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Steady-State and Unsteady-State Flow of Non~Newtonian Fluids Through Porous Media
Abstract
Non-Newtonian fluids may be injected into a reservoir during secondary recovery operations. The non-Newtonian fluid used in this work is a power-law type of fluid; that is, the viscosity of the fluid decreases as the flow rate or rate of shear increases.
This paper presents equations for steady-state, linear and radial flow of such fluids, transient behavior results from a finite difference model of a radial system, and transient behavior results from a field test.
The equations that describe the flow of a non-Newtonian fluid are non-linear and are solved numerically. Finite difference solutions are presented as curves of dimensionless pressure drop at the wellbore vs dimensionless time for a constant injection rate. Solutions were obtained for 5-percent, 10-percent and 100-percent PV of a non-Newtonian fluid for injection rates of 1, 10, 100 and 1000 cc/sec and for a 5 percent PV of non-Newtonian fluid located at r = rw, 3, 10, 20, 50 and 100 ft for a flow rate of 1 cc/sec. The buildup curves do not exhibit a straight-line portion as is the case for Newtonian flow through porous media.
Correlations also are shown for the productivity index vs rate for the computer model study and the field tests.
INTRODUCTION
During various secondary recovery operations non-Newtonian fluids are injected. Such fluids, in general, include thickened water and gelled fluids. The term "non-Newtonian" implies that the viscosity is not only dependent upon temperature and pressure, but also on the rate of shear that is applied to the fluid. For example, water, which is a Newtonian fluid, will have essentially the same viscosity no matter what rate of shear is applied. A pseudoplastic fluid exhibits a decreasing viscosity when higher rates of shear are applied; a dilated fluid has an increasing viscosity with increasing rates of shear.
The objective of studies performed and described in this report is to obtain relationships and mathematical and empirical descriptions of the flow of non- Newtonian fluids through porous media. Simple mathematical relationships, computer studies that include the unsteady- state behavior of such fluids, and field tests are used.
PREVIOUS LABORATORY STUDIES
Laboratory studies have been performed by several investigators. The fluids used in one investigation1 were surfactant-stabilized dispersions of water in hydrocarbons. Its Fig. 6 is reproduced as our Fig. 1. This effective-viscosity vs frontal-velocity diagram shows that this fluid is of the power-law type over a fairly large range of frontal velocities. Gogarty1 developed an equation for the effective viscosity as a function of shear rate that reduces to Eq. 2 for frontal velocities greater than about 2 ft/D. The fluid characteristics used in the present study are similar to those reported by Gogarty.
... The complex network of pores within the media promotes the mixing of fluids, leading to improved heat and mass transfer rates. This mixing helps in distributing heat or mass more uniformly throughout the medium [25,26]. ...
... Expressions (26), (29), (32) and (38) give the dimensionless temperature, concentration, velocity and shear stress in the p-domain. The required results in the t-domain can be obtained by taking the inverse LT of these expressions, which is indeed a tedious job. ...
The heat and mass transfer phenomenon in the presence of a moving magnetic field has a wide range of applications, spanning from industrial processes to environmental engineering and energy conversion technologies. Understanding these interactions enables the optimization of various processes and the development of innovative technologies. This manuscript is about a non-integer-order heat-mass transfer model for Maxwell fluid over an inclined plate in a porous medium. The MHD flow of non-Newtonian fluid over the plate due to the natural convection of the symmetric temperature field and general motion of the inclined plate is investigated. A magnetic field is applied with a certain angle to the plate, and it can either be fixed in place or move along with the plate as it moves. Our model equations are linear in time, and Laplace transforms form a powerful tool for analyzing and solving linear DEs and systems, while the Stehfest algorithm enables the recovery of original time domain functions from their Laplace transform. Moreover, it offers a powerful framework for handling fractional differential equations and capturing the intricate dynamics of non-Newtonian fluids under the influence of magnetic fields over inclined plates in porous media. So, the Laplace transform method and Stehfest’s numerical inversion algorithm are employed as the analytical approaches in our study for the solution to the model. Several cases for the general motion of the plate and generalized boundary conditions are discussed. A thorough parametric analysis is performed using graphical analysis, and useful conclusions are recorded that help to optimize various processes and the developments of innovative technologies.
... Many studies in petroleum and chemical engineering and rheology have focused on non-Newtonian fluid behavior though porous formations. To mention some: Hirasaki and Pope (1974), Ikoku (1979), Ikoku and Ramey (1979a), Odeh and Yang (1979), Savins (1969), andVan Poollen andJargon (1969). Several numerical and analytical models have been proposed to study the transient behavior of non-Newtonian fluid in porous media. ...
... Many studies in petroleum and chemical engineering and rheology have focused on non-Newtonian fluid behavior though porous formations. To mention some: Hirasaki and Pope (1974), Ikoku (1979), Ikoku and Ramey (1979a), Odeh and Yang (1979), Savins (1969), andVan Poollen andJargon (1969). Several numerical and analytical models have been proposed to study the transient behavior of non-Newtonian fluid in porous media. ...
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 possess non- Newtonian nature and these fluids are often approximated by Newtonian fluid flow models. In the field of well testing, several analytical and numerical models based on 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 and homogeneous formations. Well test interpretation is conducted via the straight-line conventional analysis or type-curve matching. Only a few studies consider the pressure derivative analysis. However, there is a need for a more practical and accurate way of characterizing such systems. So far, there is no methodology to characterize heterogeneous formation bearing non-Newtonian fluids through well test analysis. In this study, an interpretation methodology using the pressure and pressure derivative log-log plot is presented for non-Newtonian fluids in naturally fractured formations. The dimensionless fracture storativity ratio, ω, and interporosity flow parameter, λ, are obtained from characteristics points found on such plot. The developed equations and correlations are successfully verified by their application only to synthetic well test data since no actual field data are available. A good match is found between the results provided by the proposed technique and the values used to generate the simulated data.
... The injection of shear-thinning fluids (i.e. fluid viscosity dependent on the shear-rate) were analytically and numerically studied using pressure transient time series[259,320,173].Laoroongroj et al. (2012) [205] determined the apparent fluid viscosity of the shear-thinning fluid and the fluid front radius by PTA techniques. ...
Hydraulic fracturing is a common technique used in a variety of fields like civil and mining engineering, oil & gas and geothermal industry. It can be used to enhance the permeability of low permeable rocks, to increase the connectivity of natural fractures, to modify the rock mass strength, or to measure the Earth’s stress field. In the context of deep geothermal energy exploitation, a heat exchanger needs to be created at depth with characteristics favorable for heat extraction i.e. sufficient permeability and heat exchanger area. The creation of the heat exchanger for geothermal heat extraction remains a critical element with high associated risks including poor reservoir performance and induced seismicity. Hence the need for a better understanding of the coupled seismic-hydromechanical processes during stimulation operations. The execution of experiments on the intermediate-scale has the advantage of a better control on the processes associated with induced seismicity and reservoir performance compared to full-scale and allow to use comprehensive real time monitoring of pore pressure, rock mass deformation and seismicity. This scale is closer to the full-scale stimulation than laboratory scale, where seismo-hydromechanical interactions are generally focused on single fractures. The decameter-scale In-situ Stimulation and Circulation (ISC) project took place between 2015 and 2018 at the Grimsel Test Site (GTS), Switzerland. The GTS is located in the Central Swiss Alps, beneath the mountains of the Grimsel Pass. Overall, the moderately crystalline fractured rock mass shows a pervasive foliation and was intersected by six major sub-vertical shear zones. For each of the two assumed stimulation endmembers, hydraulic shearing and hydraulic fracturing, six experiments were conducted. Prior to the experiments, the test volume was characterized in great detail with respect to geology, geophysics, hydrogeology and in-situ stress field. This doctoral thesis aims at better understanding tensile fracture growth. It includes study of fracture toughness and fracture process zone on laboratory scale and the investigation of the seismo-hydromechanical coupled processes during in-situ hydraulic fracturing experiments.
The tested intact Grimsel Granodiorite samples indicate that the resistance against material failure is significantly higher across the foliation plane than along it. The results from Digital Image Correlation (DIC) confirm the development of a semi-elliptical fracture process zone (FPZ) with an average length to width ratio of about two for both principal directions. This agrees well with the available results in the literature. The experimental results of the length of the FPZ give supporting evidence to the fact that a nonlinear cohesion stress distribution provides an accurate cohesive model that agrees well with the experimental results. Additionally, the conformity of the ratio of the FPZ length in two principal directions with the theoretical predictions gives supporting evidence to the proportionality of the FPZ length with respect to the square of fracture toughness to tensile strength.
At the decametric scale during the in-situ experiment, the hydromechanical coupled responses of the rock mass and its fractures were captured by a comprehensive monitoring system installed along the tunnels and within dedicated boreholes. At the borehole scale, these processes involved newly created tensile fractures intersecting the injection interval while at the cross-hole scale, the natural network of fractures dominated the propagation process. The six HF experiments can be divided into two groups based on their injection location (i.e., south or north to a brittle ductile shear zone), their similarity of injection pressures and their response to deformation and pressure propagation. The experiments executed north of the shear zone, show smaller injection pressures and larger backflow during bleedoff phases. In addition, we observe re-orientation of the seismic cloud as the fracture propagated away from the wellbore. The re-orientation during propagation is interpreted to be related to a strong stress heterogeneity and the intersection of natural fractures striking different from the propagating hydraulic fracture. This leads in the details to complex geometry departing from theoretical mode I fracture geometries. The seismic activity was limited to about 10 m radial distance from the injection point. In contrast, strain and pressure signals reach further into the rock mass indicating that the process zone around the injection point is larger than the zone illuminated by seismic signals. Furthermore, strain signals indicate not just single fracture openings but also the propagation of multiple fractures.
Various methods to estimate the fracture opening and fracture contact pressure were applied and compared from single injection borehole observations with the strain gauge in distance from the injection point. The results show, that the fracture opening pressure was also observed at the strain gauge, associated with a strong increase of fracture transmissivity. The combination of injection pressure and strain observation allows to define an aperture-stress relationship with a general trend toward decreasing normal fracture stiffness during fracture opening. The fracture contact pressure can be estimated, but hydromechanical superposition of pump shut-in and corresponding pressure loss and interaction of the connected surrounding fractures make this task very challenging and error-prone. The pore pressure data set differentiate two distinct responses based on lag time and amplitudes. This allow to distinguish a near- and far-field response. The near-field response is due to pressure
diffusion and the far-field response is due to stress perturbation. The far-field pore pressure response is consistent for all experiments, indicating the dominant failure mechanism. This change in the far-field are very sensitive and can be used as a complementary method to seismic monitoring during hydraulic stimulations. The exceptional hydromechanical dataset allow to test numerical stimulations and can help to improve injection strategies, the monitoring design and the numerical modelling.
... Due to the above non-Newtonian nature of the polymer, the well test model used for the water flooding system is no longer suitable for the polymer flooding system. For the viscous polymer, a simplified single-phase model solved by the finite difference method was presented by Poollen and Jargon [29] to investigate the pressure transient behavior of a field test. Then, Ikoku and Ramey [30] and Odeh and Yang [31] developed analytical solutions of single-phase models to introduce new methods of pressure transient analysis for the viscous polymer solution. ...
Viscoelastic polymer solution shows shear thinning behavior at low shear rates and shear thickening behavior at high shear rates in reservoirs. However, models that ignored shear thickening behavior were commonly employed to interpret transient pressure data derived from tested wells in viscoelastic polymer flooding systems; although, viscoelastic polymer solutions show shear thickening behavior in the near-wellbore region due to high shear rate. To better characterize the oilfield with pressure transient analysis in viscoelastic polymer flooding systems, we developed a numerical model that takes into account both shear thinning behavior and shear thickening behavior. A finite volume method was employed to discretize partially differential flow equations in a hybrid grid system including PEBI mesh and Cartesian grid, and the Newton-Raphson method was used to solve the fully implicit nonlinear system. To illustrate the significance of our model, we compared our model with a model that ignores the shear thickening behavior by graphing their solutions on log-log plots. In the flow regime of near-wellbore damage, the pressure derivative computed by our model is distinctly larger than that computed by the model ignoring shear thickening behavior. Furthermore, the effect of shear thickening behavior on pressure derivative differs from that of near-wellbore damage. We then investigated the influence of shear thickening behavior on pressure derivative with different polymer injection rates, injection rates, and permeabilities. The results can provide a benchmark to better estimate near-wellbore damage in viscoelastic polymer flooding systems. Besides, we demonstrated the applicability and accuracy of our model by interpreting transient pressure data from a field case in an oilfield with viscoelastic polymer flooding treatments.
... Theoretical and experimental studies show that some fluids have the characteristics of Bingham type non-Newtonian behavior in porous media (Barenblatt et. al., 1968Pruess and Zhang, 1990;Ikoku and Ramey, 1978;Odeh and Yang, 1979;Savins, 1969, Van Poollen et al., 1969Gogerty, 1967). In this type of flow, fluid only moves when the applied pressure gradient is greater than the minimum pressure gradient, i.e., the threshold pressure gradient . ...
The rheological study of heavy crude oil is important in the field of petroleum engineering. The rheological properties of heavy oil (e.g., shear stress, shear rate, viscosity, etc.) depend on several factors including temperature, pressure, surface tension, diluent type and diluent composition, pH, shear stress and thermal histories, memory, and shear conditions during the analysis. The investigation of the rheology of heavy crude oil flow is a critical issue for both upstream and downstream operations. The objective of this study is to perform an investigation on the rheological properties of heavy crude oil to show the effect of shear rate, temperature, and pressure on the viscosity and the shear stress. The aim of this work was to broaden current knowledge of the rheological behavior and flow characterization of heavy crude oil. This paper takes a new look at the shear stress-strain relationship by considering the memory effect along with temperature effect on the shear rate.
It is considered that the viscosity of the heavy crude oil is a function of pressure, temperature, and shear rate. As the heavy crude oil is considered as a Bingham fluid, Bingham model is employed here for the analysis. The experimental data from previous studies are used to complete the analysis. To develop the model, a modified Darcy’s law that employs the effect of memory on the Bingham model is considered. The effect of temperature has been incorporated by the Arrhenius equation for the development of a new model to study the heavy crude oil rheological behaviors. The relationship between shear stress and viscosity has been shown at different fractional derivative order and time. The validation and the simulation of the model are performed by using the experimental and the field data from the literature. The numerical simulation of this model is conducted by using the MATLAB simulation software.
From the sensitivity analysis, it is found that the temperature has the highest impact on the viscosity over the pressure and the shear rate. On the other hand, the pressure shows a strong effect on the shear stress-shear rate relationship over the temperature. In the model analysis, it is found that the fluid memory affects in the Bingham model due to nonlinear behavior of heavy crude oil. The shear stress increases with decreasing viscosity at different fractional derivative order and time. The change in shear stress is high at large fractional derivative. The range of fractional derivative order is from 0.2 to 0.8. When fractional derivative order, 𝛼1=0.8, it shows a big change in the viscosity and shear stress relationship compared to 𝛼1=0.2. The shear stress also increases with viscosity as the time passes. Generally, it happens during the heavy crude oil production. The proposed model is validated with experimental data that shows a good match at 𝛼1=0.3. The simulation results show that the trends are the same for the viscosity-shear stress relationship when the memory effect is considered. In this context, we try to compare the effects of pressure, temperature, and shear rate on the viscosity and the shear stress-shear rate relationship, and to develop a model by considering temperature and memory effects for heavy crude oil. The study of heavy oil rheology has a great effect on transportation and processing standpoints. The different rheological measurements through quantitative experimental simulation of shear and thermal effects can significantly affect the pipeline design. We believe that we found an innovative solution to model the heavy crude oil rheology.
... However, there have been limited studies regarding pressure transients during polymer flooding, mainly because of the complexity in the coexistence of Newtonian fluids (oil and water) and non-Newtonian fluids (polymer). Some studies investigated the analytical and numerical pressure transient solutions in the polymer phase without consideration of the oil region, including Van Poollen and Jargon (1969), Ikoku and Ramey (1978), Odeh and Yang (1979), Vongvuthipornchai and Raghavan (1987), Katime-Meindl and Tiab (2001), Mahani et al. (2011), and Li and Delshad (2014). Studies of transient solutions with oil and Newtonian displacing fluids (i.e., water) were reported by Abbaszadeh and Kamal (1989). ...
Keywords:
Pressure Transient Analysis, Well Testing, Polymer Flooding, Non-Newtonian Fluids, Enhanced Oil Recovery.
Summary:
Improved-oil-recovery (IOR) and enhanced-oil-recovery (EOR) methods are used to increase recovery from proven reserves, mainly after waterflooding. Monitoring and managing the progress of flood in IOR and EOR operations is currently a challenge to the oil industry, especially in situations with large well spacing and cost-prohibitive measures such as drilling observation wells (e.g., in offshore and deepwater applications). Falloff tests have proved to be successful under waterflooding operations to determine the reservoir properties in various banks around injection wells and the location of flood fronts. In this paper, we present a new development that extends transient testing and analysis technology to IOR and EOR operations during polymer flooding. With the expanded use of permanent downhole-pressure gauges (PDHGs), the newly developed technique can be used without additional testing cost or interruption of field operations.
In this paper, the effects of polymer are described by shear-rate-dependent viscosity (non-Newtonian flow). We developed an analytical solution of wellbore pressure by combining the non-Newtonian fluids and the multicomposite reservoir models. The solution addresses the polymer region, where the fluids follow either the power law or Meter’s model (Meter and Bird 1964), and the Newtonian flow in the oil or water regions ahead of the polymer, with varying Newtonian- and non-Newtonian-fluid saturations in both regions. The developed solution was validated by analyzing synthetic data generated using a commercial numerical reservoir simulator. In secondary-recovery operations, the Newtonian fluid ahead of the polymer bank is usually oil, and in tertiary-recovery operations, the Newtonian fluid is usually the water used in waterflooding. The solution provides a deeper understanding of the physics behind the pressure transient behaviors during polymer flooding, and can be applied to guide a better implementation of well tests. An interpretation method for falloff tests using the new solution and the conventional Bourdet derivative and Horner plots is presented, indicating that existing commercial well-testing software is sufficient to analyze data with the recent development. The new solution allows us to obtain reservoir properties such as fluid mobilities in various banks and the location of the flood front.
The developed solution was applied to field data. The pressure behavior expected from the new solution was observed in the field data, validating our developed technique and yielding the characterization of reservoir parameters in various banks. Field-application results are shown in the paper.
The novelty of this method of characterizing the dynamic properties of the various banks during injection of non-Newtonian fluids and the location of the flood fronts is that an analytical solution of pressure transient behavior in two-phase flow of non-Newtonian fluids and Newtonian fluids was developed, validated, and used to analyze field data. This is the first analytical solution published to address this situation.
... For the unsteady flow of non-Newtonian fluids in a multilayered reservoir, van Poollen and Jargon [8] studied non-Newtonian power-law fluid unsteady flow in porous media and showed that the transient pressure response characteristics are different from that of Newtonian fluid. Ikoku and Ramey [9] studied non-Newtonian power-law fluid unsteady flow characteristics in porous medium, and the consideration of wellbore storage and skin effect is obtained in homogeneous infinite reservoir model in Laplace space solution. ...
A new well-test model is presented for unsteady flow in multizone with crossflow layers in non-Newtonian polymer flooding reservoir by utilizing the supposition of semipermeable wall and combining it with the first approximation of layered stable flow rates, and the effects of wellbore storage and skin were considered in this model and proposed the analytical solutions in Laplace space for the cases of infinite-acting and bounded systems. Finally, the stable layer flow rates are provided for commingled system and crossflow system in late-time radial flow periods.
... Many studies in petroleum engineering, chemical engineering and rheology have focused on non-Newtonian fluid behavior through porous formations, among them, we can name Hirasaki and Pope (1974); Ikoku (1979); Ikoku and Ramey (1979); Odeh and Yang (1979); Savins (1969) and Van-Poollen and Jargon (1969). Ikoku (1978) also presented several analytical solutions including finite systems, which are used in this paper, for non-Newtonian fluids in homogeneous and heterogeneous reservoirs. ...
Since conventional oil is almost depleted, oil companies are focusing their efforts on exploiting heavy oil reserves. A modern and practical technique using the pressure and pressure derivative, log-log plot for estimating the well-drainage area in closed and constant-pressure reservoirs, drained by a vertical well is presented by considering a non-Newtonian flow model for describing the fluid behavior. Several synthetic examples were presented for demonstration and verification purposes. Such fluids as heavy oil, fracturing fluids, some fluids used for Enhanced Oil Recovery (EOR) and drilling muds can behave as either Power-law or Bingham, usually referred to as the non-Newtonian fluids. Currently, there is no way to estimate the well-drainage area from conventional well test analysis when a non-Newtonian fluid is dealt with; therefore, none of the commercial well test interpretation package can estimate this parameter (drainage area).
... Savins (1969) presented a comprehensive review of the rheological behavior of non-Newtonian fluids and their flow behavior through porous media. van Poollen and Jargon (1969) presented a finite-difference solution for transient-pressure behavior, while Odeh and Yang (1979) derived an approximate closed-form analytical solution of the problem. Chakrabarty et al. (1993) presented Laplace-space solutions for transient pressure in fractal reservoirs. ...
The displacement of non-Newtonian power-law fluids in communicating stratified reservoirs with a log-normal permeability distribution is studied. Equations are derived for fractional oil recovery, water cut, injectivity ratio, and pseudorelative permeability functions, and the performance is compared with that for Newtonian fluids. Constant-injection-rate and constant-total-pressure-drop cases are studied.
The effects of the following factors on performance are investigated: the flow-behavior indices, the apparent mobility ratio, the Dykstra-Parsons variation coefficient, and the flow rate. It was found that fractional oil recovery increases for nw>no and decreases for nw<no, as compared with Newtonian fluids. For the same ratio of nw/no, oil recovery increases as the apparent mobility ratio decreases. The effect of reservoir heterogeneity in decreasing oil recovery is more apparent for the case of nw>no. Increasing the total injection rate increases the recovery for nw>no, and the opposite is true for nw<no. It also was found that the fractional oil recovery for the displacement at constant total pressure drop is lower than that for the displacement at constant injection rate, with the effect being more significant when nw<no.
... It is notable that van Poollen and Jargon [33] have provided an analytical solution of steady-state non-Newtonian fluid flow. They have also provided a numerical solution of transient non-Newtonian fluid flow. ...
We have developed a new method to analyze the power law based non-Darcian flow toward a well in a confined aquifer with and without wellbore storage. This method is based on a combination of the linearization approximation of the non-Darcian flow equation and the Laplace transform. Analytical solutions of steady-state and late time drawdowns are obtained. Semi-analytical solutions of the drawdowns at any distance and time are computed by using the Stehfest numerical inverse Laplace transform. The results of this study agree perfectly with previous Theis solution for an infinitesimal well and with the Papadopulos and Cooper’s solution for a finite-diameter well under the special case of Darcian flow. The Boltzmann transform, which is commonly employed for solving non-Darcian flow problems before, is problematic for studying radial non-Darcian flow. Comparison of drawdowns obtained by our proposed method and the Boltzmann transform method suggests that the Boltzmann transform method differs from the linearization method at early and moderate times, and it yields similar results as the linearization method at late times. If the power index n and the quasi hydraulic conductivity k get larger, drawdowns at late times will become less, regardless of the wellbore storage. When n is larger, flow approaches steady state earlier. The drawdown at steady state is approximately proportional to r1−n, where r is the radial distance from the pumping well. The late time drawdown is a superposition of the steady-state solution and a negative time-dependent term that is proportional to t(1−n)/(3−n), where t is the time.
... It is notable that van Poollen and Jargon [33] have provided an analytical solution of steady-state non-Newtonian fluid flow. They have also provided a numerical solution of transient non-Newtonian fluid flow. ...
In this study, we have obtained numerical solutions for non-Darcian flow to a well with the finite difference method on the basis of the Izbash equation, which states that the hydraulic gradient is a power function of the specific discharge. The comparisons between the numerical solutions and the Boltzmann solutions and linearization solutions have also been done in this study. The results indicated that the linearization solutions for both the infinitesimal-diameter well and the finite-diameter well agree very well with the numerical solution at late times, while the linearization method underestimates the dimensionless drawdown at early and moderate times. The Boltzmann method works well as an approximate analytical solution for the infinitesimal-diameter well. Significant differences have been found between the Boltzmann solution for a finite-diameter well and the numerical solution during the entire pumping period. The analysis of the numerical solution implies that all the type curves inside the well for different dimensionless non-Darcian conductivity kD values approach the same asymptotic value at early times, while a larger kD leads to a smaller drawdown inside the well at late times. A larger kD results in a larger drawdown in the aquifer at early times and a smaller drawdown in the aquifer at late times. Flow approaches steady-state earlier when kD is larger.
In oil and gas engineering, non-Newtonian fluids are universal. Thus, studies on the seepage laws of non-Newtonian power-law fluids are of great importance and significance. In this paper, we introduced the definition of elastic coefficient based on the establishing elastic outer boundary conditions. We then set up a homogeneous well test model for non-Newtonian power-law fluids under elastic outer boundary conditions, considering wellbore reservoir and skin effect. The Laplace space solution of the bottom hole pressure was obtained using the similar structure method, and the double logarithmic characteristic curve was drawn using the Stehfest numerical inversion technique. Furthermore, the main parameters affecting the characteristic curve were analyzed. The experimental results showed that the early phase of the homogeneous well test curve can be affected by the wellbore storage coefficient. In addition, the mid-term seepage is greatly influenced by the power-law flow exponent and the skin factor, while the elastic coefficient predominantly acts on the later phase of fluid flow. We concluded that the introduction of the elastic outer boundary expands the traditional fixed form of ideal outer boundary conditions, enabling a more general representation of the seepage model. This approach provides a more effective theoretical basis for further exploration of the seepage law of the reservoir. Meanwhile, we simplify the calculation process of the solution model by the use of the similar structure method of the solution and point out a new research direction for solving the more complex seepage model.
The oil-water two-phase flow pressure-transient analysis model for polymer flooding fractured well is established by considering the comprehensive effects of polymer shear thinning, shear thickening, convection, diffusion, adsorption retention, inaccessible pore volume and effective permeability reduction. The finite volume difference and Newton iteration methods are applied to solve the model, and the effects of fracture conductivity coefficient, injected polymer mass concentration, initial polymer mass concentration and water saturation on the well-test type curves of polymer flooding fractured wells are discussed. The results show that with the increase of fracture conductivity coefficient, the pressure conduction becomes faster and the pressure drop becomes smaller, so the pressure curve of transitional flow goes downward, the duration of bilinear flow becomes shorter, and the linear flow appears earlier and lasts longer. As the injected polymer mass concentration increases, the effective water phase viscosity increases, and the pressure loss increases, so the pressure and pressure derivative curves go upward, and the bilinear flow segment becomes shorter. As the initial polymer mass concentration increases, the effective water phase viscosity increases, so the pressure curve after the wellbore storage segment moves upward as a whole. As the water saturation increases, the relative permeability of water increases, the relative permeability of oil decreases, the total oil-water two-phase mobility becomes larger, and the pressure loss is reduced, so the pressure curve after the wellbore storage segment moves downward as a whole. The reliability and practicability of this new model are verified by the comparison of the results from simplified model and commercial well test software, and the actual well test data.
The oil field enters the stage of ultra-high water cut development, especially after the third infilling, the well spacing is small. The water injection advances quickly, the water cut rises quickly, the control of natural decline becomes more and more difficult. According to the traditional oil field development management method, the oil field steady production difficulty will increase continuously. Therefore, we must constantly explore new methods of oilfield development to improve the quality and benefit of oilfield development.
In this paper a model is established for unstable seepage flow with polymer concentration and pressure diffusion coupling, considering the effects of polymer molecular diffusion, adsorption, and viscoelasticity of polymer solution in the formation. The factors are close to the actual seepage parameters of the injected reservoir. For the nonlinear adsorption, the combined variable and the analytical iterative method are used to obtain the approximate analytical solution of the model. According to the concentration model, the relationship between concentration and pressure distribution is obtained. Using the model theory curve to fit the well test data, the seepage parameters of the formation are obtained, and the reflection characteristics of the unstable wellbore pressure derivative curve are analyzed.
In this study, a 3-Dimensional non-linear partial differential equation (PDE) describing flow of non-Newtonian fluid in porous media was developed for a no-flow boundary reservoir. Non-Newtonian fluid flow in porous media has direct applications in polymer flooding for secondary oil recovery operations and flow of heavy crude in the reservoir. This novel work presents the pressure behavior of horizontal wells with non-Newtonian fluid flow in porous media as well as the methodology for analyzing pressure transient data from non-Newtonian reservoirs.
The main assumptions in the mathematical modeling of the differential equation are; permeability anisotropy with directional permeabilities kx, ky and kz horizontal well is in the y-direction perpendicular to direction of maximum permeability kx effects of gravity, skin and wellbore storage were neglected and the reservoir fluid was considered to be a non-Newtonian pseudo plastic fluid that obeys power law model in an isothermal condition.
The derived equation was discretized using finite difference approach; A 3D numerical simulator was developed with the aid of MATLAB to solve the system of linear equations obtained from the discretization of a 15 X15 × 15 grid system to obtain pressure transient data. Type curves in terms of PwD and tD were generated for different power law flow index n ranging from 0.1 to 1 for horizontal well length of 600ft, 1000ft and 1200ft.
The developed type curves in this study were validated with a Newtonian case using Tiab Direct synthesis (TDS) technique to analyze the radial flow regime for the determination of average permeability as well as the early linear flow for determining kx The results obtained from the Newtonian fluid case were very close to the actual property been determined.
Keywords:
pressure transient analysis, well test analysis, polymer flooding, non-Newtonian fluids, EOR, IOR.
Abstract:
Improved and enhanced oil recovery methods (IOR and EOR, respectively) are used to increase recovery from proven reserves mainly following waterflooding. Monitoring and managing the progress of flood in IOR /EOR operations is currently a challenge to the oil industry especially in situations with large well spacing and cost prohibitive measures like drilling observation wells (e.g., offshore and deep water applications). Falloff tests have been proven successful under waterflooding operations to determine the reservoir properties in various banks around the injection wells and the location of the flood fronts (Abbaszadeh and Kamal 1989, Yeh and Agarwal 1992, and Kamal 2009). In this paper we share new development that extends transient testing and analysis technology to IOR and EOR operations during polymer flooding. With the expanded use of Permanent Downhole Pressure Gauges (PDHG), the new developed technique can be used without additional operational cost or interruption of field operations.
In this paper, the effects of polymer are described by shear rate dependent viscosity (non-Newtonian flow). We developed an analytical solution of wellbore pressure by combining the non-Newtonian fluids and the multi-composite reservoir models. The solution dose not only address the polymer region where the fluids follow either the power-law (Ostwald 1929) or the Meter model (Meter and Bird 1964), but also the Newtonian flow in the oil region with varying oil and polymer saturations in both regions. The developed solution was validated by analyzing synthetic data generated using a commercial numerical reservoir simulator. The solution provides a deeper understanding about the physics behind the transient pressure behaviors during polymer flooding, and can be applied to guide a better implementation of well tests. Interpretation method for falloff tests using the new solution and the conventional Bourdet derivative and Horner plots is presented indicating that existing commercial well testing software are sufficient to analyze data with the recent development. The new solution allows us to obtain the reservoir properties such as fluid mobilities in various banks and the location of the flood front.
The developed solution was applied to field data. The pressure behavior expected from the new solution was observed in field data validating our developed technique and yielding the characterization of reservoir parameters in various banks.
The novelty of this method of characterizing the dynamic properties of the various banks during injection of non-Newtonian fluids and the location of the flood fronts is that an analytical solution of pressure transient behavior in two phase flow of non-Newtonian fluids and Newtonian fluids was developed, validated and used to analyze field data. This is the first analytical solution developed for this type of problem.
As a special kind of non-Newtonian fluid, Bingham fluids (or plastics) exhibit a finite yield stress at zero shear rates. There is no gross movement of fluids until the yield stress, y, is exceeded. Once this is accomplished, it is also required cutting efforts to increase the shear rate, i.e. they behave as Newtonian fluids. These fluids behave as a straight line crossing the y axis in = y, when the shear stress, plotted against the shear rate, in Cartesian coordinates. The characteristics of these fluids are defined by two constants: the yield, y, which is the stress that must be exceeded for flow to begin, and the Bingham plastic coefficient, B. The rheological equation for a Bingham plastic is,
The chapter summarizes in a comprehensive manner, the existing literature on non-Newtonian effects in multiphase particulate systems. The chapter sets out to elucidate the influence of non-Newtonian fluid behavior on momentum, heat, and mass-transport processes as encountered in packed beds, liquid-solid and three-phase fluidized beds, and hindered settling in concentrated suspensions. Non-Newtonian fluids are classified into three broad categories—namely, time-independent, time-dependent, and viscoelastic fluids. The available literature on multiphase particulate systems can conveniently be divided into three categories depending on the configuration—namely, packed or fixed beds, fluidized beds, and hindered settling of concentrated suspensions. The flow of time-independent fluids in unconsolidated packed beds and porous media has been studied most extensively. The chapter examines the frictional pressure drop incurred in the flow of purely viscous fluids through packed beds of spherical and nonspherical particles under most conditions of practical interest, provided no anomalous effects are present.
In this paper, influences on the reservoir permeability, the reservoir architecture and the fluid flow pattern caused by hydraulic fracturing are analyzed. Based on the structure and production fluid flow model of post fracturing high-rank coal reservoir, Warren-Root Model is improved. A new physical model that is more suitable for post fracturing high-rank coal reservoir is established. The results show that the width, the flow conductivity and the permeability of hydraulic fractures are much larger than natural fractures in coal bed reservoir. Hydraulic fracture changes the flow pattern of gas and flow channel to wellbore, thus should be treated as an independent medium. Warrant-Root Model has some limitations and can’t give a comprehensive interpretation of seepage mechanism in post fracturing high-rank coal reservoir. Modified Warrant-Root Model simplifies coal bed reservoir to an ideal system with hydraulic fracture, orthogonal macroscopic fracture and cuboid matrix. Hydraulic fracture is double wing, vertical and symmetric to wellbore. Coal bed reservoir is divided into cuboids by hydraulic fracture and further by macroscopic fractures. Flow behaviors in coal bed reservoir are simplified to three step flows of gas and two step flows of water. The swap mode of methane between coal matrix and macroscopic fractures is pseudo steady fluid channeling. The flow behaviors of methane to wellbore no longer follow Darcy’s Law and are mainly affected by inertia force. The flow pattern of water follows Darcy’s Law. The new physical model is more suitable for post fracturing high-rank coal reservoir.
Polymer solution is known as non-Newtonian Fluid. Hence, when a well is injected by polymer solution, the well test data analysis using Newtonian fluid flow model will be erroneous. However, the analysis results usually were inaccurate when generalized non-Newtonian fluid model which considering polymer solution as power law fluid and taking no account of physical and chemical behaviors. These results clearly suggest the need for a study to come up with a new model considering both physical and chemical behaviors when polymer solution flowing in the reservoirs. At first, this study modified two parameter models: viscosity model and permeability decreasing coefficient model, all of them considering diffusion, conduction and IPV (inaccessible pore volume). Then, those models were applied to set up the new well testing model of a well located in an infinite reservoir. The log-log plots of the pressure and pressure derivatives have been prepared through numerical solutions. A further study has been done about the characteristics of the new type curves considering different parameters.
In this work, steady-state flow experimental data have been analyzed for two commonly used polymers representing two generic classes: polysaccharides (xanflood), and partially hydrolyzed polyacrylamides (pusher-700) flowing inside bead packs and Berea sandstone. Oscillatory flow measurements have been used to compute the polymer solution's longest relaxation time (qf1). Steady-state flow experimental data for the two polymers combined with measured polymer viscous properties have been converted to average shear stress-shear rate data inside porous media. An average power-law exponent () is therefore obtained for the polymer flow inside the porous medium. Using qf1 and , rock permeability (k) and porosity (f) and fluid flow velocity (u), a dimensional number Nv (viscosity number) is calculated, and found to strongly correlate with the pressure gradient inside porous medium. This correlation is the basis for defining a viscoelastic model for polymer flow in porous media. The capillary-tube model is found to be adequate for only flow of viscous polymers with insignificant elasticity.
The transient flow behavior of non-Newtonian fluids in petroleum reservoirs is studied. A new partial differential equation is derived. The diffusivity equation is a special case of the new equation. The new equation describes the flow of a slightly compressible, non-Newtonian, power-law fluid in a homogeneous porous medium. This equation should govern the flow of most non-Newtonian oil-displacement agents used in secondary and tertiary oil-recovery projects, such as polymer solutions, micellar projects, such as polymer solutions, micellar solutions, and surfactant solutions.
Analytical solutions of the new partial differential equation are obtained that introduce new methods of well-test analysis for non-Newtonian fluids. An example is presented for using the new techniques to analyze injection well-test data in a polymer injection project. project. Graphs of the dimensionless pressure function also are presented. These may be used to investigate the error when using Newtonian fluid-flow equations to model the flow of non-Newtonian fluids in porous media.
Introduction
Non-Newtonian fluids, especially polymer solutions, microemulsions, and macroemulsions, often are injected into the reservoir in various enhanced oil-recovery processes. In addition, foams sometimes are circulated during drilling. Thermal recovery of oil by steam and air injection may lead to the flow of natural emulsions and foams through porous media. Some enhanced oil-recovery projects involving the injection of non-Newtonian fluids have been successful, but most of these projects either failed or performed below expectation. These results suggest the need for a thorough study of the stability of non-Newtonian fluids at reservoir conditions, and also a new look at the flow of non-Newtonian fluids in porous media. porous media. Many studies of the rheology of non-Newtonian fluids in porous media exist in the chemical engineering, rheology, and petroleum engineering literature. In 1969, Savins presented an important survey on the flow of non-Newtonian fluids through porous media. In some cases, he interpreted porous media. In some cases, he interpreted published data further and compared results of published data further and compared results of different investigators. van Poollen and Jargon presented a numerical study of the flow of presented a numerical study of the flow of non-Newtonian fluids in homogeneous porous media using finite-difference techniques. They considered steady-state and unsteady-state flows and used the Newtonian fluid-flow equation. They considered non-Newtonian behavior by using a viscosity that varied with position. No general method was developed for analyzing flow data. Bondor et al. presented a numerical simulation of polymer presented a numerical simulation of polymer flooding. Much useful information on polymer flow was presented, but transient flow was not considered.At present, there is no standard method in the petroleum engineering literature for analyzing petroleum engineering literature for analyzing welltest data obtained during injection of non-Newtonian fluids into petroleum reservoirs. However, injection of several non-Newtonian oil-displacement agents is an important oilfield operation. Interpretation of well-test data for these operations should also be important. Obviously, procedures developed for Newtonian fluid flow are not appropriate.
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A polymer solution is classified as non-Newtonian Fluid. The results of pressure data analysis for polymer flooding reservoirs would be inaccurate when generalized using non-Newtonian fluid flow models, which treat polymer solution as power law fluid. To improve analysis results, a study to create new pressure interpretation models for polymer flooding reservoirs is needed. First, this study presents a new viscosity model by considering diffusion and conduction. Then the model is applied to set up a new model of an injection well located at an infinite polymer flooding reservoir. The modeling solutions have been prepared through numerical iterations. An extended study has been done about the characteristics of the new pressure response curves.
We analyze the transient motion of a non-Newtonian power-law fluid in a porous medium of infinite extent and given geometry (plane, cylindrical or spherical). The flow in the domain, initially at constant ambient pressure, is induced by fluid withdrawal or injection in the domain origin at prescribed pressure or injection rate.Previous literature work is generalized and expanded, providing a dimensionless formulation suitable for any geometry, and deriving similarity solutions to the nonlinear governing equations valid for pseudoplastic, Newtonian and dilatant fluids. A pressure front propagating with finite velocity is generated when the fluid is pseudoplastic; no such front exists for Newtonian or dilatant fluids. The front rate of advance depends directly on fluid flow behavior index and inversely on medium porosity and domain dimensionality.The effects and relative importance of uncertain input parameters on the model outputs are investigated via Global Sensitivity Analysis by calculating the Sobol’ indices of (a) pressure front position and (b) domain pressure, by adopting the Polynomial Chaos Expansion technique. For the selected case study, the permeability is the most influential factor affecting the system responses.
Steady-state flow experimental data have been analyzed for two commonly used polymers representing two generic classes: polysaccharides (xanflood), and partially hydrolyzed polyacrylamides (pusher-700) flowing inside bead packs and Berea sandstone. Oscillatory flow measurements have been used to compute the polymer solution's longest relaxation time (θf1) which is referred to as the characteristic relaxation time in this paper. Steady-state flow experimental data for the two polymers combined with measured polymer viscous properties have been converted to average shear stress-shear rate data inside porous media. An average power-law exponent (n¯) is therefore obtained for the polymer flow inside porous media. Using and θf1, and n¯, rock permeability (k) and porosity (ϕ) and fluid flow velocity (u), a viscoelasticity number (NV) is calculated, and found to strongly correlate with the pressure gradient inside porous media. This correlation is the basis for defining a viscoelastic model for polymer flow, analogous to Darcy's law. The proposed model asserts a non-linear relationship between fluid velocity and pressure gradient. It accounts for polymer elasticity, and for pore geometry changes due to molecular adsorption and mechanical entrapment.
A new type curve for well test analysis for non-newtonian fluids in petroleum reservoir is developed. The general analytical solution in Laplace variable presented by Ikoku and Ramey Jr.2 forms the mathematical basis of the proposed type-curves. The equation for the type curve was developed, following similar procedure to that of Bourdet et al10. The Bessel functions involved in the final solution of the non-newtonian case cannot be approximated by logarithm function as in the Newtonian case. Hence, the dimensionless group of the skin factor and wellbore storage coefficient used in the Bourdet et al's case was not used in this study. Instead, only the skin factor is retained. The dimensionless wellbore storage coefficient is grouped with the dimensionless time as usual.
The log-log plot of the pressure derivatives at the infinite acting radial flow lie on a straight line for each flow behavior index. This straight line intersects the Newtonian infinite acting pressure derivative line at tD/CD = 1. In addition to the unit slope line of the wellbore storage region, this point of intersection provides a fulcrum point for proper curve matching. Thus the characteristic mobility can be computed using this intersection point.
Apart from the conventional type-curve-matching method of analysis, the Tiab's direct synthesis (TDS) technique is developed for the evaluation of well test data in non-newtonian fluid flow. This is based on the long time solution as in the conventional case and the characteristic line of the type curve. The process does not involve type curve matching, but provides a direct method of evaluating the well test data from the log-log plot of the pressure and pressure derivatives. Two examples from the references 3 and 6 were used to validate the type curves and satisfactory results were obtained.
This study simulates the pressure transient behavior of flow of Newtonian/Newtonian and nonNewtonian/Newtonian fluid composite systems in porous media with a finite-conductivity vertical fracture. The main objective of this study is to determine the location of the flood front, reservoir rock and fluid properties and fracture parameters via history match ing of the test data. It is assumed that the inject fluid front propagates in the form of an ellipse who focal length is equivalent to the fracture length, and single-phase flow conditions exist behind and ah of the flood front. The viscosity of the non-Newton fluid in the flooded region was formulated as a function of the average flow rate. Both the injected and the reservoir fluids are treated as being slightly compressible.
It is assumed that flow inside the matrix is two-dimensional, and one-dimensional inside the fracture. Two different fluid flow equations, one for the matrix and the other for the fracture, are used to describe the single-phase flow conditions. The equations are coupled through a fluid loss term from the fracture into the matrix and/or from the matrix into the fracture. Four different fluid loss models in three different flow geometries (elliptical, linear and radial) are considered. These two non-linear equations are linearized numerically, and the resulting two systems of linear equations are solved simultaneously using the alternating direction implicit procedure.
Extensive testing and validation of this generalized formulation was conducted by using it to study nine special cases previously reported in the literature. A parametric investigation was conducted using dimensionless groups such as non-Newtonian power-law fluid exponent, fracture conductivity, elliptical interface location and mobility and diffusivity ratio. The results indicate that as the power-law index increases, the dimensionless bottom-hole pressure increases for a given injection rate. In cases of mobility ratios greater than unity, smaller elliptical interface locations, which represent initial amounts of injected fluid in the reservoir, give greater dimensionless bottomhole pressures, and in cases of mobility ratios less than unity, smaller elliptical interface locations give smaller dimensionless bottomhole pressures. Greater mobility ratios yield greater dimensionless bottomhole pressures; increasing dimensionless fracture conductivities result in decreased bottomhole dimensionless pressures. It is observed that this decrease is not significant for dimensionless conductivities larger than five hundred.
Introduction
Same enhanced oil recovery processes such as polymer flooding involve the injection of non-Newtonian fluids into a reservoir to increase oil recovery. The primary function of such a fluid injection scheme is to improve the mobility ratio. Uniform flood front, less pronounced viscous fingering and decrease in by-passing of the least permeable sections of the reservoir are some of the attributes of the non-Newtonian fluid injection. Thus, both the areal sweep efficiency and the displacement efficiency can be significantly improved. The injection of polymer solutions, and plain water injection processes are examples of non-Newtonian/Newtonian and Newtonian/ Newtonian fluid composite systems, respectively, considered in this study. Problems involving a non-Newtonian fluid have been studied in the literature/
Other EOR techniques such as steam injection and in-situ combustion invariably result in Newtonian/ Newtonian fluid composite system in the reservoir.
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Non-Newtonian fluids are very common during drilling, fracture operations and enhanced oil recovery processes. When a reservoir contains a non-Newtonian fluid such as those injected during polymer flood or the production of heavy oil, well test data cannot be interpreted using Newtonian fluid flow models. The resulting analysis would be erroneous because non-Newtonian fluids behave rather differently.
These results suggest the need for a thorough study of the behavior of non-Newtonian fluids in the reservoir and also a new look at the flow of those fluids in porous media.
This study presents an interpretation technique for pressure behavior of non-Newtonian fluid flow in a homogeneous reservoir without type-curve Matching. The inclusion of a no-flow and/or a constant pressure line is also investigated.
First, the TSD (Tiab’s Direct Synthesis) technique was applied for analyzing the pressure behavior of a well located in (1) an infinite reservoir and, (2) near a linear boundary where wellbore storage and skin effects were considered. The analysis required the generation of type-curve sets for different wellbore storage and skin values.
A step-by-step procedure is presented for the calculation of the reservoir parameters: the permeability/viscosity ratio, wellbore storage coefficient, skin factor and the distance to the nearest boundary without the use of type-curve Matching. The procedure is illustrated by an example.
This study investigates pressure falloff testing in non-Newtonian/Newtonian fluid composite reservoirs. The non-Newtonian fluids of interest exhibit dilatant behavior. Initial water saturation is accounted for. Application of non-Newtonian well test analysis techniques and conventional Horner (Newtonian) techniques is investigated. The effects of different injection times before shut-in, external radii, flow behavior indexes and non-Newtonian fluid consistencies on the pressure transient behavior constitute the salient features of this work.
It is shown that early time falloff pressure data can be analyzed by non-Newtonian techniques while the late shut-in data, under certain conditions, can be analyzed by the conventional Horner method. The time when the Newtonian fluid starts influencing the non-Newtonian falloff curves and the location of the non-Newtonian fluid front can be estimated by using the radius of investigation equation for power-law fluids and volumetric balance equation respectively. Rheo-logical consideration is made to illustrate the pressure transient behavior.
A model recently presented by Ikoku and Ramey for non-Newtonian power-law flow in porous media was extended to flow in finite circular reservoirs. A constant flow rate was stipulated at the wellbore, and two boundary conditions were considered: no-flow outer boundary and constant-pressure outer boundary. The results were used to derive a new expression for the stabilization time for power-law flow in porous media.
Wellbore storage and skin effects always distort the transient pressure behavior of wells in petroleum reservoirs. It is important to investigate the consequences of these phenomena and be able to interpret real well test information. This paper considers the effects of skin and wellbore storage on the transient flow of non-Newtonian power-law fluids in petroleum reservoirs.
A new numerical wellbore storage simulator was used to study the effects of skin and wellbore storage during the transient flow of power-law fluids in infinitely large and finite circular reservoirs. Results are presented both in tabular form and as log-log graphs of dimensionless pressures vs dimensionless times. The log-log graphs may be used in a type-curve matching procedure to analyze short-time well test data.
The early period is dominated by wellbore storage effect. A new expression was obtained for the duration of wellbore storage effect when skin exists for infinitely large reservoirs. This criterion is not valid for finite circular reservoirs with no-flow outer boundary or constant-pressure outer boundary. Results indicate that there is no apparent end of wellbore storage effect for the no-flow outer boundary condition for the values of external radius presented. New relationships were derived for skin presented. New relationships were derived for skin factor and "effective well radius" for power-law flow.
This work presents a theoretical study of the flow and displacement of a Bingham fluid in porous media. An integral method of analyzing the single-phase flow of this type of fluid is developed. The accuracy of a newly developed approximate analytical solution for transient-flow problems is confirmed by comparison with numerical solutions. The flow behavior of a slightly compressible Bingham fluid is discussed, and a new well-test-analysis method is developed by use of the integral solution. To obtain some understanding of the physics of immiscible displacement with Bingham fluids, a Buckley-Leverett analytical solution with a practical graphic evaluation method was developed and applied to the problem of displacing a Bingham fluid with water. Results revealed that the saturation profile and displacement efficiency are controlled not only by the relative permeabilities, as in the case of Newtonian fluids, but also by the inherent complexities of Bingham non-Newtonian behavior. In particular, we found that in the displacement process with a Bingham fluid, a limiting maximum saturation exists beyond which no further displacement can be achieved.
Steady-state flow experimental data have been analyzed for two commonly used polymers representing two generic classes, polysaccharides (Xanflood), and partially hydrolyzed polyacrylamides (Pusher-700), flowing inside bead packs and Berea sandstone. Oscillatory flow measurements have been used to compute the polymer solution's longest relaxation time (θf1), which is referred to as the characteristic relaxation time in this paper. Steady-state flow experimental data for the two polymers combined with measured polymer viscous properties have been converted to average shear stress−shear rate data inside porous media. An average power-law exponent (n̄) is therefore obtained for the polymer flow inside the porous medium. Using θf1, n̄, rock permeability (k), porosity (φ), and fluid flow velocity (u), a viscoelasticity number (Nv) is calculated and found to strongly correlate with the pressure gradient inside porous media. This correlation is the basis for defining a viscoelastic model for polymer flow, analogous to Darcy's law. The proposed model asserts a nonlinear relationship between fluid velocity and pressure gradient. It accounts for polymer elasticity and for pore geometry changes due to molecular adsorption and mechanical entrapment.
Gas flow in porous media differs from liquid flow because of the large gas compressibility and pressure-dependent effective permeability. The latter effect, named after Klinkenberg, may have significant impact on gas flow behavior, especially in low permeability media, but it has been ignored in most of the previous studies because of the mathematical difficulty in handling the additional nonlinear term in the gas flow governing equation. This paper presents a set of new analytical solutions developed for analyzing steady-state and transient gas flow through porous media including Klinkenberg effects. The analytical solutions are obtained using a new form of gas flow governing equation that incorporates the Klinkenberg effect. Additional analytical solutions for one-, two- and three-dimensional gas flow in porous media could be readily derived by the following solution procedures in this paper. Furthermore, the validity of the conventional assumption used for linearizing the gas flow equation has been examined. A generally applicable procedure has been developed for accurate evaluation of the analytical solutions which use a linearized diffusivity for transient gas flow. As application examples, the new analytical solutions have been used to verify numerical solutions, and to design new laboratory and field testing techniques to determine the Klinkenberg parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs from The Geysers geothermal field. We show that this new approach and the traditional method of Klinkenberg yield similar results of Klinkenberg constants for the laboratory tests; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.
This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.
An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity
is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation
of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing
the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness.
The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under
the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability
variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability
of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian
fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings.
The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index.
For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability
is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity
is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.
KeywordsNon-Newtonian-Power–law fluid-Porous media-Heterogeneity
A problem of current interest in geothermal energy extraction is the injection of cold water into a porous medium containing superheated vapor. Such injection will cause a boiling front to move away from the injection point. When flow is approximated as being one-dimensional radial, it can be shown from similarity variable concepts that temperature, pressure, and boiling rate at the front are constant, independent of time. From heat and mass balance considerations an analytical solution is obtained for front temperature and evaporation rate. Comparison with detailed numerical simulations of the injection process shows excellent agreement.
Flow and displacement of non-Newtonian fluids in porous media occurs in many subsurface systems, related to underground natural resource recovery and storage projects, as well as environmental remediation schemes. A thorough understanding of non-Newtonian fluid flow through porous media is of fundamental importance in these engineering applications. Considerable progress has been made in our understanding of single-phase porous flow behavior of non-Newtonian fluids through many quantitative and experimental studies over the past few decades. However, very little research can be found in the literature regarding multi-phase non-Newtonian fluid flow or numerical modeling approaches for such analyses.
Flow of non-Newtonian fluids through porous media occurs in many subsurface systems and has found applications in certain technological areas. Previous studies of the flow of fluids through porous media were focusing for the most part on Newtonian fluids. Since the 1950s, the flow of non-Newtonian fluids through porous media has received a significant amount of attention because of its important industrial applications, and considerable progress has been made. However, our understanding of non-Newtonian flow in porous media is very limited when compared with that of Newtonian flow. This work presents a comprehensive theoretical study of single and multiple phase flow of non-Newtonian fluids through porous media. The emphasis in this study is in obtaining some physical insights into the flow of power-law and Bingham fluids. Therefore, this work is divided into three parts: (1) review of the laboratory and theoretical research on non-Newtonian flow, (2) development of new numerical and analytical solutions, (3) theoretical studies of transient flow of non-Newtonian fluids in porous media, and (4) demonstration of applying a new method of well test analysis and displacement efficiency evaluation to field problems.
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