Transient Pressure and Pressure Derivative Analysis for Non-Newtonian Fluids

1. Introduction

Conventional well test interpretation models do not work in reservoirs containing non-Newtonian fluids such as completion and stimulation treatment fluids: polymer solutions, foams, drilling muds (this should not be considered as a reservoir fluid, since before testing we should clean the well to remove all the drilling invasion fluids, however it obeys the power-law), etc., and some paraffinic oils and heavy crude oils. Non-Newtonian fluids are generally classified as time independent, time dependent and viscoelastic. Examples of the first classification are the Bingham, pseudoplastic and dilatant fluids, Figure 1, which are commonly dealt by petroleum engineers.

As a special kind of non-Newtonian fluid, Bingham fluids (or plastics) exhibit a finite yield stress at zero shear rates. There is no gross movement of fluids until the yield stress, _y, is exceeded. Once this is accomplished, it is also required cutting efforts to increase the shear rate, i.e. they behave as Newtonian fluids. These fluids behave as a straight line crossing the y axis in = _y, when the shear stress, plotted against the shear rate, in Cartesian coordinates. The characteristics of these fluids are defined by two constants: the yield, _y, which is the stress that must be exceeded for flow to begin, and the Bingham plastic coefficient, _B. The rheological equation for a Bingham plastic is,

$$\tau ={\tau }_y+{\mu }_B\gamma .$$E1

The Bingham plastic concept has been found to approximate closely many real fluids existing in porous media, such as paraffinic oils, heavy oils, drilling muds and fracturing fluids, which are suspensions of finely divided solids in liquids. Laboratory investigations have indicated that the flow of heavy-oil in some fields has non-Newtonian behavior and approaches the Bingham type.

Figure 1.

Schematic representation of time-independent fluid

Pseudoplastic and dilatant fluids have no yield point. The slope of shear stress versus shear rate decreases progressively and tends to become constant for high values of shear stress for pseudoplastic fluids. The simplest model is power law,

$$\tau =k{\gamma }^n;n<1.$$E2

k and n are constants which differ for each particular fluid. k measures the flow consistency and n measures the deviation from the Newtonian behavior which k = and n = 1.

Dilatants fluids are similar to pseudoplastic except that the apparent viscosity increases as the shear stress increases. The power-law model also describes the behavior of dilatant fluids but n > 1.

Currently, unconventional reservoirs are the most impacting subject in the oil industry. Shale reservoirs, coalbed gas, tight gas, gas hydrates, gas storage, geothermal energy, coal – conversion to gas, coal-to-gas, in-situ gasification and heavy oil are considered unconventional reservoirs. In the field of well testing, several analytical and numerical models taking into account Bingham, pseudoplastic and dilatant non-Newtonian behavior have been introduced in the literature to study their transient nature in porous media for a better reservoir characterization. Most of them deal with fracture wells, homogeneous and double-porosity formations and well test interpretation is conducted via the straight-line conventional analysis or type-curve matching and recently some studies involving the pressure derivative have also been introduced.

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.

The purpose of this chapter is to provide the most updated state-of-the-art on transient analysis of Non-Newtonian fluids and to present both conventional and modern methodologies for well test interpretation in reservoirs saturated with such fluids. Especial interest is given to the use of the pressure and pressure derivative for both homogeneous and double-porosity formations.

2. Non-Newtonian fluids in transient pressure analysis

Non-Newtonian fluids are often used during various drilling, workover and enhanced oil recovery processes. Most of the fracturing fluids injected into reservoir-bearing formations behave non-Newtonianly and these fluids are often approximated by Newtonian fluid flow models. In the field of well testing, several analytical and numerical models taking into account Bingham and pseudoplastic non-Newtonian behavior have been introduced in the literature to study the transient nature of these fluids in porous media for a better reservoir characterization. Most of them deal with fracture wells and homogeneous formations and well test interpretation is conducted via the straight-line conventional analysis or type-curve matching. Only a few studies consider pressure derivative analysis. However, there exists a need of a more practical and accurate way of characterizing such systems.

Many studies in petroleum and chemical engineering and rheology have focused on non-Newtonian fluid behavior though porous formations, among them, we can name [6, 9, 10, 18, 20, 23]. Several numerical and analytical models have been proposed to study the transient behavior of non-Newtonian fluid in porous media. Since all of them were published before the eighties, when the pressure derivative concept was inexistent; interpretation technique was conducted using either conventional analysis or type-curve matching.

It is worth to recognize that Ikoku has been the researcher who has contributed the most to non-Newtonian power-law fluids modeling, as it is demonstrated in the works of [9,10,11,13]. All of these models have been used later for other researchers for further development of test interpretation techniques. For instance, reference [24] presented a study of a pressure fall-off behavior after the injection of a non-Newtonian power-law fluid. [14] presented a study using the elliptical flow on transient analysis interpretation in Polymer flooding EOR since polymer solutions also exhibit non-Newtonian rheological behavior such as in-situ shear-thinning and shear-thickening effects.

[25] used for the first time the pressure-derivative concept for well test analysis of non-Newtonian fluids, and later on, [12] presented the first extension of the TDS (Tiab’s Direct Synthesis) technique, [21] to non-Newtonian fluids. [7] used type-curve matching for interpretation of pressure test for non-Newtonian fluids in infinite systems with skin and wellbore storage effects. Recent applications of the derivative function to non-Newtonian system solutions are presented by [1] and [15] who applied the TDS technique to radial composite reservoirs with a Non-Newtonian/Newtonian interface for pseudoplastic and dilatants systems, respectively.

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 and successfully tested with synthetic data.

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 power-law 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:

$$\frac{{\partial }^{2}P}{\partial r^2}+\frac{n}{r}\frac{\partial P}{\partial r}=c_{t}fn{\left(\frac{{\mu }_{eff}}{k}\right)}^{1/n}{\left(-\frac{\partial P}{\partial r}\right)}^{(n-1)/n}\frac{\partial P}{\partial t}$$E3

(1)

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

$$\frac{1}{r^n}\frac{\partial }{\partial r}\left(r^n\frac{\partial P}{\partial r}\right)=G{r}^{1-n}\frac{\partial P}{\partial t}$$E4

(2)

Where:

$$G=\frac{3792.188nf{c}_t{\mu }_{eff}}{k}{\left(96681.605\frac{h}{qB}\right)}^{1-n}$$E5

(3)and,

$${\mu }_{eff}=\left(\frac{H}{12}\right){\left(9+\frac{3}{n}\right)}^n{\left(1.59344\times {10}^{-12}kf\right)}^{(1-n)/2}$$E6

(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

$$P_{DNN}=\frac{\Delta P}{141.2{\left(96681.605\right)}^{1-n}{\left(\frac{qB}{h}\right)}^n\frac{{\mu }_{eff}{r}_w^{1-n}}{k}}$$E7

(5)

$$t_{DNN}=\frac{t}{G{r}_w^{3-n}}$$E8

(6)

$$P_{DN}=\frac{k_{}h\Delta P}{141.2q{\mu }_{N}B}$$E9

(7)

$$t_{DN}=\frac{0.0002637k_{}t}{f{\mu }_N{c}_t{r}_w^2}$$E10

(8)

$$r_D=\frac{r}{r_w}$$E11

(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]:

$${\overline{P}}_D\left(\tilde{s}\right)=\frac{K_{\nu }\left(\beta \sqrt{\tilde{s}}{r}_D^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}\right)+s\sqrt{\tilde{s}}{K}_{\beta }\left(\beta \sqrt{\tilde{s}}\right)}{\tilde{s}\left(\sqrt{\tilde{s}}{K}_{\beta }\left(\beta \sqrt{\tilde{s}}\right)+\tilde{s}{C}_D\left[K_{\nu }\left(\beta \sqrt{\tilde{s}}\right)+s\sqrt{\tilde{s}}{K}_{\beta }\left(\beta \sqrt{\tilde{s}}\right)\right]\right)}$$E12

(10)

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

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

$${\left(t_D*P_D^{}\text{'}\right)}_{rNN}=0.5t_{DNN}^{\alpha }$$E13

(11)

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

$$\frac{k}{{\mu }_{eff}}={\left[0.5\frac{t_r^{\alpha }}{C^{\alpha }{\left(t*\Delta P\text{'}\right)}_r}\frac{{\left(2\pi h\right)}^{n\left(\alpha -1\right)}{r}_w^{\left(1-n\right)\left(1-\alpha \right)}}{q^{n\left(\alpha -1\right)-\alpha }}\right]}^{\frac{1}{1-\alpha }}$$E14

(12)

where

$$\alpha =-0.1486n^2-0.178n+0.3279$$E15

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:

$$t_i=\frac{\left(3.13e^{-1.85n}\right)C}{{\left(2\pi h\right)}^n}\frac{{\mu }_{eff}}{k}{\left(\frac{q}{r_w}\right)}^{n-1}$$E16

(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:

$$\frac{k}{{\mu }_{eff}}={\left[70.6{\left(96681.605\right)}^{\left(1-\alpha \right)\left(1-n\right)}{\left(\frac{0.0002637t_r}{nf{c}_t}\right)}^{\alpha }{\left(\frac{qB}{h}\right)}^{n-\alpha \left(n-1\right)}\left(\frac{1}{{\left(t*\Delta P\text{'}\right)}_r}\right)\right]}^{\frac{1}{1-\alpha }}$$E17

(14)

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

$$\alpha =\frac{1-n}{3-n}$$E18

(15)

$$s_{rNN}=\frac{1}{2}\left(\frac{{\left(\Delta P\right)}_{rNN}}{{\left(t*\Delta P\text{'}\right)}_{rNN}}-\frac{1}{\alpha }\right){\left(\frac{t_{rNN}}{G_{}{r}_w^{3-n}}\right)}^{\alpha }$$E19

(16)

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

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

$$k_2=\frac{70.6q{\mu }_{N}B}{h{\text{(}t*\Delta P\text{'}\text{)}}_r}$$E20

(17)

$$s_2=\frac{1}{2}\left[{\left(\frac{\Delta P}{t*\Delta P\text{'}}\right)}_{r2}-\text{ln}\left(\frac{k_2{t}_{r2}}{f{\mu }_N{c}_t{r}_w^2}\right)+7.43\right]$$E21

(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, t_{iN_NN}:

$$k={\left[{⟨\left(\frac{H}{12}\right){\left(9+\frac{3}{n}\right)}^n{\left(1.59344\times {10}^{-12}f\right)}^{\left(1-n\right)/2}{\left(96681.605\frac{h{r}_w}{qB}\right)}^{1-n}⟩}^{1-1/\alpha }\frac{{\mu }_N^{1/\alpha }f{c}_t{r}_w^{2}n}{0.0002637t_{irN\_NN}}\right]}^{1/2}$$E22

(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:

$$r_a={\left[\frac{G{\left(0.\text{2258731}{n}^3-0.\text{2734284}{n}^2+0.\text{5}0\text{646}0\text{2}n+0.\text{5178275}\right)}^{1/\alpha }}{t_{MAX}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(n-3\right)$}\right.}$$E23

(20)

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

$$r_a={\left[\Gamma \left(\frac{2}{3-n}\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(n-1\right)$}\right.}{\left[\frac{{\left(3-n\right)}^{2}t}{G}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(3-n\right)$}\right.}$$E24

(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-n.

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:

$$r_a={\left[\frac{G{\left(0.46811e^{0.76241n}\right)}^{1/\alpha }}{t_{e\_rNN}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(n-3\right)$}\right.}$$E25

(22)

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.

P_R = 2500 psi r_e = 2625 ftr_w = 0.33 fth = 16.4 ft

= 20 % k = 100 mdq = 300 BPDB = 1.0 rb/STB

c_t = 6.89x10^-6 1/psi r_a = 131.2 ftH = 20 cp*s^n-1_N = 3 cp

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:

t_r1 = 0.3P_r1 = 541.54 psi(t*P’)_r1 = 105.45 psi

t_r2 = 120P_r2 = 991.5 psi(t*P’)_r2 = 39.02 psi

t_MAX = 1.3t_{irN_NN} = 0.0008 hr

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/(ft^3-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 r_a = 116 ft from conventional analysis.

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:

$$P_D\left(t_D\right)=\frac{{(3-n)}^{2v}{t}_D^v}{\left(1-n\right)\text{Γ}(1-\text{ν})}-\frac{1}{1-n}$$E26

(23)

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

[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,

$$P_D={\left(\frac{\pi }{2}\right)}^{\frac{n-1}{2}}\sqrt{\pi *t_{Dxf}}$$E27

(24)

$$t_{Dxf}=\frac{0.0002637kt}{f{c}_t{\mu }^{*}{x}_f^2}$$E28

(25)

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

$${\mu }^{*}={\mu }_{eff}{\left(96681.605\frac{h}{qB}\right)}^{1-n}$$E29

(26)

And the derivative of Equation 24 is:

$$t_{Dxf}*P_D\text{'}=0.5{\left(\frac{\pi }{2}\right)}^{\frac{n-1}{2}}\sqrt{\pi t_{Dxf}}$$E30

(27)

And the dimensionless fractured conductivity is;

$$c_{fD}=\frac{k_f{w}_f}{k{x}_f}$$E31

(28)

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

$$k_f{w}_f=\frac{3.31739k}{\frac{e^s}{r_w}-\frac{1.92173}{x_f}}$$E32

(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 infinite-conductivity 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, t_RLi, 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

$$x_f={\left[0.028783\frac{{\left(1.570796\right)}^{\frac{n-1}{2}}}{{\left(\frac{0.0002637k{t}_{LRi}}{f{c}_t{\mu }^{*}}\right)}^{\alpha }}\sqrt{\frac{t_{LRi}k}{f{c}_t{\mu }^{*}}}\right]}^{\alpha }$$E33

(30)

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

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

$$t_D*P_D^{}\text{'}=2\pi t_{DA}$$E34

(31)

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

$$A=\pi {\left[\frac{t_{iLPSS}}{0.0625{\left(\frac{\pi }{2}\right)}^{n-1}G}\right]}^{2/(3-n)}$$E35

(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.

n = 0.4h = 70 ftk = 0.65 mdq = 507.5 BPD

= 10 %B = 1 rb/STB μ^* = 0.00065 cpc_t = 0.00001 psi^-1

r_w = 0.26 ftH = 20 cp*s^n-1

Figure 8.

Pressure and pressure derivative for example 2

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

t_LRi =0.4495 hrt_r =0.7217 hrP_r = 762 psi(t*P’)_r = 522.06 psi

Using Equation 15, a value of 0.23 is found for . Reservoir permeability, skin factor, half-fracture 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/(ft^3-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:

$$P_{DNN}\left(r_D,0\right)=0$$E36

(33)

$${\left(\frac{\partial P_{DNN}}{\partial r_D}\right)}_{r_D=1}=-1_{}fo{r}_{}{t}_{DNN}>0$$E37

(34)

$${\left(\frac{\partial P_{DNN}}{\partial r_D}\right)}_{r_{{}_{eD}}=1}=0_{}fo{r}_{}{t}_{DNN}$$E38

(35)

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

$$\overline{P}(\tilde{s})=\frac{\left\{K_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]•I_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]+I_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]•K_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]\right\}}{\left({\tilde{s}}^{3/2}\left\{I_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]•K_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]-K_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]•I_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]\right\}\right)}$$E39

(36)

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

$$P_{DNN}\left(r_{eD},t_{DNN}\right)=0$$E40

(37)

And the analytical solution for such case is:

$$\overline{P}(\tilde{s})=\frac{\left\{I_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]•K_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]-K_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]•I_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]\right\}}{\left({\tilde{s}}^{3/2}\left\{I_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]•K_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]+K_{2/\left(3-n\right)}\left[\frac{2}{3-n}\sqrt{\tilde{s}}\right]•I_{\frac{1-n}{3-n}}\left[\frac{2}{3-n}\sqrt{\tilde{s}}{r}_{eD}{}^{\left(3-n\right)/2}\right]\right\}\right)}$$E41

(38)

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:

$$t_{DA}=\frac{t_{}}{G\left(\pi r_e{}^{3-n}\right)}$$E42

(39)

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

$$A=\pi {\left[\frac{t_{rpiNN}}{G}*{\left(\frac{1}{4}\right)}^{1/\alpha -1}\right]}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3-n$}\right.}$$E43

(40)

Where t_rpiNN 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.

Figure 9.

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

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

$$t_{D{A}_{NN}}=-0.003n^2+0.0337n+0.052$$E44

(41)

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

$$A=\pi {\left[\frac{t_{rsiNN}}{G\pi \left(-0.003n^2+0.0337n+0.052\right)}\right]}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{${}^{3-n}$}\right.}$$E45

(42)

For dilatant fluids the correlation found is:

$$t_{D{A}_{NN}}=0.9175n^3-3.7505n^2+5.1777n-2.2913$$E46

(43)

In a similar fashion as for the pseudoplastic case,

$$A=\pi {\left[\frac{t_{rsiNN}}{G\pi \left(0.9175n^3-3.7505n^2+5.1777n-2.2913\right)}\right]}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{${}^{3-n}$}\right.}$$E47

(44)

t_rsiNN 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:

n = 0.5h = 16.4 ftk = 350 mdq = 300 BPD

= 5 %B_o = 1 rb/STB μ_eff = 0.014833 cpc_t = 0.0000689 psi^-1

r_w = 0.33 ftH = 20 cp*s^n-1 r_e = 2000 ftP_i = 2500 psi

Solution. From Figure 11, the intercept point, t_rpiNN, 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

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:

$${\tilde{P}}_{DNN}=\frac{K_{\frac{1-n}{3-n}}\left(\frac{2}{3-n}\sqrt{\tilde{s}f(\tilde{s})}\right)}{\tilde{s}\left(\sqrt{\tilde{s}f(\tilde{s})}{K}_{\frac{2}{3-n}}\left(\frac{2}{3-n}\sqrt{\tilde{s}f(\tilde{s})}\right)\right)}$$E48

(45)

The Laplacian parameter, f($\tilde{s}$) is a function of the model type and fracture system geometry and is given by:

$$f(s)=\frac{\omega \left(1-\omega \right)s+\lambda }{\left(1-\omega \right)s+\lambda }$$E49

(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:

$$m=\frac{n-1}{n-3}$$E50

(47)

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

$$n=-1.8783425-7.8618321m^3+0.19406557m^{0.5}+2.8783425e^{-m}$$E51

(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:

$$\begin{array}{l}\frac{1}{\omega }=|3180.6369+551.0582{\left(\ln \frac{t_{\min }}{t_{b2}}\right)}^2-\frac{2053.5888}{x^{0.5}}+\frac{75.337547}{x}-\frac{1.4787073}{x^{1.5}}-\\ -\frac{910.05377}{n^{0.5}}+\frac{988.80592}{n}-\frac{459.61296}{n^{1.5}}+\frac{73.93695}{n^2}|\end{array}$$E52

(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 %.

$$\begin{array}{l}\omega =0.019884508-\frac{1.153351}{y}+\frac{43.428536}{y^2}-\frac{555.85387}{y^3}+\frac{3232.6805}{y^4}-\frac{6716.9801}{y^5}\\ -\frac{0.0093613189}{n}+\frac{0.0042870178}{n^2}+\frac{0.00027356586}{n^3}-\frac{0.0005221335}{n^4}+\frac{0.000072466135}{n^5}\end{array}$$E53

(50)

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

$$\omega =\frac{-0.098427346+0.00046337048y+0.000025063353y^2-0.00000050316996y^3+0.0036057682n-0.0073959605n^2}{1-0.36468068y-0.064934748n-0.047596083n^2}$$E54

(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:

$$\lambda =\frac{\left(6.9690127\times {10}^{-7}+3.4893658\times {10}^{-8}n-3.2315082\times {10}^{-8}{n}^2-5.9013807w+21571690w^2+3.6102987\times {10}^{12}{w}^3\right)}{\left(1+0.0099353372n-3740035.1w+6.7143604\times {10}^{12}{w}^2\right)}$$E55

(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:

$$\begin{array}{l}\lambda =-0.00082917155-0.0014247498n-0.00028717451\\ z-0.00077173053n^2-3.2538271\times {10}^{-5}{z}^2-0.0003203949nz-\\ -0.0001423889n^3-1.212213\times {10}^{-6}{z}^3-1.7831692\times {10}^{-5}n{z}^2-\\ -8.6457217\times {10}^{-5}{n}^{2}z\end{array}$$E56

(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 %.

$$\begin{array}{l}\ln \lambda =-2.1223034-0.09473309n+0.077489686n^{0.5}\ln \left(n\right)-\frac{0.010651118}{n^{0.5}}-\frac{0.043958503}{w^{0.5}}\\ +\frac{1.5653137\times {10}^{-5}\ln w}{w}+\frac{0.00024143014}{w}+\frac{8.7148736\times {10}^{-9}}{w^{1.5}}-\frac{4.0331364\times {10}^{-13}}{w^2}\end{array}$$E57

(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

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

t_min = 272.6 hr t_b2 = 14480 hrt_US,i =2129.4 hr (t*P’)_min = 10 psi

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

t_Dmin =32000 t_Db2 = 17000000 t_DUS,i =250000 (t_D*P_D’)_min =0.31

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

(t)_r1 = 35724.9 hr(t*P’)_r1 = 67.292 psi(t)_r2 =56169.5 hr(t_D*P’)_r2 = 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:

Equation	(	Equation	(
49	0.052	52	5.01E-06
50	0.05	53	5.043E-06
51	0.05	54	3.66E-06

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:

$$\frac{k}{r}\frac{\partial }{\partial r}\left[\frac{\rho \left(P\right)}{{\mu }_B}r\left(\frac{\partial P}{\partial r}-G\right)\right]=\frac{\partial }{\partial t}\left[\rho \left(P\right)f\left(P\right)\right]$$E58

(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:

$$\frac{1}{r}\frac{\partial }{\partial r}\left[r\left(\frac{\partial P}{\partial r}-G\right)\right]=\frac{f{\mu }_B{c}_t}{k}\frac{\partial P}{\partial t}$$E59

(56)

The initial condition is:

$$P\left(r,t=0\right)=P_i,r\ge r_w$$E60

(57)

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

$$q=2\pi rh\frac{k}{{\mu }_B}{\left(\frac{\partial P}{\partial r}-G\right)}_{r=r_w}$$E61

(58)

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

$$G_D=\frac{G{r}_{w}kh}{141.2q{\mu }_{B}B}$$E62

(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 G_D and the reservoir radius, Figure 16.

Figure 16.

Dimensionless pressure and derivative pressure for reD = 9375

8. Conclusion

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

B	Volumetric factor, RB/STB
ct	System total compressibility, 1/psi
C	Wellbore storage, bbl/psi
CfD	Dimensionless fracture conductivity
h	Formation thickness, ft
H	Consistency (Power-law parameter), cp*s^n-1
G	Group defined by Equation 3
G	Minimum pressure gradient, Psi/ft
GD	Dimensionless pressure gradient
k	Permeability, md
k	Flow consistency parameter
kfwf	Fracture conductivity, md-ft
m	Slope
n	Flow behavior index (power-law parameter)
P	Pressure, psi
q	Flow/injection rate, STB/D
t	Time, hr
r	Radius, ft
t*P’	Pressure derivative, psi
tD*PD’	Dimensionless pressure derivative
ra	Distance from well to non-Newtonian/Newtonian front/interface
w	(/tDmin
$\tilde{s}$	Laplace parameter
x	tmin/tb2
xf	Half-fracture length, ft
y	tUSi/tmin
z	ln [(tD*PD’)min/tDmin]

Table 2.

Nomenclature of main variables

	Change, drop
(	Porosity, Fraction
(	Shear rate, s^-1
(	Shear stress, N/m
(	Dimensionless interposity parameter
(	Viscosity, cp
(eff	Effective viscosity for power-law fluids, cp*(s/ft)^n-1
(B	Bingham plastic coefficient, cp
(^*	Characteristic viscosity, cp/ft^1-n
(	Shear stress, N/m
(	Dimensionless storativiy coefficient

Table 3.

Greeks

1	Non-Newtonian region
2	Newtonian region
app	Apparent
D	Dimensionless
DA	Dimensionless based on area; Based on area
Dxf	Dimensionless based on half-fracture length; Based on area
e	External
eff	Effective
i	Initial
LPi	Intersect of linear and pseudosteady-state lines
M	Maximum
N	Newtonian
NN	Non-Newtonian
r	Radial (any point on radial flow)
RLi	Intersect of radial and linear lines
rpiNN	Intersect of radial and pseudosteady-state lines
rsiNN	Intersect of radial and steady-state lines
w	Wellbore

Table 4.

Suffices