Report Dra. Edith Perrier 2013, January 17th-May 16th Advising on the Project 168638, SENERCONACYT-Hidrocarburos Yacimiento Petrolero como un Reactor Fractal leaded by Dra. K.Oleshko: 3rd 4monthsreport. Estimating oil-water capillary pressure curves and relative oil and permeability curves in naturally fractured and vuggy oil reservoirs from multiscale geological data.

1. Introduction ............................................................................................................................3   2. The triple porosity VFM model in 2D....................................................................................6   2.1. Fracture/vugs shape factor...............................................................................................7   2.2. Fracture porosity and F objects .......................................................................................9   Definition............................................................................................................................9   Porosity.............................................................................................................................10   Local permeability............................................................................................................11   Fracture networks .............................................................................................................12   2.3. Vug porosity and V objects : .........................................................................................12   Definition..........................................................................................................................12   Porosity.............................................................................................................................13   Permeability......................................................................................................................13   2.3. Matrix porosity and M objects : ....................................................................................14   Porosity.............................................................................................................................14   Permeability:.....................................................................................................................15   Visualization and spatial variability: ................................................................................15   3. Intrinsic permeability, model and images. ...........................................................................18   3.1. Network modelling........................................................................................................18   3.2. Illustrations with real images.........................................................................................21   Sample 1H ........................................................................................................................22   Sample 2H ........................................................................................................................23   Sample 3H ........................................................................................................................24   3.3. SEM Image....................................................................................................................25   3.4. Other possible numerical experiments ..........................................................................28   Percolation threshold ........................................................................................................28   Pseudo 3D.........................................................................................................................29   4. Non-Newtonian fluids ..........................................................................................................30   4.1. Theory............................................................................................................................30   3.2. Park’s data .....................................................................................................................32   3.3. Numerical experiment ...................................................................................................34   3.4. Discussion and perspectives ..........................................................................................39   Analysis of our inverse simulation:..................................................................................39   General representation of non-Newtonian fluid in our network model. ..........................40   Perspectives ......................................................................................................................41   5. Unsaturated case...................................................................................................................42   1

5.1. Classical modelling .......................................................................................................42   Simulation of fluid invasion & capillary pressure curves ................................................42   Relative permeability curves ............................................................................................48   5.2. Investigating complex phenomena at the interface of two non-miscible fluids. ...........49   Films of wetting fluids......................................................................................................49   Emulsions .........................................................................................................................50   6. Conclusion and perspectives ................................................................................................52   7. References ............................................................................................................................53   Appendix A : Ricci balls .........................................................................................................55   A.1 Curvatures..................................................................................................................55   A.2. Interaction of spheres with different curvatures (radius)..........................................57   A.3. Ricci flow under constraints .....................................................................................59   Appendix B : Code to calculate the permeability of the VFM network..................................61  

Note: 4 movies were included in this report but because some people may have no access to the latest version of Acrobat reader, the address where to download them has been added in the appropriate figures http://edith.alice.perrier.free.fr/MovieNewtonian.mp4 (Figure 4.9.a) http://edith.alice.perrier.free.fr/MovieNonNewtonian.mp4 (Figure 4.9.b) http://edith.alice.perrier.free.fr/MovieSEMWaterInvasion.mp4 (Figure 5.4) http://edith.alice.perrier.free.fr/MovieEmulsions.mp4 (Figure 5.11)

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1.  Introduction     The present third 4 months report involves previous partial reports written during this period. The main goal is to develop mathematical models and computer tools to investigate the relationships between structural geological data and fluid properties in naturally fractured vuggy oil reservoirs. The handbook for the last version of the SIMPOR software is provided separately. Vugs and fractures are encountered at almost all the scales of the studied reservoirs. The scale-free VFM model has been conceived to work using the same principles at any scale (Figure 1).

Figure 1.1: Images at the meter scale on the left, at the millimeter scale on the right. The simulated scale is plotted on the top of each simulated sample. Let us recall that one of our first goals was to estimate the permeability of the porous media as a function of its porosity, since the permeability is a key parameter of all reservoir dynamical models. Many empirical formulas exist but it is obvious that they have to be calibrated by means of statistical analysis of real data for each particular study, since the value of the total porosity of a rock sample alone cannot predict its permeability in a deterministic way. If we had reliable information on the spatial distribution of the porosity in a 3D porous rock sample, the prediction of the permeability would be possible, at least theoretically. Nevertheless exhaustive information about the porosity of a 3D sample involves technical problems to scan all the dimensions and scales of the sample, methodological problems to extract the porosity from noisy information, as well as computer problems both to store huge 3D matrix and to solve the flow equations. Anyway original methods are investigated by other researchers in the present project to extract useful information from 3D imaging analysis. In any case, the unavoidable variability of the results from one rock sample to another one, or from one scale to another one leads to the search for simplified assumptions to model the rock multiscale geometry, namely by using fractal concepts. In the work presented here, we investigate the possibility to mix information given by 2D raw grayscale images and idealized models in order to progress with the spatial variability plus upscaling issues. 3

At the scale L of the given sample (see Figure 2), the permeability K can be measured using different methods based on the empirical definition of K, as follows : 1A Q=K ΔP (Darcy’s Law) µL (a) When a pressure gradient ΔP is imposed between two opposite sides of an anisotropic rock sample saturated with a Newtonian and € incompressible fluid (such as pure water), the € measured fluid flow Q (m3/s ) is proportional to ΔP (Pa). K is defined as the part of the proportionality constant which is an intrinsic property of the € independent of the fluid€viscosity µ . porous media, K is expressed in m2 (or in Darcy 1D ≈ 10−12 m 2 ) If A is the crossed porous area in m2, the actual (a) mean fluid velocity V in m/s through the pore space € Kρg is V = . € µΦ

€

(b) The upscaling process aims at finding ways to estimate the macroscopic value of K from information obtained at a smaller scale, for example from information obtained from images revealing the rock geometry at a (relatively) microscopic scale .

(b)

(c) Our modeling approach aims at building an abstract network linking 3 types of interconnected objects (3 “colors” on Fig.2.c) created from the vug and fracture objects identified at the resolution scale, plus matrix objects integrating the microporosity at the following scale

(c) Figure 1.2: Permeability measurement (a), investigation (b), and modeling (c). In Section II, we describe the VFM network model.

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In section III, we show how the VFM model can be used to investigate porosity-permeability relationships from observed or modeled data, in 2D or pseudo 3D numerical experiments. In section IV we address the case of the non-newtonian fluids, such as many types of oils. The actual oil velocity is still a function of the medium intrinsic permeability but we have to account also for the fact that oil viscosity varies with the flux rates generating shear stress. In section V, we move to the non-saturated case: In most oil recovery processes, another fluid (water or gas) is injected in the rock to enhance oil recovery by “pushing” the oil out of the rock pores and towards the production well. The simultaneous presence of two non-miscible fluids leads to further modeling: We first describe the simulation of the injected fluid invasion in the oil reservoir, assuming successive equilibrium stages in the invasion process and derive the capillary pressure curve at the two fluids interface. Then we estimate relative relative permeability curves from the VFM model as a function of the amount injected fluid. Some additional original modeling of the key processes at the interfaces is then proposed, to go beyond the simple classical assumption of pores full or empty with a given fluid in a binary way. We model the formation of thin films for the wetting fluid on vugs boundaries, or the emergence of bubbles for the non-wetting fluid in the center of the vugs, in order to prepare the investigation of the effect of natural or artificial emulsions for enhanced oil recovery. First attempts to model the evolution of the curvature at the interfaces are also depicted in the Ricci balls appendix. In section VI, we conclude with a list of tasks that remain to be done in the short term plus some perspectives.

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2.  The  triple  porosity  VFM  model  in  2D    

The shape of any VFM object is defined by an associated polygon in the plane. Let us note that this is the more common way to define the boundary line of any object in a discrete way. Even if some particular shapes can be defined by the mathematical equation of their boundaries (like a disk in 2D whose boundary can be defined by a circle equation), the most general way is to use a polygon defined by a list of points or segments, even in the case of a simple circle, whose polygonal approximations are widely used. Thus we decided that all the VFM computer objects (defined in VFM.hh) will inherit from the class polygon (defined in the file geometrie.hh) as shown in the following C++ class definition: class VFM_object:public polygon{ list_VFM* VFM_neighbors; }

A network is defined by a set of objects and their links to connected objects. Here VFM_neighbors is a list of so-called “pointers” to the addresses in the memory computer of the neighboring objects, since the connections in the VFM model are based on spatial relationships. Two polygons are neighbors if and only if they overlap in the simulated plane, and the neighboring relationships are calculated using dedicated geometrical algorithms related to intersection or inclusion of polygonal shapes. One of the many advantages of the object oriented programming style (e.g; Schach 2011), is that new attributes can be adding at any stage of the simulator development, without modifying the initial program structure, since all the attributes of a given object con be “encapsulated” in the class definition : that is all the other parts of the program have only access through the class methods and any modification of the class attributes and methods remain local. The first attribute added to a VFM object is an integer registering its “type” as follows: class VFM_object:public polygon{ list_VFM* VFM_neighbors; int type; } TYPE is coded by: type=1 //vugs type=2 //fractures type=3 //matrix

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2.1.  Fracture/vugs  shape  factor   Vugs and fractures are rather easily identified visually by experts in naturally fractured vuggy media. But to discriminate automatically between these two main types of porosities is not an easy task. Several shape factors have been used since the beginning of the past century and we used here the classical perimeter(P) versus area(A) ratio first defined by Cox (1927) and now widely used, that is the following “circularity ratio” used for objects embedded in a 2dimensional plane:

γ=

4 πA

P2 This ratio has the advantage to be scale-independent and dimensionless, contrary to the simplest A/P ratio, while being rather simple to calculate and to interpret. Some studies use 4 πA € (1933), or other shape, elongation, roundness, …, γ= as suggested by Wadell P2 indicators (see Blott&Pye, 2008 for an extended review ). More specific indicators could be later easily tested on the specific images provided within the present Pemex project.

€ Let us recall that, for a circle of radius R, one gets γ =

4 π . πR 2

(2πR) 2

=1 ,

and for other shapes 0  γ ≤ 1 An example of the calculations done by the SIMPOR code are given below with an example € drawn “by hand” (using the mouse, see SIMPOR where γ has been calculated on 22 objects € user-guide).

€

Figure 2.1: Vugs or fractures identified on a given image (22 objects). See calculated circularity ratio values below and histogram in Figure 3bis. The outputs values for P, A and the ratio γ were listed in the command line as follows: Number of drawn objects : 22 Porosity: 0.0817583 perim 1584.656411 surf 49444.000000 ratio 0.247305 perim 3966.888459 surf 266665.500000 ratio 0.212842 perim 8209.722444 surf 291387.500000 ratio 0.054300

€

7

perim perim perim perim perim perim perim perim perim perim perim perim perim perim perim perim perim perim perim

14034.668927 surf 1359721.500000 ratio 0.086703 3445.500472 surf 223332.000000 ratio 0.236285 2918.792808 surf 93054.500000 ratio 0.137189 4409.655338 surf 127221.500000 ratio 0.082175 5469.867900 surf 259443.500000 ratio 0.108913 3332.347424 surf 192638.000000 ratio 0.217887 2400.553015 surf 260971.000000 ratio 0.568800 866.300488 surf 32915.500000 ratio 0.550875 1353.604404 surf 79443.500000 ratio 0.544584 1343.987305 surf 102359.500000 ratio 0.711750 624.884823 surf 21666.000000 ratio 0.696897 3782.198494 surf 871385.500000 ratio 0.765088 1013.484535 surf 65414.500000 ratio 0.799888 4476.967939 surf 1191246.500000 ratio 0.746489 5415.797516 surf 1207215.000000 ratio 0.516951 4068.072546 surf 1228052.500000 ratio 0.932029 1195.695533 surf 85832.000000 ratio 0.754046 467.280334 surf 14026.000000 ratio 0.806805 2217.338987 surf 152399.000000 ratio 0.389321

Figure 2.1.bis. Histogram of the ratio γ (Plot of 100.γ for easier visualization) In the histogram (automatically drawn in the user interface) one can detect the (6+2) more or less circular vugs indentified on the image with γ  6 and the (5+5+4) shapes which can be identified as fractures with . γ  0.6 € € € €

Figure 2.2: 31 circular vugs and 50 rectangular fractures modeled on a test image. The histogram of the ratio γ shows the perfect discrimination between idealized objects.

€

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Then a new attribute is introduced for the VFM objects, their aperture , which will be the key parameter to model their fluid properties. As explained with more details in the following sections, the aperture of a fracture F will be its width, the aperture of a vug V will be its radius, and the aperture of a matrix area M will be its equivalent hydraulic radius. class VFM_object:public polygon{ list_VFM* VFM_neighbors; int type; double aperture; }

2.2.  Fracture  porosity  and  F  objects    Definition   Fractures can be defined by the user as polygons close to rectangular shapes of length L and aperture or width W. Their depth d in the 3rd dimension is not represented on the screen, but contribute to the flux intensity, the default depth is equal to the fracture length. When fractures are identified by the eye expert but are very thin, another way to define a fracture is to only draw on the real image a segment of length L (see userguide) and to assign a mean value for the width W, using the software options window. Then the depth is P − 2W calculated by d = where P is the perimeter of the fracture object. 2

€

(a) segments drawn by the user

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(b) fractures generated by the program

Figure 2.3; Fractures identified on a real image by the user drawing 1 or several segments. The main width is here W mean =30 simulations units, that is 0.87 mm once the 29 cm size of the experimental sample has been taken into account. € allocated in the way than for theoretical modeled fractures; that is using the The apertures are software interface to define the mean L and W values Lmean and W mean .

€ €

Then for each segment i, one has: Wi = ±(var* W mean * random) , random ∈ [0,1] € € Li = ±(var* Lmean * random) , random ∈ [0,1] If var=0, all the fractures have the same aperture and length else one generate distributions around the mean value€such as the one drawn in Figure 6 where, ⎡ 1 ⎤ ⎡ 1 ⎤ 3 3 With var=0.5 , one has: € Wi ∈ ⎢ W mean , W mean ⎥ and Li ∈ ⎢ Lmean , Lmean ⎥ . ⎣ 2 ⎦ ⎣ 2 ⎦ 2 2 Porosity   € € The total porosity is the sum of the porosity of the drawn fractures plus that of the modeled fractures if any.

For a squared simulated sample of size L, one has: ⎛i=n _ F _ drawn i=n _ F _ mod eled ⎞ ΦF = ⎜⎜ ∑ LiWi + ∑ LiWi ⎟⎟ / L2 ⎝ ⎠ i=1 i=1

€

10

Same drawn fractures as those shown in Figure 34, colored red with white boundaries with the addition of 50 modeled fractures, colored red with red boundaries Lmean =2000 (Length F), W mean =30 (Aperture F). € €

Once the porosity has been created, the user drawn and the modeled fractures are handled in the same way as VFM objects of type 2.

(number size distribution for 1000 simulated fractures of mean size 30) Figure 2.4: Superimposition of fractures whose location has been drawn by the user on the image and modeled fractures with random locations, all of them with mean size 30 simulation units. Local  permeability   The local permeability of a fracture is classical modeled using the following Poiseuille law:

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Fracture  networks   Two fractures are neighbors if and only if they intersect and two examples of fracture networks are shown below and some numerical experiments about their properties for fluid flow will be described in following sections.

100 fractures. Lmean=3000 Wmean=30

500 fractures. Lmean=2000 Wmean=0.0001

Figure 2.5 : Fracture networks

2.3.  Vug  porosity  and  V  objects  :   Again we can mix real information and modeled vugs. Definition   When vugs are visible at a given image resolution, they can be drawn by the user as shown in Figure 8. They could be later imported from automatic recognition of contours on images (Example Entropy method and possible coupling of the softwares Prognoz and Simpor)

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Figure 2.6 : Real vugs detected on a 2D image. It is theoretically possible to simulate any theoretical vug distribution. At the present stage we introduced only a very simple fractal number size distribution of circular vugs defined by its fractal dimension Dvug, the maximum vug radius Rmax, and the number of iterations Niter At each iteration i, 0 ≤ i ≤ Niter , 2i vugs are generated with a radius equal to Ri = Rmax 2

€

€

−i Dvug

.

€

Figure 2.7 : Modeled circular vugs (127) with a simple fractal number size distribution. , Dvug=1.89, Rmax=300,Niter=7 Porosity   At any scale L, the vug porosity is the sum of the areas calculated using a computerized discrete method for the user-drawn vugs but the sum of the modeled circular vugs. i=n _ V _ mod eled ⎛i=n _ V _ drawn ⎞ ΦV = ⎜⎜ πRi2 ⎟⎟ / L2 ∑ area(Pi ) + ∑ ⎝ ⎠ i=1 i=1 Permeability  

€

The permeability of the vugs is, as for any other porous object, a function of their size. We do have explicit physically-based formula available for more or less spherical pores. In the present version, we consider that the Poiseuille laws known for cylindrical or parallepipedic shapes might be used a as a first approximations;

Approximation using Poiseuille law for a fracture

Approximation using Poiseuille law for a cylinder Figure 2.8

We selected in the computer program:

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qvug

( =

Ai

)

3

ΔP 10 µ and we may modify this local law later according to fits to experimental data or to new theoretical results, for example Mortensen étal., 2005. Let us note that this is a novelty to account for vugs in a pore network model. Classical pore € are thousands of other models quoted in the Web of Science) networks (Let us note that there deals with fracture networks only (ex : Murzenko et al., 2011) or with « throat-body » networks where spherical pores (the bodies) are linked through cylindrical capillary tubes (the throats) (cf. most of the models quoted in the previous reports, and Balhoff&Thompson 2006, present report). In these models the pore bodies are used to store the fluids, but their local permeability is assumed to be infinite, only the throats are conducting. Some researchers try to overcome this limitation. For example Raoof&Hassanizadeh (2012) write in the abstract of their recent paper “Results of many studies show a discrepancy between calculated relative permeability and corresponding measured values. Often, calculated values overestimate the measured values. An important feature of almost all pore network models is that the resistance to flow is assumed to come from pore throats only; i.e., the resistance of pore bodies to the flow is considered to be negligible compare to the resistance of pore throats. We contend that this simplification may considerably affect the results for relative permeability curves ». The results of their investigation is not fully convincing and my own assumption is also questionable. Nevertheless the present assumption is that local permeability varies in a non-linear way as a function of the pore size or aperture, as for all the pores of irregular shape encountered in natural porous medium. From a practical point of view, as soon as the vugs aperture are much larger than those of the other connected objects in the network, we can check by simulation that their effect on the mean permeability is not very significant. From a theoretical point of view, this is due to the fact that when any two objects or permeabilities k1 and k2 are connected, if k1>>k2 the mean permeability is close to k2 (namely as regards two connected tubes of different sizes, see the general assumption of harmonic mean between k1 and k2 done later in this report §3). On the other hand, if we had only vug porosity in the porous media, it is very likely that they would be totally disconnected and they do not conduct at all. This is one of the reasons why we introduce now the matrix porosity. Another reason is to be ready to handle multiscale porous structures.

2.3.  Matrix  porosity  and  M  objects  :   Porosity   Contrary to the fracture and vug porosity, the matrix porosity is not calculated from real or modeled objects, but the matrix objects are here inversely defined from the matrix porosity denoted ΦM . Knowing the total porosity ΦT , one has: ΦM = ΦM − ΦF − ΦV For example ΦM = 0.21 when the following Mporosity = 21.00 has been selected in the € € €

program: € Then M objects are created by splitting the space around the vugs into matrix polygonal areas.

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using a dedicated Voronoi algorithm using seeds on the vug boundaries (see previous reports and the Simpor userguide). Here we give details on the way the matrix porosity is handled from a more formal point of view, by introducing its “hydraulic radius” As underlined in the introduction we can have different values of the permeability for the same porosity value ΦM . Permeability: So another parameter is needed to assign a local permeability k to the micropores: this is the € so-called hydraulic radius Rmean , that is the size of a conceptual cylindrical tube which would have the given permeability k. € Rmean = 10 (corresponding to a real number rMatrix = 10.00 in the code).

The local permeability is proportional to the square of the hydraulic radius 2

€

k ∝ ( Rmean ) The default unit is the micrometer µm , and with a default simulated sample of size 10000 µm =10 cm and area A=10cm2, Rmean = 10 models tubes of equivalent radius 10 µm and 4

€

π ( Rmean ) € permeability of 0.03925 mdarcy using the assumed following equivalence k = € 8 A Inside the computer code, polygonal zone of € the local permeability calculated for each matrix € the network has been set proportional to the proportion of the total area occupied by its zone, in order to preserve scale consistency. Further theoretical work should be done to fully € address this upscaling issue. Visualization  and  spatial  variability:   The smaller Rmean the larger the (integer) number ni of micropores of radius Rmean used to visualize the same microporosity ϕM (Figure 2.9.a): By the way if Rmean is very small, the visualization may be very long, since a great number of circular area have to be calculated an ni € € drawn € € €

€

⎛ ⎞ ϕM Ai ⎟ ⎜ ni = Int 2 ⎟ ⎜ π ( R ⎝ mean ) ⎠ Ri = Rmean or Ri = Rmean Figure 2.9: Visualization of the microporosity with several “dots” of size Ri

€ In the code some variability is introduced around the mean hydraulic radius Rmean as follows: € € Ri = ±(var* Rmean * random) , random ∈ [0,1] ⎡ 1 ⎤ 3 With var=0.5 , one has: Ri ∈ ⎢ Rmean , Rmean ⎥ . ⎣ 2 ⎦ 2 €

€

€

€

15

j=n i

2

i=n M

One has: ϕi = ∑ π ( Ri ) ≈ ϕM and Φ = ∑ ϕi Ai / AM ≈ ϕM j=0

€

€

i=0

Special case with large Rmean values or small ϕM values: Rmean and ϕM are no more independent If the number of modeled micropores in each matrix area is less than 1, that is € € ⎛ ⎞ €Ai ⎟ € ϕM ni = Int⎜  1, the algorithm automatically creates a single micropore (Figure 2 € ⎜ π ( R ⎟ ⎝ mean ) ⎠ 2.9bis) : For each polygonal matrix zone Pi of area Ai, one defines 1 circular matrix pore pi of a ϕM Ai area ai such as ϕM = i : its radius is set equal to: Ri = . ( Ri ∝ Ai ) Ai π

ni = 1 ϕM Ai Ri = π

€

€

€

€ Figure 2.9.bis : Visualization of the microporosity of large hydraulic radius by using only 1 “dot” of radius Ri € One can easily check that the corresponding total porosity Φ for all the n M matrix areas is: i=n M

∑ π ( Ri )

1 i=n M ϕ i=n M ∑ ϕM Ai = M ∑ Ai€= ϕM € AM AM i=0 AM i=0 In this special case, Rmean becomes just an upper bound for the actual mean hydraulic radius Ri since one can check that ⎛ ⎞ 2 ⎞ 2 2 ⎞ ⎛ ⎜ ϕM Ai ⎟  1 ⇔ ⎛⎜ϕ A ≤ π ( R mean ) ⎟⎠ ⇔ ⎜⎝ π ( Ri ) ≤ π ( Rmean ) ⎟⎠ ⇔ ( Ri ≤ Rmean ) €2 ⎟ ⎝ M i ⎜ π ( R ⎝ mean ) ⎠ ΦM = i=0

€

€

€

2

=

So the mean hydraulic radius is used both for the sake of visualization, using circular micropores of various sizes, (which is not very important, squares or any other shapes could have been used as well) and also for the calculation of the local permeability of the matrix. Because the visualization is very long for very small hydraulic radius, others graphical means could be used in the future to decouple the selection of the microporal permeability and its representation. Finally let us recall that the matrix polygonal areas are created in the computer code using a Voronoi tesselation using the vertices of the vug polygonal areas, as already explained in the previous report but an illustration is again given here (Figure 2.10) to conclude about the mathematical description of the V,F,M objects.

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(a) The vertices of the vug polygonal areas (dots colored red are the seeds of a Voronoi tessellation used to generate automatically matrix areas of low permeability around the vugs.

(b) rather small dots to visualize a microporosity of 12% and small hydraulic radius (1 simulation units)

(b) large dots to vizualize a microporosity of 12% and small hydraulic radius Rmean =50 simulation units)

€

Figure 2.10 Visualization of the microporosity in the polygonal areas around the vugs

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3.  Intrinsic  permeability,  model  and  images.   3.1.  Network  modelling   The principle of our network modeling approach was already described in our previous reports. The code had to be rewritten for the VFM model in order to account for the new structure of the network, a namely for a variable number of neighbors So the new systems of equations is based on the same principle as before, but the way of writing the system is more general. The calculation of each local flux is given by an equation similar to Eq.1b in following Table 3.1 used for capillary tubes : pi − p j 1 qi→ j = −kij lij µ(r) The length lij of the link between any two nodes i and j is calculated as the distance between the gravity centers d(Gi,Gi). For fractures, the calculation of the length is refined as d(Gi,pij)+d(pij, Gj) where pij is the intersection point. €

The permeability kij of the link between any two nodes of the network is classically calculated as the harmonic mean between the two local permeabilities (cf. Eq.5b3 Table 3.1). 2 kij = 1 1 + ki k j So, when two connected objects have the same local permeabilities (ki=kj) the connection between the two objects is assigned the same permeability (kij=ki=kj), and when have very different local permeabilities, the flow is controlled by the smallest one (When ki->ε (ε->0) € kij->ε ) As a result, there is no direct connections between the vugs, the global permeability is, as expected, mainly controlled by the smallest pores of the matrix. For the calculation of the intrinsic permeability, one equation is written for each internal conducting node i in the saturated case (cf. Eq.5b1 Tabl e3.1) where pi and pj are unknowns: p −p ∑ qij = 0 = ∑ kij j i lij j j plus one equation for each node connected to the boundaries where the pressure is imposed (cf. Eq.5b2 Table1), P1 at the top side of the simulated sample and P2 at the bottom side. j 2 =n 2 j1 =n1 €= Q2 = ∑ ki P2 − pi = Q1 = ∑ ki P1 − pi Q 2 1 li 2 li1 j =1 j =1 2

1

Once the system has been solved, the program double check that Q2 = Q1 then K is calculated as (cf. Eq.7 Table 3.1), :

€ €

18

K=

Qµ L ΔP A

One can double checked using the code that the permeability calculated for a medium saturated with oil is identical to the permeability calculated for a medium saturated with oil, € in the calculation of Q disappears with the calculation of K since the viscosity term included which is an intrinsic property of the medium defined from the flow of any Newtonian fluid. (See section 3. for non-Newtonian fluids). One can also check that the permeability value is also the same whatever the arbitrary pressure gradient imposed. Let us give an example of the matrix written by the code to solve the system of equations for a simple example with 9 nodes and 9 equations (For more than 10 equations, the matrix is not printed, and for the last example in this section we got 5403 equations, so more than 25 billions numbers in the matrix). 24.68 -6.65 0.00 0.00 0.00 -5.58 -9.70 0.00 0.00

-6.65 33.04 -7.27 0.00 0.00 0.00 -10.40 -5.74 0.00

0.00 0.00 0.00 -5.58 -9.70 0.00 -7.27 0.00 0.00 0.00 -10.40 -5.74 49.84 0.00 -19.27 0.00 0.00 -8.97 0.00 41.36 -21.77 0.00 0.00 0.00 -19.27 -21.77 61.61 0.00 0.00 0.00 0.00 0.00 0.00 43.78 -19.84 0.00 0.00 0.00 0.00 -19.84 104.65 -36.21 -8.97 0.00 0.00 0.00 -36.21 68.82 -3.20 -5.35 -7.73 0.00 0.00 -5.46

0.00 0.00 -3.20 -5.35 -7.73 0.00 0.00 -5.46 28.70

13756.47 14942.08 55663.52 71207.86 64231.45 36710.39 56994.34 24866.41 13923.13

(a) The simulated network with the centers of gravity of the porous objects are respectively colored dark grey (Vugs), black (Fractures) and light gray (Matrix) (b) Plot of the solution of the system of equations, colored red, for a pressure P1=4000 on the top yellow nodes and P2=0 on the bottom green nodes. The local fluxes are very heterogeneous and cannot be plotted altogether at the same scale, they are colored green for the input fluxes and yellow for the output fluxes Figure 3.1. Calculation of microscopic and macroscopic fluxes in a VFM network represented in the abstracted network used for the calculation of the permeability

19

“Microscopic level”: Poiseuille law in a cylindrical tube pi − p j 1 πr dP 1 (1.b) q(r) = −kij q(r) = − (1.a) lij µ(r) 8 dL µ(r) q : m 3 / s (r : m, L : m, µ : Pa.s) q : m 3 / s (r : m, L : m, µ : Pa.s) 4

where P€= H"g (2a), p = h"g (2b), P, p : Pa(kg / m.s€2 );H, h : m; ρ : kg / m 3 ;g : m / s2 “Macroscopic level”: Flux Q through a bundle of parallel capillary tubes

€ € rmax

rmax

! Q = ∫ q(r)dN(r) = ∫ q(r)n(r)dr (3.a) € rmin rmin

!

3

rmax

Q = ∑ q(ri )N(ri ) (3.b) rmin 3

Q=m /s Q=m /s “Macroscopic level”: Flux Q, Bundle of parallel capillary tubes with tortuosity

€ €

rmax

Q = ∫ q(r)n(r)τ (r)dr (4.a) rmin

€ €

rmax

Q = ∑ q(ri )N(ri )τ (ri ) (4.b) rmin

(eg. Rieu&Sposito 1991)

(eg. Li et al. 2008)

€

“Intermediate level”: Network of capillary tubes with connectivity € p j − pi few attempts: for example… very conceptual (5.b.1) ∑ qij = 0 = ∑ kij lij j j results such as Aρg Q ≈ rc2 where rc is defined as a critical path

µ

€

length imagined using percolation theory in a € 3 − D p rm 3 −1−D p r r dr = Vc … fractal pore space 3−D ∫ rm p rc € p.111… (Hunt&Ewing 2009) (5.a)

j 2 =n 2 P − pi j1 =n1 P1 − pi (5.b.2) Q = ∑ ki 2 2 = ∑ ki1 li 2 li1 j 2 =1 j1 =1 2 kij = (5.b.3) 1 1 + (Present report) ki k j

“Macroscopic level”: actual velocity € V = Q (6) AΦ V : m / s (Q : m 3 / s, A : m 2 )

€

€

“Macroscopic level”: permeability € Qµ(V ) L µ(V ) (7) K= = VΦ ΔP A ΔP K : m2

(units m,s,kg… used instead of L,T,M… for consistency checking)

Table 3.1: Main € formula for continuous&discrete calculations of the permeability. € Let us note that the formula for the continuous case on the left hand side are not commented in the present report and are given to illustrate the correspondence between the continuous and the discrete approaches.

20

3.2.  Illustrations  with  real  images   We will show in this section illustrations of our model compared to real data where for some samples where both an image and permeability experimental data are available. The data are the following ones: Numero

Profundidad (métros)

Porosidad (%)

Permeabilidad Klinkenberg (md)

Permeabilidad Aire (md)

1H 3220.19 1.40 0.00036 0.00106 2H 3220.52 3.73 0.01666 0.02317 3H 3220.84 2.39 3.73820 4.01024 4H (no image 3221.48 1.58 0.16318 0.18901 available) 5H 3221.05-3221.34 no data no data No data Table 3.2. Experimental data Formación KM, Núcleo - 2 (3220.00 - 3225.00) métros. JUNIO 2007

Figure 3.2 Experimental images Formación KM, Núcleo - 2 (3220.00 - 3225.00) métros. JUNIO

21

 Sample  1H   Figure 3.3. Sample1H: IMAGINING FRACTURES+MICROPOROSITY

(a) Simulation of a network of small fractures plus microporosity . In this example, the simulated permeability is 0.0001 mD and the porosity 0.14 to fit experimental data given in Table 3.2. It is easy to fit a network matching the given experimental permeability then to add rather impermeable microporosity to fit the experimental porosity!

(b) The links between fractures are drawn colored brown, the fractures themselves are almost invisible. We selected 200 random fractures of mean length 3000 simulation units, and mean aperture 0.02 simulation units, that respectively 87mm and 0,58 µm , since the size of the simulated sample is 29 cm (see scale on (c)). The network is purely invented to show that we can provide an equivalent€network, but let us note that the order of magnitude of the modeled fractures is plausible.

c) For a given pressure gradient (4000 on top side -0 bottom), after solving of the system, The simulated local fluxes are plotted with input flows from the bottom colored green and output flows through the top side colored yellow, using a large value for the plotting parameter scale_q since the actual values are very small. Of course many other networks could be fitted to match the experimental values!

22

Sample  2H   Figure 3.4. Sample2H:IMAGINING VUGS+MICROPOROSITY

(a) Simulation of a network of small vugs identified manually on the image plus microporosity . In this example, the simulated permeability is 0.00002 mD and the porosity 3.73 to fit experimental data given in Table 3.2. The identified vugs do not contribute vry much to the total porosity nor to the permeability It is easy to fit the microporosity volume then to fit the microporosity aperture!

€

(b) The links of the M,V network are drawn colored brown, the microporosity is almost invisible. We selected a total microporosity value ϕ = 3.205 of mean radius Rmean = 1.8 simulation units, that respectively 52,2 µm , since the size of the simulated sample is 29 cm (see scale on € (c)). The network is purely invented to show that € we can provide an equivalent network, but let us note that the order of magnitude of the modeled micropores is plausible.

c) For a given pressure gradient (4000 on top side -0 bottom), after solving of the system, The simulated local fluxes are plotted with input flows from the bottom colored green and output flows through the top side colored yellow, using a large value for the plotting parameter scale_q since the actual values are very small. Of course many other networks could be fitted to match the experimental values!

23

Sample  3H   Figure 3.5. Sample1H:IMAGINING V, F, &M (a) Simulation of a network of small vugs identified manually on the image + fractures +microporosity. The microporosity identified manually is φV = 0.9% in this example, the fracture porosity is φ F = 0.64% and the total porosity is matched by adding φ F = 0.85% The identified vugs do not contribute vry much to the € total porosity nor to the permeability, and once € permeability has been identified, the the fracture micropores hydraulic€radius is fitted as in previous examples the simulated permeability is 0.00380 mD and the porosity 2.39% to fit experimental data given in Table 3.2.

€ €

(b) The links of the V,F,M network are drawn colored brown, the microporosity is almost invisible. We selected a total microporosity value ϕ = 0.85 of mean radius RVmean = 4.7 , 100 random fractures of mean length LFmean = 2000 , and mean aperture W Fmean = 3 simulation units, that is € € respectively 136.3 µm , 58mm ,znd 87 µm , since the size of the simulated sample is 29 cm (see scale on (c)). The network is purely invented to show that we € an equivalent network, € can provide but let us note that the order of magnitude of the modeled micropores and microfractures is plausible

(c) For a given pressure gradient (4000 on top side -4000 bottom), after solving of the system, The simulated local fluxes are plotted with input flows from the bottom colored green and output flows through the top side colored yellow. Of course many other networks could be fitted to match the experimental values!

24

3.3.  SEM  Image   We describe here recent work which has been done from the following SEM images with vugs colored blue identified by other members of the project.

Figure 3.6. Original data A dedicated software CONTOUR and very important new algorithm work has been done by Thibault Rieutord to extract the polygonal vug contours in order to couple with the SIMPOR software. SIMPOR imports the files which are created using CONTOUR.

Figure 3.7. Extracted vug contours

25

Figure 3.8. Voronoi Tesselation around the vugs to crate matrix areas

Figure 3.9. Extraction of the V,M network

26

Figure 3.10. Simulated porous media with vugs a matrix areas

Figure 3.11. Estimating the permeability The value of the calculated vug porosity φV = 30% . When we select a microporosity φ M = ϕ = 12% and an hydraulic radius Rmean = 10 , that is here 14,4 µm , then the simulated conductivity is: K= 0.006mD.

€ €

€

€

27

The estimated porosity when we add the vugs is multiplied by about 2 compared to the simulation of the same microporosity without the vugs. Other images of the same series should be soon analyzed, using different assumptions for the microporosity, then compared in a statistical way.

3.4.  Other  possible  numerical  experiments   Other experiments were began but are not finished at the time of this report, we just mention them here for information. Percolation  threshold   We could investigate the theoretical precolation values for a fracture network in a way to that studied in Mourzenko, Thovert and Adler (2011). A very simple example in Figure 3.12 just illustrates how the the percolation threshold is reached when the in fracture density incrases. For small networks the variability is high and for the same density (N=100 in Figure 3.12) the network can percolate or not, that this moving to the impermeable case to the conducting case.

(N=80)

(N=100)

(N=100)

(N=100) (N=100) (N=500) Figure 3.12. Random fractures simulations. Percolation (and thus non-zero permeability) is reached by increasing the number N of fractures Lmean = 2000 Rmean = 0.05 Only a few ewamples with N= 80,100 (4 random realization), N=500 well above the threshold.

€

€

28

Pseudo  3D   The goal is to be able to introduce explicit assumptions on the 3D connectivity from a set of several planar images. class VFM_object:public polygon{ list_VFM* VFM_neighbors; int type; double aperture; int no_plan; }

Another attribute is given to the VFM_object, no_plan, which acts as a coordinate in the third dimension. Beyond the simplicity of this first modification, a lot of programming work was done to begin the development of a multiplane version for the VFM model We just give here a screencopy of what could be completed in the future.

Figure 3.13. Computer developments. Beginning the multi-plane version of the VFM model

29

 

4.  Non-­‐Newtonian  fluids  

 

4.1.  Theory   The Darcy and Poiseuille law are widely used for Newtonian fluids such as water, providing a lot of simplified assumptions (non turbulent flow under the critical Reynolds number, non deforming media, incompressible fluid, etc…) : namely the fluid viscosity is considered as a constant (depending only on the fluid, on the temperature, etc.), and µ(V ) in Table 3.1 is a constant. For a non-newtonian fluid, one looks for some kind of extensions of these laws. Namely one ! has to account for the fact that the viscosity is no more a constant but varies as a function of the applied shear stress. Neglecting some other phenomena linked to the history of shear rates, and any other more complex phenomena, let us consider only the so-called generalized non-newtonian fluids where the only difference with the a function of the shear stress. The 3 most widely used models for the variation of the viscosity of non-newtonian fluids follow, using either the expressions given as a function of the shear stress τ or the shear rate γ˙ where τ = µγ˙ : 1) The Ellis Model €

€

€

α −1 ⎞ ⎛ 1 1 ⎜ τ ⎟ where µ is the viscosity at zero shear and τ = 1+ 0 1/ 2 is the shear stress at ⎟ µ µ0 ⎜⎝ τ1/ 2 ⎠ µ which the viscosity is 0 2 € €

€

2) The Carreau model

€

where µ0 and µ∞ are limiting viscosities at high and µc = µ∞ + ( µ€0 − µ∞ ) 1+ λ2γ˙ low shear rates to get not only 1 Newtonian plateau as the Ellis model but 2 ones. µ∞ is usually very small and often neglected so this 4-parameter model becomes a 3 parameter model with only one plateau. € €

[

]

( n −1) / 2 c

3) The standard power law model

€

€

€

n −1 µ p = µ0 γ˙ ( p ) Let us not that if n p  1 the fluid is shear thinning since the viscosity tends towards infinity µ p →∞ when the shear rate γ˙ →0 , whereas if n p  1 it is shear thickening since µ p →0 when γ˙ →0 .

€ Many other such so-called constitutive models exist (Malek J.2011). The three previous € € models are the more widely used and € are very well described by Myers (2005), but they have € been used for years by other researchers and they are still used in more recent studies (eg. Bakhoff,2005; Sochi, 2007). Myers is quoted as one of the more important references in Li et al. 2008. Li and co-workers (Professor Boming’s team) are interested in fractal porous media and they write in their introduction that “ The flow of non-Newtonian fluid through porous

30

media has been the subject of extensive investigations for years because of the wide variety of applications, such as in soil mechanics, groundwater hydrology, industrial filtration of polymer solutions and slurries, and the movement of aqueous polymer solutions through oil reservoirs in secondary oil operations. Non-Newtonian fluids are well described by the power-law model, Ellis model, … ». I agree with this introduction but I disagree with Li et al. 2008 conclusions about their main equation (30) about actual velocity V of non-newtonian fluids in fractal porous media.

Li et al. 2008 write that: “There is no empirical constant and the tortuosity of capillaries is taken into account in the present fractal model Equation (30). Moreover, Equation (30) relates the average velocity to the microstructural parameters and the fractal dimensions, and therefore Equation (30) is more fundamental for understanding the Ellis fluid flow in porous media. Note that the average flow velocity by Equation (30) does not have any empirical constant ». Because their results were used in Turcio et al. (2013) for petroleum engineering applications, where it is made clear that several fractal parameters of the Li&Yu model were not easy to measure, my following contribution aims first at providing additional arguments to show that I am not convinced by Li et al. 2008 analysis for the following reason: very long mathematical calculations on very simple models are done to upscale from the pore scale to the macroscopic scale, whereas their main result appears to be a non-explicit fit obtained using the data measured by Park at a macroscopic scale.

Figure 4.1. Li et al. 2008’s Figure 1 showing the good fit obtained with their fractal model (Equation 30) compared to a classical fitted macroscopic model

31

€

Li et al. 2008 consider that their model is validated because « there is good agreement between the present fractal model predictions and Park’s experimental data » (see their Figure 1 in present Figure 4.1.), and there is no other validation with other experimental data in the paper. I consider that their model is mainly based on the 3 microstructural parameters denoted α, τ1/ 2 , µ0 in their equation (30), that is those given by Park in the original table shown in Figure 3.2, and which were used to fit the original data using the Ellis model for the viscosity). All the other parameters are not necessary to obtain the good macroscopic fit presented. Since there are many uncertainties in the measure of all the other parameters, so fitting is very easy and do not prove the validity of the fractal model. This is also why other macroscopic models can be used, as shown in Figure 4.1 I show in the following subsections: -

that I can also reproduce the same Park’s old experimental data (1972), which are still used as reference data by several other researchers (Bakhoff, 2005; Sochi, 2007), by explicit inversion in very simple way. that the mean effective viscosity derived from the experimental data is well described by a power-law of the mean actual velocity V that the value of effective viscosity depends only on the value of the permeability of the porous media. Despite the title of their paper, Li et al. 2008 do not estimate the permeability but they derive a theoretical expression for the effective permeability and they use the permeability estimation done by Balholf and Thompson (2006) using simulation of pore networks in artificial sets of packed spherical particles (see Figure 4.1)

3.2.  Park’s  data   Park’s PhD experiment (1972) was done for flow in packed beds of shear-thinning fluids that can be characterized as Ellis fluids in a limited shear-rate range. They were obtained using polyacrylamide solutions with different weight concentrations (Figure 4.2). The nonNewtonian fluid is pumped at a constant rate through a porous medium and the resulting pressure drop is measured. The porous media a packed bed of spherical particles of average diameter 0.1621 cm and total porosity of 42,3%. As mentioned before, the permeability was not measured but estimated by Balhoff and Thompson (2006). The experimental data can be illustrated in a simple plot of Darcy velocity versus pressure gradient, where the velocity increases nonlinearly with the pressure gradient (Figure 4.2).

Figure 4.2. (a) Original Park data (Table 5.2, reported from Sochi, 2007)

32

Figure 4.2.(b) Original Park data (reported from Sochi, 2007) : Plot of the observed velocities versus the pressure gradients. As Li et al. 2008 did, we worked only with 1 of the 4 curves, the one obtained for the polyacrylamide solution at 0.50% concentration. We first reproduced the original data in an excel file as shown on Figure 4.3 (red data points reproduced from the available graphics). If the fluid was Newtonian, the fluid velocity would be proportional to the pressure gradient (see Table 3.1): ⎛ −dP ⎞ Vnewtonian = a⎜ ⎟ ⎝ dL0 ⎠ For a non-newtonian fluid, as shown with the expriment, the fluid velocity varies in a nonlinear way with the pressure gradient, and we looked for a statistical fit of the data that we assumed to be in the form : € ⎛ −dP ⎞ ⎛ −dP ⎞α Vnon−newtonian = a⎜ ⎟ + b⎜ ⎟ ⎝ dL0 ⎠ ⎝ dL0 ⎠ The blue line on Figure 4.3 used a = 10−6 , b = 8.10−9 , α = 2.47 is just to get a reasonable approximation of data to be used for extrapolation in the following numerical experiments where any dP pressure gradient can be used. The form of the function €⎛ −dP ⎞ € € Vnon−newtonian = f ⎜ in section 4.3. ⎟€will be discussed ⎝ dL0 ⎠ €

€

33

Figure 4.3. Approximating Park’s experimental data using the data provided in Figure 3.1. ⎛ ⎞ ⎛ ⎞ 2.47 −6 −dP −9 −dP Blue line: 10 ⎜ ⎟ + 8.10 ⎜ ⎟ ⎝ dL0 ⎠ ⎝ dL0 ⎠

3.3.  Numerical  experiment   One can see in Figure 4.4 a€screen copy of the numerical experiments, carried out an explicit inverse method whereas I consider that Li et al. 2008 used an implicit inverse method which is not the result of their previous theoretical calculations. This explicit inverse method works as follows For each given imposed pressure gradient dP / dL0 , one calculates successively : €

1.

€

€

Vexp : The actual velocity Vexp is known from experiment (or one calculated the extrapolated

⎛ −dP ⎞ ⎛ −dP ⎞α values for any gradient pressure using Vnon−newtonian = a⎜ ⎟ + b⎜ ⎟ ). ⎝ dL0 ⎠ ⎝ dL0 ⎠ 2. QDarcy : € the Darcy flow QDarcy is calculated by network simulations using any constant fluid viscosity µref as a starting value € for a first simulation denoted sim1. From Equation 1 in Table 3.1 and from the summation of the local fluxes one derives the inverse ξ € proportionality between QDarcy and µref , thus QDarcy = . µ ref € € 3. Vsim1 :

€ €

€

€

34

Then from Eq.6 Table 3.1 one has : Vsim1 =

QDarcy AΦ

=

ξ ξ where = µref Vsim1 AΦ AΦµref

4. µeff : To get Vexp simulated values instead of the first Vsim1 value obtained at step 1 using the viscosity value µ , one € just has to modify the viscosity€according to : ref

€

€

€

Vsim1 µeff = µref Vexp € € 5. Vsim2 : € One can double check that, using µeff in the simulation, Vexp Vexp ξ ξ that is Vsim2 = Vexp Vsim2 = = = Vsim1 AΦµeff AΦµref Vsim1 Vsim1

€ Only step2 involves simulations. In the art of code given below, SIMEXPERIMENTVISCOSITY implements the loop € for dP / dL0 varying between 0 and € 1000 Pa/cm (the associated with an interface button, the SimVelocityVersusGradP button…). This function calls the FULL_WITH_OIL function with deals with all the issues of simulated multiscale units and real units. This function in turn calls the CONDUCTIVITE function which calculates the permeability of a network of€pores. The code of this function is given in the appendix. void SIMEXPERIMENTVISCOSITY() { SimVelocityDH=1; int i,imax=50; int DeltapL0max=1000; for(i=1;i