Percolation theory in research of oil-reservoir rocks

VIETNAM JOURNAL Petro ietnam An Official Publication of The Vietnam National Oil and Gas Group Vol 10 - 2009 ISSN-0866-854X Percolation theory in research Comprehensive CO2 EOR study - Study on Applicability of CO2 EOR to Rang Dong field PETROVIETNAM JOURNAL IS PUBLISHED MONTHLY BY VIETNAM NATIONAL OILAND GAS GROUP Contents Editor - in - chief Dr.Sc. Phung Dinh Thuc Deputy Editor-in-chief Dr. Nguyen Van Minh Dr. Vu Van Vien Dr. Phan Tien Vien Members of the Editorial Board Eng. Vu Thi Chon Dr. Hoang Ngoc Dang Dr. Nguyen Anh Duc BSc. Vu Xuan Lung Dr. Hoang Quy Eng. Hoang Van Thach Dr. Phan Ngoc Trung Dr. Le Xuan Ve Secretaries of Editorial board Dr. Pham Duong MSc. Nguyen Van Tuan BSc. Vu Van Huan Editorial office D4A, Thanh Cong Collective zone, Ba Dinh District, Ha Noi Tel: (84.04) 37727108 Fax: (84.04) 37727107 Mobile: 0988567489 Email: tapchidk@vpi.pvn.vn 01Percolation theory in research of oil-reservoir rocks Distribution rule of lower Miocene sandstone in Cuu Long basin 18 Determination of fractured basement permeability in White Tiger oil field from well log data by artificial neural network system using zone permeability as desired output 24 Comprehensive CO2 EOR study - Study on Applicability of CO2 EOR to Rang Dong field 34 Novel surfactans for high temperature, high salinity emhanced oil recovery applications Quasi-dynamic and dynamic random analysis of mooring system of FPSO installed at White-Tiger field using hydrostar and ariane-3D softwares Prediction of aquatic organism impact on rig submerged structures of oil and gas field At Cuu Long basin Designed by Le Hong Van Publishing Licences No. 170/GP - BVHTT dated 24/04/2001; No. 20/GP - S§BS 01, dated 01/07/2008 THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 1 petroleum EXPLORATION & PRODUCTION Percolation theory in research of oil-reservoir rocks Ass. Prof. Dr. Nguyen Van Phon Hanoi University of Mining and Geology Abstract Following the articles about fractal geometry in the research of oil-reservoir rocks [1, 2], in this article, the author will introduce the application of percolation theory in researching the permeability process of fluid in void space in general, and fractured rock in particular. Percolation theory is a mathematical method which has been introduced since the early 1950s, and it has been applied widely in social and human sciences, and technological sciences since 1970s. Through this work, the author would like to suggest applying the percolation theory in researching the layers of oil-reservoir rock, based on the similarity between geometrical forms of percolation process and physical nature of permeability process of fluid in void space. In the final part of this work, the author proposes the procedure of calculating the permeability in fractured rocks according to well-log datas, based on applica-tion of percolation theory. Introduction In reality, the reservoir rock space is a very complex metamerism; however when caculating according to the common way, in many cases, we consider the void structure in the rocks as similar fractal, and use suitable statistical approximate for-mula to demonstrate the space in form of effective homogene. When researching the layers, we take the rock samples from one layer with different col-lector parameter. To get a parameter value (grain density, porosity, permeability, saturation etc.) of a researched object, we calculate the average value of parameters measured from samples of the same object. Therefore, the real space is inhomogeneous (metamerism) when we consider it in a small scale (core sample), however we consider it homogene-nous in large scale (formation, layer) with an aver-age value according to a way of calculation. For example: In a volume V, with distribution of parameter values Xi we can get the average value Xi of the effective space according to: , or if the values Xi are distributed standard. That way of calculation will not be suitable when there is a strong inhomogene in the research-ing space as in the case of fractured basement of Cuu Long basin. Percolation Theory will help much in calcula-tion of permeability characteristic as an accidental process in complex structures. This theory was introduced more than half of centuries ago, and has been applied widely and developed strongly since middle 1970s in many fields: Matter formation, material technology, transport and forest-fire protec-tion, etc. In this series of articles, the author only would like to introduce the application of percolation theory in reasearching the percolation process (per-meation, disffusion) of fluid in void space with com-plex structure. Introduction to Percolation Theory In this writing, the meaning of the term “perco-lation” is only limited within the permeation or the penertration of the fluid into the solid matters with voids. When percolating into solid objects, the fluid penetrates into sites which has capability of contain-ing fluid or it flows in bonds, capillary segments con-necting the sites in the space. Sites, bonds and types of percolation Starting from simple cells, for example net of squares (Figures 2a). Cells with black round spot are called reservoir sites, white cells are called empty sites (no reservoir). If we call p the probabili- PETROVIETNAM JOURNAL VOL 10/2009 1 THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 2 petroleum EXPLORATION & PRODUCTION ty of a reservoir site in the net, the probability of an empty site will be (1 - p). Squares with shared bor-der are called contiguous sites, squares with shared angular vertex are called adjacent sites. The fluid can only penertrate from one cell in the net to the contiguous cell (if that is a reservoir site), it can not penertrate to a adjacent cell. In a net with 2 or more contiguous reservoir cells, these cells form a clus-ter. That way of fluid penertration is called site per-colation (Figure 2a). If all the squares cells are reservoir, a channel allowing a connection between two contigous cells is called a bond. Bond is a conduit allowing the fluid to penertrate into the space, it is also a conduit between 2 contiguous reservoir sites. If we call p the probability allowing 2 contiguous sites to connect with each other through a bond, (1 - p) is the prob- Table 1 The Figure 2a shows the layers of reservoir, the white cells are solid rocks, with no capability of fluid containing, cells with round black spot are void space that can contain fluid, then the probability p is considered the common void ratio of the rock. If cells (sites) are connected with each other through ability ensuring no connection between them a bond, the propotion of void connected are called (clogged, disconnected bond). When there are 2 or more bonds connect contiguous sites continuously, they form bond group (Figure 2b). That way of fluid penertration is called bond percolation. In space with voids such as oil-reservoir rock, these groups are sites connected with each other through a bond. Therefore the percolation in oil-reservoir has the characteristisc of both site percolation and bond percolation. Percolation threshold and unlimited group For low value p, there are only groups with dif-ferent sizes. When p increases the number of reservoir sites or the number of bonds also increases, creating a chance for groups in the net to increase their size. If p continues to increase, the groups also grow gradually, and they can inte-grate to each other through a common bond to form a bigger group. Until reaching an ultimate value p = pc, the big groups will become unlimited-size group and the ultimate probability pc is called percolation threshold. The percolation threshold pc is an ultimate probability enabling an unlimited group to form in a large net. With p > pc unlimited group are enlarged more and more, extend form margin to margin (in 2D) or from face to face (in 3D) of a large net. With p < pc there is no unlimit-ed group in the net. Percolation threshold pc depend on type of cell (square, triangle, hexagonal, etc.), number of dimension and type of percolation. Value of perco-lation threshold pc stated in Table 1 is the calcula-tion result of (2003) with different types of cell net. open void ratio or connection void ratio P. Then P is the probability ensuring that any site or bond belonging to a largest group, P ≤ p. In layers of reservoir, the value P determines the permeability of the space. From above: When p < pc, there is only fluid in connection group with small size. In that situation, if a well is designed to put at any site, it can easily penertrate into a small group, the exploiting capaci-ty of this well will decrease rapidly. To get much product, and long-term stable exploiting capacity, the reservoir layer with p > pc should be chosen to put the exploiting well, and the well has to pener-trate an unlimited group. A new problem is raised here: With the proba-bility p in the square net, how we calculate the aver-age size (average number of sites and bonds) of the group and the proportion of sites belong to unlimit-ed group P? The quantity of groups, average size and space of group correlation In net of squares, identify the probability so that a random cell (site) is a group which has the mini-mum size s = 1, which means that it is a reservoir site and independently standing among nonreser-voir sites. The reservoir site has its own probability, and around it is 4 adjacent nonreservoir sites with a probability of (1-p) for each site. These five sites cells (sites) are independent so they are cooperat-ed in terms of probability by the product of probabil-ity: n1 = p(1 - p)4. For the case of 2 reservoir sites standing 2 PETROVIETNAM JOURNAL VOL 10/2009 THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 3 PETROVIETNAM among 6 adjacent nonreservoir sites, these two Then, the ratios are equivalent to one another. This sites can be arranged in vertical or horizontal direc-tion; therefore, n2 = 2p2(1 - p)6. It is easy to conclude that in a net of squares includes 3 aligned sites, there will be n3 = 2p3(1 - p)8, for around 3 aligned sites are 8 nonreservoir sites with the probability of (1 - p) for each site. We call n1, n2, n3,… the number of groups which have 1, 2, 3,… aligned sites on the net of squares. More generally, the number of site includ-ing S aligned sites, n is the probability so that these groups can be formed in the net of squares. We write: nS = 2pS(1 - p)2S+2 (1) For p < 1, if S , nS 0 is the probability for a group which has sites S aligning in net of works is very low, nearly reaching 0. In 3D, on a simple net of squares, each aligned group including S will have (4S + 2) adjacent non-reservoir blocks and sites which can be aligned in 3 perpendicular directions , the number of average of groups (for a net of sites) is calculated as follows: means that adjacent to level pc is a fractal which has the similar structure with scale D~2.5 in 3D [9]. This explains why at this level, the description of active space becomes unsuitable for space which has strong homogene. Around percolation pc, correlation length x is calculated as follows x ~ |p - pc|-x, (4) In which ultimate exponent does not depend on the arrangement of net. In 3D, x 0.88, 2D, x 1.33, [7, 11]. At the level pc small groups can connect to each other, widen the size, increase correlation dis-tance. In the net, there are sites under different cor-relation groups and formed unlimited groups. Point (crack) density in network of limitless group Assume P (L) is billion parts of point in a net-work belonging to limitless group, and also average density of points in limitless group. In square net having area L2, this density is identified: nS = 3pS(1 - p)2S+2 (2) For the case of hypercubic d-dimensions, each site has 2d adjacent boxes; for internal sites of a S group, sites creating lines will have (2d - 2) non-reservoir sites. If two ends are considered, Group of S-sites in this case will have (2d - 2)S + 2 adjacent nonreservoir sites. In this case, the number of groups are calculated as follows: nS = dpS(1 - p)(2s-2)S+2 (3) The expression (3) is true for d = 1, 2, 3. The expression above is only accurate for sim-ple case; however, natural world is so complicated! They will not be true for cases in unaliged groups in the net, for cases of 3 unaligned sites, the alterna-tives of arrangement is abundant. The Figure below (Figure 3) shows that group S = 4 sites has 19 dif-ferent arrangements. If number of sites S of one group increases, the number of arrangement (configuration) is increasing rapidly. For instance, if S = 5, there will be 63 alter-natives of arrangement; if S = 24, there will be 1023 different alternatives. Back to 2D case, for the probability p < pc on the net of squares, there will be only groups of aver-age size S. The size S of the group is nearly equal to the correlation length x, average distance between two sites under a correlation group. If p pc, nearly equal to percolation, the scale (ratio level) for typical average computation (volume in 3D, area in 2D) is getting bigger to the scale “mini” around pc. In which M (L) is number of center points in the same group in area L2 (L is positive integral odds 3, 5, 7, 9,…, because it is necessary to have odd num-ber in length of square to have a square in the mid-dle of net from which the others is symmantric). It is clear that M (L) increase gradually in accordance with area L2, P does not depend on L but only depends to p; p is propositional to P. Therefor M (L) is L’s function, at ~pc, it is proportion-al to L2. P is the probability for any point (crack) belonging to limitless group, when p is probability for any point (crack) to contain (connect). If p is con-sidered as common porosity inaccordance with sur-veying terms, P is connecting porosity or opening porosity (P ≤ p). When logM(L) and logL are represented in loga couple chart for net having large number of points, Staufer (2003) found that chart was a line having angle factor D = 1.9 (Fingure 4). D 1.9 is fractal integral number of limitless group in 2D presenta-tion space. Fractal dimensional numbers of limitless group do not depend on arranging form of network (triangle, square…) and only denpend on Euclid position dimension. In 3D scale D 2.5. In Figure 4, line chart shows that: M (L) ~ L1.9, (6) Meams that M(L) grows with L1.9, average den-sity (5) is not a constant number but decrease L-0.1 PETROVIETNAM JOURNAL VOL 10/2009 3 ... - tailieumienphi.vn