**3. Pseudoplastic infinite-acting radial flow regime in homogeneous formations**

Interpretation of pressure tests for non-Newtonian fluids is performed differently to conventional Newtonian fluids. During radial flow regime, Non-Newtonian fluids exhibit a pressure derivative curve which is not horizontal but rather inclined. As shown by [12], the smaller the value of *n* (flow behavior index) the more inclined is the infinite-acting pressure derivative line, see Figure 2.

A partial differential equation for radial flow of non-Newtonian fluids that follow a powerlaw relationship through porous media was proposed [11]. Coupling the non-Newtonian Darcy's law with the continuity equation, they derived a rigorous partial differential equation:

$$\frac{\stackrel{\frown}{\circlearrowleft}^{2}P}{\stackrel{\frown}{\circlearrowleft r}^{2}} + \frac{n}{r}\frac{\stackrel{\frown}{\circlearrowleft P}}{\stackrel{\frown}{\circlearrowleft r}} = c\_t \phi \text{n} \left(\frac{\mu\_{eff}}{k}\right)^{1/n} \left(-\frac{\stackrel{\frown}{\circlearrowleft P}}{\stackrel{\frown}{\circlearrowleft P}}\right)^{(n-1)/n} \frac{\stackrel{\frown}{\circlearrowleft P}}{\stackrel{\frown}{\circlearrowleft P}}\tag{1}$$

This equation is nonlinear. For analytical solutions, a linearized approximation was also derived by [11]:

$$\frac{1}{r^n} \frac{\partial}{\partial r} \left( r^n \frac{\partial P}{\partial r} \right) = Gr^{1-n} \frac{\partial P}{\partial t} \tag{2}$$

Where:

$$\mathbf{G} = \frac{\mathbf{3792.188n\phi c\_t\mu\_{eff}}}{k} \left( \mathbf{96681.605\frac{h}{qB}} \right)^{1-n} \tag{3}$$

and,

$$
\mu\_{eff} = \left(\frac{H}{12}\right) \left(9 + \frac{3}{n}\right)^n \left(1.59344 \times 10^{-12} \, k\phi\right)^{(1-n)/2} \tag{4}
$$

**Figure 2.** Pressure derivative for a pseudoplastic non-Newtonian fluid in an infinite reservoir – After Reference [12]

The dimensionless quantities were also introduced by [10] as

156 New Technologies in the Oil and Gas Industry

and successfully tested with synthetic data.

**formations** 

equation:

derived by [11]:

Where:

and,

derivative line, see Figure 2.

As far as non-Newtonian fluid flow through naturally fractured reservoirs is concerned only a study presented by [19] is reported in the literature. He presented the analytical solution for the transient behavior of double-porosity infinite formations which bear a non-Newtonian pseudoplastic fluid and his analytical solution also considers wellbore storage effects and skin factor; therefore, [2] used the analytical solution without wellbore storage and skin introduced by [19] was used to develop an interpretation technique using the pressure and pressure derivative, so expressions to estimate the Warren and Root parameters [26] (dimensionless *storage coefficient and interporosity flow parameter)* were found

**3. Pseudoplastic infinite-acting radial flow regime in homogeneous** 

Interpretation of pressure tests for non-Newtonian fluids is performed differently to conventional Newtonian fluids. During radial flow regime, Non-Newtonian fluids exhibit a pressure derivative curve which is not horizontal but rather inclined. As shown by [12], the smaller the value of *n* (flow behavior index) the more inclined is the infinite-acting pressure

A partial differential equation for radial flow of non-Newtonian fluids that follow a powerlaw relationship through porous media was proposed [11]. Coupling the non-Newtonian Darcy's law with the continuity equation, they derived a rigorous partial differential

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

*P nP μ P P*

*r rr k r t* 

This equation is nonlinear. For analytical solutions, a linearized approximation was also

1 *n n* <sup>1</sup>

*P P r Gr r rr t*

<sup>1</sup> 3792 188 96681 605 . .

.

*μ k n*

*k qB* 

1 2 <sup>3</sup> <sup>12</sup> 9 1 59344 10

*<sup>n</sup> <sup>n</sup>*

*t eff n c <sup>μ</sup> <sup>h</sup> <sup>G</sup>*

*t*

*n*

12

*H*

*eff*

*c n*

/ ( )/ *<sup>n</sup> n n eff*

(2)

*n*

( )/

(1)

(3)

(4)

2

$$P\_{DNN} = \frac{\Lambda P}{141.2 \left(96681.605\right)^{1-n} \left(\frac{qB}{h}\right)^n \frac{\mu\_{eff} r\_w^{1-n}}{k}}\tag{5}$$

$$t\_{\rm DNN} = \frac{t}{Gr\_w^{3-n}}\tag{6}$$

$$P\_{DN} = \frac{k \text{ } h\Delta P}{141.2q\mu\_N B} \tag{7}$$

$$t\_{DN} = \frac{0.0002637k \text{ t}}{\phi \mu\_N c\_t r\_w^2} \tag{8}$$

$$r\_D = \frac{r}{r\_w} \tag{9}$$

Where suffix *N* indicates Newtonian and suffix *NN* indicates non-Newtonian. The dimensionless well pressure analytical solution in the Laplace space domain for the case of a well producing a pseudoplastic non-Newtonian fluid at a constant rate from an infinite reservoir is given in reference [11]:

$$\overline{P}\_{D}\left(\hat{s}\right) = \frac{\mathsf{K}\_{\upsilon}\left(\boldsymbol{\beta}\sqrt{\tilde{\boldsymbol{s}}}r\_{D}^{\mathcal{N}\_{\beta}}\right) + s\sqrt{\tilde{\boldsymbol{s}}}\mathsf{K}\_{\beta}\left(\boldsymbol{\beta}\sqrt{\tilde{\boldsymbol{s}}}\right)}{\tilde{s}\left(\sqrt{\tilde{\boldsymbol{s}}}\mathsf{K}\_{\beta}\left(\boldsymbol{\beta}\sqrt{\tilde{\boldsymbol{s}}}\right) + \tilde{s}\mathsf{C}\_{D}\left[\boldsymbol{K}\_{\upsilon}\left(\boldsymbol{\beta}\sqrt{\tilde{\boldsymbol{s}}}\right) + s\sqrt{\tilde{\boldsymbol{s}}}\mathsf{K}\_{\beta}\left(\boldsymbol{\beta}\sqrt{\tilde{\boldsymbol{s}}}\right)\right]\right)}\tag{10}$$

Being = 2/(3-*n*) and = (1-*n*)/(3*-n*).

The dimensionless pressure derivative during radial flow regime is governed by:

$$\left(t\_D \stackrel{\*}{\*} P\_D \stackrel{\cdot}{)}\_{r \text{NN}} = 0.5 t\_{D \text{NN}}^{\alpha} \tag{11}$$

[12] presented the following expression to estimate the permeability,

$$\frac{k}{t^{\mu}\_{eff}} = \left[ 0.5 \frac{t\_r^{\alpha}}{\mathcal{C}^{\alpha} \left( t^\* \,\Delta P^\circ \right)\_r} \frac{\left( 2 \pi h \right)^{\eta \left( \alpha - 1 \right)} r\_w^{\left( 1 - \eta \right) \left( 1 - \alpha \right)}}{q^{\eta \left( \alpha - 1 \right) - \alpha}} \right]^{\frac{1}{1 - \alpha}} \tag{12}$$

where <sup>2</sup> α 0 1486 0 178 0 3279 . .. *n n*

being *n* the flow behavior index which may be found from the slope of the pressure derivative curve during radial flow regime. [12] also introduced another expressions and correlations to find permeability, skin factor and wellbore storage coefficient using the maximum point (peak) found on the pressure derivative curve during wellbore storage effects which are not shown here. The point of intercept between the early unit-slope line and radial flow regime is used to estimate wellbore storage:

$$t\_i = \frac{\left(3.13e^{-1.85n}\right)\mathbb{C}}{\left(2\pi h\right)^n} \frac{\mu\_{eff}}{k} \left(\frac{q}{r\_w}\right)^{n-1} \tag{13}$$

Parameters in both Equations 11 and 12 are given in CGS (centimeters, grams, seconds) units.

[1] presented more practical expressions for the determination of both permeability and skin factor:

$$\frac{k}{\mu\_{eff}} = \left[ 70.6 \left( 96681.605 \right)^{(1-\alpha)(1-\eta)} \left( \frac{0.0002637 t\_r}{n \phi \mathbf{c}\_t} \right)^{\alpha} \left( \frac{qB}{h} \right)^{n-\alpha\left(n-1\right)} \left( \frac{1}{\left(t^\* \ \Delta P' \right)\_r} \right) \right]^{\frac{1}{\left[1-\alpha\right]}} \tag{14}$$

Where is the slope of the pressure derivative curve and is defined by:

$$\alpha = \frac{1 - n}{3 - n} \tag{15}$$

Transient Pressure and Pressure Derivative Analysis for Non-Newtonian Fluids 159

$$s\_{r\text{NNN}} = \frac{1}{2} \left( \frac{\left(\Delta P\right)\_{r\text{NN}}}{\left(t^\* \,\Delta P'\right)\_{r\text{NN}}} - \frac{1}{a} \left| \left(\frac{t\_{r\text{NN}}}{G \,\,r\_w^{3-n}}\right)^a \right. \tag{16}$$

#### **4. Well pressure behavior in non-Newtonian/Newtonian interface**

158 New Technologies in the Oil and Gas Industry

= 2/(3-*n*) and

Being 

units.

factor:

Where  *D*

*P s*

where <sup>2</sup> α 0 1486 0 178 0 3279 . .. *n n*

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

[12] presented the following expression to estimate the permeability,

*tP t* (11)

1

(10)

(12)

(13)

1

(14)

(15)

1 1

 

*α n α n α α*

1

1 1 1 1

*β D ν β*

*s sK β s sC K β s s sK β s*

*K β sr s sK β s*

1 *β ν β D*

\*' . 0 5 *<sup>α</sup> D D DNN rNN*

2

being *n* the flow behavior index which may be found from the slope of the pressure derivative curve during radial flow regime. [12] also introduced another expressions and correlations to find permeability, skin factor and wellbore storage coefficient using the maximum point (peak) found on the pressure derivative curve during wellbore storage effects which are not shown here. The point of intercept between the early unit-slope line

1 85 <sup>1</sup> 3 13

. . *<sup>n</sup> <sup>n</sup> eff*

*k r πh* 

Parameters in both Equations 11 and 12 are given in CGS (centimeters, grams, seconds)

[1] presented more practical expressions for the determination of both permeability and skin

*k t qB*

*eff t r*

*μ nc h t P*

*α*

is the slope of the pressure derivative curve and is defined by:

1 1 0 0002637 <sup>1</sup> 70 6 96681 605 . . . \* '

1 3 *n*

*n*

 

 

*<sup>α</sup> <sup>n</sup> <sup>α</sup> <sup>n</sup> <sup>α</sup> <sup>α</sup> <sup>n</sup> <sup>r</sup>*

*r w*

 

The dimensionless pressure derivative during radial flow regime is governed by:

*<sup>α</sup> <sup>n</sup> α α eff <sup>r</sup> k t πh r*

*<sup>i</sup> <sup>n</sup> <sup>w</sup> e C <sup>μ</sup> <sup>q</sup> <sup>t</sup>*

2

0 5. \* '

and radial flow regime is used to estimate wellbore storage:

*μ Ct P q*

In many activities of the oil industry, engineers have to deal with completion and stimulation treatment fluids such as polymer solutions and some heavy crude oils which obey a non-Newtonian power-law behavior. When it is required to conduct a treatment with a non-Newtonian fluid in an oil-bearing formation, this comes in contact with conventional oil which possesses a Newtonian nature. This implies the definition of two media with entirely different mobilities. If a pressure test is run in such a system, the interpretation of data from such a test through the use of conventional straight-line method may be erroneous and may not provide a way for verification of the results obtained. Then, [13] proposed a solution for the system sketched in Figure 3 which was solved numerically by [17].

[15] presented for the first time the pressure derivative behavior for the mentioned system, Figure 4. Notice in that plot that the pressure derivative shows an increasing slope as the flow behavior index decreases. Also, the derivative has no slope during infinite-acting Newtonian behavior, as expected.

During the non-Newtonian region, region 1 in Figure 3, Equations 13 to 15 work well. For the Newtonian region, region 2, the permeability and skin factor are estimated with the equations presented by Tiab (1993) as:

**Figure 3.** Composite non-Newtonian/Newtonian radial reservoir

**Figure 4.** Dimensionless pressure derivative behavior for *ra* = 200 ft. Case Non-Newtonian pseudoplastic

$$k\_2 = \frac{70.6q\mu\_N B}{h(t^\* \ \Delta P')\_r} \tag{17}$$

$$s\_2 = \frac{1}{2} \left[ \left( \frac{\Delta P}{t^\* \ \Delta P^\circ} \right)\_{r2} - \ln \left( \frac{k\_2 t\_{r2}}{\phi \mu\_N c\_t r\_w^2} \right) + 7.43 \right] \tag{18}$$

Suffix 2 denotes the non-Newtonian region.

[15] also found an expression to estimate the non-Newtonian permeability using the time of intersection of the non-Newtonian and Newtonian radial lines, *tiN\_NN*:

$$k = \left[ \left\langle \left( \frac{H}{12} \right) \left( 9 + \frac{3}{n} \right)^{\mathrm{n}} \left( 1.59344 \times 10^{-12} \,\mathrm{\phi} \right)^{(1-\mathrm{n})/2} \left( 96681.605 \frac{hr\_w}{qB} \right)^{\mathrm{l}-\mathrm{n}} \right\rangle^{\mathrm{l}-\mathrm{l}/\mathrm{a}} \frac{\mu\_{\mathrm{N}}^{\mathrm{l}/\mathrm{a}} \phi\_{\mathrm{l}} r\_w^2 n}{0.0002637 t\_{\mathrm{i}\mathrm{N}\\_\mathrm{NN}}} \right]^{\mathrm{l}/2} \tag{19}$$

The radius of the injected non-Newtonian fluid bank is calculated using the following correlation (not valid for *n*=1), obtained from reading the time at which the pressure derivative has its maximum value:

$$r\_a = \left[\frac{G\left(0.2258731n^3 - 0.2734284n^2 + 0.5064602n + 0.5178275\right)^{1/\alpha}}{t\_{MAX}}\right]^{1/\alpha} \tag{20}$$

[13] found that the radius of the non-Newtonian fluid bank can be found using the radius investigation equation proposed by [10]:

160 New Technologies in the Oil and Gas Industry

0.1

pseudoplastic

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10

*n* 1 0.8 0.6 0.4 0.2

2

intersection of the non-Newtonian and Newtonian radial lines, *tiN\_NN*:

*Gn n n*

*s*

Suffix 2 denotes the non-Newtonian region.

derivative has its maximum value:

*a*

*r*

*t DNN*

70 6. ( \* ') *N r*

*<sup>q</sup><sup>μ</sup> <sup>B</sup> <sup>k</sup>*

*P k t*

*t P μ c r*

[15] also found an expression to estimate the non-Newtonian permeability using the time of

12 0 0002637

*<sup>H</sup> hr <sup>μ</sup> cr n <sup>k</sup>*

. . .

The radius of the injected non-Newtonian fluid bank is calculated using the following correlation (not valid for *n*=1), obtained from reading the time at which the pressure

*n qB t* 

.2258731 .2734284 .5 646 2 .5178275 *<sup>α</sup> <sup>n</sup>*

 

*MAX*

*t*

<sup>1</sup> <sup>3</sup> 3 2 0 0 00 0 0

<sup>1</sup> 7 43

ln . \* '

 

*r N tw*

1 2 1 1 <sup>1</sup> 1 2 1 2 <sup>3</sup> <sup>12</sup> 9 1 59344 10 96681 605

*<sup>α</sup> <sup>n</sup> <sup>n</sup> <sup>α</sup> <sup>n</sup> <sup>w</sup> N tw*

2 2

*r*

*ht P* (17)

/ / / /

(18)

\_

1

/

*irN NN*

(19)

(20)

**Figure 4.** Dimensionless pressure derivative behavior for *ra* = 200 ft. Case Non-Newtonian

2

2 2 2

1

*t D\*P D'*

10

$$r\_a = \left[ \Gamma \left( \frac{2}{3-n} \right) \right] \bigvee^{(n-1)} \left[ \frac{\left( 3-n \right)^2 t}{G} \right] \bigvee^{(3-n)} \tag{21}$$

where *t* is the end time of the straight line found on a non-Newtonian Cartesian graph of *P* vs. *t*1-*n*/3-*<sup>n</sup>*.

Later, [16] found that Equations 13, 14, 15 and 22 also worked for dilatant systems. This is the case when 2 < *n* < 1. The pressure derivative behavior is given in Figure 5. Notice that for this case the slope decreases as the flow behavior index increases. For dilatant-Newtonian interface the position of the front obeys the following equation:

**Figure 5.** Dimensionless pressure derivative behavior for *ra* = 200 ft. Case Non-Newtonian dilatant

**Example 1.** A constant-rate injection test for a well in a closed reservoir was generated by [13] with the data given below. It is required to estimate the permeability and the skin factor in each area and the radius of injected non-Newtonian fluid bank.


n = 0.6

**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. From Figure 6 the following information was read:


First, 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 is found with Equation 16 to be 179.7.

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

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 resulted to be 120 ft.

Equations 17 and 18 were used to estimate permeability and skin factor of the Newtonian zone. They resulted to be 100 md and 4.5.

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