Quantitative Analysis of Non-Newtonian Fluid Flow Through Naturally-Fractured Reservoirs
Authors: Keshavarz M. , Moreno R. B. Z. L. 
Published: September 2023

ABSTRACT
This study examines the displacement of non-Newtonian fluid invasion in naturally fractured reservoirs (NFR) by employing the evaluated diffusivity solution proposed by Keshavarz, M., & Moreno, R. B. Z. L. (2023)1,for a comprehensive quantitative analysis. The solution is capable of considering not only drilling fluid pseudo-plasticity throughout the fractured but also the matrix system under constant wellbore pressure assumptions. The effects of non-Newtonian fluid parameters and NFR characteristics on fluid flow, leak-off phenomenon, and cumulative volume are examined. The developed solution generates type-curves for quantitative analysis, it validated by field data of loss measurements in a fractured well in the Gulf of Mexico. Parametric analysis reveals that drilling fluid pseudo-plasticity significantly impacts cumulative volume and ROI (radius of invasion) in NFRs with higher differential pressure, leak-off (particularly at earlier times), and larger fracture apertures. Additionally, the procedure allows operators to determine the ROI for equivalent starting and ending times of the transient period. The proposed solution, applicable to well-completion, enhanced oil recovery (EOR) processes, NFR characterization, and drilling operations, facilitates quantitative analysis of reservoir properties and cumulative volume in NFRs. It also assists operators in optimizing drilling fluid rheology in each period, reducing mud loss or enhancing volumetric sweep efficiency in polymer flooding cases.

INTRODUCTION
Non-Newtonian fluid flow through naturally fractured reservoirs (NFR) has received considerable attention over the last decades in both experimental and modeling research (Yi, 2004; Adenuga et al., 2019), as it controls important technological processes in petroleum and geotechnical engineering. The viscosity of such fluids has a non-linear relationship between shear stress and shear rate. Polymer solutions, heavy (waxy) crude oil, and drilling fluid are common examples of such fluids in petroleum; thus, they have been used for many years as fracturing agents, drilling fluids, and in some enhanced oil recovery processes (Ikoku & Ramey, 1979; Odeh & Yang, 1979; Garcia-Pastrana et al., 2017; Elkatatny et al., 2020; Albattat & Hoteit, 2021; Albattat et al., 2022; Pang et al., 2022). Even though their application is not novel, the subject of transient flow of non-Newtonian fluids in NFR is critical for improving the accuracy of total cumulative volume measurement and well testing (Ikoku & Ramey, 1979; Escobar et al., 2011; Dokhani et al., 2020). When fluids, flow through porous rocks, pressure behavior is affected by fluid rheology, reservoir architecture, reservoir type, and wellbore condition. NFR has been idealized as double porosity reservoir. The governing equation of the fluid for this type of reservoir is modeled considering flow regime and the geometry of the reservoir. While well test analysis involving the flow of Newtonian and non-Newtonian fluids in single and double-porosity reservoirs under wellbore constant rate has been extensively studied (Odeh & Yang, 1979; Da Prat, 1990; Dake, 2001; Escobar et al., 2011), none of these studies considered wellbore condition constant pressure, to measure non-Newtonian fluid rate and cumulative volume. The basic partial differential equation for newtonian fluid flow in NFRs are originally presented and solved under wellbore constant rate by Warren & Root et al (1963). The case was then studied and solved by Da Prat (1990), under constant wellbore producing pressure. Ikoku & Ramey et al. (1979) proposed a model characterizing homogeneous reservoirs with presence of a non-Newtonian (pseudoplastic) fluid. Olarewaju et al. (1992), was the first to develop an analytical solution for the transient behavior of dual-porosity formations containing a non-Newtonian pseudoplastic fluid. More than a decade later, Escobar et al. (2011) developed and successfully tested a methodology (based on solution presented by Olarewaju et al. 1992) for the interpretation of pressure tests in heterogeneous formations bearing a non-Newtonian fluid, based on the pressure and pressure derivative plot; as a result, they presented new equations for estimation of NFR parameters, which was later modified by Garcia-Pastrana et al (2017). Therefore, the focus of this research is on developing the applicability of evaluated solution presented by Keshavarz, M., & Moreno, R. B. Z. L. (2023) for the diffusivity model, which is solved under wellbore constant pressure and is applicable for cumulative fluid measurement. This evaluated solution has been applied as reference model due to considering dimensionless matrix contribution in pseudosteady-state inter-porosity transfer as the assumption of the dual porosity model. To the best of the Author's knowledge, no study has utilized an evaluated non-Newtonian dual porosity model to measure cumulative volume through NFR while taking leak-off into account. In this regard, evaluated dimensionless rate and cumulative volume solutions in Laplace domain were applied for radial flow of non-Newtonian pseudoplastic fluids in infinite dual-porosity reservoirs. The solutions were numerically inverted by Stehfest et al (1970) algorithm, followed by the formulation of dimensionless terms and generation of type curves to facilitate quantification of parameters in real time.