**2. Non-Newtonian fluids in transient pressure analysis**

154 New Technologies in the Oil and Gas Industry

Slope *p*

Newtonian fluid

Dilatant 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

; . 1 *<sup>n</sup> τ kγ n*

*k* and *n* are constants which differ for each particular fluid. *k* measures the flow consistency

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

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

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

Pseudoplastic

Slope 

Slope *a*

 fluid

and *n* = 1.

Bingham Plastic fluid

**Figure 1.** Schematic representation of time-independent fluid

pseudoplastic fluids. The simplest model is power law,

pressure derivative have also been introduced.

verification of the results obtained.

fluids but *n* > 1.

and *n* measures the deviation from the Newtonian behavior which *k =* 

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.
