Th P06 05 Permeability Estimation Using CFD Modeling in Tight Carboniferous Sandstone
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1 Th P06 05 Permeability Estimation Using CFD Modeling in Tight Carboniferous Sandstone P.I. Krakowska (AGH University of Science and Technology in Krakow), P.J. Madejski* (AGH University of Science and Technology in Krakow) & J.A. Jarzyna (AGH University of Science and Technology in Krakow) SUMMARY Permeability, besides the standard laboratory measurements (permeameters), may be determined using combination of computed microtomography and Computational Fluid Dynamics methods. Generation of the geometric model of the pore space network is possible thanks to computed microtomography (micro- CT). In calculations, the Knudsen number should be considered, as a condition, which classifies the modelling approach and allows the use proper method (correct flow regimes). Using the CFD modelling the flow of the gas (nitrogen) and water through the porous system was analysed. Different pressure drops and meshes were used in simulation to verify the CFD modelling results which are depended on the pressure difference (boundary conditions). Permeability is calculated on the basis of simulation results using specially modified Darcy equation. Determined permeability values are similar for different fluids and pressure drops used in simulations. Compatibility between the permeability value calculated from the standard laboratory measurements and CFD modelling was observed for the Carboniferous tight sandstone sample.
2 Introduction Permeability determination is one of the most significant tasks for petrophysicists. Most, discovered conventional reservoirs reach permeability values in the range of thousands of md. Nowadays, unconventional resources (tight gas, shale gas) are under careful consideration because of permeability values range in nano-level. Permeability, besides the standard laboratory measurements permeameters, may be determined using Computational Fluid Dynamics (CFD) fluid flow simulations (Bielecki et al. 2012, Arns et al. 2004, Fredrich et al. 2006). Firstly, the Knudsen number should be consider as a condition which classifies the modelling approach. On the basis of Knudsen number the correct flow regimes should be chosen. Finally, the fluid flow simulation and the permeability calculation may be carried out. Generation of the geometric model of the pore space network is possible thanks to computed microtomography (micro-ct). Micro-CT is as a non-destructive method allowing to examine the object as a 3D image in micro scale. Numerical mesh was generated reflecting the pore space network. Using the Finite Volume Method (FVM) the flow of the gas and liquid through the porous system was analysed. FVM is the most popular discretization method of Navier-Stokes equation in CFD modelling. The results of the modelling are the flow parameters depended on the pressures difference (boundary conditions). Permeability is calculated on the basis of the simulation results, fluid parameters and appropriately modified Darcy equations. 3D geometric model of the pore space network 3D micro-ct image was used to create geometric model of the pore space network. Carboniferous sandstone sample, cored from the well located on the Pomeranian Anticlinorium in N Poland, was the object of the analysis. Sandstone sample is an example of the tight, low porosity and low permeability formation, buried at the depth of about 3154 m. Analysed sample, according to the quantitative X-ray analysis, is built in 93% from quartz, 2% from dolomite and anhydrite and about 1% from clay minerals. Total porosity, effective porosity and permeability values, obtained from the standard laboratory measurements, are equal to 13.3%, 2.2% and 94.4 md, respectively. Total porosity values, obtained from the micro-ct images, reached the value of 12.6%. Figure 1 and 2 presents the microtomographic image of Carboniferous sandstone sample. Resolution of micro-ct measurement in this case is 5.6x5.6x5.6 µm. For the fluid flow simulation purpose the part of the micro-ct image was extracted (Figure 1) with the size of 100x100x100 voxels, from the center of the 3D microtomographic image. Theory and Method To estimate the permeability values the water and nitrogen flow simulation was performed. Firstly, the Knudsen number (Kn) was calculated in order to classify the flow regimes. It is necessary to estimate the flow regime by Kn in case of modelling the gas flow through porous media. The possible modelling approaches according to Knudsen number (Civan 2011) are presented in Table 1. Table 1 Modelling approaches according to Knudsen number. Modelling approach Knudsen number (Kn) Models Continuum flow Kn Boltzmann, Euler, no-slip Navier-Stokes equations Slip flow 0.001<Kn<0.1 Boltzmann, slip Navier-Stokes equations Transition flow 0.1<Kn<10 Boltzmann, Burnett equations Free molecular flow 10 Kn Boltzmann equation Assuming the mean free path λ of nitrogen in standard temperature and pressure conditions is equal to 10*10-8 m (Couture & Zitoun 2000) and representative physical length scale L (diameter) in the smallest pore channel recorded from micro-ct image, the Knudsen number reaches value close but less than Kn 0.001, what allows to use the no-slip Navier-Stokes equations (Kn=0.0008). If Kn reaches higher value, then slip Navier-Stokes equations should be applied to the fluid flow modelling. Continuum flow regime is not preserved. It is caused i.e. by the Klinkenberg effect existence which is
3 connected with molecules slip on the pore walls at low mean pressures resulting in overestimation of permeability values (Tiab 2012). In case of liquid flow simulation the slip effect does not exist and has no influence on permeability estimation. Reynolds number (Amao 2007) was also determined in order to check if the simulation may be performed in range of laminar flow conditions (Navier-Stokes equations for laminar flow in CFD simulation, Darcy law regime in permeability estimation). Applied approach used to permeability estimation is detailed described in Narsillio et al. (2009), Zalewska et al. (2010) and Krakowska et al. (2013). Equation 1 was used to determine the permeability values: Φ μl vda A k out A (p p ) p 1 2 where: k [md] permeability, Φ [frac] effective porosity, A p [m 2 ] cross-section area of pore space network at the outlet, p 1, p 2 [Pa] pressure at the inlet and outlet of the sample, L [m] sample length, μ [Pa s ] dynamic viscosity of the fluid, v [m/s ] local velocity at the outlet, A [m 2 ] crosssection area at the outlets. Fluid flow simulations were performed for two cases: water and nitrogen (Figure 2) in Star CCM+ software. Liquid (water) characterised with density ρ w =997 kg/m 3 and dynamic viscosity μ w =8.887*10-4 Pa*s, while for nitrogen: ρ n =f(p n, T n ), where p 2 =p n = 100 kpa and T n =273.15K, ρ n =1.123 kg/m 3 and μ n =1.663*10-5 Pa*s. The nitrogen flow simulation was set without taking into consideration the molecules slip on the pores walls. Following parameters were also applied in simulations: p 1 =60 or 600 kpa, p 2 =100 kpa, L=560*10-6 m, A p value was obtained from the geometric model, as an outlet cross-section area of the pore network. Fluid flow simulations for water were performed for the two different meshes (number of cells) for pressure drop Δp =60 Pa and for the same mesh but different pressure drops Δp =60 and 600 Pa (Figure 3). For the nitrogen simulation pressure drop was set to Δp =600 Pa. Results Permeability values for Carboniferous sandstone sample were determined using CFD modelling (Andersson 2012). Four different simulations for different fluid flow parameters were performed in order to confirm the reliability of the simulation results and hence, permeability estimations. Table 2 shows the permeability calculations for four fluid flow simulations. Table 2 Results of permeability calculations for assumed modelling parameters. Symbols: ε r relative error. Lp. Fluid Number of cells Δp = p 1 p 2 k ε r [Pa] [md] [%] 1 water nitrogen Permeability values differ with the type of fluid used in simulations. For nitrogen, permeability reached the highest value, and close to the value obtained from standard laboratory measurements, carrying out in permeameter. Very small difference in permeability values is observed for water flow simulations. The best match between permeability obtained from laboratory measurements and modelling is received for nitrogen simulations. Relative error for calculated permeability oscillates around 20%. The permeability values obtained for different fluids (water and nitrogen), for different inlet pressures (p 1 ) and cell numbers are very similar. It is a confirmation that the simulation was carried out properly. Permeability is a parameter which depends on the geometric properties of pore space network. Also two water flow simulations for the same flow parameters but different meshes were performed. (1)
4 Figure 1 Microtomographic image of the Carboniferous sandstone sample. Figure 2 Geometric model of the pore space network from micro-ct. Figure 3 Division of the geometrical model into finite volumes. Figure 4 Velocity distribution of water for pressure difference, p=60 Pa. Figure 5 Water velocity streamlines for pressure Figure 6 Water pressure streamlines for difference, p=60 Pa. pressure difference, p=60 Pa. The goal of these simulations was to examine the influence of cells number in mesh on the simulation results. Because of the fact that the tight, low porosity and low permeability rock was analysed the accuracy of permeability estimation had to be set with the minimum error. Simulation results for two different numbers of cells in the model are similar, differing in about 1 md.
5 Figures 4 and 5 present the velocity field distribution for water flow simulation. The highest velocities are visible in the centre of the sample, in the maximum throat (minimum channel diameter). Velocity streamlines run informs where is located the main fluid stream. Pressure distribution is showed in Figure 6, reaching the highest values at inlets areas and minimum at outlets areas. Conclusions Permeability calculations were carried out using CFD modelling. Water and nitrogen flow simulation was analysed in terms of quantity (obtained parameters) and quality (parameters distribution in pore space). Difference between the permeability calculated from the standard laboratory measurements (permeameter) and CFD modelling is around 20% for the Carboniferous tight sandstone sample. The differences between permeability values obtained for water and nitrogen are very similar, as well as for pressure drops and number of cells in geometric model. It is a proof that the CFD simulation can provide proper results for liquids and also for gases, if the Knudsen number is below or close to limit value equal to CFD approach in permeability calculation would not be possible without the tool which allows to create 3D geometrical model of pore space network. Computed microtomography or nanotomography gives the possibility to calculate permeability in non-invasive way. In case of low porosity and low permeability formations permeability calculations using micro-ct and CFD approaches are very sensitive to firstly, the proper pore space extraction from micro-ct images and secondly, the proper choice of pore space network, being representative for the whole sample. Hence, the difference between permeability from laboratory measurements and CFD modelling may result. Combination of micro-ct results with CFD modelling is a powerful tool in the analysis of fluid low in pore space and permeability calculations. Acknowledgements Authors thank Polish Ministry of Environment for the core samples. Laboratory measurements of core samples were carried out in Oil and Gas Institute in Krakow, Poland. Research project is funded by the National Science Centre, project no. AGH References Amao A.M. [2007] Mathematical model for Darcy Forchheimer flow with applications to well performance analysis. Texas Tech University, Misato. Andersson B., Andersson R., Hakansson L., Mortensen M., Sudiyo R., Wachem B. [2012] Computational Fluid Dynamics for Engineers. Cambridge University Press, Cambridge, United Kingdom. Arns Ch. H., Knackstedt M. A., Val Pinczewski W., Martys N. S. [2004] Virtual permeametry on microtomographic images. Journal of Petroleum Science and Engineering, 45, pp Bielecki J., Bożek S., Dutkiewicz E., Hajduk R., Jarzyna J., Lekki J., Pieprzyca T., Stachura Z., Szklarz Z., Kwiatek W. M. [2012] Preliminary investigations of elemental content, microporosity and specific surface area of porous rocks using PIXE and X-ray microtomography techniques. Acta Physica Polonica, 121/2, pp Civan F. [2011] Porous media transport phenomena. John Wiley & Sons Inc., New Jersey. Couture L.,Zitoun R. [2000] Statistical Thermodynamics and Properties of Matter. Gordon and Breach Science Publishers, Amsterdam. Fredrich J. T., DiGiovanni A. A., Noble D. R. [2006] Predicting macroscopic transport properties using microscopic image data. Journal of Geophysical Research, 111, B03201, pp Krakowska P., Madejski P., Jarzyna J. [2013] Fluid flow modelling in tight Carboniferous sandstone. EAGE EartDoc database, 75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013, June, London, United Kingdom, extended abstract. Narsilio G., Buzzi O., Fityus S., Yun T., Smith D. [2009] Upscaling of Navier- Stokes equations in porous media: Theoretical, numerical and experimental approach. Computers and Geotechnics, 36, Elsevier, pp Tiab D., Donaldson E. [2012] Petrophysics. Elsevier Inc., Oxford, United Kingdom. Zalewska J. (eds.) [2010] X-ray computed microtomography in carbonates analysis. Oil and Gas Research Papers, 171, Oil and Gas Institution Publishing, Krakow, pp
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