Cubic Law Research Articles

Analytical and visual assessment of fluid flow in fractured medium

Most reservoirs consist of natural and artificial fractures, including isolated microscopic fissures. These fractures form complicated paths for reservoir characterization and fluid movement that ultimately impacts production performance and ultimate recovery. That's why recovery estimations of fractured reservoirs are considered to be extremely challenging due to complexity and heterogeneity of the geological patterns.Oil production from fractured reservoirs results in varying saturation values throughout the reservoir. This is due to the microscopic fissures and heterogeneity of the fracture environment, which could not be swept thoroughly. Higher production/injection ratios also enhance the fingering effect by passing oil through the reservoir.In this study, naturally and artificially fractured cores were used. Analytical and experimental calculations were performed in order to understand the physical structure of the cores. Moreover, in the literature the analytical solutions of the fractures were defined by cubic law (CL), local cubic law (LCL) and modified cubic law (MCL). However, due to the lack of consideration of fracture properties with limited experimental work, there needs still work on this field.Equivalent fracture aperture (EFA) measurements were done by microscope and compared with the analytical calculations. Laboratory results were defined by the equation developed for predicting equivalent fracture apertures as Improved Cubic Law (ICL). Shrinkage in equivalent fracture aperture was also defined by ICL observed by microscope.Using ICL, analytical calculations were done in different environments. Equivalent fracture apertures were calculated for all the experimental flow rates under laminar flow. ICL has worked with the fractured cores flow definition whereas not for homogeneous cores.That's why due to higher calculated values and the 15% error on EFA, a correction coefficient (C) included to the CL equation for the tortuosity or the roughness effect. This result was in th error range of previous results and gave more realistic estimation with respect to the previous studies carried out. Finally, C coefficient was calculated in a range of 0.53–0.65.

A Modified Cubic Law for single-phase saturated laminar flow in rough rock fractures

The purpose of the study presented was to develop a Modified Cubic Law (MCL) for single-phase saturated laminar flow in rough rock fractures. Based on the fundamental assumptions made for the Cubic Law (CL), the proposed MCL incorporates modifications to the aperture field by considering flow tortuosity, aperture variation and local roughness effects. We assess the performance of the MCL by applying it to synthetic fractures with different surface roughness and varying aperture and compare the outputs to numerical simulation results from solving the Navier-Stokes Equations and results from previous versions of CL-based models. In general, the MCL performs well in predicting the volumetric flow rate in synthetic fractures with deviations (D) from simulation results ranging from −11.1% to 8.4% and an average effective deviation of 4.7%. The proposed model retains the simplicity of CL models and improves the accuracy of flow prediction in terms of single-phase saturated laminar flow in rough rock fractures and it can be extended to analyse other hydro-related problems.

Assumptions of the analytical solution for solute transport in a fracture–matrix system

This study focuses on numerical modeling of fluid flow and solute transport in rough-walled rock fractures and fracture-matrix systems, with the main aim to investigate the impacts of fracture surface roughness on flow and transport processes in rock fractures. Both 2D and 3D fracture models were built from laser-scanned surface tomography of a real granite rock sample, to consider realistic features of surface tomography and potential asperity contacts. The flow was simulated by directly solving the Navier-Stokes equations (NSE) and the transport was modeled by solving the advection-dispersion equation (ADE) in the entire domain of fracture-matrix system, including matrix diffusion process. Such direct simulations provided detailed flow and concentration fields for quantitatively analysis of flow and transport behavior. The detailed analysis of surface roughness decomposition, complex flow patterns (i.e., channeling, transverse and eddy flows), effective advective flow apertures, effective transmissivity, effective dispersivity, residence time, transport resistance and specific surface area demonstrated significant impacts of realistic fracture surface roughness on fluid flow and solute transport processes in rock fractures. The results show that the surface roughness and shear displacement caused asperity contacts significantly enhance nonlinearity and complexity of flow and transport processes in rough-walled fractures and fracture-matrix systems. The surface roughness also causes invasion flows in intersected fractures which enhance solute mixing at fracture intersections. Therefore, the fracture surface roughness is an important source of uncertainty in application of such simplified models like cubic law (CL) for fluid flow and analytical solutions for solute transport in rock fractures. The research conducted advances our understanding of realistic flow and transport processes in natural fractured rocks. The results are useful for model validation/extension, uncertainty analysis/quantification and laboratory experiments design in the context of various applications related to fracture flow and transport.

Modification of the Local Cubic Law of fracture flow for weak inertia, tortuosity, and roughness

AbstractThe classical Local Cubic Law (LCL) generally overestimates flow through real fractures. We thus developed and tested a modified LCL (MLCL) which takes into account local tortuosity and roughness, and works across a low range of local Reynolds Numbers. The MLCL is based on (1) modifying the aperture field by orienting it with the flow direction and (2) correcting for local roughness changes associated with local flow expansion/contraction. In order to test the MLCL, we compared it with direct numerical simulations with the Navier‐Stokes equations using real and synthetic three‐dimensional rough‐walled fractures, previous corrected forms of the LCL, and experimental flow tests. The MLCL performed well and the effective errors (δ) in volumetric flow rate range from −6.7% to 13.4% with an arithmetic mean of |δ| (<|δ|>) equal to 3.7%. The MLCL is more accurate than previous modifications of the LCL. We also investigated the error associated with applying the Cubic Law (CL) while utilizing modified aperture field. The δ from the CL ranges from −14.2% to 11.2%, with a slightly higher <|δ|> = 6.1% than the MLCL. The CL with the modified aperture field considering local tortuosity and roughness may also be sufficient for predicting the hydraulic properties of rough fractures.

A New Geometrical Model of Fluid Flow in Rock Fractures for Valid Application of the Cubic Law

The cubic law (CL) is one of the most commonly applied physical laws for flow through rock fractures and fractured media, but many studies indicate that the CL is often not adequate. We investigate the establishment conditions of valid applying the “cubic law” to flow in fractures. A dimensional analysis of the N-S equations yields three conditions for the applicability of cubic law, as a leading order approximation in a local fracture segment with parallel walls. These conditions may not be met in many natural rock fractures as whole and demonstrate that the “cubic law” should be exactly applied in local segments (local cubic law, LCL). In this way, a new 2D discrete equivalent model for natural rough fractures is introduced, and its equivalent aperture and flow formulation are derived. By comparing the developed model and experimental results of different fractures, good accuracy was found, and the model was validated. The model could be useful for theory studies of flows through real rock fractures.