Percolation theory in research of oil-reservoir rocks ppt

ISSN-0866-854X An Official Publication of The Vietnam National Oil and Gas Group Vol 10 - 2009 PETRO PETRO Petro ietnam VIETNAM JOURNAL ENMP TROVIET A Percolation theory in research of oil-reservoir rocks Comprehensive CO2 EOR study - Study on Applicability of CO 2 EOR to Rang Dong field Comprehensive CO2 EOR study - Study on Applicability of CO2 EOR to Rang Dong field Publishing Licences No. 170/GP - BVHTT dated 24/04/2001; No. 20/GP - S§BS 01, dated 01/07/2008 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 Designed by Le Hong Van PETROVIETNAM JOURNAL IS PUBLISHED MONTHLY BY VIETNAM NATIONAL OIL AND GAS GROUP Contents 01 11 41 Percolation theory in research of oil-reservoir rocks Distribution rule of lower Miocene sandstone in Cuu Long basin 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 18 Comprehensive CO2 EOR study - Study on Applicability of CO 2 EOR to Rang Dong field 24 Novel surfactans for high temperature, high salinity emhanced oil recovery applications 34 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 66 PETROVIETNAM JOURNAL VOL 10/2009 1 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 X i we can get the average value − X i of the effective space according to: , or if the values X i 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- Ass. Prof. Dr. Nguyen Van Phon Hanoi University of Mining and Geology Percolation theory in research of oil-reservoir rocks 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. petroleum EXPLORATION & PRODUCTION THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 1 PETROVIETNAM JOURNAL VOL 10/2009 2 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- ability ensuring no connection between them (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 = p c , the big groups will become unlimited- size group and the ultimate probability p c is called percolation threshold. The percolation threshold p c is an ultimate probability enabling an unlimited group to form in a large net. With p > p c 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 < p c there is no unlimit- ed group in the net. Percolation threshold p c depend on type of cell (square, triangle, hexagonal, etc.), number of dimension and type of percolation. Value of perco- lation threshold p c stated in Table 1 is the calcula- tion result of (2003) with different types of cell net. 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 a bond, the propotion of void connected are called 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 < p c , 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 > p c 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: n 1 = p(1 - p) 4 . For the case of 2 reservoir sites standing petroleum EXPLORATION & PRODUCTION THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 2 PETROVIETNAM PETROVIETNAM JOURNAL VOL 10/2009 3 among 6 adjacent nonreservoir sites, these two sites can be arranged in vertical or horizontal direc- tion; therefore, n 2 = 2p 2 (1 - p) 6 . It is easy to conclude that in a net of squares includes 3 aligned sites, there will be n 3 = 2p 3 (1 - p) 8 , for around 3 aligned sites are 8 nonreservoir sites with the probability of (1 - p) for each site. We call n 1 , n 2 , n 3 ,… 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: n S = 2p S (1 - p) 2S+2 (1) For p < 1, if S  ∞, n S  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: n S = 3p S (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: n S = dp S (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 10 23 different alternatives. Back to 2D case, for the probability p < p c 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 ξ, average distance between two sites under a correlation group. If p  p c , 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 p c . Then, the ratios are equivalent to one another. This means that adjacent to level p c 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 p c , correlation length ξ is calculated as follows ξ ~ |p - p c | -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 p c 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 L 2 , this density is identified: In which M (L) is number of center points in the same group in area L 2 (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 L 2 , P does not depend on L but only depends to p; p is propositional to P. Therefor M (L) is L’s function, at ~p c , it is proportion- al to L 2 . 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) ~ L 1.9 ,(6) Meams that M(L) grows with L 1.9 , average den- sity (5) is not a constant number but decrease L -0.1 THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 3 PETROVIETNAM JOURNAL VOL 10/2009 4 times in rate grade. The larger scope is, the larger the difference is. For example, average density P (amount of workable oil) counted on an area of reservoir with porosity of approximate p c and edge L = 100km shall be smaller than area counted in sam- ple with edge L = 10cm, in accordance with space coefficient (10 6 ) -0.1 ~ 0,25. 75% of remained amount of oil in reservoir is not connected directly with exploiting wells located in central point. In 3D, corre- sponding coefficient is much smaller: (10 6 ) -0.5 ~ 10 -3 . In fact, the counting result shall not always bad because density P that is gradually constant to L and p is larger than p c , at that time there is a corre- lated length ξ(p), a limitation so that: M(L) L 1.9 to L < ξ and M(L) L 2 to L > ξ. Limitation ξ is the limitation of the farest well in the packing, it shall decrease similar to increasing p than p c . Therefore, explorer shall use a sample with L that is larger than ξ to calculate amount of oil which may be exploited more exactly. Of course, the amount of oil take from reservoir layer depends on many other factors relating to fluid flow in pore space and dynamtics characteristics in osmotic packages such as diffission of fluid in mixed space and force osmosis which shall be discussed in another works. Bethe net In order to have exact solution for complex structures, problem above is studied in form of branch separated tree – Bethe net. Bethe net (or Cayley tree) is tree shaped net with unlimited dimensionals. Approximated calculation Bethe is used to give anwser for tree problems. Therefore complex structures with unlimited dimensional d are Bethe net. In order to understand structures with unlimited dimensional d, we will start with d = 2: Area of circle with radius r is πr 2 , its circle is equal to 2πr. Area S of sphere (3D) radius r is 4πr 2 , and volume V is propotional with r 3 . In d- volume dimensional of ball shall be propotional r d , and surface area S is propo- tional with r d-1 . General calculation: (7) (Symbol is used to count rate between val- ues. In several cases, this rate means approximate limitation; d  ∞) Expression (7) shows that when dimension up to unlimited point (d  ∞), then area of the ball outer will approach to the cubic content. This remark is true even with grid, cube, multi-cube etc., Construction of Bethe network To construct a simple Bethe network (Figure 5) to conduct as follow: To point of O origin site, pass- ing four origin points (Z = 4) adjacent to A. From A site, four bonds are generated, one connecting to O, the other three ones connect to B site. From B site, it connects to (Z - 1) = 3 of new site C, and more lengthened by this way. Bethe Net is an unclosed net, which has no branch connecting to O origine site by any form. Continuation with this process of branching, we will have an unlimited network, of which the sites will increase in the distance from the outside site to the O origin site with a structure d- its dimension will be incresed: (distance) d . In example in Figure 5: Z = 4, original site is covered by 4 sites A (the first system), second sys- tem (or layer) will have 12 B site, the third will be 36 C sites therefore, the point network consisting of the first system to the last system of 4 x 3 r -1 site is the outer site. Then, the network expand to r, the last system consist of 4 x 3 r-1 / 2 x 3 r -1 = 2/3 of the total sites on Bethe net. This is equal to and correct to (Z-2) / (Z-1) any Bethe net with Z at random. From this point of view, we can expand to the 3D case, ratio of area of internal side and volume of the ball whose radius is of r will reach the approxi- mation ~1 when Z  ∞. This is fitted with the expression (7) when 1/d  0. Now we can see that Bethe net is an abnormal model; thus, when men- tioning to the percolation of the Bethe net, one very important thing is to imagine that it only occurs inside the net but not effect the outerest surface. The evaluations above imply that probability by which an infinite unit spreads all over the net is zero, and the percolation threshold p c is given by: (8) This calculus is suitable with the site percola- tion case and bond percolation case. For further understanding about the percolation threshold p c , take a look back Figure 5. There are four rays at each site A (Z = 4) among which one connects with O, (Z - 1), the three remained rays reach the site B, and the same for the case of site C. etc. Hence, (Z - 1) -1 will be ultimate probability for the creation of infinte unit, and called percolation threshold of the Bethe net. Average size (S) of the unit when probability approximately reach the level p c In order to calculate S, we assume T is the average size of each unit at four branches. T is the petroleum EXPLORATION & PRODUCTION THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 4 PETROVIETNAM PETROVIETNAM JOURNAL VOL 10/2009 5 average number of sites connected with original site to form each branch A. Each of these separate branch is continuously divided into three smaller infinite branches, T will still be average size of the unit in each branch. Next to site A, site B may be the containing with probability p or non-containing with the probability (1 - p). The non-containing sites will not be very meaningful while the containing sites contribute (1 + 3T) point for this branch in which one is point B and 3T is three branches extended from this site. Therefore: (9) The size of group originating from the site O is 0 if this site is non-reservoir site or (1+4T) if it is reservoir site. Therefore: (10) Referring to (8), so S can be adjusted according to (p c - p) -1 for p < p c . For p > p c , S will branch off. If p  p c with the ultimate exponent x ≈ 1. S = (p c - p) -1 (11) Relating to the expression (4) we find that if p  p c , average size S of group and the correlation length ξ are equal and reach infinity. By similar inference, it is possible to identify ratio P of sites under unlimited group in the space which has p greater than permeability p c . P, as ana- lyzed above, is the probability for the original site O under unlimited group. It can arrange different abili- ties so that original site O is connected to 4 neigh- boring sites (Figure 6). On the drawing, each arrow is an unlimited daisy chain connected from original site O Figure 6. We consider Q as the probability connecting from O to adjacent site A which is discontinuous (congestion). According to Figure 6, we find that probability P is identified referring to the probabili- ties of three final alternatives c, d or e. If each arrow is an unlimited daisy chain, O must be of two unlim- ited chains which consider them as the connection part of a permeability group. The probability to exist the arrow between O and site A is (1 - Q). For case (c), we have probabil- ity 6Q 2 (1 - Q) 2 (including 6 probabilities of arrange- ment so that from O there are two arrows and 2 nonarrows which rotate indifferent directions). For case (d) it will be 4Q(1 - Q) 3 ; and for the case (e), it will be (1 - Q) 4 . Total probability will be: (12) In fact, probability has function relation with probability p. In fact, a chain line connecting from O to site A is discontinuous if O and A are not connect- ed (probability1 - p), or if O and A are connected but fragmented on connecting to A (probability pQ 3 ), it can be computed as follows: Q = (1-p) + pQ 3 (13) Equation (12) has the simplest result Q = 1, P = 0, which means that at that time the system is under the percolation threshold. Furthermore, this equation also has two different results, but these result mentioned below have physical meaning: (14) Q reduces from 1 to 0 if p increases from to 1. In the range p < p c , Q = 1, P ≡ O. From two depend- ent relations between P(Q) and Q(p) it is possible to find out P(p). Around the threshold p c we find that P change referring to the form (p - p c ) 2 : p  p c ; P ≈ (p - p c ) 2 (15) It is possible to express (15) as Figure 7. Picture 7 shows that the permeation probabili- ty P or the site density (bond) of the unlimitted group increases from 0 to 1 when the probability p of the site increases from p c to 1. In the space of p < p c the set status is below the permeation threshold P =0. This means the reservoir rock space has the critical void ratio (p c ), the space will have the perme- ation or the permeability will occurs at that time. The finner the grain of the clastic rocks is, the higher the critical void ratio value (p c ) is; the void rate of the fractured rock with the kinetic penetration is usually lower than that of the crumb rock. This relates to the specific surfaces and the channel bend of the two mentioned above rocks. In the basement of Bach Ho oil field and other fields in Cuu Long basin, the hydrodynamic penetra- tion occurs at fractures spaces (F f ) and macrofrac- tures while the capillary penetration occurs in microfractures. The result of 270 granitoit fractured samples analysis (2001) in basement of Bach Ho oil field by Mr. P. A. Tuan showed that their average gen- eral void ratio is 3.1% while the average open void ratio is only 1.88%, it means that the close non-con- nected void ratio makes up nearly 40% of the gener- al void ratio. The equivalent numbers in the analysis of the well-logs in a well of Rang Dong field are 5.455%, 0.617% and 89% Percolation through fractured rock space Percolation theory gives us a decription method of strongly heterogeneous space in reser- voir rocks. Here meaning of threshold is often THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 5 petroleum EXPLORATION & PRODUCTION PETROVIETNAM JOURNAL VOL 10/2009 6 research carefully. Threshold effect can be seen clearly in many phenomena occuring in the nature and human society. The permeability of fractured rocks space are emphasized here. Fractured rock space The process of development and mature of the rocks result in the appearance of interrupted frac- tures within rocks due to various reasons: volume shrinkage (the freezeing process of igneous rock ), load reduction (weathering and erosion), mechani- cal stress (tectonic activities), corrosion and elutria- tion (thermalization activities), etc. Fractures are the results of the disruption of the initial uninterrupted structure. To simulate the fractured rock space, we will look through dished fractures and evaluate them in terms of dip anglar, strike, aperture, radius, filling – up level of secondary minerals and density in rocks. We consider space as rocks with different frac- tures of random distribution. If there are not many fractures in the space, it is unlikely that such frac- tures cut each others, low connectivity, zero perme- ability.The higher the density of fractures is, the greater the probability for such fractures cut. each others. If the critical density is to be outnumbered, there will be a ratio f representing intersecting frac- tures in the space, which forms the “unlimited group” (Figure 8) and enables the fractured space to let the fluid through – that means the non-zero permeability. Using the model Bethe net (Figure 5) with Z = 4 to demonstrate the fracture net , we have: f is den- sity P, and p c = 1/3, and p is the probability so that two random intersecting fractures cut At different value of p, which is greater than the critical proba- bility p c , then p would be directly proportional to P, which is the ratio of intersecting fractures and belong to unlimited group within the scope of stud- ied volume. As P increases, the probability of leting fluid through the space also increases. This princi- ple is also applicable for the conductance of fracture net if the carrying fluid follows the saturated fluid (water) in the empty space of fractures. In this case the Ohm Law and Darcy Law is compatible. For the purpose of calculation, we demonstrate the fractured rock space as in Figure 9 and assume that all fractures have the dished form, radius c, aperture 2w and density N ≈ 1/ℓ 3 , in which ℓ is the average distance between fractures (Figure 9). Estimating pressure p according to c, w and density N Assume that p is the pressure to have two ran- dom intersecting fractures. As the density N and radius c increase, p also increases. The number of fractures over a partial volume, the greater the aver- age dimension c of fractures is, the higher the prob- ability that these fractures cut each others. The product Nc 3 is the non-dimensional quantity, so p=0 as one of the two parameters (N or c) is equal to 0, so it can be assumed that p changes accordingly with Nc 3 : (16) To calculate the pressure p and determine the percolation zone and permeability, we introduce a new concept: Peripheral volume V ex is the maximal volume containing a random fracture with center O to have a second fracture O’ ( with the same radius c), which is arranged randomly in the volume. The result is that the two fractures will cut each others. In the fluid crystal physics (De Gennes, 1976), this volume is defined as: V ex = π 2 c 3 (17) At a density of fracture N, the average number of intersecting instances of each fracture is v = NV ex . The Bethe net in Figure 5 shows that the probability for an isolated fracture (does not have any fracture) is p o = (1 - p) 4 . This probability can be presented according to v as follows. Assume V o is wide volume, in which there are disorder distribution of centre O of fractures with the density N. Probability for a random point in V o falls into a volume V which is smaller (V⊂ V o ) will be V /V o . Assume that n is a random in V o the probability p m for m points (m<n) falls into V will be calculated as follows: (18) In which: If we calculate the limit of the expression (18) as n and V o will reach the unlimited pole n /V o =N we will have: (19) If there is no point falling into V, m=0 then the limit (19) p o = e -n will be the probability for a fracture to be isolated (does not cut any fracture). But then p o = (1 - p) 4 . So we have: (20) According to (20) we can see if the density N  ∞, that means v  ∞ then the probability for p so that two random intersecting fractures will be nearly THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 6 PETROVIETNAM PETROVIETNAM JOURNAL VOL 10/2009 7 equal to an unit If N is so small, n << 1 that p ~ v /4. Because so (20) express the connection between p, c, ℓ and N. Take and , the conditions for percolation threshold p c = 1/3 will divide (mặt) (ℓ, c) into two domains (Figure 10), the percolation domain in the right side of the dividend line, corre- sponding to values and . There will not be percolation if ℓ is too great, density N is too small or c is too small (small frac- tures). This is suitable for (15) as p moves upwards to p c from the greater value (p > p c ), the pemermabil- ity will be proportional to (p - p c ) 2 . This is also cor- rect for the conductance if the electric current is the ion current flowing through the saturated fluid in the fractured voids. Percolation effect In fact the intersections between fractures in the fractured rock space is at random. The fractured space has the permeability as the intersections between fractures form an unlimited group and the number of groups or average dimension of the group stretchs out, number of springs P belonging to the unlimited group is greater. To calculate the permeability of the fractured rock space, let’s get back to the model (Figure 9), and apply Darcy Law. In mechanics, call q as the Darcy speed of the fluid, which is equal to the volume of the fluid through the cross-section S perpendicular to the speed direction over an area unit in a time unit: The volume of fluid is equal to q. If the fluid has the viscosity h and gra- dien with pressure , then the Darcy Law will be presented as: (21) In which k is the permeability with the perme- ability factor [m 2 ]. Darcy speed is the volume flux (not the actual speed of the fluid) and can present the connection between it with the average speed of the fluid in the porous hole F according to Dupuit-Forcheimer Law. q = vΦ (22) In the fractured space, the average speed ― v of the fluid between the two parallel sides will apply Landau-Lifshitz Law (1971): (23) From (23) and (21) we can easily conclude that: (24) Take the approximate porosity F of the frac- tured rocks as an replace (24) we can calcu- late that: (25) Here, once again it is proved that in the frac- tured rock space, permeability k depends on three micro-structure factors: c, w and ℓ. Expression (24) and (25) are true for the per- meability which all fractures will connect with each others completely, that’ means p ≡ P as the model of Warren – Root (see instruction documents of chapter [6], chapter7). But in reality, such cases occur rarely, because the intersections between the fractures in the fractured rock space are at random and we have to apply the permeability theory to evaluate the intersecting level between the frac- tures in the net. From part 3.2 result, peripheral volume V ex = π 2 c 3 and estimated , we can see: If, the possibility of intersections between any fractures is low, the space does not have permeability, k = 0; in contrast, if p > p c , unlimited group is created and the space is capable of permeability, and increase in multiplication factor f = (p - p c ) 2 (see (15) and Figure 10.) This factor is permeability probability P(p) shown in Figure 7. In calculation, permeability factor calculated as (24) and (25) need to be multiplied with f factor because of permeability effect: (26) In which, W is aperture of fracture, p is com- mon porosity, p c is porosity limen for fuildl passing permeability space, and Φ is leaky porosity or open porosity, including carven and fracture porosity which are called secondary one in some docu- ments. Define permeability basing on well log datas According to the porosity result in bore well, common porosity (p) in fractured rocks is calculate by average of porosity Φ D and Φ N at the same depth; secondary porosity Φ is calculated as following: (27) In which Φ S is calculated basing on sonic THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 7 petroleum EXPLORATION & PRODUCTION PETROVIETNAM JOURNAL VOL 10/2009 8 method at matrix rock without fracture. Dimensions w, c and ℓ or fracture density in the space is identified from analysis results of FMI and FMS datas. For fractured basement at White Tiger and Dragon oilfield, Institute of Marine Research, Vietsovpetro considered porosity threshold perme- able in fractured rocks p c = 0,01 in calculation of hydrocarbon in inplace and oil recovery factor; aver- age aperture of fractures. From that statistics, we could calculate permeability of fractured base based on: (26) For example, result of open porosity Φ = 0,018, common porosity p = 0,031 to be replaced in (26), we have: Permeability of fractured rocks depends on aperture of fractures. Aperture changes twice, per- meability will changes four times. In addition, per- meability of fluid in fractured rocks depend on draught of fractures. The draught of fractures increase, the permeability decrease. Up to now, many authors research more this projects. Suggestion and Conclusion Reservoir is pore space which its microstruc- ture is complex and changes along with rock devel- opment process. Each kind of rock contains pore microstructure with typical characteristics but asyn- chronous. Strong asynchronous state is characteris- tic of fractured stocks: They have two porosities, two permeability abilities. They are fractured and internuclear porosity (or block porosity); kinematical permeability in large fractures and caves, capillary in internuclear porosity and fractures. Large frac- tures have small ratio in common porosity but play an important and decisive role in effective perme- ability. Minimum fractures and porosity of particles play an important role in determining the ability of product area. The penetration of fluid into an space with 2 porosity is a complex process. In the porosi- ty space of fractures, they are up to gradient pres- sure of fluid, the minimum fractures and porosity among particles are determined by wettability and capillary force. The physical nature of permeability in multi-fracture space are suitable with shape of permeability. The analogy is the base to apply the theory of permeability in an unsuitable space such as fracture stones in Bach Ho oil field and other fields in Cuu Long basin. The evaluation and use of the permeability density P(p) as the permeability effect factor is the specific result of this construction to overcome the disadvantage of Warren- Root model in order to determine the k permeability in the fracture rock space with two void and two permeabilities objects. The author would like to thank fellows for help- ing and exchanging experiences and ideas in the work implementation… This work is the result of the research project KHCB 7.1.5206 sponsored by Ministry of Science and Technology. References [1]. Nguyen Van Phon (2007). Fractal geome- try for researching reservoir (I). Petrovietnam Rev. Vol. 2 - 2007, pp 23-26. [2]. Nguyen Van Phon (2007). Fractal geome- try for researching reservoir (II). Petrovietnam Rev. Vol. 3 - 2007, pp 14-21. [3]. Hoang Van Quy, Phung Dac Hai and Borixov A.V. (1997) Collecting data of geologic structure, determining oil & gas an condensat con- tent of Dragon oilfield. Report of Institute of Scientific Research and Statistics – Vietsovpetro. Vung Tau 1997. [4]. Pham Anh Tuan (2001). Physical features, heterotrophic features and hydrodynamics of oil- reservoir rocks of complicated structures in the con- ditions of modeling pressure and temperature of the formation. Ph.D. thesis – University of Mining and Geology. [5]. De Gennos P.G. (1976) The physics of fluid crystals. Oxford University Press. [6]. Golf-Racht Van T.D. (1982) Fundamentals of Fractured Reservoir Engineering. Elsevier Scientific Publishing Co. [7]. Guéguen V., and Palciauskas V., (1994). Introduction to the Physics of Rocks. Princeton University Press. [8]. Landau L., and Lifshitz (1971) Fluid Mechanics. Moscow Edit. “Mir” . [9]. Mandelbrot B.B. (1982) The Fractal Geometry of Nature. San Francisco Freeman. [10]. Snarskii A.A. (2007). Did Maxwell know about the percolation threshold? Uspekhi Fizicheskikh Nauk. 177(12). 1341 – 1344. [11]. Stauffer D., and Akarony A., (2003). Introduction to Percolation Theory. Taylor and Francis (2003). THANG 10:Dien VN SO 24.qxd 11/25/2009 4:23 PM Page 8 [...]... the grain size of BI.2 sands varies from very fine-fine ( . 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. probabili- Ass. Prof. Dr. Nguyen Van Phon Hanoi University of Mining and Geology Percolation theory in research of oil-reservoir rocks Abstract Following the articles