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R. Lenormand

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Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 5–8, 2003

Paper Number: SPE-84543-MS

Abstract

Abstract The relative permeability curves (Kr) control the production and are of primary importance for any type of recovery process. In the case of production by displacement (waterflood or gasflood) the Kr curves obtained in laboratory can be used in numerical simulators to predict hydrocarbon recovery (after upscaling to account for heterogeneity). In the case of reservoirs produced under solution gas drive (depressurized field, foamy oils), the experiments conducted in laboratory depend on the depletion rate and cannot be used directly for reservoir simulations. We have developed a novel approach for calculating representative field-relative permeabilities. This new method is based on a physical model, which takes into account the various mechanisms of the process: bubble nucleation (preexisting bubbles model), phase transfer (volumic transfer function), gas displacement (bubbles flow). In our model, we have identified a very few number of "invariant" parameters that are not sensitive to depletion rate and are specific of the rock-fluids system (mainly the preexisting bubble size distribution and a factor between gas velocity and oil velocity in the dispersed phase regime). These invariant parameters are determined by history matching of one experiment at a given depletion rate. The calibrated model is then used to generate synthetic data at any depletion rate, and especially at very low depletion rates representative of the reservoir conditions. Relative permeability are derived from these "numerical" experiments in the same way as real experiments. The calculated Kr are finally introduced in commercial reservoir simulators. We have tested our model by using several series of published experiments with light and heavy oils. After adjusting the invariant parameters on one or two experiments, we are able to predict other experiments performed at different depletion rates with a very good accuracy. Finally, we present an example of determination of relative permeabilities at reservoir depletion rates. Introduction In the case of conventional recovery processes (waterflood, gasflood), the experiments that are conducted in the laboratory can mimic the conditions that prevail in the reservoir. Hence, the Kr data that are derived from these experiments can be used practically strait forward for field simulation purpose (upscaling is often needed to account for heterogeneities). The problem is more complicated for recovery by gas drive. In this case, the laboratory experiments fail in reproducing the reservoir conditions. In reservoirs, depletion rates are at least several times lower than what can be obtained in the laboratory. As the depletion rate controls the gas topology (bubble density), the diffusion of gas from solution (out of equilibrium) and the gas displacement (dispersed flow), it affects also dramatically the shape of the Kr curves. Hence, the depletion experiments can not be used to derive directly field Kr data. Macroscale approach In the literature, mainly two types of approaches are currently used to interpret the laboratory data in terms of relative permeability. The first one consists in interpreting the experiments with a reservoir simulator. It is then possible to history match each experiment individually but not a series of experiments. This is due to the physics that is implemented in conventional simulator and that can not handle out of thermodynamic equilibrium behaviors. Since the supersaturation and the diffusion phenomena are not taken into account in conventional simulator, Kr curves have to be modified from one experiment to another in order to compensate differences in terms of depletion rate (Kumar 1 , Kamath 2 , Kamp 3,4 ). The field data are then derived by extrapolation from series of experiments. Hence, this macroscale approach is very empirical, not predictive and time consuming.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 29–October 2, 2002

Paper Number: SPE-77563-MS

Abstract

Abstract Permeability is one of the most important petrophysical parameter for reservoir characterization but the most difficult to obtain. Logs provide a good estimate of porosity and saturations, but the accuracy on permeability derived from NMR is rather poor. So far, reliable values of permeabilities are only obtained from laboratory measurements on core samples for local measurements and well testing for a larger scale averaged determination. We present an original method for measuring the permeability of drill cuttings without any specific laboratory conditioning (cleaning, coating, etc.). A volume of about 100 cc of cuttings is placed in a pressure vessel. The cell is then filled with a viscous oil. The process of oil invasion into the cuttings always traps a certain amount of gas. When a pulse of pressure is applied on the cell, the oil enter into the cuttings thanks to the gas compressibility. The permeability is then derived from the dynamic of the oil invasion by using a simple model. The method was tested by using various samples of crushed rock samples of known permeability. An excellent reproducibility and a good agreement between cores and cuttings permeabilities were found for many decades of permeabilities. This method presents many advantages. The measurements can be performed in a few minutes, leading to the possibility of operating on-site during drilling. The limitations of the method are mainly related to the size and the representativity of the drill cuttings. In developing this method, our purpose is not to replace core analysis, but rather to provide an additional cheap and quick information on reservoir characterization. Introduction When a new well is drilled, the main concerns of operating companies is to answer quickly two key questions: what are the reserves (porosity, saturation) and what is the well deliverability (permeability)? Most of the time, the logs provide a good estimate of porosity and saturations along the well. In this paper, we will focus on the evaluation of the permeability profile, which is much more difficult to obtain because this parameter refers to a flowing property of the reservoir rock. We present an original method to perform a direct measurement of permeability from cuttings, which may be suitable during the drilling operation. Routine analysis for permeability Core analysis at laboratory is the most reliable technique to measure permeaability but is rather expensive (coring, rig time, transportation, measurement). The data are available several weeks after drilling. Well testing provides information on the extension and the connectivity of the reservoir and gives an average value of permeability. A more accurate permeability profile can also be obtained with the MDT technique (Modular Dynamics Tester), run with wireline, with a spatial resolution of the order of the meter. The NMR log is now widely used to derive a fast evaluation of the permeability profile along the wells. However, the NMR tool is sensitive to the pore size whereas permeability is sensitive to throat size. Hence, the permeability evaluation is obtained through empirical laws, which need to be calibrated according to the reservoir rock and also the fluids in place (Fleury 2001). Routine cuttings analysis The cuttings are routinely used by the mudloggers to build the "masterlog", where the geological description of the drilled formation is reported. Hydrocarbon indices are also detected from cuttings to identify the reservoir levels. Although the cuttings rock material is coming directly from the reservoir, few applications of permeability characterization are reported in the literature. The published works can be divided into two categories: the direct and indirect evaluations. Routine analysis for permeability Core analysis at laboratory is the most reliable technique to measure permeaability but is rather expensive (coring, rig time, transportation, measurement). The data are available several weeks after drilling. Well testing provides information on the extension and the connectivity of the reservoir and gives an average value of permeability. A more accurate permeability profile can also be obtained with the MDT technique (Modular Dynamics Tester), run with wireline, with a spatial resolution of the order of the meter. The NMR log is now widely used to derive a fast evaluation of the permeability profile along the wells. However, the NMR tool is sensitive to the pore size whereas permeability is sensitive to throat size. Hence, the permeability evaluation is obtained through empirical laws, which need to be calibrated according to the reservoir rock and also the fluids in place (Fleury 2001). Routine cuttings analysis The cuttings are routinely used by the mudloggers to build the "masterlog", where the geological description of the drilled formation is reported. Hydrocarbon indices are also detected from cuttings to identify the reservoir levels. Although the cuttings rock material is coming directly from the reservoir, few applications of permeability characterization are reported in the literature. The published works can be divided into two categories: the direct and indirect evaluations.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 29–October 2, 2002

Paper Number: SPE-77457-MS

Abstract

Abstract The modeling of primary production of heavy oils by solution gas drive is an active area of research. All the models, either written at Darcy scale or at pore scale (capillary network, population balance), account for the following mechanisms: bubble formation, bubble growth and gas flow. The first stage of bubble formation, also called bubble nucleation, is still controversial. In this paper, we discuss the existing nucleation models and demonstrate that the preexistence of bubbles is the only theory that is justified physically and can explain the ensemble of experimental observations. The preexisting bubbles are stabilized either by surfactants (models used for cavitation studies) or capillarity in crevices (models used in boiling). In both models, a given number of bubbles are activated at a given pressure drop. The only adjustable parameter is the distribution of diameters of the preexisting bubbles. This distribution is a property of the rock/fluid system that can be experimentally determined. The other models used in literature are based on the formation of a stable nucleus by thermal fluctuations. They lead to the notion of nucleation rate that is in contradiction with experimental results. We also discuss the terminology used in recent papers. Especially the terms of "instantaneous nucleation" and "progressive nucleation" are irrelevant if a mechanism of preexisting bubbles is assumed. They are also misleading since they lump the mechanism for bubble formation (statistics or preexistence) and the mechanism for pressure decline (either step or constant rate). Introduction The mechanism of bubble formation is involved in many domains such as boiling 1–6 and cavitation 7–9 and an important bibliography exists either for applications in bulk 10–18 or in porous media 8,19–32 . Nucleation is also important in petroleum applications for solution gas drive. In this process, a reservoir is depleted and oil is produced by expansion of the gas released from the crude. The process of solution gas drive can be described in three steps: Gas nucleation: corresponding to the release of the dissolved light components into a free gas phase when pressure is decreased below the bubble point. Bubble growth: corresponding to mass transfer by molecular diffusion of the dissolved light components to the free gas phase. Until equilibrium concentration is reached, the liquid is always supersaturated and the system tends to the equilibrium state by transfer to the gas phase of the dissolved light components. Gas mobilization: Above a given gas saturation, called critical gas saturation Sgc, the gas phase becomes connected and is produced preferentially because of its higher mobility in comparison of oil. In some cases, flow of dispersed gas is also considered. In heavy oil for example, high oil viscosity may maintain gas in a dispersed form. In this paper we will discuss the first step of nucleation. Two types of models exist to describe the mechanism of nucleation: models based on thermodynamics and models based on the preexistence of microbubbles. A lot of experimental works in different domains have shown that thermodynamic models are not relevant to describe nucleation processes in common laboratory experiments 2,7,14,15,23–25 . However, the thermodynamical approach is still used in the domain of oil production. A justification is sometimes given by the agreement between models and experiments. But generally a given experiment can be fitted by different models. A good agreement does not mean that the physics is relevant and the extrapolation at reservoir scale can be wrong.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 30–October 3, 2001

Paper Number: SPE-71511-MS

Abstract

Abstract In this paper, we study the dissolution process in carbonates for the design of acidizing jobs. The difficulty is the modelling of the formation and the propagation of the dissolution patterns (the wormholes) which result from an instability very similar to viscous fingering. In reservoir simulators, the flow equations must generally be written at a scale larger than the standard Darcy equations in order to incorporate the heterogeneities of the formation (upscaling). In multiphase displacements without acidizing, the upscaled properties such as relative permeabilities and capillary pressure, are derived from properties on homogeneous core samples by numerical simulations using the standard Darcy's equation ("Kyte and Berry" methods). For acidizing, the upscaling process must also incorporate the effect of the dissolution which is similar to heterogeneity which change with time. To derive the upscaled properties, we will use a similar approach based on numerical simulations. However, we need a numerical simulator that can account for worm-hole formation at Darcy scale. We have built a 2D numerical simulator which calculates the dissolution pattern in a core. The originality of the simulator is the introduction of the physics at the pore scale: The flow equation takes into account Darcy flow in the matrix and Stokes law in the wormhole. The dissolution equation is derived from a volume averaging of the diffusion and reaction equations written at the pore scale. Comparison between the numerical simulations and experimental results demonstrates that adequate physics is introduced in the model: The various dissolution regimes observed in experiments are also obtained in numerical simulations: compact, wormholing and homogeneous An optimum flow rate corresponding to the maximum penetration of the wormhole is also reproduced with the simulator. For the effect of concentration and core length, the simulations exhibit the same trends as those observed with the experiments In addition, the simulator as been used to study the effect of parameters which are difficult to change experimentally. The larger number of "numerical experiments" will be used to build the upscaled model for reservoir simulations. Introduction Acid injection is a process widely used for stimulation of petroleum wells to increase rock permeability 1,2 . The dissolution of the porous matrix is an unstable process similar to viscous fingering that leads to the formation of dissolved channels called wormholes. Many experimental and theoretical studies have tried to describe the roles of the various factors on the formation of the wormholes: injection rate, acid volume, permeability, reaction kinetics, etc…. Today, most of the mechanisms can be considered as understood but the models available in literature are still qualitative. The dissolution kinetics at the level of the solid walls of the grains is well understood. The dissolution process can be separated into three successive steps: the acid transport by diffusion and advection to the solid surface, the chemical reaction at the solid surface, and the transport of the products of the reaction away from the surface. If the chemical reaction characteristic time is very short compared to the mass transfer kinetics, the process is called mass transfer limited. This is the case for limestone dissolution with HCl at temperatures larger to 0°C 3 . On the other hand, if mass-transfer kinetics is slow, then the process is reaction rate limited. It is the case for dolomite dissolutions with HCl at temperatures inferior to 50°C 4 . In this paper we will study only the case of mass transfer limited dissolution processes.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 30–October 3, 2001

Paper Number: SPE-71504-MS

Abstract

Abstract Because of the existence of large reserves, the production of heavy-oils is the object of increasing interest. Some heavy oil reservoirs show anomalous behavior in primary production, with rates of production better than predicted. Interpretations based on the formation of foam (or bubbles dispersion) have been proposed. However there is no evidence for foam production in reservoirs and a mechanism based only on high viscosity has also been proposed. In order to determine the real mechanism, we have studied the "foamability" of heavy oil from South America. The experiments consist of standard measurements in foam studies: qualitative measurements (foam volume, thin film lifetime), and physical properties (volume viscosity, dynamic and static surface tension, elastic modulus). Due to the high viscosity of the crude oil, measurements were performed on solutions of resins and/or asphaltenes in toluene. Asphaltenes were extracted by the HPLC method, and resins by adsorption on silica gel. First, solutions with only asphaltenes were used with concentrations in weight from 1% to 20%. The effect of resins was then investigated using solutions with various resin and asphaltene concentrations. The main results are: The presence of asphaltenes enhances foamability and film lifetime, with a large variation in all measurements at around 10% asphaltene concentration. A very long characteristic time of dynamic tension and a large increase of elastic modulus were observed. Added resins have no effect on foam volume. However, the threshold in properties around 10% asphaltene concentration is no longer observed with resins. The results can be interpreted with existing models for polymer clustering and interfacial behavior of proteins. As proteins, asphaltene molecules can reorganize themselves at the interface, leading to a long time of tension decrease and a high increase of elastic modulus. The effect of resins is analyzed by comparison with crude-water emulsion. In conclusion, with the systems we have studied, asphaltene fraction enhances significantly foam properties (foamability, single film lifetime, surface rheology) above 10% in weight. Because problematic is flow of heavy oil in porous media. Then we have also performed experiments in glass micromodel with the same fluids, but no proofs of interfacial effects were observed. Introduction Extensive use of fossil energy has been made in the recent years. Oil production is expected to begin to decline before 2010 1 . Because of the existence of still important reserves, the production of heavy oils 2 , which are highly viscous oils is the object of much interest. The high viscosity of these oils is attributed to the large quantity of asphaltenes, which are polyaromatic compounds of high molecular weight, ranging from 1,000 to over 50,000 g/mol. They are a solubility class defined as the oil soluble fraction in toluene and insoluble fraction in alkanes, in practice pentane or heptane. They contain a large variety of chemical species, with functional groups including acids and bases. Their composition depends on the alkane nature and amount, and on the origin of the crude oil 3 . Some of these groups are hydrophobic, whereas the polyaromatic skeleton is more polar and as a result, asphaltenes are surface active 4 . In the bulk oil, they pack into stacks of layers and form extended aggregates, fractal-like when dispersed in good solvents and more globular although large (several microns) in marginal solvents such as mixtures of toluene and heptane 5 .

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 27–30, 1998

Paper Number: SPE-49006-MS

Abstract

Abstract Two-phase flow properties are required for modeling fluid flows and pollutant transport through fractured media. Several laboratory experiments have been reported in the literature but so far there is no general model to predict the two-phase conductivity of a fracture for a given pair of flowing fluids. In this paper we propose a simple model based on viscous coupling between two fluids flowing simultaneously in a single fracture. This viscous coupling model leads to simple analytical relationships between the relative permeabilities (Kr) and either saturation or fluid velocities. This model also explains the dependence of the non-wetting fluid Kr on the viscosity ratio. Results obtained with this model were compared to several series of data available in the literature with air and water. The fit between predicted and measured Kr is good when the ratio of flow rates is used as variable (Lockhart-Martinelli parameter). For use in numerical simulations, the saturation can be derived from the flow rates using the same model. The proposed model can improve the relative permeability curves that are used in the numerical simulations. P. 253

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 27–30, 1998

Paper Number: SPE-49007-MS

Abstract

Abstract Numerical simulation of recovery in fractured reservoirs by simulators requires determination of transfer parameters between fracture and matrix. These large-scale transfer parameters accounts for capillary, diffusive or gravity effects and can be derived by up-scaling the local physical mechanisms. The transfer driven by capillary or gravity forces is easy to model, but there is no simple model for diffusion. Our study was aimed to calculate the transfer of a component by diffusion as a function of fracture geometry, fluid velocity in the fracture, and fluid compositions in fracture and matrix. The diffusion flux is calculated assuming a uniform fluid saturation at the interface between matrix and fracture and a laminar flow in the fracture. The coupled flow and concentration equations are solved along the width of the fracture and then integrated over its length. The resulting diffusion flux is given by a simple analytical formula. Calculations were then compared to experiments performed at reservoir conditions with pure methane injected in the fracture and binary mixtures in the matrix (Cl, C5). The results were in very good agreement, without any adjustable parameters. Finally, it is shown how the calculations can be implemented in a reservoir simulator. The fracture is considered as a porous block but then cannot directly account for the diffusion flux as a boundary condition. The only way to model the diffusion transfer is to calculate an effective diffusion coefficient which accounts for the diffusion transfer but also the grid discretization. A good agreement is obtained without any adjustable parameters between numerical simulations and experiments. P. 259

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 5–8, 1997

Paper Number: SPE-38899-MS

Abstract

Abstract A new method for construction of hysteresis capillary pressure relationships for use in reservoir simulation models is presented. The method is based on the experimental drainage-imbibition bounding curves and the history of the saturation changes. By scaling of the measured drainage and imbibition bounding curves into the saturation ranges in question for hysteresis scanning loops, any family of hysteresis curves may be constructed. The method is well suited for use in reservoir simulation models. Results of highly accurate laboratory measurements of capillary pressures on a gas-oil system including re-imbibition and re-drainage are presented. Predictions of hysteresis behavior of the laboratory system by the new method show satisfactory agreement with the experiments, while prediction by the much used Killough's method fail to match the experiment, primarily because it does not scale saturations. It is also observed that the commonly used Land equation for prediction of residual saturations is not representative for the system under investigation. Since the Land equation do not distinguish between rock types, it is not recommended used unless experimental support of its applicability exists. Our results show that the relationship between the residual saturation and the initial saturation for a hysteresis imbibition process is approximately linear. Introduction In a paper published in 1965, Morrow and Harris presented experimental results and a comprehensive discussion of capillary behavior of porous materials. They show that a hysteresis curve departing from one of the bounding drainage or imbibition curves is uniquely defined by the departing point on the curve. By the same token, virtually an infinite number of families of hysteresis curves may result from saturation reversals, and each branch is defined by the departing point and the history of saturation reversals. In order to define an imbibition hysteresis curve, the residual saturation in addition to the departing point must be known. Commonly used for prediction of residual saturation is the semi-empirical relation presented by Land based on matching of experimental data. He found that for a given sand the difference in reciprocals of residual and initial saturations remains constant. Several authors have discussed hysteresis behavior of porous media. Of particular interest is representation of hysteresis in reservoir simulation. Model input data normally includes complete drainage and imbibition curves. The simulation model then applies some method to predict hysteresis residual saturations and hysteresis capillary pressures and relative permeabilities. Both Killough and Carlson presented methods for predicting hysteresis in relative permeability. For prediction of hysteresis in capillary pressures, the method presented by Killough is frequently employed. His method computes hysteresis capillary pressures by weighting of the complete drainage and imbibition curves. However, as pointed out by Tan, Killough's method was specially formulated for the case where the drainage and imbibition curves meet at the residual saturation. Because of that, the method is often inadequate. Recently, very accurate laboratory measurements of gas-oil capillary hysteresis have been made in the laboratories of IFP in a cooperation with Total and Elf Aquitaine. Measurements of capillary pressures including complete drainage and imbibition curves and intermediate drainage-imbibition and drainage-imbibition-drainage loops were made. The results are presented in this paper and used for evaluation of hysteresis prediction methods. Experiment Gas-oil drainage and imbibition capillary pressure cycles were measured using the Porous Plate Method. A schematic of the laboratory setup is shown in Fig. 1. The laboratory setup includes: P. 597^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 5–8, 1997

Paper Number: SPE-38921-MS

Abstract

Physical Mechanisms for Bubble Growth during Solution Gas Drive. Abstract A series of experiments were performed in transparent micromodels to understand the mechanism of nucleation by pressure decline in a CO2-water solution. All our observations can be interpreted by a process that consists in the following succession of steps: pre-existence of stabilized microbubbles (order of magnitude of one micron), growth of the microbubbles in a very large number of sites and trapping by capillary forces, activation of some of these sites when pressure drawdown balances capillary trapping, growth of these bubbles by gas diffusion to a size allowing observation (around 30 microns). The above process is similar to the one found in literature explaining boiling on heated surfaces. Introduction In petroleum engineering, there is a need for modelling gas production in reservoirs when pressure decreases below the bubble point. Modelling of gas production requires an accurate description of the three physical mechanisms involved in the process: the rate of appearance of gas bubbles, the growth of bubbles inside the pores, the formation of a continuous gas phase which will invade the medium. The physical mechanisms governing the second and third steps are quite well understood. For instance, bubble growth inside pores is described by molecular diffusion with some correction factors to account for the presence of the porous medium (Li and Yortsos). However, the understanding of the first step, bubble formation inside a porous medium, is quite poor. The few existing models found in petroleum literature are contradictory. The classical model is derived from "homogeneous nucleation" (Wilt, Li and Yortsos). A recent paper by Firoozabadi and Kashiev, propose a completely different approach based on instantaneous nucleation. In order to improve the understanding of bubble appearance, we performed visualisations and measurements in transparent micromodels. The conclusion of our work is that nucleation is instantaneous, but not so simple than the model proposed by Firoozabadi and Kashiev. In fact, there is an additional step of trapping due to capillary force that explains the observed threshold and the progressive appearance of the bubbles. Heterogeneous nucleation. We will first recall the principle of "homogeneous" nucleation, when the liquid is pure, without any solid or liquid interface. When pressure is decreased under the bubble point, a transient gas "nucleus" appears by thermal fluctuations. This nucleus will grow in order to form a bubble only if its radius r is larger than a critical value. Otherwise, the nucleus will collapse under capillary pressure proportional to l/r (Laplace's law). The appearance of bubbles is therefore a random process such as photon emission, characterized by a rate of nucleation J (number of bubble per unit time and unit volume of liquid). J is an exponential function of the supersaturation P=P(equilibrium)-P (1) where N is the number of molecules per unit volume, m the mass of a molecule, kT is thermal energy, interfacial tension and B a parameter close to 2/3. It is now well established that homogeneous nucleation requires a very high supersaturation ratio P/P. Wilt gave the example of 1100 to 1700 for CO2 solutions near room conditions. Consequently, homogeneous nucleation cannot occur during pressure decline in porous media (Li and Yortsos). Instead, gas formation with low supersaturation rates can only be explained by "heterogeneous" nucleation. In heterogeneous nucleation, the nucleus is formed on a solid interface. The nucleus is stabilized by the combined effect of surface roughness and wettability (Wilt, Li and Yortsos). For instance, it is simple to show that in a site of conical geometry with an angle, the interface between liquid and gas is flat when the contact angle is equal to /2- /2. There is no curvature of the interface and therefore there is no capillary pressure to collapse the nucleus. Consequently, hydrophobic sites reduce the energy involved for nucleus formation and can explain gas nucleation at low supersaturation rate. P. 805^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 23–26, 1990

Paper Number: SPE-20475-MS

Abstract

Abstract This paper depicts an original model for the description of heterogeneous media, based on fractal and multifractal geometries. Rather than using, as in Hewett's fractal approach, a random noise to generate heterogeneous patterns, we have developed an approach much closer to geostatistics and based on multifractal approach. Multifractal is a characterization of spatial distribution of a continuous variable, such as permeability or porosity. The construction rule is based on a multiplicative porosity. The construction rule is based on a multiplicative process which gives a multifractal square structure with the size process which gives a multifractal square structure with the size 2n × 2n. The main property of such a construction is the correlations it gives at all length scales. This method is then modified to generate anisotropies in the x and y directions by using rectangular rather than square multifractal blocks. A wide variety of patterns is obtained by tuning only two dimensionless correlation lengths. Consequently, any pattern can be located on a 2D diagram with the extreme cases being the homogeneous medium, the horizontal and vertical layering, and the multifractal. Two-phase displacement patterns are obtained by a network flow simulator coupled to this anisotropic multifractal generator. The results are in good agreement with experimental patterns obtained by CT-Scanning during gas injection in layered Berea sandstone. The model contains the main features of the rock sample, which play a role in immiscible displacements, such as randomness, all ranges of correlations and anisotropy. Introduction The need to describe reservoir heterogeneities and, more generally, multiphase flow in heterogeneous porous media has appeared in the last few years. This paper describes an original model for heterogeneities based on multifractals. Recently, fractal geometry, has appeared as a complement of standard geostatistics (see for instance Kinzelbach and Yortsos for reviews on this approach). Hewett and coworkers have proposed a model of reservoir derived from the construction of fractal landscapes in which the elevation was replaced by the porosity. This approach is based on a generalization of random porosity. This approach is based on a generalization of random noise, called fractional noise, which introduces correlations at all length scales. The level of correlations depends on a parameter H, called the intermittency parameter (or Hurst exponent). parameter H, called the intermittency parameter (or Hurst exponent). Log analysis indicates an exponent H from 0.6 to 0.9 (H = 0.5 for random noise). Weatcraft and Tyler have presented another approach that is more suitable for tracer injection. It consists in modeling the trajectory of the particles as a fractional random walk. The main result is the ability to model dispersion with variance in t2H. These new techniques based on fractional random walk are successful because they introduce correlations at all length scales, which seems to be a property of geological structures. However these methods are quite complicated and the analogy with random walks is difficult to justify. For these reasons, we prefer to use a multifractal approach, which is a more direct way of characterizing non-uniform distributions. The concept of multifractal has been proposed by Mandelbrot in relation with turbulence intermittency. p. 121

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 23–26, 1990

Paper Number: SPE-20525-MS

Abstract

Abstract By means of an ultrasonic technique, the nucleation rate J in depletion experiments has been measured. An array of 8 ultrasonic transducers, fixed along a rock sample. continuously scans the sound velocity and the attenuation in the sample. Due to the great difference in sound velocity between the gas and the liquid, bubbles are easily detected and located by transducers. The sample is also set up with capillary tube at one end for the observation of production by the displacement of a meniscus. The main advantage of ultrasonic technique is the capacity to work at high pressure and temperature. An improvement in the accuracy over the standard techniques is observed. Introduction Many oil reservoirs produce oil by primary depletion that leads to the formation of a gas phase when the pressure falls below the bubble point. Depletion experiments in reservoir pressure and temperature conditions need to be carried out pressure and temperature conditions need to be carried out to determine the influence of parameters such as fluid properties, nature of the rock, depletion rate, etc. The most properties, nature of the rock, depletion rate, etc. The most important parameter governing the nucleation and the critical gas saturation in oil reservoirs is the nucleation rate J expressed by: (1) where Vo is the volume of the fluid contained in the rock sample and At the time necessary for the first bubble to appear under a given supersaturation. In this paper we present an original method to determine the bubble nucleation rate during depletion experiments in rock samples by means of an ultrasonic technique. This technique presents several advantages: the capacity to work at high pressure and temperature without a complicated equipment; the possibility of locating the bubble formed in the sample; the ability of detecting very small volumes u compared to standard methods based on meniscus displacement. Several authors have studied the heterogeneous nucleation by evaluating the nucleation rate. Kennedy and Olson report visual observations of bubbles nucleating at the surface of silica and calcite crystals in a mixture of methane and kerosene. Wood has studied nucleation of oil in a porous medium by the displacement of a meniscus in a capillary tube following the volume increase. Wieland and Kennedy have redesigned the apparatus of Wood to allow greater precision at low supersaturation so that a volume change as precision at low supersaturation so that a volume change as small as 0.05 mm can be detected by electric bellows under pressures in the range of 80 - 100 × 10- Pa. Lubetkin and Blackwell have developed a technique for counting the number of bubbles released out of a supersaturated solution of carbon dioxide in water. However, this technique, which consists in detecting the bursting of the bubbles when they reach the liquid surface, cannot be applied to porous media. Many problems arise with the methods based on volume change measurements. Firstly, this volume change has two different contributions which are difficult to discern: the volume variation of the rock sample due to its elasticity and the volume change due to bubble formation. Secondly, it Is impossible to locate the bubble in the sample. Finally, there are many problems for the visualization of the meniscus in high pressure and temperature experiments. We present in this paper an acoustical technique which overcome these difficulties. P. 551

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 27–30, 1987

Paper Number: SPE-16954-MS

Abstract

Abstract In the past, many network simulators have been presented to describe transport processes through porous media, such as deep filtration, immiscible or miscible displacements, blob mobilization, acidizing, etc. In this paper we compare the results of physical and stochastic network simulators with experiments in micromodels. We first describe the patterns obtained by displacing a wetting fluid by a non-wetting fluid in a 2-dimensional porous medium, when both viscosity ratio (M) and capillary number (Ca) are varied independently over many orders of magnitude. The main result is the good agreement between experimental and simulated patterns (trapping, fingering, connectivity.. even for extreme values of M and Ca (high viscosity contrast or very strong capillary effects, for instance). In the second part, both experiments and simulations are compared to the results of stochastic models. The classical models of "Invasion percolation" and "diffusion limited aggregation" are shown to represent the limiting cases when the dimensionless numbers Ca (for I.P.) and 1/M (for DLA) tend to zero. Finally, some comments are presented on a recent model based on fluid compressibility and particle diffusion. Introduction Two main approaches are used for the study of transport properties through porous media. The first is based on the continuum description of the porous medium associated with macroscopic laws (Darcy's law, relative permeabilities…). The second is based on a microscopic description of the pore geometry and on the physical laws of flow and transport within the pores. This latter approach is generally adopted to explain the macroscopic behaviour from the pore level properties, by means of computer simulations on a network of pores and throats representing the porous medium. In addition to these physical simulators, stochastic network models such as percolation or "diffusion limited aggregation" have been recently proposed. This paper presents a validation of such physical and stochastic network simulators by comparing them with experiments. This study has been restricted to the 2-dimensional case of immiscible displacements, and to the case of drainage (the non-wetting fluid displacing the wetting fluid). Studies using computer simulations and experiments in 2-dimensional etched networks (micromodels) can be found in the literature, but very few comparisons between the two approaches have been proposed. Network simulators have been used to describe a wide range of phenomena: transport of tracers deepbed filtration, dissolution of a porous matrix, two-phase flow, etc. The first use of a network of capillaries by Fatt was to describe the shape of the relative permeability curves for two-phase flow. The principle of this study was to calculate the overall conductance of a network of capillaries filled at random. P. 213^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 27–30, 1987

Paper Number: SPE-16955-MS

Abstract

Abstract In this paper, we study fluid displacements in layered porous media and focus particularly on the effects of viscous instability and of capillarity. We first present several series of experiments of miscible and immiscible displacements in a three-layer cell with two different permeabilities. We cover a large range of viscosity ratios in the two cases of unstable and stable miscible displacements. The fluid boundaries are followed by visualizations and quantitative saturation data obtained by ultrasonic mapping. The second part of the paper is devoted to the description of a pore network simulator and a stochastic model which are suitable for describing viscous instability. Both models not only reproduce the experimental patterns very closely but are also in good quantitative agreement with experimental data for the ratio of the volumes of fluid injected in the two different layers. Introduction One of the most important, yet still poorly understood, aspects of flow in porous media is the displacement of one fluid by another in a heterogeneous permeable medium. Permeability variations affect both the small-scale behaviour of laboratory models as well as the large-scale dynamics of field tests. Misunderstanding of the mechanisms governing the flow in a laboratory model can lead to an incorrect scale-up of the process. Thus an understanding of heterogeneity effects at laboratory scale is essential to provide the framework to improve oil recovery or water control techniques at reservoir scale. Among the geological features representative of heterogeneity, stratification in the vertical plane is the simplest and most frequently encountered. Several model studies, made on systems containing contiguous layers of different permeability, can be found in the literature. Fluid flow in stratified porous media is influenced by flow from one stratum to another in directions perpendicular to the bulk flow. This crossflow may be caused by any or all of the following forces: viscosity, capillarity, gravity, and dispersion. These forces interact with each other in real displacements, and it is not always easy to determine which mechanism is causing the crossflow. A theoretical analysis of viscous crossflow mechanisms has been performed by Zapata and Lake. They used the method of characteristics to solve the flow equations for a two layered system under the idealised circumstances of vertical equilibrium. This essentially means that crossflow takes place instantaneously so there is no vertical pressure drop. Their analysis can only deal with piston-like displacement within each layer and hence both waterfloods and polymer floods are treated in a similar manner. Another model for viscous crossflow has been proposed by Wright and Dawe taking into account interlayer crossflow, but its validity is restricted to the cases where the viscosity contrast between injected and displaced fluid is not too high. An analytical treatment of capillary crossflow in the absence of viscous and gravitational crossflow has been presented by Yokoyama and Lake They introduce dimensionless longitudinal and verse capillary numbers which govern the size of the capillary effect. P. 223^

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 5–8, 1986

Paper Number: SPE-15390-MS

Abstract

Abstract Multiphase reservoir properties are generally extrapolated from laboratory measurements, and scaling laws are based on dimensionless numbers obtained from macroscopic flow equations. This approach is not valid when fingering occurs on the macroscopic scale because of an unfavorable viscosity ratio or capillary effects. We describe a more general s technique taking this fingering into account. Introduction The purpose of this paper is to provide better understanding of the relevant mechanisms which control the displacement of a wetting fluid (labeled 1) by a non-wetting fluid (labeled 2) in a porous medium when both capillary and focous forces are present. In a previous paper, we have described experimental results showing the possibility of three basic mechanisms: capillary fingering Ashen capillary forces are very strong compared to viscous forces, and, at high flow rate, viscous fingering when a less viscous fluid is displacing a more viscous one and piston-type or stable displacements in the opposite case. We have also shown how these displacements could be described by statistical theories: invasion percolatzon and diffusion limited aggregation (D.L.A.). These basic percolatzon and diffusion limited aggregation (D.L.A.). These basic mechanisms have also been observed and studied separately by other authors, for viscous fingering"' arid for capillary fingering'. The originality of our approach is to include all these mechanisms in a general model. Previously, in a short note 5, we calculated approximately the boundaries between these three different mechanisms by using only two parameters, the capillary number Ca: ..........................................(1) where E is the cross-section area of the sample q/ being the mean velocity of the fluid in the channels, and M= u2/ u1 the viscosity ratio between the two fluids. This approach led to an original display of the domains on a general diagram with axes representing Ca and M and we called this result the "phase-diagram" for capillary and viscous displacements in porous media. This study, we propose a 3-Dimensional version of this phase-diagram and demonstrate the utility of this approach phase-diagram and demonstrate the utility of this approach for scaling experiments to field condition. The model is based on the calculation of the various forces during a displacement in a 3-dimensional porous medium, assuming a simple form for the fingers. Using very naive assumptions we estimate the boundaries of the various domains as functions of the fluids properties (viscosity ratio M, capillary number Ca), the sample size L and parameters related to the pore geometry (permeability, aspect ratio…). The intersections of the boundaries determine three points which are only functions of the geometrical properties of the medium (microscopic structure and sample size). Since these points are riot located on the axes and are not functions of the same parameters, it is impossible to calculate simple scaling laws. However, assuming these three points are a characteristic of the porous medium we can calculate a more general 2-dimensional scaling. Consequently, the structure and the properties of a displacement can be determined simply by locating its position on the diagram. Furthermore this method can be used to match flow conditions in laboratory experiments to those in the field.