Applying Tiab ’ s direct synthesis technique to dilatant non-Newtonian / Newtonian fluids Aplicación de la técnica TDS a un yacimiento compuesto con fluidos dilatantes no newtoniano / newtoniano

Non-Newtonian fluids, such as polymer solutions, have been used by the oil industry for many years as fracturing agents and drilling mud. These solutions, which normally include thickened water and jelled fluids, are injected into the formation to enhanced oil recovery by improving sweep efficiency. It is worth noting that some heavy oils behave non-Newtonianly. Non-Newtonian fluids do not have direct proportionality between applied shear stress and shear rate and viscosity varies with shear rate depending on whether the fluid is either pseudoplastic or dilatant. Viscosity decreases as shear rate increases for the former whilst the reverse takes place for dilatants. Mathematical models of conventional fluids thus fail when applied to non-Newtonian fluids. The pressure derivative curve is introduced in this descriptive work for a dilatant fluid and its pattern was observed. Tiab’s direct synthesis (TDS) methodology was used as a tool for interpreting pressure transient data to estimate effective permeability, skin factors and non-Newtonian bank radius. The methodology was successfully verified by its application to synthetic examples. Also, comparing it to pseudoplastic behavior, it was found that the radial flow regime in the Newtonian zone of dilatant fluids took longer to form regarding both the flow behavior index and consistency factor.


Introduction
Okpobiri and Ikoku (1983) presented the standard work on well test analysis for dilatant non-Newtonian fluids.They studied the behavior of both flow pattern index and fluid consistency in pressure fall-off tests and conducted reservoir characterisation by using the conventional straight-line method.The pertinent oil sector literature contains several papers on the behaviour of non-Newtonian fluids in porous media.Odeh and Yang (1979) derived a partial differential equation regarding a power-law for fluid flow through porous media; they used a power-law relating viscosity to shear rate.The power-law viscosity function was coupled to the variable viscosity diffusivity equation and a shear-rate ratio proposed by Savins (1969) to give a new partial differential equation and an approximate analytical solution.Ikoku has been the most outstanding researcher in the field of non-Newtonian power-law fluid modelling, as shown by Ikoku (1979), Ikoku and Ramey (1979a, 1979b, 1979c) and Lund and Ikoku (1981) with pseudoplastic non-Newtonian fluids.
Interpreting pressure tests for non-Newtonian fluids is differently performed to conventional Newtonian fluids.Non-Newtonian fluids have a pressure derivative curve during radial flow regime which is not horizontal but rather inclined with a negative slope for non-Newtonian dilatant fluids.

Palabras clave:
Palabras clave: Palabras clave: Palabras clave: fluidos dilatantes, consistencia, viscosidad, ley de potencia, flujo radial field.The fluid was considered to be non-Newtonian dilatant obeying power law behaviour.The Newtonian fluid had constant viscosity, as usually considered in well test analysis.A piston -like behaviour of the Newtonian fluid by the non-Newtonian fluid was also assumed.Figure 1 sketches the composite reservoir being considered.
Since the pressure wave was travelling through a porous medium saturated with a non-Newtonian fluid, the pressure derivative had a certain degree of inclination as the flow behaviour index increased its value.Once, the travel wave arrived in the Newtonian zone, the pressure derivative formed a plateau, as expected, after an obvious transition period.Figure 2 illustrates such a situation for different n values.Notice than for n = 1, two different plateaus were observed since two types of mobility were being dealt with.The bigger the n, the longer the time required to obtain horizontal radial-flow line during the Newtonian zone.

Dimensionless parameters
Dimensionless pressure, P D , and dimensionless time, t D , for each region were expressed as: rect synthesis (TDS) technique regarding a composite reservoir with dilatant non-Newtonian/Newtonian interface.The model previously developed by Lund and Ikoku (1981) was thus solved numerically to obtain pressure and pressure derivative behaviour, so new expressions for characterising the reservoir by means of the TDS technique were developed and verified with synthetic data.Vongvuthipornchai and Raghavan (1987) used the pressurederivative method for well test analysis of non-Newtonian fluids; however, the very first application to non-Newtonian behaviour by the TDS technique was reported by Katime-Meindl and Tiab (2001).Escobar et al., (2010) conducted a numerical study to observe transient pressure behaviour when a non-Newtonian fluid is injected into a conventional reservoir containing Newtonian oil.They formulated an interpretation methodology using pressure and pressure derivative functions without type-curve matching.Escobar et al., (2011) first applied the pressure derivative concept to non-Newtonian fluids in double-porosity formation, in which an extension of the TDS technique was presented.The most recent power-law non-Newtonian flow behaviour work was by Mahani et al., (2011) for interpreting fall-off tests in polymer fluids using type-curve matching.Martinez et al., (2011) performed a numerical experiment for studying transient pressure behaviour and develop an interpretation methodology for non-Newtonian Bingham fluids.Ikoku and Ramey (1979b) proposed a partial differential equation for non-Newtonian fluids' radial flow following a power-law relationship through porous media.Coupling the non-Newtonian Darcy's law with the continuity equation, they derived a rigorous partial differential equation.The non-linear form of the partial differential equation was solved using the Douglas-Jones predictor/corrector method for numerical solutions of non-linear partial differential equations.

Mathematical model
A linearised approximation (Eq. 1) was also derived by Ikoku and Ramey (1979b) for analytical solutions.Linear and non-linear equation solutions were compared and found to fit very well.The errors introduced by the approximate linear equation were small and decreased as both the value of the flow behaviour index, n, and time increased.
where: and, The system being considered assumed radial flow of a non-Newtonian and a slightly compressible Newtonian fluid through porous media.It was assumed that the reservoir was homogeneous and isotropic.It also had constant thickness.The reservoir was cylindrically shaped with a finite outer radius.The non-Newtonian fluid was injected through a well in the centre of the

TDS formulation
The interpretation methodology followed the TDS technique philosophy introduced by Tiab (1995) which was based upon the definition of characteristic features found on the logarithmic plot of pressure and pressure derivative versus time.This meant that several specific regions on that plot were dealt with.
1.According to Katime-Meindl and Tiab (2001), the dimensionless equation for radial-flow regime derived in the non-Newtonian region without considering wellbore storage effects was: The value of this derivative was: Combination of Eq. 9 and 10 led to obtaining: Substituting Eq. 3 into Eq.11 and solving for k 1 yielded: α being pressure derivative curve slope on the non-Newtonian region defined as: 2. The governing pressure equation during non-Newtonian radial flow regime may be expressed by: 3. The skin factor, s 1 , was obtained by dividing Eq. 14 by Eq. 9: 4. The radius of the injected non-Newtonian fluid bank was calculated using the following correlation (valid for n>1), obtained from reading the time at which the radial flow non-Newtonian ended (t e_rNN ): 5. For the Newtonian region the permeability and skin factor were presented by Tiab (1995) as: 6. Non-Newtonian/Newtonian radial flow line intercept.

Examples
Example 1 An injection test was simulated, using the information from Table 1.Pressure and pressure derivative data are shown in Figure 3. Permeability and skin factor in each region and the radius of non -Newtonian fluid bank had to be estimated.

Solution.
The log-log plot of pressure and pressure derivative against injection time is given in Figure 3. From that plot the following information was read: First, a was evaluated with Eq. 13 to be -0.25.Non-Newtonian effective fluid permeability was estimated with Eq. 12, resulting in 94.16 md (9.29x10 -14 m 2 ).Eq. 3 was then used to find effective viscosity, resulting in 3,098.61cp(s/ft) n-1 (4.984 Ns n /m 2 ).Skin factor in the non-Newtonian region was then found with Eq. 15 to be -1.94.

Example 2
Another injection test was simulated using the information from Table 1.Pressure and pressure derivative data are shown in Figure 4. Permeability and skin factor in each area and the radius of non-Newtonian fluid bank had to be estimated.

Analysis of results
The simulation experiments showed that the pressure derivative pattern during the non-Newtonian zone for a dilatant-type fluid had a negative slope which increased as the flow behaviour index also increased.When n = 1 (Newtonian), the pressure derivative became a flat, horizontal line, as expected.The opposite occurred for pseudoplastic behaviour in which the pressure derivative straight-line had an increasing slope as flow decreased.
The estimation of effective permeability and skin factors matched very well with the simulated input data.It was found that the transition period, after the non-Newtonian zone, took longer to reach than for pseudoplastic behaviour and also bank zone estimation was more accurate for dilatants.

Figure 3 .
Figure 3. Pressure and pressure derivative for example 1

Table 1 .
Reservoir and fluid data for the synthetic examples Escobar et al., (2010)time of non-Newtonian and Newtonian radial line intersection.The above equations for permeability and skin factors were also found and verified byEscobar et al., (2010)for 5