Report Dra. Edith Perrier May 17th-September 16th Advising on the Project 168638, SENERCONACYT-Hidrocarburos Yacimiento Petrolero como un Reactor Fractal leaded by Dra. K.Oleshko: How to estimate oil-water capillary pressure curves and relative oil and permeability curves in naturally fractured and vuggy oil reservoirs from multiscale geological data?
1. Introduction .....................................................................................................................................................2 2. Modelling fractal fractured porous media .........................................................................................4 3. Modelling fractured and vuggy porous media: first attempts....................................................6 4. The key issue of connectivity addressed through network modeling. ...................................9 5. Perspective 1: Towards a new network model with triple porosity ....................................12 6. Perspective 2: Towards direct modeling of porous media images .......................................17 7. Fractal and percolation modeling........................................................................................................20 8. First conclusions..........................................................................................................................................21 9. References......................................................................................................................................................22 10. Appendices (description of already delivered codes)..............................................................23 10.1. SIM-‐POR ...............................................................................................................................................23 10.2. MF...........................................................................................................................................................26
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1. Introduction This contribution aims at investigate new concepts and tools to be developed then applied to model oil recovery in multiscale, naturally fractured vuggy reservoirs, in the context of data acquisition and modeling done by the whole team involved in the project leaded by Dra. Oleshko. From the available literature in the domain, a preliminary conclusion is that research in the domain is still in its infancy. As mentioned by Wu et al. (2011), even if the existence of vugs or cavities in naturally fractured reservoirs has long been observed, and even though these vugs are known for their large contribution to reserves of oil, natural gas, or groundwater, few quantitative investigations of fractured vuggy reservoirs have been conducted. Arbogast and Gomez (2009) conducted numerical simulations at the interface between the Stokes and Darcy flow domains, they studied the macroscopic effective permeability of a vuggy medium, and showed that « the influence of vug orientation, shape, and, most importantly, interconnectivity determine the macroscopic flow properties of the medium ». Mayur Pal (2012) states that the « traditional » numerical modeling of flow through vuggy porous media using coupled Darcy–Stokes equations poses several numerical challenges and proposes an alternative method, the upscaled Stokes-‐Brinkman model. Such attempts at the level of fluid dynamics modeling are still very theoretical and cannot predict the behavior of a given reservoir from field data, they mainly investigate the shape of relative permeability curves to be fitted on direct measurements. Lv et al. (2011) consider that there is an urgent need to research the oil-‐water flowing law and influence mechanism of fractured-‐vuggy medium so as to provide the basis for accurately prediction of reservoir performance… and they carry out a series of numerical fluid flow experiments on fracture network models firstly, then adding some vugs with different position to assess the influence of vug density and vug-‐ fracture connectivity on the shape of the relative permeability curves as a function of water saturation, without any comparison with real data at this stage. In this report, I present first some similar work (see following section 2), starting from programs initially developed to model fractal pore networks (Perrier et al., 1995) and to simulate associated retention curves and water conductivity in aggregated soils. As mentioned by Gringarten (2008) many analysis techniques used in Petroleum Engineering originated from subsurface hydrology. The programs were adapted to simulate the oil-‐water capillary pressure curve and the oil and water relative permeability curves from a network of embedded fractures similar to the pore network defined by a widely used mass fractal model of porous media (Rieu and Sposiro, 1991). These programs were also updated to provide a new interface for any end-‐user to easily understand the principles of network pore-‐scale modeling now widely used in Petroleum Research (e.g. Al-‐Kharusi and Blunt, 2007, 2008). Then the initial model was modified to include vugs (section 3). First simulations confirm that connectivity, which is an important issue whatever the model as regards the prediction of permeability, is a particularly key issue for vugs. Their more or less spherical shape is a from pure geometrical reasons a priori associated to disconnected holes, where trapped oil from ancient days could not be recovered unless by pumping directly in the filled hole! In fact, we know that they may be connected, either because
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they are crossed by fractures or because they are linked trough smaller fissures, as those exhibited in section 3. More generally, apparently isolated vugs at a given scale of observation can be connected through (micro)pores that would be revealed at a smaller scale through what is called the matrix at the observation level. It appears necessary to build first a triple porosity model accounting for the three types of porosity (fractures, vugs, matrix), then to address the upscaling issue. Connectivity is difficult to assess from 2D information, but 3D simulators are very time and memory consuming. Section 4 presents a pseudo-‐3D simulator where one can consider several planes, including intra-‐plane connections and inter-‐planes connections, using first the same type of models as those presented in section 2 and 3 in each plane. The focus is on network modeling considering that the detailed physically based assumptions made as regards fluid flow in each type of porosity are so far less important than finding ways to account for the local connections which rules the macroscopic connectivity thus a major component of the permeability. The part of the present available code dealing with the simulation of retention curves and relative permeability curves from the input of any set of interconnected conducting objects is quite general, and can be quickly adapted to deal with any type of network, either in 2D or 3D, since only the links or connections between objects are important in a network model. But before launching the actual development of new computer codes that can be used by end-‐users, which is always consuming a lot of time, we first investigated different options from the conceptual design of possible algorithms. The underlying question is: how to build both realistic and tractable triple porosity network models? In section 5 and 6, I propose two directions for future work, which may complement each other. The goal is obviously to calibrate models from real data. But there are at least two ways to proceed: 1) Either we consider some kind of inverse modeling: That is we start from theoretical models, and whatever the type of selected conceptualization (either mathematical models based on fluid dynamic modeling including different conceptual parameters linked to different types of porosities, or computer models based on networks involving also parameters associated with different types of porosities), the model parameters will be fitted a posteriori to match real measured retention or permeability curves. We will consider this perspective, which is the most classical in academic research, in section 5. A first proposal would be to work on extensions of the PSF model developed initially by Perrier, Bird and Rieu (1999) for soils and which appears quite convenient to model fractal size distributions of vugs and matrix areas in rocks, and could be coupled with a fracture model. 2) Or we consider some kind of direct modeling: We look at the available geological data and measurable variables, that is in the present projects, images in 2D, or maybe in 3D, and we build models integrating a priori such data, which seems more appealing in applied research but has also some shortcomings. We will address this second option in section 6. We refer to previous work done par Delerue and Perrier (2002) initially to extract pore networks from soil tomographical 3D images, but a new code would also have to be developed to account for the matrix since this code was working on binarized images of the pore space where vugs would have been detected as pure disconnected holes, Finally we will address briefly in section 7 the issue of multiscale fractal modeling where some theoretical developments mail help to deal with upscaling from micro to meso then macro scale of a large reservoir.
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2. Modelling fractal fractured porous media A multiscale algorithm was developed to generate a network of fractures according to the Rieu and Sposito(1991) fractal model. An example is given if Fig.2.1: any given studied area can be divided into embedded polygonal subareas according to seeds given at each observation level. In the example below, N=10 random points are the initial seeds at level 1 (Fig.2.1.a), the same number of divisions are used at level 2 (Fig.2.1.b), and at level 3 we get 1000 polygons. Applying the same similarity ratio k=96% to each polygon generates the mass fractal medium of dimension D (D=1+LogN/(LogN-2Logk)). colored in brown (Fig.2.1.c) and a fractal number size distribution of fractures colored in blue and following a power-‐law involving the same D exponent. N and k can vary at each scale to obtain non fractal but multiscale structures.
(a)
(b)
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(d) (e) (f) Figure 2.1. Example of construction of a fractal network of fractures with N=10 k=96% D=2.9315… A geometrical algorithm was designed to tessellate the pore space according to local apertures (the result is shown on Fig.2.1.d) since the pore or fracture aperture is the key parameter to use physically-‐based laws: 1) To estimate the capillary pressure curve, a simple capillary model is used to model the distribution of two non miscible fluids in a porous medium (the wetting fluid is water, while the non wetting fluid is oil), assuming Laplace’s law to be valid throughout the pore distribution range in the following simplified manner. For a given capillary pressure h and the corresponding fluid equilibrium state, a pore is filled with water (or oil) if its aperture is smaller than r=alpha/h (resp. larger than r=alpha/h), where alpha is a constant that can be calculated from the liquid-‐solid contact angle and the liquid-‐
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water-‐oil surface tension. The value of alpha depends on the experimental conditions, namely the type of oil, the temperature, the results presented in Fig 2.2.a are so far qualitative, and the numerical values would be proportional to the plotted ones. Simulations take into account not only the pore size but also its accessibility, here from the top of the sample: On Figure 2.1.h one observes that water could reach only the smallest pores connected to the top side. Accounting for the network connectivity explains the hysteresis shown here on Fig. 2.2 only for the pressure curve, where injection of water in an oil-‐filled network corresponds to the lowest curve in the plot. h K
W%
W%
(b) Figure 2.2. (a) Simulation of the water-‐oil capillary pressure curve (b) Simulation of water and oil permeability curves, where water permeability is the highest for the highest water content W%. Further plots will present relative permeabilities normalized by the maximum value. 2) To estimate the relative permeability curves, K is calculated by analogy with electrical transport in a classical manner for pore network. Each pore or fracture is given an elementary resistance varying as a power law of its aperture (in a fracture of aperture r and length l, the flow is proportional to r3/l, according to the simplest formula of Poiseuille’s law in a cylinder. Other local values derived from local integration of Navier-‐Stokes equation could be used for different types of pores, namely for spherical There is one equation per node, vugs). At a given water content, if a pressure so for example we have a linear gradient is prescribed between two opposite sides, system of 3061 equations in the water can flow through the subnetwork of pores case of Fig.2.1.c. to calculate the filled with water and connected both to the inlet and relative water permeability at the outlet of the sample. The sum of the local flows water saturation shown in at each node of this Kirchoff network must be zero. Fig.2.2.b. This condition leads to a linear system to solve, For the equilibruim state imposing a precise local pressure at each point. The represented in Fig.2.1.f. water-‐ macroscopic equivalent relative permeabilities are filled pores are not connected then derived, respectively for water-‐filled and the permeability value is subnetworks and for oil-‐filled subnetworks. zero.
(a)
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3. Modelling fractured and vuggy porous media: first attempts The Pore Solid Fractal model is a generalization of the previous model. It was developed by Perrier et al. (1999) to overcome the limitations of the mass fractal models, whose mass tends towards zero and porosity towards infinity when iterated over an arbitrary large number of scales. The theoretical model involves a probability for the fractal set to be decomposed into smaller solid and voids at each iteration (pF=9/16 in Figure 3.1.)
(a) (b) Figure 3.1. The PSF model can represent a fractal number size distribution of vugs. (a) The proportion of vugs kept at each scale is 3/16 with this example iterated 3 times on a 16x16x16 squared grid. (b) an example iterated 3 times on 20x20x20 polygonal tessellation, with an incomplete fragmentation process inspired from the PSF model, where 90% of the space is occupied by vugs at each level. The simulator presented in section 2 was adapted to investigate the behavior of a fractal distribution of vugs inspired by the PSF model. It only “PSF-‐inspired” because no solid impermeable blocks are kept at each iteration, future work might use the black areas to model the matrix (see section 5). In the present investigation the vug-‐model was only coupled with the fracture model to show some first qualitative results. On following Fig.3.2, numerical experiments confirm that, when one considers a set of disconnected vugs (Fig.3.2.a), only those connected to the water injection front can release oil. When micro-‐fractures are added, water can enter first the smallest fractures connected to the top side (Fig.3.2.b), but further filling will decrease the capillary pressure at the oil-‐water interface, and finally most of the porosity will become reachable : At the end of the simulation, only small disconnected vugs will remain full of black oil (Fig.3.2.c). On Fig.3.2.c one can also observe the associated simulated normalized oil and water relative permeabilities curves, the values are 0 in the case without fractures shown in Fig.3.2.a. Fig.3.2.d shows the simulated pore size distribution, with a great of amount of the porosity distributed on micro vugs or micro fractures. The simulated capillary pressure curve (Fig.3.2.c) is linked to this pore size distribution, the decrease of the capillary pressure is very sharp for low water content, since water can go in the large vugs as oil going out the same vugs when the pressure at the two fluids interface becomes negligible.
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(a) With only disconnected vugs, water injected from the top can reach only part of the (micro-‐scale) reservoir
(b) When micro-‐fractures are added to connect vugs, water injected from the top can reach first the smallest connected fractures then most of the pore space (see following Figure 3.3)
Figure 3.2. Part 1. Simulation of water injection in a vuggy porous medium, without fractures (a) and with fractures (b, c, d).
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(c)
(d) Figure 3.2. Part 2. Simulation of water injection in a vuggy porous medium, without fractures (a) and with fractures (b, c, d).
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4. The key issue of connectivity addressed through network modeling.
Previous sections confirm that the connectivity of the pore space is a key issue. The best way to address it is to work in 3D, since apparent connectivity in 2D is always biased, but there is a lack of 3D reliable data, and when such data exist, for example with X-‐ray tomographical images, they cover a limited range of scales linked to the resolution of the acquisition system (Delerue and Perrier, 2002). Many researchers try to extrapolate from 2D to 3D. At the London Imperial College Blunt’s team reconstruct 3D porous media having the same statistical properties than those observed on 2D images, then they extract a network model from the reconstructed samples to simulate the fluid properties. The reconstruction and network extraction may be very long (several days) for a virtual sample of some microns length (Al-‐Kharusi and Blunt, 2008). Moreover the geometrical features disappear in the statistical reconstruction. We investigated here a pseudo 3D approach where several planar networks of objects can be connected in the third dimension. This option would have the advantage to be more tractable in terms of computational memory and time, and the artificial transversal links could be used to fit experimental pressure or permeability curves using real data. Inverse modeling could be achieved by optimization methods similar to those developed par Peter Matthews and co-‐workers (Laudone, Matthews et al., 2007) who estimated permeability curves from a Por-‐Cor Model calibrated using retention curves. Extensions could account for three different types of porosity in three planes, one with fractures, a second one with vugs, a third one with the matrix. Unfortunately this would assume that the matrix is continuous whereas some fractured or vuggy media might well act as important continuities preventing easy access to some oil-‐filled areas. I would advise to develop a new network model including from the very beginning three types of conducting objects (see section 5).
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(a)
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Figure 4.1 A 3-‐plane realization of a vuggy porous medium Part 1. (a) if there are no connecting paths between planar sections, oil remains trapped in disconnected vugs, and the fluid properties are just the mean of those obtained with several independent realizations. (b) if water can flow through (invisible) paths between sections in a pseudo 3D way, the fluid properties are strongly modified (see results in Figure 4.1 Part 2). In the present simulations, a 3D network has been build by connecting arbitrary each node in a given plane to a neighboring one in the following plane.
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(c)
(d) Figure 4.1 Part 2: Simulation of the fluid properties for the model shown on Figure 4.1.c. (c) Simulation of the water-‐oil capillary pressure curve (d) Simulation of water and oil relative permeability curves, where water permeability is the highest for the highest water content, and the other values are normalized by the maximum value to get results ranging from 0 to 1.
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5. Perspective 1: Towards a new network model with triple porosity Until now, previous codes were adapted to account for vugs (Section 1,2,3). As mentioned before, we propose now to build an improved, more realistic model. The PSF (Pore Solid Fractal) was conceived (Perrier at al.,1999) to allow the representation of both solids and voids in space in a more realistic way than with a mass fractal model (which represents mainly a power law distribution of pores sizes included in a solid phase that vanishes when a broad range of scales is considered). The same PSF can model solid particles or grains at a given scale or the matrix at another scale. Section 3 was only inspired by the PSF model to account for vugs, here we propose to develop a new simulator actually based on the PSF to account first for vugs and matrix, then later in this section for fractures. Several papers using the PSF model have already been published. The mathematical calculations were done so far on squared on cubes representations equivalent to those illustrated in Figure 5.1.a or Figure 3.1.a (or on random spatial distributions to study percolation properties in 2D or 3D). At each iteration of the model, a vanishing fractal set, colored yellow on Figure 5.1.a, is subdivided in N subareas: Each subarea can be a solid area (colored brown), with a probability pS, or a pore area (colored white), with a probability pP, or a new (yellow) area, with a probability pF, which is again divided into solids (now matrix) and pores at the following iteration. On has obviously : pF + pS + pP = 1 The fractal dimension of the pore-‐solid interface is given as a function of the Euclidean dimension d (2or 3), N and pF as follows: €
D =d
(
Log N.pF LogN
)
The PSF model can be fitted from the knowledge of the total porosity and a pore size distribution, or from a capillary € pressure curve which is given by the following equation: ⎧ D−d ⎫ ⎪ pP ⎪ ⎛ h ⎞ θ =Φ− ⎨1− ⎜ ⎬ for h min ≤ h ≤ hmax ⎟ pS + pP ⎪ ⎝ hmin ⎠ ⎪⎭ ⎩ pP When the model is developed ad infinitum one has: Φ = . € pS + pP € For the present extension including an additional porosity Φ M in the matrix ( with pore
sizes smaller than those of the pores identified to vugs, that is smaller than the value rmin € associated to hmax), it has been shown (Perrier, PhD 1995) that the same analytical relationship holds for the capillary pressure on the [€ hmin, hmax] range. In this case we will have: pP Φ = ΦM + pS + pP
The PSF has been widely used in soil science (quoted about 80 times), it explains links between structural and fluid r€etention properties, its connectivity has been recently studied using percolation theory, and we propose now to extend it to naturally fractured and vuggy oil reservoirs at multiple scales.
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(a)
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Figure 5.1 A PSF example F=8/16 P=2/16 S=6/16
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We will first move from squares to polygons (see Figures 5.1.b&c, not only to generate more realistic pictures, but to get the smoothed properties of statistical fractal (The theoretical regular model generates step-‐functions, with one value for each scale, whereas the statistical realizations smooth the functions). This first step is straightforward in 2D, and a 3D Voronoi algorithm should be used to generate polyedras in 3D. At the last level of a given simulation (Figure 5.1.d), the iterations are stopped, and the yellow areas “disappear” to be replaced by vugs and matrix in the same proportion pP and pS. Several examples of PSF realizations representing multiscale fractal vugs and matrix areas can be observed in Figure 5.2. , for different values of this 3-‐parameter model (N, pF,pP).
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(b)
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(c) N=50, pF=0.50, pP=0.25 pS=0.25
(e) N=100, pF=0.5, pS=0.4, pP=0.1, 2 levels
(d)
(f) N=100, pF=0.75, pS=0, pP=0.25, 2 levels
Figure 5.2: Examples of PSF random realizations a) a fragmentation skeleton after 2 iterations b) a porous medium associated to previous skeleton. c) and d) : 3 iterations, with and without plotting the constructions lines, note that the matrix and vug areas have the same mean value but that their distribution is random. e) a PSF model with more matrix than vugs f) a particular case of the PSF: pS=0, where one gets again a fractal power-‐law distribution of vug sizes but also a mass solid fractal for the matrix.
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The PSF has been widely used in soil science (1999’s initial paper quoted about 80 times), it explains links between structural and fluid retention properties, its connectivity has been recently studied using percolation theory, and we propose now to calculate its predictions as regards oil and water relative permeability. This was not done before, since it requires the development of a new network simulator similar to the one used in section 1 and 3. Most of previous algorithms can be reused to carry out numerical simulation, first in 2D, later in 3D. The first stage consists in extracting the network. This could be done as explained later in following section 6 for real images. But for virtual images, one can take into account the actual construction process of the model, to get exact calculations of the required neighboring links. This is possible because the Voronoi tesselation is, for theoretical reasons, equivalent to a Delaunay triangulation. We show In Figure 5.3 the network obtained for the first level of the model construction. Extra work has to be done to handle the network hierarchy of the multiscale case.
Figure 5.3: Extracting a network of connected objects at the first iteration of a PSF model
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Finally the last step of the model construction will deal with the addition of fractures, to model a triple porosity medium. This can be done : -‐ either by simple superimposition of a classical fracture network on the PSF vug-‐ matrix model, using the multiple planes software to find inter-‐plane connections. This first option is illustrated as in Figure 5.4. -‐ or by including fractures whose position in space is linked to the natural fragmentation process as illustrated as in Figure 5.5.
Figure 5.4
(a)
(b)
Figure 5.4
These 2 options have to be discussed further with the geologists and considering the specificities of the present field of investigation.
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6. Perspective 2: Towards direct modeling of porous media images First ideas presented here have been developed from 2D images, such as obtained at a microscopic scale from the field of investigations. They could be used in pseudo 3D simulations or generalized if 3D images are available. As usual we will proceed first with examples. In example 1 (Figure 6.1), linear and spherical features were identified from image analysis and stored in a multi-‐layered GIS (Figure 6.1.b). Adding the automatic detection of spatial neighboring connections in such a set of spatial objects is not an easy task. Moreover, in this example as in many other ones, the extracted network of fractures and vugs would be non-‐conducting. This is why the surrounding matrix has to be taken into account. We propose an original way to both identify the connections between the fracture and vug nodes of a future network, and to account for nodes representing matrix areas in an orginal way also based on the Voronoi tessellation algorithm (Figure 6.1.c), using an ingenious way to select seeds on the object boundaries.
(a) Oleskko 2012
(b) Cherkasov 2012
Figure 6.1: Example 1 A Voronoi tessellation of a porous space to accounting to identify neighboring relationships between different types of idealized porous objects. (c) 1 idealized fracture, 1 vug and several surrounding porous matrix areas).
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The proposed method is inspired from the work done by Delerue and Perrier (2002) to extract pore networks from 3D binarized images. Let us consider a porous space colored black in Figure 6.2.a. Points are selected on the boundary of the pore space, as indicated on Figure 6.2.b, to be used a the seeds of a Voronoi tessellation. From theoretical reasons, the closer the space between these dots, the better is the determination of the skeleton of the pore space which is also the set of the centers of the maximum balls covering the pore space. The method is also very close to the one developed by Phillips et al. (2012) to design optimal filling of shapes in theoretical physics and mathematics. The new idea proposed here is to generalize the method to find the centers of optimal balls covering the pore space but also to find the centers of the optimal spheres covering the dual space in order to build a network accounting also for the matrix. (Embedded tessellations such as those used in section 1 could take into account of new, smaller objects revealed by increasing the resolution)
(a) (b) Figure 6.2. A Voronoi tessellation with seeds selected on the boundary of vug-‐objects From this type of space tesselation, it will become possible to extract the network liking both matrix polygonal areas and pore objects. Vugs or fractures will be handled in the same way at this stage (see Figure 6.3), they differ in according to their more or linear or spherical shapes, but both types are represented by polygons. Such polygons could be read from the GIS-‐ files obtained by Cherkasov. The expected resulting network is shown colored red on Figure 6.3. An automated algorithm can be developed by accounting from the neighboring relationships calculated by the Voronoi algorithm plus specific geometrical-‐based algorithms similar to those presented in section 1.
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Fig. 6.3 Extracting a network from the specific space partition shown in Figure 2.b.
(a)
(b)
Devopping an algorithm which can be used by end-‐users would involve different steps : 1) either reading the original grey-‐scale image and developing a computer-‐assisted method to detect the significant objects or reading associated binarized images where the objects have already been automaticall detected. 2) Selecting seeds on the interfaces at the highest possible resolution (1 pixel in 2D or 1 voxel in 3D) 3) running a voronoi tesselation in 2D or 3D 4) extracting the associated network with 3 types of nodes (fractures, vugs, matrix) and generic links 5) simulating the fluid pressure and permeability properties as previously, either in 2D, in pseudo 2D or in 3D, depending on the available data.
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7. Fractal and percolation modeling The tools that we propose to develop are based on fractal geometrical concepts, which are a special case of multiscale modeling. The fractal framework is useful to design the model. Then computer simulations can be easily adapted to modify the parameters at each scale (selecting different values for the number of sub-‐areas, for the number of matrix versus vug subareas, etc.) and thus to simulate multiscale properties going beyond the mere pure fractal case. When the fractal assumption is verified, fractal dimensions calculated on images will help calibrating the models. A dedicated algorithm has been provided (see Appendix 2) to estimate not only fractal dimensions, but also the multifractal parameters, the Renyi dimensions, which can be considered as geostatistical indicators of the spatial variability when used on spatial data. Some mathematical calculations can be done on pure fractals to estimate the capillary pressure curve, where the connectivity effect can be neglected in several cases. Some other theoretical investigations were carried out to evaluate their connectivity on random realizations of square or cubic grids. This was done using percolation theory and probabilistic renormalization function, which can help to investigate the type of expected connectivity on random distributions. Let us just give hints about possible applications of such a theoretical approach. Consider a fractal power-‐law distributions of vugs included in a mass fractal matrix (such as illustrated on Figure 5.2.f). Providing the assumption that the vugs follow a power-‐law distribution of sizes, percolation theory could estimate, for a total vug-‐porosity given value, the minimum number of fractal iterations to be considered for the whole set of vugs to be connected or not (in 2D or 3D). This required minimum number of iterations corresponds to a critical vug size Rmin. If the vug network is connected, the critical permeability could be calculated from Rmin. Above this critical value above, the permeability will be “vug-‐driven”, and calculated from the vug size distribution. Below this transition point, the permeability would be “matrix driven”, that is low valued, and calculated from the mean size of the microscopic porosity. Such qualitative theoretical arguments could prove useful to constraint the simulations by theoretical reasoning. We will not develop futher this point because the present report focuses on the extraction of networks, whereas the fractal dimensions give more useful information on the scaling behavior than on the type of geometrical network model to be build to match realistic spatial arrangements.
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8. First conclusions We propose to develop new original tools to account for the triple porosity media encountered in naturally fractured vuggy reservoirs. We consider that the keys issues as regards estimation of their permeability and oil accessibility is the connectivity of the different spatial areas involved in natural rock formations. We propose to undertake the extraction of realistic networks linking these areas and enabling the simulation of relative permeabilities at a given scale, from local permeabilities of each area estimated at smaller scale. We propose to develop algorithms based on computational geometry to capture the key components of natural geometries as regards fluid flow. Our previous work was already based on such ideas at the pore-‐scale level. Similar work is widely used with fracture network models. To our knowledge no similar work has been done so far on triple porosity models. The algorithms that we propose to develop are potentially usable at any scale from spatialized data. But whatever the exponential increase computer power, it will be always impossible to handle a large broad of scales in a single simulation. By explicitly introducing matrix areas as a new type of nodes in the network, we will also provide ways to allocate porosity and permeability values integrating smaller scales in larger scale simulations. What appears as a matrix at the scale of large fractures and vugs can reveal fissures and small vugs at a higher resolution : we can either assume similarity through a large range of scale or just consider a way to cope with multiscale properties. Fractal geometry gives hints for the upscaling issue since mathematical calculations can be derived in the case of idealized self-‐similar media to validate the computer simulations. If the reservoir was a perfect self-‐similar fractal trough the whole range of scale from microns to kilometers, one could focus on the micro scale then looking for some kind of renormalization functions to extrapolate towards the meso and macroscale. We believe that fractal theories and fractal dimensions apply only over limited ranges of scale, and that several scales must be investigated as regards the acquisition of experimental data. Empirical relationships between porosity and permeability have already been established in the past, but the fitted parameters have to be estimated again when investigating a new reservoir, which requires the acquisition of large data sets of both porosity and permeability to carry out reliable statistics. My point of view consists in advising to take more time to test new ideas for the estimation of permeability directly from data about the porosity spatial distribution in a network. Some direct measurement of permeability and capillary pressure curves would be also needed at different scales, first to calibrate and validate the model. The proposed model will need further implementation and this would delay the production of expected entregables at the first step. But any new ideas involve both some risk and some hope as well in applied research as in pure research and progressive knowledge of all types of data available on the field of investigation will also drive the model construction. As regards the modeling of fluid properties from, the work done for the micro-‐scale will be re-‐used at meso and macro scale providing appropriate varying functions for local behavior of fluids in fractures, vugs, or matrix.
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9. References Al-‐Kharusi Anwar and Martin Blunt 2007, Network extraction from sandstone and carbonate pore space images, Journal of Petroleum Science and Engineering, Volume: 56 Issue: 4, Pages: 855-‐860 Al-‐Kharusi Anwar and Martin Blunt 2008, Multiphase flow predictions from carbonate pore space images using extracted network models, Water Resour. Res., 44, W06S01, doi:10.1029/2006WR005695 Arbogast Todd and Mari San Marin Gomez, 2009, A discretization and multigrid solver for a Darcy–Stokes system of three dimensional vuggy porous media, Computational Geosciences Volume : 13 Issue : 3, pages : 331-‐34. Delerue J.F and E.Perrier, 2002, DXSoil, a library for 3D image analysis in soil science, Computer and Geosciences, Volume: 28 Issue: 9 Pages: 1041-‐1050. Gringarten A., 2008. From straight lines to deconvolution: the evolution of the state of the art in well test analysis, SPE Res Eval & Eng11 (1) pages : 41-‐62. Huiyun Lu, 2007. Investigation of Recovery Mechanisms in Fractured Reservoirs, PHhD dissertation, http://www3.imperial.ac.uk/earthscienceandengineering/research/perm/porescalemo delling/phd%20theses Laudone, Giuliano M.; Matthews, G. Peter; Gane, Patrick A. C.; et al, 2007. Estimation of structural element sizes in sand and compacted blocks of ground calcium carbonate using a void network model, Transport in porous media, Volume: 66 Issue: 3 Pages: 403-419 Lv Aimin, Jun Yao, Wei Wang, 2011, Characteristics of Oil-Water Relative Permeability and Influence Mechanism in Fractured-Vuggy Medium, Procedia Engineering 18, pages : 175183 Mayur Pal, 2012, A unified approach to simulation and upscaling of single-‐phase flow through vuggy carbonates, Int. J. Numer. Meth. Fluids ; 69:1096–1123 Phillips C, J. Anderson, G.Huber,and S. Glotzer,2012, Optimal Filling of Shapes, Physical Review Letters 108, 198304 (2012) Perrier E, C.Mullon, M. rieu and G. de Marsily, 1995. Computer construction of fractal soil structures: Simulation of their hydraulic and shrinkage properties, Water Resources Research, Volume 31, Issue 12, pages 2927-‐2943 Perrier, E; Bird, N; Rieu, M, 1999. Generalizing the fractal model of soil structure: the poresolid fractal approach, Geoderma Volume: 88 Issue: 3-4 Pages: 137-164 Rieu M. and G. Sposito, 1991, Fractal fragmentation, soil porosity and soil-‐water properties. 1. Theory, Soil Science Society of America Journal, Volume: 55 Issue: 5 pages: 1231-‐1238 Wu, Yu-‐Shu; Di, Yuan; Kang, Zhijiang; Fakcharoenphol, Perapon ; 2011. A multiple-‐ continuum model for simulating single-‐phase and multiphase flow in naturally fractured vuggy reservoirs, Journal of Petroleum Science and Engineering, Volume: 78 Issue: 1, Pages: 13-‐22
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10. Appendices (description of already delivered codes)
10.1. SIM-‐POR A simulator of virtual porous structures to calculate water retention curves and relative water and oil permeability curves. The interface of the simulator, using choice buttons, is straightforward (see screencopy Figure 10.1.1.). Buttons or triangular icons give access to additional options.
Figure 10.1.1. The description is done here using a series of examples. Example 1: a classical mass fractal fractured media (Figure 10.1.2). After the first example, all the parameters can be modified to try other configurations.
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Select “Fragment number all level”= 10 then click “init” once, then click “Frag.next level” 3 times (you get any similar random realization)
Click on “identify nodes”. The number of allocated nodes (3043 here) is printed in the main xterm frame.
You can fill the pore space with oil by selecting “fill with oil” in the “drain” menu
select “automatic pressure pressure decreasing” in the “inject” menu. Redo it by clicking on “yes” in the “Calc.permeability” button (it will be slower)
Select the similarity ratio k% to be equal to 95 for all levels then click on “create porous medium”.
Click on any of the item of Histograms/curve to plot and to store data in files RES1, RES2, RESK,
Figure 10.1.2 Example 2: a vuggy porous media (Figure 10.1.3) Select first a different model by going into the “General Options” subwindow: Select PSF instead of RS and a fragmentation probability equals to 0.90 in this example.
Select, “Fragment number all level”= 50 then click “init” once, then click “Frag.next level” twice. Then click on “identify nodes”.
Select in “similarity ratios” level 2 only and for the ratio k2 a value of 1. then click on “create porous medium” (only vugs above)
Select now in “similarity ratios” level 1 only and for the ratio k1 a value of 98. then click on “create porous medium” :vugs + fissures
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Then in each case select “automatic pressure pressure decreasing” in the “inject” menu as in previous example 10.1.2
Figure 10.1.3 The code developed for such a software includes different files. 1) random.h and random.cc are very small programs to generate random numbers. The present version relies on the standard function provided by C++ (gcc). Other generators could be used if necessary, when the type of randomization is a key issue (namely for random fracture network or any type of virtual porous structure-‐ 2) liste.h and liste.cc are short programs providing functions to handle lists of any computer objects. Lists are particularly useful to navigate within connected graphs and standard librairies could have been used. 3) geometrie.h and geometrie.cc are customized programs to handle not only standard points, lines, polygons, the calculation or areas, centers of gravity, intersections, translations, similarities,etc, but also polygons whose edges include a pointer towards neighboring polygons, useful to handle connections in a network of geometrical objects. Geometrie.cc includes a third file named voronoi.cc, which implements a very useful tessellation algorithm to divide a given area around seed points. The originality here is that the voronoi algorithm works in a recursive way, which is useful to handle multiscale geometries. graphique.cc generates arrays ready to use for graphical plots of curves or histrograms. 4) simule.h and simule.cc This is the core of the object oriented simulator used to create specific objects such as pores, aggregates, nodes and to handle a hierarchy of embedded and connected objects. It includes the generation of fractal structures according to two types of fractal models, the RS (Rieu&Sposito, 1991) mass fractal model and the initial version of the PSF model (Perrier, Bird and Rieu, 1999). The program includes specific routines for generation of a pore network (linked so far to the type of geometrical model used, but could be generalized). The generation of networks linking multiple planes is also included in these files It includes two general routines: i) to calculate water retention curves from a pore size distribution accounting for the pore size and for the connectivity of the pore network. ii) to calculate water and oil relative permeability curves by solving the set of n linear equations associated to each of the n nodes of a network. Each equation is based on the assumption of local flows in each pore according to the Poiseuille law and equilibria at each nodes of input and output flows. 5) vorosol.h and vorosol.cc To associate to any user action a function defined in the core program simule.cc and to handle the graphical plots sent to subwindows. 6) vorosol_ui.h and vorosol_ui.cc Only used for the User Interface (UI) with menus, buttons, windows, etc, it is based on the Xview librairies running on the X11 windowing system (Unix, Linux, included in Mac OS and possible Cygwin unix emulator on Windows)
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10.2. MF A tool to calculate fractal dimensions or generalized multifractal Renyi dimensions D(q) on 2D images.
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q= q= q= q= q= q= q= q= q= q= q= q= q= q= q= q= q= q= q= q= q=
0 1 2 3 4 5 6 7 8 9 10 -‐1 -‐2 -‐3 -‐4 -‐5 -‐6 -‐7 -‐8 -‐9 -‐10
Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq= Dq=
1.933 1.934 1.937 1.939 1.942 1.945 1.947 1.949 1.951 1.952 1.953 1.929 1.921 1.923 1.923 1.923 1.922 1.920 1.919 1.918 1.917
R2=0.9997078840639666, R2=0.9999841252975528, R2=0.9999986306665456, R2=0.999990116178825, R2=0.9999794464215885, R2=0.9999698482318073, R2=0.9999616538159831, R2=0.9999546393134002, R2=0.9999485251756162, R2=0.9999430832247717, R2=0.9999381470051176, R2=0.9895262354225199, R2=0.9336743702031681, R2=0.8841167109470233, R2=0.8508342108189841, R2=0.8278568244143374, R2=0.8111482132543008, R2=0.7984717820380415, R2=0.7885353794347528, R2=0.7805435225624624, R2=0.7739799723471931,
ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7, ddl=7,
(Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999) (Sign.1%ifR2>0.6363252899999999)
One just show here a few illustrations, using screencopies of the software used on 3 real images coming from the field of investigation. The Reyni dimension of order zero Dq(0) reduces to the classical fractal dimension. When D(0) is equal to the Euclidean dimension (here d=2 ), the image is not fractal. When the spectrum of D(q) values is flat, the medium is not multifractal.
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