VNU Journal of SciencG, E arth Sciences 28 (2012) 173-180 Calculation the Irreducible water saturation Swi and determination Capillary pressure curve Pc from Well Log data Dang Song Ha* H a n o i U niversity o f M in in g a n d G e o lo g y Received 12 September 2012; received m revised form 28 September 2012 Abstract Calculation o f irreducible water saturation s ^ ị and determination Capillary pressure curve Pc is very important in the Oil and gas exploration and production The com plex reservoừs always represent a quite challenge to geologist and engineers to calculate [1] Capillary pressure curvers are usually determined in the laboratory in core analysis and only can perform when we known the iưeducible water saturation s ^ ị It is very difficult in the fact This research gives a method: Calculate s ^ ị and determine o f p ^ curve from obtain’s data set (5 • /> ) with: j = (which is easyly to collect for every reservoừs) The method o f this study can use for both the carbonate reservoirs and the Sandstone reservoirs These reservoirs consict o f 90% oil and gas in the world The declared actual testing result from data in variety o f diefferent huge oil fields around the world which are found on website and PVEP data has affirmed the appropriateness o f this method challenge calculate Introduction Calculation o f irreducible water saturation S - is an important step in D reservoir modeling saturation to s ị geologist [1] and engineers to The capillary phenomem na occurs in ^ ^ porous media when more immiscible fluids are ^2], In the interface ! ^ ^ betw een the two phases, Capillary pressureis defined as the difference in pressure betwwen the wetting and nonw etting phases [2]: studies The iưeducible water S - disfribution will dictate the original oil in place (STOIP) estimation and influence to the subsequent steps in establishment o f dynamic modeling The complex reservoirs always represent a quite p =P R^ w ^ n w ^ c where is w etting phase capillary and non wetting phasecapillary ‘ Tel: 84-934277116 E-mail: blue_sky27216@yahoo.com 173 Í1) V ‘/ is 174 D s Ha / V N U journal o f Science, Earth Sciences 28 (2012) 173-180 Because the gravity forces are balanced by the Capillary foces, so that Capillary pressure at a point in the reservoir can be estimated from the hight above the oil-w ater contact and the diffrrence in fluid densities For the oilwater media, we have: P c = (P -P o )g h (2) Capillary forces are reflected by Capillary pressure curvers affect the recovery efficiency o f oil displaced by water, gas or different chemicals, thus, Capillary pressure functions are need for perform ing reservoir simulation studies o f the different oil recovery processes In [3] and more other studies suggest methods to plot curvers by em piricalism and analise the relationships parameters betw een it and other hand, the measurement o f it in the laboratory by core analysis is very difficult, espensive and time-consuming The declared actual testing result from data in variety o f diefferent huge oil fields around the world which are found on website and PVEP data has affirmed the appropriateness of this method Nomenclature: S rVI-: irreducible water saturation Well log data : : Capillary pressure orther The theoretical basic of the method Interpretation o f Capillary pressure curvers may yield useful informations regarding the petrophysical properties o f rocks and the fluid rock interaction Relative permeability, absolute permeability and pore disừ*ibution to the nonwetting and w etting phases can obtain from the Capillary pressure curvers This research gives a method: obtain’s data set(5 ^ ; with: j = From we 5'„■ W| and determ ine function o f curve , determine three constants: s^ị ,a \ b 2.1 The empirical method: Capillary pressure curvers are usually determined in the laboratory in core analysis by the mercury injection method The determination o f Capillary pressure using reservoir fluids is usually done by the restored “ state method or using a centrifuge, and according to the coưeclation coefficient to obtain the reservoir pressure [4] calculate The object o f this study is both the carbonate reservoirs and the Sandstone reservoirs These reservoirs consist 90% reserver oil and gas in the world Verification for both these resen^oirs The m ost important result o f this study is calculation o f the irreducible water saturation S^ị and plot /Ị curve as ứie graph o f a continuous function from data, which is easyly to collect in the fact On the 2.2 The method o f this study: Capillary pressure curvers are presented by the equation [5]: a Where : a, b, (3) are three constants with < a < and < b =1 = - 2± [ y , - ( A x , * B ) >=1 , we ( X i) + of or: p curve , determine three constants: 5^, ,a \b , j=\ { Ỳ ^ j).A = Ỳ y jj=\ >1 (X x ,).B +( X x / ) A ^ X x ^ y , and plot the graph o f (3) j=\ j =\ j =\ in the matrix form: Calculate the Irreducible water saturation Swi and determination Capillary pressure curve Pc from Well Log data : The capillary pressure represented by the equation P = curvers a p The problem is that: From the collection data: (6*^; w ith: j = \ ,p we calculate , •• ■(4} are (3) ■ p i , y , /=1 U sing the liner regression method represents in [5] to solve equation (4), we find out A and B The constants a , b are calculated as following: a = 10^ - b ^ - A S ^ ị, curver, determine three constants: s^ị Obviously that: and a\ b , plot the graph o f function by (3) consecutively: Taking common logarithms o f both sides of (3) gives; , we give s^ị = 0; Ẳ, 2Ắ (n - Ì)Ả with Ẩ = 0.0025 and: nĂ = m m ( V tS J For every value : s^ị = 0; Ẳ, 2Ẳ .(n - l)Ẫ lg P ,= lg a - Ă lg ( ,- „ ,) we we calculate F , then choose F m in is the smallest value in the series n values o f F, we determine im m ediately three constants: 5^- the and linear regression analysis to determine A and B (to infer a \b ) a;b Denote: have:_y = ^ y = ^gPc \ A ~ - b \ B = \ga +5 Consider s^ị is constant, perform In order to minimize the mean squared eưor a \b in (3) W ith parameters s^ị and plot the graph o f (3) Programming by the MATLAB language Application conditions Differentiate (3) we have : 176 D s Ha / V N U journal o f Science, Earth Sciences 28 (2012) 173-180 p' = - A , so function p becausee is with: (5^; only big enough 2) The accuracy o f the calculation result can be evaluated by analising and interpretation the constants a\ b and s^ j-\ ,p (exemple: satisfy condition: < a < l ; < ố < ) and the variable behaviour o f the function F In the MATLAB Programming we plot the grapth F-iS^ Consider the theory and testing must be unit value and monotono degreeing It is mean ứiat : Data /J,) must satisfy condition: If ( s ) < ( s ) t h e n This condition is satisefied easily Notice: 1) calculates while pc —> +O0 and S^ị measured value with degenerated and non uniform continuous on (5^,; 1]; thus application’s conditions is the collection data: W| The value 5^-calculates by this study usually smaller than the measured value a little, on the practical data, we see that: The F curve reflects the accuracy o f calculation result The result is good if the minimum o f function is reflected clearly., It is mean that function F decreases quickly to the minimum and increases quickly as the following figure (on the right): DANG SONG HA Lop DVL K 52 Sien thien cua Fmin f n _ I I I I I ( 0 05 0.1 15 0.2 0.25,,^, ,3 , 0.35 04 Verification: 1) On the Internet: Consist o f S a m p le s from the big oil and gas fields in tìie world Reader can find tìiem in [1,4,6] on the Internet Sam ple 1: VtPc=[8.00 4.56 2.78 2.15 1.64 1.40 1.30 1.15]; D s Ha / V N U Journal of Science, Earth Sciences 28 (2012) 173-180 V t S w = [0.37 0.41 48 0.54 61 0.65 70 80] ; Sample 2: V t P c = [0.867 16 1.45 73 02 31 2.89 7.8 V t S w = [0.90 80 70 60 0.50 45 0.40 0.35 Sample 3: VtPc= [0.15 30 0.50 V t S w = [1 982 883 0.771 0.698 0.664 0.648 Sample 4: VtPc= [0.15 0.30 0.50 VtSw= [1 0.917 815 699 616 0.581 0.566 Sample 5: VtPc= [0.15 0.30 50 V t S w = [0.984 942 868 786 0.72 0.687 0.673 Sample 6: V t P c = [0.15 30 0.50 V t S w = [0.983 0.929 823 0.708 648 617 0.603 177 21.7] ; 0.30] / 10 ]; 0.639] ; 10 ]; 560] ; 10 ]; 0.665] ; 10 ]; 0.596] ; Calculation results from the samples: Results in Sample the Internet Results of this study S WỊ a 0.330 0.300 0.607 0.539 0.633 0.575 0.3300 0.2850 0.6050 0.5375 0.6325 0.5725 Comparision: According to the data from “Calculations o f Fluid Saturations from Log-Derived JFunctions in Giant Complex Middle-East 0.5925 0.6012 0.0555 0.0803 0.0401 0.0631 B 8085 0.8476 1.5359 1.2703 1.6138 1.3509 Carbonate Reservoir”[l] We calcalate and compare: The result o f this stady is ploted by MATLAB on the rig h t, the result o f the auther [4]on the left in the following figure: ■, I 178 D s Ha Ị V N U Journal o f Science, Earth Sciences 28 (2012) 173^180 Results and Discussions Conclusions 1) The fact that objective testing data in range of the reservoirs in the world which is found on website and data from PVEP, the appropriateness of parameter a, b and variation o f F curve is good has shown that: This research method is used for determining s^ị The empirical method only can plot the empirical capillary pressure curvers but can not calculate iưeducible water saturation and and (3) Not only s^ị but olso two parameters : a\ b have their peừophysical meaning In the reservoirs we usually have more data sets Pc).y so we have more data sets curve establishment Good variation of F curve proves that mathematical basic about minimum condition o f fuction is it’s derivation equals zeros is enttusted theorical foundation 2) The most important result o f this study is the calculation o f the iưeducible water saturation s^ị from ; p^) .data The influence o f s^ị on /Ị is mentioned detailly in [3] and other documents, but none o f them has mentioned about the calculation o f s^ị from The P^).data detemiines s^ị calculation of from Value s^ị p^) then from (5^,; p^), is reasonable value S^ị can be found out by solving reverse mathematical problem which is used frequently in geology 3) As well as other problem in Petrol and geology, the calculation for s^,ị and establishment for curve in this research two parameters a,b The method o f this study can calculate S^ị , two parameters a,b and plot the graph o f S^j;a;b I comparision data sets the reseavoir Thus, this study is an approaching way together with other ones make a solution for sifnificant as well as difficult problem in peừol exploration and geology Analysis, { s^ ; a ; b ] may yield useful informations regarding the reservoior Acknowledgments The auther would like to thank doctor Lê Hải An, Hanoi University o f Mining and G eo lo g y , en gin eer Đ an g D u e N han et all PVEP in for helping to the auther finish this study References [1] Lê Hải An, Vật ỉý thạch học, giảng cho sinh viên đại học mỏ địa chất Hà Nội [2] Nguyễn Đức Nghĩa, Tỉnh toán khoa học, giảng cho sinh viên khoa CNTT, Đại học Bách khoa Hà Nội [3] Michael Holmes, Capillary pressure & Relative Permabiỉity Petrophysỉcaỉ Reservoir Models, Derives, Colorado USA May 2002 [4] I'awfic A.Obfcdia, Yousef S.Ai-Mehin, Karri Suryanarayana: Calculations o f Fluid Saturations from Log - Derived J-Functions in Giant Complex Middle-Easi Carbonate Reservoir should not be rewiewd separately but analytical comparation s^ị o f ửiis study with other paramerters as Permeability K, porosity ẹ of respectively investigation [5] Noaman El-Khatib: Development o f a Modified Capillary Pressure J-Function, KingSaud University [6] Crain’s pressure Petrophysical Handbook-Capillary D s Ha / V N U Journal o f Science, Earth Sciences 28 (2012) 173-180 Appendix Programm calculation for 5^, and establishment for curve VtSw=[0.37 0.41 0.48 0.54 0.61 0.65 0.70 0.80]; VtPc=[8.00 4.56 2.78 2.15 1.64 1.40 1.30 1.15]; x = m i n(V tSw ); l = r o u n d ((x/0.0025)); p =le ngt h( VtS w); Kqua =zeros(l,4); for n=l:l Swi=0.0025 *( n-1 ); Kqua(n,1)=Swi; H s o = z e r o s (2,2); N g h i e m = z e r o s (2,1)/ T u d o = z e r o s (2;1); for j=l:p Sw =Vt Sw (j); %thay dong can % thay dong can Pc=VtPc(j); H s o (1,2)= H s o {1,2)+ log lO(Sw-Swi); Hso (2 ,2)= H s o { ’2 ) + {loglO(Sw-Swi))^2; T u d o (1,1)=Tudo(l,1)+lo glO (P c); T u d o (2,1)=Tudo(2,1)+loglO(Sw-Swi)*loglO(Pc); end Hso(l,l)=p; Hs o( , 1)=Hso(l,2) / Nghiem=inv(Hso)*Tudo; a l = N g h i e m (1,1); a = ^ a l ; bl=N ghi em {2, 1); b=-bl; Kqua(n,2)=a; Kqua(n,3)=b; Fmin=0; for j=l:p Sw =Vt S w ( j ) ; P c = V t P c (]); Fmin=: Fmin+ [Pc- (a/ ( (Sw-Swi) ^b) ) ] ^2; end Kqua(n,4)= Fmin; end F m i n = K q u a (1,4); for n=2:l x= Kqua(n,4); if (x