A comparative study of the tensor and upscaling methods for evaluating permeability in fractured reservoirs

Transcription

1 IAMG Proceedings F1203 A comparative study of the tensor and upscaling methods for evaluating permeability in fractured reservoirs R. FLÓRIO 1* and J. ALMEIDA 2 1 Departamento de Ciências da Terra Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Portugal 2 GeoBioTec, Departamento de Ciências da Terra Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Portugal *presenting author Abstract This work presents a comparative study of the Oda tensor and upscaling methods for evaluating permeability in fractured fields. This comparison enables the results obtained using the Oda method to be calibrated with those obtained using the upscaling method, avoiding the drawbacks of the Oda method mentioned in the literature. As the upscaling method is not an alternative for making estimates across the entire reservoir, the approach taken is to run both methods for only a small number of blocks, and then to evaluate the relationship between the results given by the two methods for several increments of fractures. An example with data conditional to synthetic statistics of orientation, density, and aperture is presented. Fractures are simulated in 3D within a grid block and the same set of fractures is evaluated with the two methods. The results confirm the bias reported in the literature and show that the upscaling method can be used for calibrating permeability values given by the tensor method. 1 Introduction Fractured reservoirs are typically geological structures where fluid flow occurs mainly within discontinuities, rather than rock pores (Nelson, 1985; Almeida et al., 2002; Wang and Park, 2002; Vitel 2007, Flório, 2015). Discontinuities exist at all scales, from large sizes of more than one hundred metres (detectable by seismic surveys), to several metres (detectable by FMI logs at well locations) or centimetres or less (only detectable in cores). This replication of discontinuities at all scales makes the characterization of fractured reservoirs a very complex issue and a hot topic of current research. The characterization of a fractured reservoir prior to flow simulation requires the allocation of equivalent permeability values for all blocks, and this evaluation is dependent on the 3D geometry, apertures, and number of fractures. The tensor method is commonly used to assess the equivalent permeability of fractured blocks (Oda, 1985). The method is efficient even for highly fractured blocks with several million fractures, and can be applied to the entire reservoir at once. For the same aperture, the equivalent permeability obtained is proportional to the area or intensity of the fractures. Due to this fact, the literature mentions drawbacks, which are overestimation of the permeability of sparsely fractured blocks and underestimation of the permeability of highly fractured blocks. If we imagine a block with one small fracture, not face to face, within a rock matrix with no permeability, the equivalent permeability of this block should be zero, however the tensor method gives a non-zero value for permeability. If we increase the number of fractures within the same block, the equivalent tensor permeability increases proportionally according to the area of fractures. In reality, the equivalent permeability remains close to zero for low fracture intensities. Increasing the number of fractures, after a critical intensity at which the fractures all intersect with each other, the equivalent permeability increases quickly in a non-linear relationship with fracture intensity. This conceptual non-relationship between fracture intensity and equivalent permeability (figure 1) is not captured by the tensor method as proposed by Oda, An alternative to the Oda method is the upscaling method (Wang et al, 2008). This involves partitioning the reservoir block into a mesh of microblocks, identifying microblocks intersected by fractures, assigning permeability values to each microblock, and finally upscaling the permeability. As the microblocks are smaller than the original block, this method gives more accurate results than the ISBN (DVD) 465

2 A comparative study of the tensor and upscaling methods for evaluating... tensor method, but it is less efficient and therefore can be applied to only a small number of blocks. In theory, if the partition of microblocks is small enough, the upscaling method captures the non-linear relationship between fracture intensity and equivalent permeability. However, as the upscaling method is computationally inefficient and can be only used in a limited number of blocks, our proposal is to use the upscaling method only for the purpose of calibrating the tensor method. Figure 1: Conceptual relationship between the intensity of fractures and equivalent permeability. 2 Methodology In order to compare the results of the two methods, tensor and upscaling of microblocks, it was necessary to develop an efficient algorithm for simulation of fractures as polygonal objects. In the second step, each simulated set of fractures is evaluated for equivalent permeability using the two methods. Software for simulation of fractures and evaluation of permeability (FROM3D-K) was written in C specially for this work. For simulation of fractures, FTRIAN software developed by Almeida and Barbosa (2008) was used as a starting point. Move software from Midland Valley was used for 3D visualization of the simulated fractures and for validation of permeability calculations. The core approach of the work can be summarized in the following four main steps: 1) Simulation of a set of fractures with intensity P32 (fracturing intensity index equal to the area of fracture per unit volume) within a volume V; 2) Calculation of the equivalent permeability tensor of the set of fractures previously simulated in step 1) by the tensor method proposed by Oda; 3) Calculation of the equivalent permeability in the orthogonal directions of the set of fractures previously simulated in step 1) by the upscaling method; 4) Comparison of the results. 2.1 Fracture Simulation The FROM3D-K program generates fractures within a region conditional to cumulative distribution functions (CDFs) of orientation, aperture and size, and fractures are represented by polygonal squares. FROM3D-K considers two cubic regions, one larger for the positioning of the centre of gravity of the fractures to be generated and a smaller one inside the first, to evaluate properties such as density, intensity and porosity of fractures and equivalent permeability. These two regions are intended to simulate the interaction between adjacent blocks in a reservoir grid of blocks and the occurrence of 466

3 IAMG Proceedings F1203 fractures crossing several blocks. In order to evaluate fractures only within the inner cube, fractures are intersected with the inner cube and the outer portions are removed. Fractures are identified by a list of vertex coordinates and properties such as orientation (azimuth and tilt), area, and aperture. The generation of fractures by polygons involves the following steps: 1) Two cubes, an outer cube (e.g. 7m x 7m x 7m) and an inner evaluation cube (e.g. 5m x 5m x 5m) are initiated; 2) Simulation of fractures, following the steps for each fracture (see figure 2): 2.1) Generation of a point randomly located within the outer cube as the centre of gravity of the fracture (x cg, y cg, z cg ); 2.2) Simulation of the fracture set by Monte Carlo; 2.3) Simulation of aperture, area, and angles (azimuth and tilt), conditioned to previously defined objective CDFs, again by Monte Carlo. Conversion of area to a distance (d) between the centre of gravity and each square vertex of the fracture (x i, y i, z i ). 2.4) Starting from the centre of gravity point (x cg, y cg, z cg ), compute the vector normal to the plane ( ) defined by the azimuth and tilt simulated angles; 2.5) Starting from the centre of gravity (x cg, y cg, z cg ), place a vertex (x u, y u, z u ) at a distance d from the centre of gravity according to the previously simulated azimuth and tilt. Rotate vertex (x u, y u, z u ) around the normal vector ( ) from a random angle between 0 and 360 and obtain the first vertex of the square (x 1, y 1, z 1 ). This rotation allows the squares to have any orientation. 2.6) Perform three consecutive 90 rotations of the segment joining the points (x cg, y cg, z cg ) and (x 1, y 1, z 1 ) around the normal to obtain the remaining three vertices of the square / fracture (x 2, y 2, z 2 ), (x 3, y 3, z 3 ) and (x 4, y 4, z 4 ). After these consecutive rotations, confirm that (x 4, y 4, z 4 ) = (x 1, y 1, z 1 ). 2.7) Save fracture information and proceed to the next fracture until an objective intensity per area or number of fractures is reached. Figure 2: Illustration of the fracture generation process, positioning of four vertices plus intersection with the evaluation cube. 3) Consider the minimum and maximum coordinates of the inner cube as (x min, y min, z min ) and (x max, y max, z max ). Intersect the polygonal fractures with the inner / evaluation cube, reshape fractures and recalculate areas. 3.1) Check if the line segments (x 1, y 1, z 1 ) - (x 2, y 2, z 2 ), (x 2, y 2, z 2 ) - (x 3, y 3, z 3 ), (x 3, y 3, z 3 ) - (x 4, y 4, z 4 ) and (x 4, y 4, z 4 ) - (x 1, y 1, z 1 ) intersect the planes y = y min and y = y max. If they do, add the intersection points to the initial square polygon. The fracture maintains its square shape but new vertices are added. 3.2) Now, remove all vertices with y i > y max or y i < y min. ISBN (DVD) 467

4 A comparative study of the tensor and upscaling methods for evaluating ) Apply steps 3.1 and 3.2 to the remaining plans x = x min and x = x max and finally to z = z min and z = z max. The resulting polygonal fractures are new polygons each with between three and eight sides, according to the positioning of the fracture and the inner cube. 3.1) Recalculate the area of the fracture, i.e. calculate the area of a 3D polygon defined by the general sequence of coordinates (x 0, y 0, z 0 ), (x 1, y 1, z 1 ),... (x n, y n, z n ) = (x 0, y 0, z 0 ) of the p vertices V i. If n x, n y, n z are the normal vector components of the plane polygon, then the area of the fracture can be calculated by (Goldman, 1994): with ; ; 2.2 Evaluation of the equivalent permeability Permeability is a direct measure of the ability of a fluid to move through a rock. Fluid movements in a porous medium are governed and quantified by Darcy s law; the cubic law explains the flow through the fractures. The flow rate Q of an incompressible fluid in a laminar flow, through a block with face area A, width W, in a fracture is given by: If A f is the fracture area with A f = W.e, the flow rate can be calculated by: By analogy of the equations (the cubic law and Darcy s Law), the permeability of a single fracture (k f ) is given by. If ф if the porosity of the fracture, the equivalent permeability of a rock block with a fracture and null permeability matrix is, in the fracture direction: ( ) In determining the equivalent permeability of a fractured bloc, it is considered that the rock matrix has very low or zero permeability. The flow occurs solely by fractures, which are generalized to several directions if there are different sets of intersecting fractures. No flow perpendicular to the fractures is considered. If a block is intersected by several fractures, the equivalent permeability of the block k B is equal to the sum of individual contributions k fi Oda method The Oda method utilizes the sum of the tensor components of each fracture weighted by a connectivity factor (Ψ) for calculation of the permeability tensor of a fractured block. This method assumes that flow only occurs in the direction of the fracture and is proportional to the area of the fracture. Estimation of the permeability tensor starts with a 3D network of fractures and a grid of reservoir blocks. For each reservoir block of volume V, N fractures are simulated and are represented by the components of the normal vector -, and, A a area and k a individual permeability. For all fractures within a block, the components of the equivalent permeability tensor can be calculated by (Oda, 1985): 468

5 IAMG Proceedings And: [ ] F1203 [ ( ) ] permeability tensor component scalar permeability of the matrix, value very low or null connectivity factor, considered equal to one in this work Upscaling method The equivalent permeability of a volume V with N fractures simulated as polygons can be determined by microblock evaluation plus upscaling (Almeida et al, 1996; Almeida et al, 2002; Vitel 2007). After simulation of fractures, a large partition of microblocks is initiated within the volume V. The greater the number of microblocks, theoretically the better the results, but calculations take more time. For an initial 5m x 5m x 5m block, it is reasonable to initiate a discretization of up to 1,000,000 microblocks, which corresponds to a unit volume of 5cm x 5cm x 5cm. Subsequently, microblocks intersected by fractures are identified and the number of fractures intersected is saved (, i= 1,...a; j= 1,...b; k= 1,...c). The permeability of each microblock kijk, has two components, permeability from the matrix Kmijk (a low or null value) and the contribution from the fractures kfijk = f(nfijk). The permeability of microblocks without fractures is the permeability of the matrix. Figure 3 illustrates the conversion procedure, from vector to raster, of intersecting microblocks with a simulated fracture network. Figure 3: Illustration of conversion step of polygon fractures to microblocks: (left) 10,000 simulated fractures; (right) permeability values of each microblock. Two issues remain: (1) to relate the number of fractures with the permeability of the microblock; (2) to pass microblock permeability values for the equivalent permeability of the original block,, and. To address the first question, a set of tests to determine the average fracture area that would be contained in a microblock with the condition of the fractures simulated in the surrounding area were performed. The average fracture area plus aperture results in an equivalent isotropic permeability value. Concerning the second question, the permeability of the microblocks was averaged to the large block by the classic approach of combining arithmetic and harmonic means. 3 Case study The case study begins with simulation of sets of fractures using the procedure explained in 2.1. The fractures were simulated with random orientations, but to simplify the demonstration presented in this ISBN (DVD) 469

6 A comparative study of the tensor and upscaling methods for evaluating... paper both area and aperture sizes were taken to be constant (1m 2 for area and 0.1 mm for aperture). It is important to note that FROM3D-K is able to import fracture size measurements as CDFs, and in this case values for each fracture are generated by Monte Carlo. For each realization of fractures, equivalent permeability was computed for both Oda tensor and upscaling methods, enabling comparison of solutions for the same realization of fractures. The permeability of the matrix was considered as darcy. The solutions were compared for several incremental numbers of fractures. In order to check the realism of the solutions obtained by FROM3D- K, a similar simulation of fractures and evaluation of permeability was performed in Move software; however, this software only provides a solution for the Oda method. 3.1 Simulation of fractures The size of the outer cube for the generation of fractures is 7m and the size of the inner cube for evaluation of permeability is 5m (125 m 3 ). The maximum number of simulated fractures was 200,000, but partial evaluations were made after 1,000, 2,000, 5,000, 7,000, 10,000, 20,000, 50,000 and 100,000 fractures. During the simulation of fractures with FROM3D-F, P10 (linear density of fracturing), P32 (fracture area per volume unit or fracture intensity), and P33 (fracture volume per volume unit or porosity) are reported. In assessing P10, 25 sample scanlines randomly positioned but parallel to the coordinate axes were used. Table 1 presents a summary of measurements and the number of fractures for a single realization. The values confirm that all relationships are linear. No. of fractures (#) Total fracture area (m 2 ) Area inside the inner cube (m 2 ) P10 (#/m) P32 (m 2 /m 3 ) P33 (m 3 /m 3 ) Porosity (%) 1,000 1, ,000 2, ,000 5,000 1, ,000 7,000 2, ,000 10,000 3, ,000 20,000 7, ,000 50,000 18, , ,000 36, , ,000 73, Table 1: Fracture measurements P10, P32, P33 and porosity as a function of the number of fractures and fracture area. The ratio between the total fracture area and the area inside the inner cube is similar to the relationship between the volumes of the outer cube and the inner cube = 7 3 /5 3 = 2, Equivalent permeability from Oda and upscaling method The simulated fractures were evaluated using both Oda and upscaling methods. For the upscaling method, the contribution of each intersection of a fracture with a microblock was considered isotropic and constant. The constant used was evaluated from the average area of an intersection of any fracture with a microblock and the volume of the microblock. Table 2 and figure 4 summarize the results obtained by the Oda method; tables 3 and 4 and figures 5 and 6 summarize the results obtained by the upscaling method. The values presented in table 3 and figure 5 display a power relationship between microblock size and porosity and permeability, meaning that if the microblock is smaller, the contribution of a fracture for each microblock is much smaller, but in compensation there are many more intersected blocks. 470

7 IAMG Proceedings F1203 No. of fractures (#) Permeability components (darcy) k xx k xy k xz k yy k yz k zz k max k min Anisotropy k max/ k min 1, , , , , , , , , Table 2: Results provided by the Oda method for the set of simulated fractures. Figure 4: Maximum and minimum values of the permeability tensor vs P32. The points in red and blue represent the results obtained by Move software. Side size (m) Microblock size Face area (m 2 ) Volume (m 3 ) Average area of the fractures that intersect the microblock (m 2 ) Maximum area of the fractures that intersect the microblock (m 2 ) Porosity for aperture of 0.1 mm (%) Equivalent permeability of a microblock with one fracture with the average area (darcy) Table 3: Results of evaluation tests for calculating the individual contribution of each fracture intersection with a microblock to the equivalent permeability of the microblock. A microblock size of 0.05m was used for all equivalent permeability evaluations. ISBN (DVD) 471

8 A comparative study of the tensor and upscaling methods for evaluating... Figure 5: Representation of the porosity and permeability values of a microblock intersected by a fracture of area equal to the average area of intersection in relation to the size of the microblock (log/log scale). The relationships are fitted by power functions. No. of Upscaling method Oda method fractures k x k y k z k máx k min Anisotropy K máx K min (#) (darcy) k máx/ k min (darcy) 1, , , , , , , , , Table 4: Equivalent permeability obtained by the upscaling method and comparison with results from the Oda method. Figure 6: Comparison of equivalent permeability obtained by the Oda and upscaling methods depending on the number of simulated fractures. 472

9 IAMG Proceedings F Discussion of results The three components K xx, k yy and k zz of equivalent permeability provided by the Oda method are all linearly proportional to the intensity of fractures P32 because the increase in permeability results from the sum of the individual contributions of each fracture, which in turn depend on the area and aperture assumed to be constant during the simulation. The three components are almost isotropic, which is in agreement with a random network of fractures (without preferential orientations). Solutions provided by Move software for two specific P32 are similar or equal to those obtained by FROM3D-K (blue and red dots on the graph in Figure 4). The Oda method is computationally very fast; the larger simulations of the permeability evaluation with 200,000 fractures were performed in a few seconds on an Intel i7 laptop computer. Making use of the upscaling method, the three components K xx, K yy and K zz are not linearly proportional to the number of fractures or fracture intensity, particularly when a small number of fractures are considered. This is in accordance with the conceptual expectation mentioned above: if the intensity of fractures is small (figure 5 shows a critical number of fractures close to 25,000), the fractures are not interconnected face to face, the higher permeability values identified in a few microblocks do not have continuity, and therefore upscaling gives very low values, close to that considered to the matrix. As fracture intensity increases (above 25,000 fractures), there are more and more intersected microblocks, and for a critical number of fractures the permeability increases very quickly, meaning that almost all fractures became connected to some other fracture. After a more rapidly increasing step, the new fractures (above 50,000 fractures) have a linear contribution. Despite the better agreement with the conceptual relationship between fracture intensity and permeability, the upscaling of microblocks is a computationally very slow method, evaluations of permeability taking several minutes, and therefore this method can only be used for calibration procedures. 4 Summary and final remarks This work reviewed computer-based calculation of equivalent permeability of a 3D fracture network, with particular application to oil reservoirs. For this purpose, a computer application, FROM3D-K, was developed and tested to simulate fractures as square polygons and equivalent permeability tensors were evaluated by the Oda and upscaling methods. The solutions provided by this new software were validated by comparison to those obtained with Move for the same intensity of fractures. Results with the Oda method from both programs are similar, and those provided by upscaling are in accordance with the conceptual expectation of results. Also, the time taken to simulate fractures and to evaluate them by the Oda method, even for a high number of fractures as 200,000 within a 125 m 3 block, is just a few seconds, but the same task for upscaling takes several minutes. This demonstrates the possibility of using the upscaling method for calibration of the Oda results. For instance, suppose that we have a reservoir with three sets of fractures and the intensity of fractures varies between blocks as usual. If we split the intensity of fractures of each set into three cutoffs, we have 27 scenarios to evaluate the relationship between fractures and permeability, which is feasible by the upscaling method. Subsequently, the evaluation can be run for the entire reservoir by fracture simulation and Oda evaluation, but after this evaluation, the value obtained should be adjusted according to the upscaling solutions. This is easy to do, and the results will certainly be more realistic. Acknowledgements This work is a contribution to the Project UID/GEO/04035/2013 funded by FCT-Fundação para a Ciência e a Tecnologia, in Portugal. References Almeida, J., Soares, A., Pereira, M., Daltaban, S. (1996) Upscaling of Permeability: Implementation of a Conditional Approach to Improve the Performance in Flow Simulation. SPE paper 35490, Proceedings of the 1st European 3D Reservoir Modelling Conference, Norway. ISBN (DVD) 473

10 A comparative study of the tensor and upscaling methods for evaluating... Almeida, J., Lopes, M., Santos, J. (2002) Equivalent permeability derived from a fractured system. Proceedings of the 8th Annual Conference of the IAMG, Almeida, J.A., Barbosa, S. (2008) 3D stochastic simulation of fracture networks conditioned both to field observations and a linear fracture density. Proceedings of the 8th International Geostatistics Congress, vol 1, Flório, R. (2015) Estimação da permeabilidade em reservatórios fraturados - Comparação entre o método do tensor e o método de upscaling, MSc thesis, FCT Universidade Nova de Lisboa (in Portuguese). Nelson, R. (1985) Geological Analysis of Naturally Fractured Reservoirs. Gulf Publishing Company, Houston, Texas, EUA. Oda, M. (1985) Permeability tensor for discontinuous rock masses, Geotechnique 35(4), Park, Y., Sung, W. (2000) Estimation of effective permeability for a naturally fractured reservoir. Geosystem Engineering, 3(1) Vitel, S. (2007) Méthodes de discrétization et de changement d échelle pour les reservoirs fractures 3D. Tese de Doutoramento, École Nationale Supérieure de Géologie, 180p. Wang, J., Park, H. (2002) Fluid Permeability of Sedimentary Rocks in a Complete Stress-Strain Process, Engineering Geology, 63, Wang, H., Forster, C., Deo, M. (2008) Simulating naturally fractured reservoirs: comparing discrete fracture network models to the upscaled equivalents, Annual Convention, San Antonio, Texas. Warren, J.E., Root, P.J. (1963) The behavior of naturally fractured reservoirs. In Pittsburgh, PA.: Gulf Research & Development CO.,