An efficient lubrication-based code for solving non-Newtonian flow in geological rough fractures

2021 ◽  
Author(s):  
Alessandro Lenci ◽  
Yves Méheust ◽  
Mario Putti ◽  
Vittorio Di Federico
Keyword(s):  
Power Law ◽  
Fractured Reservoirs ◽  
Numerical Code ◽  
Symmetric System ◽  
Finite Volume Scheme ◽  
Single Fracture ◽  
Newtonian Flow ◽  
Shear Rates ◽  
Ellis Model ◽  
Low Shear

<p>The study of the flow in a single fracture is the starting point to understand the complex hydraulic behaviour of geological formations and fractured reservoirs, whose comprehension is of interest in many natural phenomena (e.g., magma intrusion) and the optimization of numerous industrial activities in fractured reservoirs (e.g., Enhanced Oil Recovery, drilling engineering, geothermal energy exploitation). Despite the considerable technical prospects of this topic, the associated mathematical complexity and computational burden have so far mostly discouraged investigations of the combined effects of fracture heterogeneity and of the complex rheology of relevant fluids. Indeed, magmas, foams, muds, and suspensions of natural colloids such as clay particles in water are complex fluids and often present in subsurface applications and natural processes. These fluids are characterized by a shear-thinning behavior, which can be well described by the Ellis model, a continuous three-parameter model that behaves as a power-law fluid at high shear rates and as a Newtonian fluid at low shear rates. The Ellis model parameters are: <em>n</em> the power law exponent, <em>μ</em><sub>0</sub> the low shear rates viscosity, and <em>τ</em><sub>1/2</sub> the shear rate such that <em>μ<sub>app</sub></em>(<em>τ</em><sub>1/2</sub>)=<em>μ</em><sub>0</sub>/2. We use this rheological description in combination with the lubrication theory, which is a depth-averaged formalism permitting us to reduce the full 3-D problem to a 2-D plane formulation. It has been applied to study Newtonian flow in a single fracture for decades and, as far as the aperture gradient remains small (∇<em>d</em>«1), the approximation error introduced by this model is limited. We present here a lubrication-based numerical code aiming at simulating the flow of an Ellis fluid in rough-walled fractures. The code is composed of two modules: a 2D FFT-based fracture aperture field generator and a lubrication-based non-Newtonian flow solver. The former module generates a random aperture field <em>d</em>(<em>x</em>,<em>y</em>) with isotropic spatial correlations, given a mean aperture ⟨<em>d</em>⟩, a coefficient of variation <em>σ<sub>d</sub></em>/⟨<em>d</em>⟩, a Hurst exponent (<em>H</em>) and a correlation length (<em>l<sub>c</sub></em>), reproducing realistic geometries of geological fractures. In the latter module, a 2-D finite volume scheme is adopted to solve the non-linear lubrication equation describing the flow of an Ellis fluid. The equation is discretized on a staggered grid, so that <em>d</em>(<em>x</em>,<em>y</em>) and the pressure field <em>p</em>(<em>x</em>,<em>y</em>) are defined at different locations. Computational efficiency is achieved by means of the inexact Newton algorithm, with the linearized symmetric system of equations solved via variable-fill-in Incomplete Cholesky Preconditioned Conjugate Gradient method (ICPCG), and a parameter-continuation strategy for the cases with strong nonlinearities. The code proves to be stable and robust when solving flow within strongly heterogeneous fractures (e.g., <em>σ<sub>d</sub></em>/⟨<em>d</em>⟩=1), even on very fine and coarse meshes (e.g., 2<sup>14</sup>×2<sup>14</sup>) and considering a wide range of power-law exponents (e.g., 0.1≤<em>n</em>≤1). The code is validated by comparing the results against analytical solutions (e.g., parallel plates model, sinusoidal profile) and full 3-D CFD simulations, considering different closures.</p>

Non-Newtonian Flow Behavior in Microchannels for Emulsion Formation

2006 ◽  
Author(s):  
Ravi Arora ◽  
Eric Daymo ◽  
Anna Lee Tonkovich ◽  
Laura Silva ◽  
Rick Stevenson ◽  
...  
Keyword(s):  
Shear Stress ◽  
Pressure Drop ◽  
Wall Shear Stress ◽  
Power Law ◽  
Flow Behavior ◽  
Scale Up ◽  
High Wall Shear Stress ◽  
Shear Rates ◽  
Wall Shear ◽  
Low Shear

Emulsion formation within microchannels enables smaller mean droplet sizes for new commercial applications such as personal care, medical, and food products among others. When operated at a high flow rate per channel, the resulting emulsion mixture creates a high wall shear stress along the walls of the narrow microchannel. This high fluid-wall shear stress of continuous phase material past a dispersed phase, introduced through a permeable wall, enables the formation of small emulsion droplets — one drop at a time. A challenge to the scale-up of this technology has been to understand the behavior of non-Newtonian fluids under high wall shear stress. A further complication has been the change in fluid properties with composition along the length of the microchannel as the emulsion is formed. Many of the predictive models for non-Newtonian emulsion fluids were derived at low shear rates and have shown excellent agreement between predictions and experiments. The power law relationship for non-Newtonian emulsions obtained at low shear rates breaks down under the high shear environment created by high throughputs in small microchannels. The small dimensions create higher velocity gradients at the wall, resulting in larger apparent viscosity. Extrapolation of the power law obtained in low shear environment may lead to under-predictions of pressure drop in microchannels. This work describes the results of a shear-thinning fluid that generates larger pressure drop in a high-wall shear stress microchannel environment than predicted from traditional correlations.

A Model for Wettability Alteration in Fractured Reservoirs

SPE Journal ◽  
2015 ◽  
Vol 20 (06) ◽  
pp. 1261-1275 ◽  
Author(s):  
Pål Østebø Andersen ◽  
Steinar Evje ◽  
Hans Kleppe ◽  
Svein Magne Skjæveland
Keyword(s):  
Oil Recovery ◽  
Enhanced Recovery ◽  
Fractured Reservoirs ◽  
Ionic Composition ◽  
Spontaneous Imbibition ◽  
Wettability Alteration ◽  
Numerical Code ◽  
Significant Feature ◽  
Rock Surface ◽  
Single Fracture

Summary We present a mathematical model for wettability alteration (WA) in fractured reservoirs. Flow in the reservoir is modeled by looking at a single fracture surrounded by matrix on both sides. Water is injected into the formation with a chemical component that enters the matrix and adsorbs onto the rock surface. These changes of the mineral surface are assumed to alter the wettability toward a more water-wet state, which leads to enhanced recovery by spontaneous imbibition. This can be viewed as a representation of “smart water” injection in which the ionic composition of injection brine affects recovery. The WA is described by shifting curves for relative permeability and capillary pressure from curves representing preferentially oil-wet (POW) conditions toward curves representing more-water-wet conditions. The numerical code was successfully compared with ECLIPSE for the specific case in which a fixed wetting state is assumed. Also, the relevance of the WA model was illustrated by modeling a spontaneous-imbibition experiment in which only a modification of the brine composition led to a change in oil recovery. The model can predict sensitivity to matrix properties such as wettability, permeability, and fracture spacing and to external parameters such as schedule of brine compositions and injection rate. Our model illustrates that one cannot use conventional reservoir modeling to capture accurately the behavior we observe. The rate of recovery and the level of recovery have a strong dependency on the component chemistry and its distribution. A significant feature of gradual WA by injecting a component is that the rate of fluid transfer is maintained between matrix and fracture. The resulting recovery profile after water breakthrough can behave close to linear as opposed to the square-root-of-time profile that is observed when the wetting state is fixed (Rangel-German and Kovscek 2002). The water will typically break through early as dictated by the initial POW state, but a higher final recovery will be obtained because higher saturations can imbibe. Improved understanding of the coupling between WA controlled by water/rock chemistry and fracture/matrix flow is highly relevant for gaining more insight into recovery from naturally fractured reservoirs.

scholarly journals Time dependent power law behaviour at low shear rates for paint systems

1971 ◽  
Vol 10 (4) ◽  
pp. 607-607 ◽  
Author(s):  
M. Camina ◽  
C. G. Roffey
Keyword(s):  
Power Law ◽  
Time Dependent ◽  
Shear Rates ◽  
Paint Systems ◽  
Low Shear

Rheological Modelling of Complex Flow Behaviour of Bitumen-Solvent Mixtures and Implication for Flow in a Porous Medium

2021 ◽  
pp. 1-34
Author(s):  
Olalekan Alade
Keyword(s):  
Shear Rate ◽  
Power Law ◽  
Rheological Model ◽  
Flow Characteristics ◽  
Solvent Mixtures ◽  
Pore Scale ◽  
Rheological Models ◽  
Shear Rates ◽  
Rate Region ◽  
Low Shear

Abstract The viscosity of extra-heavy oils including bitumen can be reduced significantly by adding solvent such as toluene to enhance extraction, production and transportation. Thus, prediction of viscosity and/or rheology of bitumen-solvent mixtures has become necessary. More so, selecting a suitable rheological model for simulation of flow in porous media has an important role to play in engineering design of production and processing systems. While several mixing rules have been applied to calculate the viscosity of bitumen-solvent mixtures, rheological model to describe the flow characteristics has rarely been published. Thus, in this investigation, rheological behaviour of bitumen and bitumen-toluene mixtures (weight fractions of bitumen WB = 0, 0.25, 0.5, 0.6, 0.75, and 1 w/w) have been studied at the flow temperature (75 °C) of the bitumen and in the range of shear rates between 0.001 and 1000 s−1. The data was fitted using different rheological models including the Power Law, Cross Model, Carreau-Yasuda Model, and the newly introduced ones herein named as Cross-Logistic and Logistic models. Then, a computational fluid dynamics (CFD) model was built using a scanning electron image (SEM) of rock sample (representing a realistic porous geometry) to simulate pore scale flow characteristics. The observations revealed that the original bitumen exhibits a Newtonian behaviour within the low shear rate region (0.001 to 100 s−1) and shows a non-Newtonian (pseudoplastic) behaviour at the higher shear rate region (100 to 1000 s−1). Conversely, the bitumen-toluene mixtures show shear thinning (pseudoplastic) behaviour at low shear rate region (0.001 to 0.01), which appears to become less significant within 0.01 to 0.1 s−1, and exhibit shear independent Newtonian behaviour within 0.1 and 1000 s−1 shear rates. Moreover, except for the original bitumen, statistical error analysis of prediction ability of the tested rheological models as well as the results from the pore scale flow parameters suggested that the Power Law might not be suitable for predicting the flow characteristics of the bitumen-toluene mixtures compared to the other models.

On the stationary quasi-Newtonian flow obeying a power-law

1995 ◽  
Vol 18 (12) ◽  
pp. 927-948 ◽  
Author(s):  
Eric Blavier ◽  
Andro Mikelić
Keyword(s):  
Power Law ◽  
Newtonian Flow

VISCOSITY OF NATURAL RUBBER SOLUTIONS AT VERY LOW RATES OF SHEAR

1957 ◽  
Vol 35 (4) ◽  
pp. 381-387 ◽  
Author(s):  
Morton A. Golub
Keyword(s):  
Experimental Data ◽  
Natural Rubber ◽  
Pressure Head ◽  
Viscosity Data ◽  
Theoretical Relation ◽  
Newtonian Flow ◽  
Shear Rates ◽  
Reduced Viscosity ◽  
Wide Range ◽  
Composite Data

The shear dependence of viscosity of benzene solutions of natural rubber was studied at rates of shear from about 500 down to less than 1 sec.−1. Measurements involved following the change of pressure head with time of the various solutions flowing in a capillary, U-tube viscometer. Curvature in the plots of the logarithm of pressure head versus time indicated non-Newtonian flow. From such curves, reduced viscosity data over the above-mentioned shear range were readily derived. As a check, data over the range 100–500 sec.−1 were also obtained with a five-bulb viscometer of the Krigbaum–Flory type, and these data overlapped those obtained with the U tube. The reduced viscosity increased very sharply with decrease in gradient, making extrapolation to the viscosity axis quite unreliable. However, a theoretical relation proposed by Bueche fitted the composite data rather well. This work furnished a nice technique for determining the zero shear reduced viscosity (ηap/c)0 without the necessity of performing an uncertain extrapolation: evaluate the parameters of the Bueche formula which best satisfies the experimental data over a fairly wide range of shear rates, and then calculate (ηap/c)0 directly.

Effect of Blood Viscosity on Thrombosis Potential Near a Step Wall Transition

2009 ◽  
Author(s):  
Scott C. Corbett ◽  
Amin Ajdari ◽  
Ahmet U. Coskun ◽  
Hamid N.-Hashemi
Keyword(s):  
Blood Flow ◽  
Vessel Wall ◽  
Coagulation Factors ◽  
Artificial Organs ◽  
Physical Factors ◽  
Local Geometry ◽  
Induce Platelet Aggregation ◽  
Shear Rates ◽  
Chemical Factors ◽  
Low Shear

Thrombosis and hemolysis are two problems encountered when processing blood in artificial organs. Physical factors of blood flow alone can influence the interaction of proteins and cells with the vessel wall, induce platelet aggregation and influence coagulation factors responsible for the formation of thrombus, even in the absence of chemical factors in the blood. These physical factors are related to the magnitude of the shear rate/stress, the duration of the applied force and the local geometry. Specifically, high blood shear rates (or stress) lead to damage (hemolysis, platelet activation), while low shear rates lead to stagnation and thrombosis [1].

scholarly journals Hydroxyurea Improves Oxygen Transport Effectiveness in Sickle Cell Anemia Patients

Blood ◽  
2014 ◽  
Vol 124 (21) ◽  
pp. 2716-2716
Author(s):  
Vivien A. Sheehan ◽  
Sheryl Nelson ◽  
Caroline Yappan ◽  
Bogdan R. Dinu ◽  
Danielle Guffey ◽  
...  
Keyword(s):  
Blood Flow ◽  
Sickle Cell ◽  
Blood Viscosity ◽  
Whole Blood ◽  
Patient Characteristics ◽  
Whole Blood Viscosity ◽  
Shear Rates ◽  
High Shear Rates ◽  
Altered Blood ◽  
Low Shear

Abstract Background: Sickle cell disease (SCD) patients have altered blood rheology due to erythrocyte abnormalities, including increased aggregation and reduced deformability, which together affect microcirculatory blood flow and tissue perfusion. At equal hematocrit, sickle cell blood viscosity is increased compared to normal individuals. The hematocrit to viscosity ratio (HVR) is a measure of red blood cell (RBC) oxygen carrying capacity, and is reduced in SCD with clinical consequences related to altered blood flow and reduced tissue oxygenation. Erythrocyte transfusions reduce HVR at low shear rates that mimic venous circulation, and do not change HVR at high shear rates that mimic arterial blood flow. Hydroxyurea is a safe and effective therapy for SCD; however, its effects on sickle cell rheology and HVR have not been fully investigated. Evaluating the effects of hydroxyurea on viscosity is especially critical, before its use is extended widely to patients with cerebrovascular disease or genotypes with higher hematocrit and higher viscosity such as Hemoglobin SC (HbSC). Methods: To determine the effects of hydroxyurea on viscosity and HVR, we designed a prospective study to measure whole blood viscosity at 45 s-1 (low shear) and 225 s-1(high shear) rates in pediatric patients with SCD using a Brookfield cone and plate viscometer under oxygenated conditions. Venous blood samples (1-3mL) were collected in EDTA and analyzed no more than 4 hours after phlebotomy; samples were run in duplicate by persons blinded to the patient’s sickle genotype and treatment status. Laboratory values were obtained using an ADVIA hematology analyzer. Samples were analyzed from three non-overlapping cohorts of patients with SCD and HbAA individuals for comparison: untreated HbSS patients (n= 43), HbSS patients treated with hydroxyurea at maximum tolerated dose (n=98), untreated HbSC patients (n=53) and HbAA patients (n=19). Laboratory parameters that differed significantly among the SCD groups were analyzed by simple linear regression. Results: Patient characteristics and viscosity measurements are shown in the Table. Within the SCD population, the viscosity was lowest among the untreated HbSS patients, presumably due to their low hematocrit, while viscosity was higher in HbSS patients on hydroxyurea and HbSC patients. When the HVR was calculated for each group, no significant difference was identified between untreated HbSS and untreated HbSC patients. However, hydroxyurea treatment significantly increased HVR at both 45s-1 and 225 s-1 (p<0.001), indicating that the slightly increased viscosity in this cohort was more than compensated by a higher hematocrit. Correlations were tested for hemoglobin (Hb), mean corpuscular volume (MCV), white blood cell count (WBC), absolute neutrophil count (ANC), absolute reticulocyte count (ARC), % fetal hemoglobin (HbF), and average red cell density in g/dL with HVR, at both shear rates. The hydroxyurea-associated HVR increase at both shear rates was independent of %HbF or MCV, but the increased HVR at 225 s-1was associated with lower WBC (p<0.001), lower ANC (p=0.002), and lower red cell density (p=.009). Conclusions: We provide prospective data on whole blood viscosity measurements in a large cohort of children with SCD. Hydroxyurea increases the hematocrit in HbSS patients more than the viscosity, and thus improves HVR. These findings imply that hydroxyurea improves RBC oxygen transport at both high and low shear rates, which should confer clinical benefits, and these effects are independent of HbF induction. Concerns about hydroxyurea increasing whole blood viscosity and reducing tissue oxygenation in children with cerebrovascular disease or HbSC patients may not be warranted, if the same beneficial HVR effects are achieved. Abstract 2717. Table 1. Patient characteristics. Viscosity was typically measured in duplicate and averaged for each patient. HVR at 45 s-1 and 225s-1 was calculated as hematocrit/viscosity. Results are presented as mean ± 2SD. HbAAn=19 HbSS, untreatedn=43 HbSS, on Hydroxyurean=98 HbSCn=53 Age (years) 15.4 ± 3.8 10.4 ± 5.1 10.7 ± 3.4 10.5 ± 4.3 Hemoglobin (gm/dL) 13.5 ± 1.7 8.5 ± 1.0 9.9 ± 1.4 11.0 ± 1.2 Hematocrit (%) 40.9 ± 5.3 25.5 ± 3.1 28.4 ± 3.7 31.3 ± 3.2 Viscosity (cP) at 45s-1 5.3 ± 0.9 4.6 ± 1.2 4.3 ± 0.9 5.5 ±0.9 HVR at 45s-1 7.5 ± 0.9 5.8 ± 1.1 6.75 ± 1.0 5.77 ± 0.7 Viscosity (cP) at 225s-1 3.8 ± 0.5 3.3 ± 0.5 3.4 ± 0.5 4.1 ± 0.5 HVR at 225s-1 10.3 ± 0.7 7.7 ± 0.8 8.53 ± 0.8 7.72 ± 0.6 Disclosures Off Label Use: Hydroxyurea is not FDA approved for use in pediatric sickle cell patients.

A Sensitivity Analysis for Effective Parameters on 2D Fracture-Network Permeability

2009 ◽  
Vol 12 (03) ◽  
pp. 455-469 ◽  
Author(s):  
Alireza Jafari ◽  
Tayfun Babadagli
Keyword(s):  
Sensitivity Analysis ◽  
Experimental Design ◽  
Fractured Reservoirs ◽  
Fracture Network ◽  
Static Model ◽  
Naturally Fractured Reservoirs ◽  
Single Fracture ◽  
Data Sets ◽  
Fracture Networks ◽  
Fractal Characteristics

Summary Fracture-network mapping and estimation of its permeability constitute two major steps in static-model preparation of naturally fractured reservoirs. Although several different analytical methods were proposed in the past for calculating fracture-network permeability (FNP), different approaches are still needed for practical use. We propose a new and practical approach to estimate FNP using statistical and fractal characteristics of fracture networks. We also provide a detailed sensitivity analysis to determine the relative importance of fracture-network parameters on the FNP in comparison to single-fracture conductivity using an experimental-design approach. The FNP is controlled by many different fracture-network parameters such as fracture length, density, orientation, aperture, and single-fracture connectivity. Five different 2D fracture data sets were generated for random and systematic orientations. In each data set, 20 different combinations of fracture density and length for different orientations were tested. For each combination, 10 different realizations were generated. The length was considered as constant and variable. This yielded a total of 1,000 trials. The FNPs were computed through a commercial discrete-fracture-network (DFN) modeling simulator for all cases. Then, we correlated different statistical and fractal characteristics of the networks to the measured FNPs using multivariable-regression analysis. Twelve fractal (sandbox, box counting, and scanline fractal dimensions) and statistical (average length, density, orientation, and connectivity index) parameters were tested against the measured FNP for synthetically generated fracture networks for a wide range of fracture properties. All cases were above the percolation threshold to obtain a percolating network, and the matrix effect was neglected. The correlation obtained through this analysis using four data sets was tested on the fifth one with known permeability for verification. High-quality match was obtained. Finally, we adopted an experimental-design approach to identify the most-critical parameters on the FNP for different fracture-network types. The results are presented as Pareto charts. It is believed that the new method and results presented in this paper will be useful for practitioners in static-model development of naturally fractured reservoirs and will shed light on further studies on modeling and understanding the transmissibility characteristics of fracture networks. It should be emphasized that this study was conducted on 2D fracture networks and could be extended to 3D models. This, however, requires further algorithm development to use 2D fractal characteristics for 3D systems and/or development of fractal measurement techniques for a 3D system. This study will provide a guideline for this type of research.

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