Solution**. The log-log plot of pressure and pressure derivative against injection time is given in Figure 6. Suffix 1 and 2 indicate the non-Newtonian and Newtonian regions, respectively.

 is evaluated with Equation 15 to be 0.17 and a value of 100.4 md was found with Equation 14 for the non-Newtonian effective fluid permeability. Equation 4 is used to find an effective viscosity of 0.06465 cp(s/ft)*n*-1. Then, the skin factor in the non-Newtonian region

<sup>2</sup> 901 5 psi *<sup>r</sup> P .*

<sup>2</sup> 39 02 psi *<sup>r</sup> t\* P .*

0.0001 0.001 0.01 0.1 1 10 100 1000

A value of 6.228x10-5 hr/(ft3-*n*) was found for parameter *G* using Equation 3. This value is used in Equation 24 to find the distance from the well to the non-Newtonian fluid bank. This

Equations 17 and 18 were used to estimate permeability and skin factor of the Newtonian

Using a time of 0.0008 hr which corresponds to the intersect point formed between the non-Newtonian and Newtonian radial flow regime lines in Equation 19, a non-Newtonian effective fluid permeability of 96 md is re-estimated. [13] obtained a permeability of the non-

[18] linearized the partial-differential equation for the problem of a well intercepted by a

Newtonian zone of 101 md and *ra* = 116 ft from conventional analysis.

vertical fracture. Their dimensionless pressure solution is given below:

*t*, hr

1 3 hr *Mt .* <sup>2</sup> 100 hr *rt* 0 0008 hr *riN \_ NN t .* <sup>1</sup> 0.3 hr *rt*

n = 0.6

First,  [24] presented two interpretation methodologies: type-curve matching and conventional straight-line for characterization of fall-off tests in vertically hydraulic wells with a pseudoplastic fluid. They indicated that at early times, a well-defined straight line with slope equal to 0.5 on log-log coordinates will be evident, then,

$$P\_D = \left(\frac{\pi}{2}\right)^{\frac{n-1}{2}} \sqrt{\pi^\* \, t\_{D\!xf}} \tag{24}$$

$$t\_{D\mathbf{x}f} = \frac{0.0002637kt}{\phi \mathbf{c}\_t \boldsymbol{\mu}^\* \mathbf{x}\_f^2} \tag{25}$$

Where the characteristic viscosity, \* , is given by:

$$
\mu^\* = \mu\_{eff} \left( 96681.605 \frac{h}{qB} \right)^{1-n} \tag{26}
$$

And the derivative of Equation 24 is:

$$t\_{D\ge f} \, \, ^\*P\_D \, ^\*= 0.5 \left(\frac{\pi}{2}\right)^{\frac{n-1}{2}} \sqrt{\pi t\_{D\ge f}} \,\tag{27}$$

And the dimensionless fractured conductivity is;

$$\mathbf{c}\_{f\mathbf{D}} = \frac{k\_f \pi v\_f}{k \,\mathbf{x}\_f} \tag{28}$$

[22] presented an expression which relate the half fracture length, *xf*, formation permeability, *k*, fracture conductivity, *kfwf*, and post-frac skin factor, *s*:

$$k\_f w\_f = \frac{3.31739k}{\frac{e^s}{r\_w} - \frac{1.92173}{x\_f}}\tag{29}$$

However, there is no proof that Equation 28 works for Non-Newtonian systems. Using Equation 23, [3] presented pressure and pressure derivative curves for vertically infiniteconductivity fractured wells. See Figure 7. They extended the *TDS* methodology, [21], for the systems under consideration. By using the intersect point of the pressure derivatives during linear flow regime, Equation 26, with the radial flow regime governing equation, Equation 11, *tRLi*, an expression to obtain the half-fracture length is presented:

**Figure 7.** Dimensionless pressure and pressure derivative behavior for a vertical infinite-conductivity fractured well with a non-Newtonian pseudoplastic fluid with *n* = 0.5

$$x\_f = \left[0.028783 \frac{\left(1.570796\right)^{\frac{n-1}{2}}}{\left(\frac{0.0002637kt\_{LRi}}{\phi c\_t \mu^\*}\right)^{\alpha}} \sqrt{\frac{t\_{LRi}k}{\phi c\_t \mu^\*}}\right]^\alpha \tag{30}$$

Where = (1-*n*)/(3-*n*).

The expression governing the late-time pseudosteady-state flow regime is:

$$\text{tr}\_{\text{D}}\,^\*P\_{\text{D}}\,^\prime = 2\pi t\_{\text{DA}}\tag{31}$$

The point of intersection of the pressure derivatives during linear flow and pseudosteadystate (mathematical development is not shown here) allows to obtain the well drainage area by means of the following expression:

$$A = \pi \left[ \frac{t\_{iLPSS}}{0.0625 \left( \frac{\pi}{2} \right)^{n-1} G} \right]^{2/(3-n)} \tag{32}$$

**Example 2.** Fan (1998) presented a pressure test of a test conducted in a hydraulic fractured well with the information given below. Pressure and pressure derivative data for this test is reported in Figure 8.


$$r\_{\overline{\mathbf{v}}} = 0.26 \text{ ft} \qquad \qquad H = 20 \text{ cp\*} \text{s}^{n-1}$$

1.E-03

= (1-*n*)/(3-*n*).

reported in Figure 8.

by means of the following expression:

= 10 % *B* = 1 rb/STB *μ\**

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03 1.E+05 1.E+07 *t Dxf*

**Figure 7.** Dimensionless pressure and pressure derivative behavior for a vertical infinite-conductivity

1 570796 2

*t*

The point of intersection of the pressure derivatives during linear flow and pseudosteadystate (mathematical development is not shown here) allows to obtain the well drainage area

0 0625

.

*<sup>t</sup> <sup>A</sup> <sup>π</sup>*

*n* = 0.4 *h* = 70 ft *k* = 0.65 md *q* = 507.5 BPD

*c μ*

1

*<sup>t</sup> LRi*

*<sup>c</sup> <sup>μ</sup> kt*

2 3

/( )

= 0.00065 cp *ct* = 0.00001 psi-1

*n*

1

*<sup>π</sup> <sup>G</sup>*

2

**Example 2.** Fan (1998) presented a pressure test of a test conducted in a hydraulic fractured well with the information given below. Pressure and pressure derivative data for this test is

*iLPSS n*

 

0 0002637 \* \*

*n*

fractured well with a non-Newtonian pseudoplastic fluid with *n* = 0.5

*x*

0 028783

. .

The expression governing the late-time pseudosteady-state flow regime is:

.

*f α*

*LRi t*

Linear flow regime Radial flow regime

Pressure derivative curve

*α*

\* ' 2 *D D DA t P πt* (31)

(30)

(32)

*LRi*

*t k*

1.E-02

1.E-01

1.E+00

*t \*P ' Dxf D*

*P* and *D*

Where  1.E+01

1.E+02

**Figure 8.** Pressure and pressure derivative for example 2

**Solution.** The following information was read from the pressure and pressure derivative plot, Figure 8,

*tLRi* =0.4495 hr *tr* =0.7217 hr *Pr* = 762 psi (*t*\**P'*)*r* = 522.06 psi

Using Equation 15, a value of 0.23 is found for . Reservoir permeability, skin factor, halffracture length were estimated with Equations 14, 16 y 30. Their respective values are 0.65 md, -13.9 and 771 ft. Reservoir permeability and half-fracture length are re-estimate by simulating the test providing values of 0.65 md and 776 ft, respectively; therefore, the absolute errors for these calculations are less 0.06 % and 0.5 %. A *G* value of 0.001241 hr/(ft3-n) was found with Equation 3.

A fracture conductivity of 868.5 md-ft was calculated using Equation 29. It is important to clarify that this equation is valid for the Newtonian case. This value was used in Equation 28 to find a dimensionless fracture conductivity of 1.73.

#### **6. Finite-homogeneous reservoirs**

For the cases of bounded and constant-pressure reservoirs, [8] presented the solutions to Equation 1. The initial and boundary conditions for the first case are:

$$P\_{\rm DNN} \left( r\_{\rm D}, 0 \right) = 0 \tag{33}$$

$$\left(\frac{\partial P\_{\rm DNN}}{\partial r\_{\rm D}}\right)\_{r\_{\rm D}=1} = -1 \quad \text{for } t\_{\rm DNN} > 0\tag{34}$$

$$\left(\frac{\partial P\_{\rm DNN}}{\partial r\_{\rm D}}\right)\_{r\_{\rm c\rm D}=1} = 0 \quad \text{for } t\_{\rm DNN} \tag{35}$$

The analytical solution in the Laplace space domain for the closed reservoirs under constantrate case is given as:

$$\overline{P}(\hat{s}) = \frac{\left| \begin{array}{c} \mathrm{K}\_{2\{(3-n)\}} \left[ \frac{2}{3-n} \sqrt{\varsigma} \varsigma\_{\varepsilon D}^{(3-n)/2} \right] \bullet I\_{\frac{1-n}{3-n}} \left[ \frac{2}{3-n} \sqrt{\tilde{s}} \right] + I\_{\frac{1}{2}\{3-n\}} \left[ \frac{2}{3-n} \sqrt{\varsigma} \varsigma\_{\varepsilon D}^{(3-n)/2} \right] \bullet K\_{\frac{1-n}{3-n}} \left[ \frac{2}{3-n} \sqrt{\tilde{s}} \right] \right|}{\left[ \varsigma^{3/2} \left[ \frac{2}{3-n} \sqrt{\varsigma} \varsigma\_{\varepsilon D}^{(3-n)/2} \right] \bullet K\_{\frac{2}{3}\{3-n\}} \left[ \frac{2}{3-n} \sqrt{\varsigma} \varsigma\_{\varepsilon D}^{(3-n)/2} \right] \bullet I\_{\frac{1}{2}\{3-n\}} \left[ \frac{2}{3-n} \sqrt{\varsigma} \varsigma\_{\varepsilon D}^{(3-n)/2} \right] \end{array} (36)$$

For the case of constant-pressure external boundary, the boundary condition given by Equation 35 is changed to:

$$P\_{\rm DNN} \left( r\_{eD'} t\_{\rm DNN} \right) = 0 \tag{37}$$

And the analytical solution for such case is:

$$\overline{P}(\overline{s}) = \frac{\left| I\_{1-n} \left[ \frac{2}{3-n} \sqrt{s} r\_{cD} \left( \frac{3-n}{3-n} \sqrt{\frac{2}{3-n}} \sqrt{\tilde{s}} \right) - K\_{\frac{1-n}{3-n}} \left[ \frac{2}{3-n} \sqrt{s} r\_{cD} \left( \frac{3-n}{3-n} \sqrt{\frac{2}{3-n}} \sqrt{\tilde{s}} \right) \right] \bullet I\_{\frac{1-n}{3-n}} \left[ \frac{2}{3-n} \sqrt{\tilde{s}} \right] \right|}{\left| \tilde{s}^{\frac{3}{2}\hbar} \left[ I\_{2\left(3-n\right)} \left[ \frac{2}{3-n} \sqrt{\tilde{s}} r\_{cD} \left( \frac{3-n}{3-n} \sqrt{\tilde{s}} \right) \right] + K\_{\frac{1}{2}\hbar \sqrt{s}} \left[ \frac{2}{3-n} \sqrt{\tilde{s}} r\_{cD} \left( \frac{2}{3-n} \sqrt{\tilde{s}} r\_{cD} \left( \frac{3-n}{3-n} \sqrt{\tilde{s}} \right) \right) \right] \right|} \right| + \overline{P}(\overline{s}) $$

Using the solution provided by [8], [4] presented pressure and pressure derivative plots for such behaviors as shown in Figs. 9 and 10. In these plots it is seen for closed systems in both pseudoplastic and dilatant cases, that the late-time pressure derivative behavior always displays a unit-slope line as for Newtonian fluids. As for Newtonian behavior, the late-time pressure derivative decreases in both dilatant or pseudoplastic cases.

[4] rewrote Equation 6 based on reservoir drainage area, so that:

$$t\_{DA} = \frac{t}{G\left(\pi r\_e^{3-n}\right)}\tag{39}$$

[4] combined Equations 11, 31 and 39 to develop an analytical expression to find well drainage area,

$$A = \pi \left[ \frac{t\_{priNN}}{G} \* \left( \frac{1}{4} \right)^{l/\alpha - 1} \right]^{2} \tag{40}$$

Where *trpiNN* is the intersection point formed by the straight-lines of the radial and pseudosteady-state flow regimes. The above equation was multiplied by ((1/-1))1/3-*n* as a correction factor. This is valid for both dilatant and pseudoplastic non-Newtonian fluids.

rate case is given as:

Equation 35 is changed to:

And the analytical solution for such case is:

( )

( )

drainage area,

*P s*

*P s*

1 0

The analytical solution in the Laplace space domain for the closed reservoirs under constant-

3 2 3 2

*n n n nn n eD eD*

 

*n n n n eD n eD n*

*<sup>K</sup> sr I s I sr K s n nn n*

 

2 22 2 3 33 3

2 22 2 3 33 3

3 3

*n n*

*sI sr K sK srI s n nn n*

For the case of constant-pressure external boundary, the boundary condition given by

 

1 11 1 3 33 3

*s I s K sr K s I sr n n n n*

 

2 3 1 1 2 3

pressure derivative decreases in both dilatant or pseudoplastic cases.

*A π*

*t*

[4] rewrote Equation 6 based on reservoir drainage area, so that:

3 2 3 2

*n n n eD n n eD n n nn n*

*I sr K s K sr I s n nn n*

2 22 2 3 33 3

3 2 3 2 3 2

Using the solution provided by [8], [4] presented pressure and pressure derivative plots for such behaviors as shown in Figs. 9 and 10. In these plots it is seen for closed systems in both pseudoplastic and dilatant cases, that the late-time pressure derivative behavior always displays a unit-slope line as for Newtonian fluids. As for Newtonian behavior, the late-time

> *DA* <sup>3</sup> *<sup>n</sup> <sup>e</sup> t*

[4] combined Equations 11, 31 and 39 to develop an analytical expression to find well

4 \* *<sup>α</sup> <sup>n</sup> rpiNN <sup>t</sup>*

 

Where *trpiNN* is the intersection point formed by the straight-lines of the radial and pseudosteady-state flow regimes. The above equation was multiplied by ((1/-1))1/3-*n* as a correction factor. This is valid for both dilatant and pseudoplastic non-Newtonian fluids.

*G*

2 1 1 <sup>3</sup> <sup>1</sup>

2 2 2 2 3 3 3 3

*n n n eD n eD n n*

3 3

 

2 3 2 3 2 3 2 3

*<sup>P</sup> for t*

*DNN*

(35)

, 0 *DNN eD DNN P rt* (37)

*n n*

*<sup>G</sup> <sup>π</sup><sup>r</sup>* (39)

(40)

(36)

(38)

*eD*

*DNN*

*D r*

3 2 3 2 3 2

2 3 1 1 2 3

*<sup>r</sup>*

**Figure 9.** Dimensionless pressure and pressure derivative behavior in closed and open boundary systems for a non-Newtonian pseudoplastic fluid with *n* = 0.5, *re* = 2000 ft

**Figure 10.** Dimensionless pressure and pressure derivative behavior in closed and open boundary systems for a non-Newtonian dilatant fluid with *n* = 1.5, *re* = 2000 ft

There is no pressure derivative expression for open boundary systems. Then, for pseudoplastic fluids the following correlation was also developed [4],

$$t\_{DA\_{\rm NN}} = -0.003n^2 + 0.0337n + 0.052\tag{41}$$

Equating Equation 41 to 39 and solving for reservoir drainage area, such as:

$$A = \pi \left[ \frac{t\_{\rm rsiNN}}{G \pi \left( -0.003n^2 + 0.0337n + 0.052 \right)} \right]^{\lambda\_{1-n}} \tag{42}$$

For dilatant fluids the correlation found is:

$$t\_{DA\_{\rm NN}} = 0.9175n^3 - 3.7505n^2 + 5.1777n - 2.2913\tag{43}$$

In a similar fashion as for the pseudoplastic case,

$$A = \pi \left[ \frac{t\_{\rm rsiNN}}{G \pi \left( 0.9175 n^3 - 3.7505 n^2 + 5.1777 n - 2.2913 \right)} \right]^{\lambda - n} \tag{44}$$

*trsiNN* in Equations 42 and 44 corresponds is the intersection point formed by the straight-line of the radial and negative unit-slope line drawn tangentially to the steady-state flow regime.

**Example 3.** [4] presented a synthetic example to determine the well drainage area. Pressure and pressure derivative data are provided in Figure 11 and other relevant information is given below:


**Solution**. From Figure 11, the intercept point, *trpiNN*, of the radial and pseudosteady-state straight lines is 60 hr which is used in Equation 40 to provide a well drainage area of 275 acres. Notice that this reservoir has an external radius of 2000 ft which represents an area of 288 acres. This allows obtaining an absolute error of 2.33 %.

**Figure 11.** Pressure and pressure derivative for example 3

#### **7. Heterogeneous reservoirs**

168 New Technologies in the Oil and Gas Industry

given below:

10

100

*Pt*\**P '*, psi

1000

For dilatant fluids the correlation found is:

In a similar fashion as for the pseudoplastic case,

3 2 0 9175 3 7505 5 1777 2 2913 . . .. *DANN*

*trsiNN* in Equations 42 and 44 corresponds is the intersection point formed by the straight-line of the radial and negative unit-slope line drawn tangentially to the steady-state flow regime. **Example 3.** [4] presented a synthetic example to determine the well drainage area. Pressure and pressure derivative data are provided in Figure 11 and other relevant information is

**Solution**. From Figure 11, the intercept point, *trpiNN*, of the radial and pseudosteady-state straight lines is 60 hr which is used in Equation 40 to provide a well drainage area of 275 acres. Notice that this reservoir has an external radius of 2000 ft which represents an area of

0.01 0.1 1 10 100 1000

*t*, hrs

*Gπ nnn*

3 2 0 9175 3 7505 5 1777 2 2913 . . ..

 

 

*Gπ n n*

*rsiNN <sup>t</sup> <sup>A</sup> <sup>π</sup>*

*rsiNN <sup>t</sup> <sup>A</sup> <sup>π</sup>*

*n* = 0.5 *h* = 16.4 ft *k* = 350 md *q* = 300 BPD

288 acres. This allows obtaining an absolute error of 2.33 %.

**Figure 11.** Pressure and pressure derivative for example 3

 = 5 % *Bo* = 1 rb/STB *μeff* = 0.014833 cp *ct* = 0.0000689 psi-1 *rw* = 0.33 ft *H* = 20 cp\*sn-1 *re* = 2000 ft *Pi* = 2500 psi

<sup>2</sup> 0 003 0 0337 0 052 ...

2 3

*t nnn* (43)

*n*

2 3

*n*

60 hr *rpiNN t*

(42)

(44)

In the well interpretation area of the Petroleum Engineering discipline a homogeneous reservoir is conceived to possess a single porous matrix while a heterogeneous reservoir has a porous matrix and either vugs or fractures. A common term used for heterogeneous systems is naturally-fractured reservoirs. However, this term is not recommended to be used since the fractures may result for either a mechanic process or a chemical process (matrix dissolution). Therefore, a more convenient term used in this book is double porosity systems in which the well is fed by the fractures and the fractures are fed by the matrix. By the same token, in a double-permeability system the well is fed by both fractures and matrix and the fractures are also fed by the matrix. This last one, however, has little application in the oil industry.

The governing well pressure solution in the Laplacian domain for a double-porosity system with a non-Newtonian fluid excluding wellbore storage and skin effects was provided by [19] as:

$$\tilde{P}\_{\text{DNN}} = \frac{K\_{\frac{1-n}{3-n}} \left( \frac{2}{3-n} \sqrt{\tilde{s} f(\tilde{s})} \right)}{\tilde{s} \left( \sqrt{\tilde{s} f(\tilde{s})} K\_{\frac{2}{3-n}} \left( \frac{2}{3-n} \sqrt{\tilde{s} f(\tilde{s})} \right) \right)} \tag{45}$$

The Laplacian parameter, *f*( *s* ) is a function of the model type and fracture system geometry and is given by:

$$f(\mathbf{s}) = \frac{\omega(1-\omega)\mathbf{s} + \lambda}{(1-\omega)\mathbf{s} + \lambda} \tag{46}$$

[2] implemented the *TDS* methodology for characterization of double-porosity systems with pseudoplastic fluids. As for Newtonian case, the infinite-acting radial flow regime is represented by a horizontal straight line on the pressure derivative curve. The first segment corresponds to pressure depletion in the fracture network while the second portion is due to the pressure response of an equivalent homogeneous reservoir. On the other hand, the transition period which displays a trough on the pressure derivative curve during the transition period depends only on the dimensionless storage coefficient, . The warren and Root parameters are defined in reference [26].

Figure 12 shows a log-log plot of the dimensionless pressure and pressure derivative for a double- porosity system with constant interporosity flow parameter, constant *n* value and variable dimensionless storage coefficient the higher the less pronounced the trough. As seen there, as the value of *n* decreases, the slope of the derivative during radial flow increases. In Figure 13 is shown the effect of variable of the interporosity flow parameter for constant values of dimensionless storage coefficient and flow behavior index. Notice in that

plot that as the value of *λ* decreases, the transition period shows up later. Finally, Figure 14 shows the effect of changing the value of the flow behavior index for constant values of and . The effect of the increasing the pressure derivative curve's slope is observed as the value of *n* decreases. Needless to say that neither wellbore storage nor skin effects are considered.

**Figure 12.** Dimensionless pressure and pressure derivative log-log plot for variable dimensionless storage coefficient, =1x10-6 and *n*=0.2 for a heterogeneous reservoir

**Figure 13.** Dimensionless pressure and pressure derivative log-log plot for variable interporosity flow parameter, *ω*=0.05 and *n*=0.8 for a heterogeneous reservoir

The infinite-acting radial flow regime is identified by a straight line which slope increase as the value of the flow behavior index decreases. See Figure 14. The first segment of such line corresponds to the fracture-network dominated period, and, the second one -once the transition effects are no longer present-, responds for a equivalent homogeneous reservoir. An expression for the slope is given [11] as:

170 New Technologies in the Oil and Gas Industry

and 

considered.

1.E-01

storage coefficient,

1.E+02

1.E-01

parameter, *ω*=0.05 and *n*=0.8 for a heterogeneous reservoir

1.E+00

P , (t \* P ')D D D

1.E+01

1.E+00

1.E+01

P , (t \* P ')D D D 1.E+02

1.E+03

plot that as the value of *λ* decreases, the transition period shows up later. Finally, Figure 14 shows the effect of changing the value of the flow behavior index for constant values of

. The effect of the increasing the pressure derivative curve's slope is observed as the value of *n* decreases. Needless to say that neither wellbore storage nor skin effects are

> 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 t D

> 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 t D

**Figure 13.** Dimensionless pressure and pressure derivative log-log plot for variable interporosity flow

**Figure 12.** Dimensionless pressure and pressure derivative log-log plot for variable dimensionless

=1x10-6 and *n*=0.2 for a heterogeneous reservoir

 0.3 0.1 0.05 0.03 0.01 0.005

> 1x10 5x10 1x10 5x10 1x10 5x10 1x10


$$nm = \frac{n-1}{n-3} \tag{47}$$

Also, the slope of the pressure derivative during radial flow regime is related to the flow behavior index by:

$$m = -1.8783425 - 7.8618321m^3 + 0.19406557m^{0.5} + 2.8783425e^{-m} \tag{48}$$

As observed in Figure 12, as the dimensionless storage coefficient decreases the transition period is more pronounced no matter the value of the interporosity flow parameter. Therefore, a correlation for 0 1 with an error lower than 3 % as a function of the minimum time value of the pressure derivative during the trough, the flow behavior index and the beginning of the second of the infinite-acting radial flow regime is developed in this study as:

$$\begin{aligned} \frac{1}{\omega} &= \left| 3180.6369 + 551.0582 \left( \ln \frac{t\_{\text{min}}}{t\_{b2}} \right)^2 - \frac{2053.5888}{\text{x}^{0.5}} + \frac{75.337547}{\text{x}} - \frac{1.4787073}{\text{x}^{1.5}} - \right| \\ &- \frac{910.05377}{\text{n}^{0.5}} + \frac{988.80592}{\text{n}} - \frac{459.61296}{\text{n}^{1.5}} + \frac{73.93695}{\text{n}^2} \right| \end{aligned} \tag{49}$$

**Figure 14.** Dimensionless pressure and pressure derivative log-log plot for variable flow behavior index, *ω*=0.03 and =1x10-5 for a heterogeneous reservoir

Another way to estimate uses a correlation which is a function of the intersection time between the unit-slope pseudosteady-state straight line developed during the transition period, the time of the trough. We also found that this correlation is also valid for 0 1 with an error lower than 0.7 %.

$$\omega = 0.019884508 - \frac{1.153351}{y} + \frac{43.428536}{y^2} - \frac{555.85387}{y^3} + \frac{3232.6805}{y^4} - \frac{6716.9801}{y^5}$$

$$-\frac{0.0093613189}{n} + \frac{0.0042870178}{n^2} + \frac{0.00027356586}{n^3} - \frac{0.0005221335}{n^4} + \frac{0.000072466135}{n^5} \tag{50}$$

A final correlation to estimate *ω* valid for 0 1 with an error lower than 0.4 % is given as follows:

$$\omega = \frac{-0.098427346 + 0.00046337048y + 0.00002506358y^2 - 0.00000059316996y^3 + 0.0036057682n - 0.0079959605n^2}{1 - 0.36468068y - 0.064934748n - 0.04796608y^2} \\ (51) \quad (52) \quad (53) \quad (54) \quad (55) \quad (56) \quad (57) \quad (58) \quad (58) \quad (59) \quad (50) \quad (50) \quad (51) \quad (51)$$

The interporosity flow parameter also plays an important role in the characterization of double porosity systems. From Figure 13, it is observed that the smaller the value of the later the transition period to be shown up. A correlation for it was obtained using the time at the trough and the dimensionless storage coefficient, as presented by next expression:

$$\lambda = \frac{\left(6.9690127 \times 10^{-7} + 3.4893668 \times 10^{-8} n - 3.2315082 \times 10^{-8} n^2 - 5.9013807 w + 21571690 w^2 + 3.6102987 \times 10^{12} w^3\right)}{\left(1 + 0.0099353372 n - 3740035.1 w + 6.7143604 \times 10^{12} w^2\right)} \tag{52}$$

Equation 51 is valid for 1x10-4 < < 9x10-7 with an error lower than 4 %. A correlation involving the coordinates of the trough is given as:

$$\begin{aligned} \lambda &= -0.00082917155 - 0.0014247498u - 0.00028717451 \\ z &- 0.00077173053u^2 - 3.2538271 \times 10^{-5} z^2 - 0.0003203949u z - \\ -0.0001423889n^3 &- 1.212213 \times 10^{-6} z^3 - 1.7831692 \times 10^{-5} n z^2 - \\ -8.6457217 \times 10^{-5} n^2 z \end{aligned} \tag{53}$$

Which is valid for 1x10-4 < < 9x10-7 with an error lower than 3.7 %. Another expression for within the same mentioned range involving the minimum time of the trough is given for an error lower than 1.3 %.

$$\ln\lambda = -2.1223034 - 0.09473309n + 0.077489686n^{0.5}\ln(n) - \frac{0.010651118}{n^{0.5}} - \frac{0.043958503}{w^{0.5}}$$

$$+ \frac{1.5653137 \times 10^{-5} \ln w}{w} + \frac{0.00024143014}{w} + \frac{8.7148736 \times 10^{-9}}{w^{1.5}} - \frac{4.0331364 \times 10^{-13}}{w^2} \tag{54}$$

**Example 4.** Figure 15 contains the pressure and pressure derivative log-log plot of a pressure test simulated by [2] with the information given below. It is requested to estimate from these data the dimensionless storage coefficient and the interporosity flow parameter.

**Figure 15.** Pressure and pressure derivative for example 4

172 New Technologies in the Oil and Gas Industry

with an error lower than 0.7 %.

A final correlation to estimate *ω* valid for 0

uses a correlation which is a function of the intersection time

2345

1 with an error lower than 0.4 % is given as

2 3 2 2

12 2

< 9x10-7 with an error lower than 4 %. A correlation

*n n ww w*

(51)

234 5

*y yyyy*

*ynn*

7 8 8 2 2 12 3

*nw w*

3 6 3 5 2

< 9x10-7 with an error lower than 3.7 %. Another expression for

1 5 2

0 5

.

5 9 13

0 010651118 0 043958503 2 1223034 0 09473309 0 077489686

.

*n z nz*

1

(50)

(52)

(54)

(53)

0 5 0 5

*n w*

. .

between the unit-slope pseudosteady-state straight line developed during the transition period, the time of the trough. We also found that this correlation is also valid for 0

1 153351 43 428536 555 85387 3232 6805 6716 9801 0 019884508

.. . .. .

... ..

0 0093613189 0 0042870178 0 00027356586 0 0005221335 0 000072466135

*n nnn n*

0 098427346 0 00046337048 0 000025063353 0 00000050316996 0 0036057682 0 0073959605 1 0 36468068 0 064934748 0 047596083

The interporosity flow parameter also plays an important role in the characterization of double porosity systems. From Figure 13, it is observed that the smaller the value of

the later the transition period to be shown up. A correlation for it was obtained using the time at the trough and the dimensionless storage coefficient, as presented by next

 

2 5 2

 

within the same mentioned range involving the minimum time of the trough is given for an

*z n z nz*

0 0001423889 1 212213 10 1 7831692 10

1 5653137 10 0 00024143014 8 7148736 10 4 0331364 10

*w w w w*

. ln .. .

. . ln . . . ln

0 00082917155 0 0014247498 0 00028717451 0 00077173053 3 2538271 10 0 0003203949

6 9690127 10 3 4893658 10 3 2315082 10 5 9013807 21571690 3 6102987 10 1 0 0099353372 3740035 1 6 7143604 10

.. . . . . . .

> ... .. .

5 2

*λ n nn*

*n z*

*λ n*

.. .

.. . . . . .. . *<sup>y</sup> y y nn <sup>ω</sup>*

Another way to estimate

*ω*

follows:

expression:

Equation 51 is valid for 1x10-4 <

involving the coordinates of the trough is given as:

8 6457217 10

*w*

.

Which is valid for 1x10-4 <

error lower than 1.3 %.

*λ*

**Solution.** From Figure 15 the following characteristic points are read:

*tmin* = 272.6 hr *tb*2 = 14480 hr *tUS,i* =2129.4 hr (*t*\**P*')*min* = 10 psi

Using Equations 5 and 6, the above data are transformed into dimensionless quantities as follows:

*tDmin* =32000 *tDb2* = 17000000 *tDUS,i* =250000 (*tD\*PD*')*min* =0.31

During the infinite-acting radial flow regime the following points were arbitrarily read:

(*t*)*r*1 = 35724.9 hr (*t\*P*')*r*1 = 67.292 psi (*t*)*r*2 =56169.5 hr (*tD\*P*')*r*2 = 61.2283 psi

With these points a slope is estimated to be *m* = 0.108. Equation 47 allows obtaining a flow behavior index of 0.76. The Warren and Root's naturally fractured reservoir parameters are estimated as follows:


**Table 1.** Summary of results for example 4

As a final remark, I would like to comment that some crude oils or other type of fluids used in the oil industry may display a non-Newtonian Bingham-type behavior. It is common to deal with Non Newtonian fluids during fracturing and drilling operations and oil recovery processes, as well. When a reservoir contains a non-Newtonian fluid, such as those injected during EOR with polymers flooding or the production of heavy-oil, the interpretation of a pressure test for these systems cannot be conducted using the conventional models for Newtonian fluid flow since it will lead to erroneous results due to a completely different behavior.

The problem considered now, presented in reference [27], involves the production of a Bingham fluid from a fully penetrating vertical well in a horizontal reservoir of constant thickness; the formation is saturated only with the Bingham fluid. The basic assumptions are: (a) Isothermal, isotropic and homogeneous formation, (b) Single-phase horizontal flow without gravity effects, (c) Darcy's law applies, and (d) Constant fluid properties and formation permeability.

The governing flow equation can be derived by combining the modified Darcy's law with the continuity equation and is expressed in a radial coordinate system as:

$$\frac{k}{r}\frac{\partial}{\partial r}\left[\frac{\rho\left(P\right)}{\mu\_B}r\left(\frac{\partial P}{\partial r} - G\right)\right] = \frac{\partial}{\partial t}\left[\rho\left(P\right)\phi\left(P\right)\right] \tag{55}$$

The density of the Bingham fluid, (*P*), and the porosity of the formation, *i = (P*), are functions of pressure only, so Equation 54 may be rewritten as:

$$\frac{1}{r}\frac{\partial}{\partial r}\left[r\left(\frac{\partial P}{\partial r} - G\right)\right] = \frac{\phi \mu\_B c\_t}{k} \frac{\partial P}{\partial t} \tag{56}$$

The initial condition is:

$$P\left(r, t=0\right) = P\_{\text{i}}, \qquad r \ge r\_w \tag{57}$$

At the wellbore inner boundary, *r = rw*, the fluid is produced at a given production rate, *q*; then, the inner boundary condition is:

$$q = 2\pi rh \frac{k}{\mu\_B} \left(\frac{\partial P}{\partial r} - G\right)\_{r = r\_w} \tag{58}$$

Parameter *G* is de minimum pressure gradient which expressed in dimensionless form yields:

$$G\_D = \frac{Gr\_w kh}{141.2 \eta \mu\_B B} \tag{59}$$

[15] solved numerically Equation 55 and provided an interpretation technique for this type of fluids using the pressure and pressure derivative log-log plot. For a Bingham-type non-Newtonian fluid, this behavior changes by observing that there is a point where the dimensionless pressure derivative is high and this increases with an increase of *GD* and the reservoir radius, Figure 16.

**Figure 16.** Dimensionless pressure and derivative pressure for *reD* = 9375

### **8. Conclusion**

174 New Technologies in the Oil and Gas Industry

behavior.

formation permeability.

The initial condition is:

yields:

The density of the Bingham fluid,

then, the inner boundary condition is:

As a final remark, I would like to comment that some crude oils or other type of fluids used in the oil industry may display a non-Newtonian Bingham-type behavior. It is common to deal with Non Newtonian fluids during fracturing and drilling operations and oil recovery processes, as well. When a reservoir contains a non-Newtonian fluid, such as those injected during EOR with polymers flooding or the production of heavy-oil, the interpretation of a pressure test for these systems cannot be conducted using the conventional models for Newtonian fluid flow since it will lead to erroneous results due to a completely different

The problem considered now, presented in reference [27], involves the production of a Bingham fluid from a fully penetrating vertical well in a horizontal reservoir of constant thickness; the formation is saturated only with the Bingham fluid. The basic assumptions are: (a) Isothermal, isotropic and homogeneous formation, (b) Single-phase horizontal flow without gravity effects, (c) Darcy's law applies, and (d) Constant fluid properties and

The governing flow equation can be derived by combining the modified Darcy's law with

<sup>1</sup> *B t P P <sup>μ</sup> <sup>c</sup> r G rr r k t* 

At the wellbore inner boundary, *r = rw*, the fluid is produced at a given production rate, *q*;

*k P q πrh G*

Parameter *G* is de minimum pressure gradient which expressed in dimensionless form

141 2. *<sup>w</sup> <sup>D</sup>*

*Gr kh <sup>G</sup>*

*<sup>w</sup> B r r*

*B*

*μ r*

*P P*

(*P*), and the porosity of the formation,

, , 0 *i w Prt P r r* (57)

(58)

*<sup>q</sup><sup>μ</sup> <sup>B</sup>* (59)

(55)

*i = (P*), are

(56)

the continuity equation and is expressed in a radial coordinate system as:

*k P <sup>ρ</sup> r G <sup>ρ</sup> r r μ r t*

*B P*

functions of pressure only, so Equation 54 may be rewritten as:

2

This chapter comprises the most updated state-of-the-art for well test interpretation in reservoirs having a non-Newtonian fluid. Extension of the *TDS* technique along with practical examples is given for demonstration purposes. This should be of extreme importance since most heavy oil fluids behave non-Newtonially, then, its characterization using conventional analysis is inappropriate and the methodology presented here are strongly recommended.

#### **Nomenclature**



**Table 2.** Nomenclature of main variables


**Table 3.** Greeks


**Table 4.** Suffices

176 New Technologies in the Oil and Gas Industry

*k* Permeability, md

*m* Slope

*P* Pressure, psi

*t* Time, hr *r* Radius, ft

*w* 

**Table 3.** Greeks

*h* Formation thickness, ft

*G* Group defined by Equation 3 *G* Minimum pressure gradient, Psi/ft *GD* Dimensionless pressure gradient

*k* Flow consistency parameter *kfwf* Fracture conductivity, md-ft

*q* Flow/injection rate, STB/D

*t\*P'* Pressure derivative, psi

*xf* Half-fracture length, ft

**Table 2.** Nomenclature of main variables

Dimensionless interposity parameter

*<sup>B</sup>*Bingham plastic coefficient, cp

*\** Characteristic viscosity, cp/ft1-n

Dimensionless storativiy coefficient

*eff* Effective viscosity for power-law fluids, cp\*(s/ft)n-1

*z* ln [(*tD*\**PD*')min/*tDmin*]

/*tDmin s* Laplace parameter

*x tmin*/*tb*<sup>2</sup>

*y tUSi*/*tmin*

Change, drop

Shear rate, s-1

Viscosity, cp

Porosity, Fraction

Shear stress, N/m

Shear stress, N/m

*tD\*PD'* Dimensionless pressure derivative

*H* Consistency (Power-law parameter), cp\*sn-1

*n* Flow behavior index (power-law parameter)

*ra* Distance from well to non-Newtonian/Newtonian front/interface
