Dendrou, B.

Houstis, Elias N.

English

Purdue E-Pubs

Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1977 Stochastic Analysis of the Effect of Random Permeability Distributions on Confine Seepage B. Dendrou Elias N. Houstis Purdue University, enh@cs.purdue.edu Report Number: 77-234 Dendrou, B. and Houstis, Elias N., "Stochastic Analysis of the Effect of Random Permeability Distributions on Confine Seepage" (1977). Department of Computer Science Technical Reports. Paper 170. https://docs.lib.purdue.edu/cstech/170 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. STOCHASTIC ANALYSIS OF THE EFFECT OF RANDOM PERMEABILITY DISTRIBUTIONS ON CONFINE SEEPAGE iV B. Dendrou", E. N. Houstls" CSD-TR 23^ April 1977 Abstract We describe two stochastic models for the determination of the statistical properties of the solution process. An error analysis of the models is considered which indicates the convergence of the overall procedure. Then a real case of a seepage encountered in an underground facility is treated based on the data provided by a field Investigation. The outcome of the analysis Is In concordance with the past engineering experience. JL Department of Soil Mechanics, Purdue University, West Lafayette, IN ^7907. Aft Department of Computer Science, Purdue University, West Lafayette, IN 47907. It is generally recognized that within any homogeneous rock mass the physical properties exhibit a variability which must be considered in a design project. This variability is due to different causes during the geologic formation of the strata, and is estimated by means of a site investi-gation. This will in turn provide the designer with the desired phys-ical properties of the rock media at particular locations where the drillings are performed, • To handle the variability of the rock media an uncertainty factor is introduced in the simplified analytical model. Two interrelated problems are thenVccountered: A. Hov the above mentioned uncertainty of the physical properties will be inferred from the field information. B. How to introduce the uncertainty of these properties in the already existing analytical model simulating the real world. Both cases are treated by assuming that the rock properties Pj(*,y,z) are spatial stochastic processes, as suggested by Cornel (1). Thus to a point in space (x,y,z) corresponds a merely probable value of Pj known also as statistical uncertainty which can be decreased at the expense of additional field informations. The two problems are in the common practice solved independently and not coupled together to provide a consistent picture of the real phenomenon. Usually the first problem is overcome by making the assumption that the statistical properties of the rock media are the same all over the I region of interest or in other words that the process reflecting the randomness of the physical property is stationary. In that case there is no need of an inference model and one can proceed to solve the second problem by applying a first-order uncertainty analysis in con-nection with the already existing analytical model. This approach was adopted by several investigators in the field of continuum mechanics. B. Cambow (z ) L. Esteva (*) and J. Pqdilla(6). However, in rock mechanics the above, procedure cannot be adopted and the first problem has to be solved exhaustively merely because of the nonstationarity of the physical properties. In this case the rock volume is assumed as made up of a number of elementary volumes within each of which the physical property at hand if treated as stationary. Moreover the properties assigned to each particular point in the two dimensional system must "be inferred from the field data of a limited number of rock samplej cfe«t"i'ly then the need of an inference model is Justified for: 1) The relatively large scale used to perform the analytical model. 2) The small amount of information of the rock media. Krumbein gives the fundamental techniques used in defining linear inference models. Such techniques are for example the method of leQ&t squares, fitting a polynomial in two variables, etc. All these techniques fail to provide an evaluation of how well the estimatron is performed, and present operational difficulties for the nonstationary case. However, G. Matheron (5) proposed an inference scheme which is attractive for our case. Indeed with every obtained estimation of the property under consideration a parameter related with each estimat 'on .s is indicating the performance level reached by the inference. This parameter is nothing else than the variance of the estimator. Moreover the nonstationarity can be treated as veil as the stationarlty case. In the following a technique similar to the one suggested by G. Mather.-on " adopted to estimate the physical, properties in the domaine of interest and to determine their spatial distribution. The results of the inference procedure aire then^ used to treat the second problem. In other words a coupling of the inference model and the analytical v ^ T jfcj model ,Ya coupling of the data field investigation and the analytical simulation of the real phenomenon, is proposed. The analytical model is handled by using a finite element technique. Implementation of the Inference model in a two-dimensional geometric space. To determine the values of a rock property z(x,y) a number of meas-urements are made on rock samples from boreholes. The set of points where observations are made is the set index 6 and Z„ represents the measured > ~ value of the random rock property at point 0. Then the estimation Z(x ,y ) o o at any particular point ( x Q > y o ) "the media Ly evaluated based on the given Zg values. A simple way that one can Imagine to perform this estimation is A to define Z in terms of the known values Z^ according to a linear combin-ation as follows: Where B and the known data points and b^ are the unknown weight coefficients to be determined by the inference model which is based on an optimal scheme o{ Ike C ^ J o h . j^riALVfi An aJtcrnato approach would. beVeutlmate thu mean value of Z(x,y), namely Z(x,y) according to a linear combination. where & are the known data points as previously and a^ are the unknown weight coefficients. The random variable Z(x,y) is characterized spatially by the ex-pression . ^ € + * * where Z(x,y) the mean and FZ(x,y) the fluctuating term around the mean. Then the two overmentioned estimation procedures to two distinctive groups of assumptions characterizing the inference model. The first group is given by 1) E l U < „ l ] = 2) t l ^ u ^ ^ t w ^ o l = + where Z(x,y) is the trend and c(n, Hj) the covariance of the variable Z( x ^ . However these assumptions are in many cases dealing with rocks, too re-strictive and need to be replaced by some^"" flexible assumptions. This • iS. realized by considering the rate of change of the ran-dom variable Z(x,y) and will lead to the second group of assumptions. The second group indeed is defined by: 1) i t i K v l - ^ f j o ] -2) E - a u ^ x f t 1 , - 2. U \ In both set nevertheless of assumptions there is a need to characterize the nature of the randomness of the variable Z(x,y). Therefore the following hypothesis concerning the randomness are added to each group. More specifically we assume: First locally at a point (x,y) the mean Z(x,y) is approximated by known functions. Indeed, a^ being some unknown weight coefficients and ^ \ the apriori known functions. Secondly the covariances c(x^,x ) are computed based on field meas-urements (See Padilla) and can be represented as approximated functions I - oi of the form c(x ,x 2) = G where r - distance between x^ and x^, ^ and k some parameters. At that point following the general trend of thought the two groups of assumptions correspond two inference models with two different goals reached in each case. In the first model the goal will be t<jinake the best estimation for the mean 2 llt>l) while in the second model the best estimation is required for the random variable Z(x,y) itself. Both models lead to the problem of identifying the best estimators among all possible functions satisfying the hypothesis covering the ran-domness of the rock media. These goals are achieved in each case by optimizing the expression of the variance using the method of lagrangian multipliers under the constraints imposed by the assumptions concerning the first moments. In appendix (I) the computations give the following results: KQR MODEL 1 concerning the mean Z(x,y) The method of Lqgragian multipliers leads to a set of + ^ equations with & unknowns namely the weight coefficients: - r ; £ A £ c where Qj the unknown weight coefficients (^ the Lcjgrangian multipli and the krovie.c.k.e»- dc'td The variance of the estimati on is given by ^ ^ Vi I ' M FOR MODEL 2 concerning the random variable Z(x,y) Similarly the following system is obtained (Appendix (l)) lers k i I ^ u + ^ " V U ^ ^ j O ... 1 z. \>" 7 * . ti o where b the unknown weight coefficients and ^ ^ the Lagrongian multipliers. The variance of estimation being: « i t r - 2 . t " 'I + U " " . . UNCERTAINTY ANALYSIS THE ANALYTICAL MODEL As mentioned previously the analytical model . i-S treated using the finite element technique, which .. provides us the transfer mechanism between a set of inputs and set of outputs generally linknown . Then the general solutions ls given by the following relation in matrix form - 4 W = i v y \ s i [k] is generally known as transfer matrix and is defined function of the random variables Z, . Z„, . . .Z . 1 2 n tte] = ( •-• Applying now the first order uncertainty analysis as described by Popoulis ( ) the following moments are obtained: FIRST MOMENT - X . . -L L The second part of the second member can be neglected being a very small quantity SECOND MOMENT + 2. ^ * u coo lUt^ + W fxjrti'al ^e^^w^i'lies Oft. "^ wt S ' ^ t i o m o^ W i ^ z d ^ A i ^ A ^ I l i L - ^ t M * \ u a , U L u ; I Coupling of the Inference Model and the Analytical Uncertainty Model In this procedure the statistical properties of the solution process are determined by the previously given expressions where the moving aver-age or spatial mfean Z (x,y), Z (x,y) and the variances a„ (x,y)j a (x,y) 1 2 . Z 1 Z 2 are provided by the inference model. By substituting these quantities in the statistical relations of the dependent random variable {ID, its co-efficient of variation is defined. This in turn is an essential statis-tical quantity used to evaluate the performance of the analytical model. The convergence of the overall procedure is considered in Appendix 2 and checked through several examples. In general the results are im-proved both in the analytical and statistical sense when the mesh becomes denser. Algorithm Description The geometric domain under investigation is divided using a rectang-ular mesh common for the Inference Model and the analytical model (Finite element mesh). The computations will be performed in each mode using a number of known realizations of the random variable Z(x,y). Therefore a zone of influence, characteristic of the media and depending on the covariance c(x^,x^) is defined at each node of the mesh. In this zone eight given points are selected for efficiency and influence the computed estimation of the random variable Z(x,y) at that particular node. Indeed at every nodal point (x,y) a system of fourteen equations A is solved and the estimator Z(x,y) computed according to the previously defined relations. The outcome of the procedure will of course depend on Lhc number and closeness of the measured information of Z(x,y) provided by the field investigation. If the informations are not enough in number for all the domaine of interest then the1 estimator violates the original assumptions^but the variance on the other hand indicates the poor performance of the estim-ation and gives the exact location in which more informations are needed. The flow chart in figure { 1 ) gives the sequence in which the com-putations will be performed by program INFMODA. Several examples were treated to test the program, the more signif-icant being the following: A square of hOOm by UOOm is examined and the random value Z(x,y) is assumed to possess a realization lying on a portion of a sphere as shown in figure ( 2, ). The domaine is divided into squares of 25 x. 25m having 289 nodes in which the computations are performed. The apriori known function characterizing the behavior of the mean Z(x,y) was taken as a quadratic function of the form: f(x,y) = a Q + a^ x y + ' ' i *1 ^ ! , On the other hand the covariance was given by c ( Ua.) = e The results are given in table ) and are conform with what was expected. An interesting point to be mentioned, is that the computed variances are more sensitive to the location of the given information than,the gradient of the mean x ^ T h e »i>tcoM« tVe. i* -Jirvftotl^ J a t the nodes of the triangular element vised in the finite element procedure. Then the uncertainty analysis is conducted in a conventional way. UUrtjtO. of GiJVu lut-oAHAT, Mt>BtU UHKeR t* Ptt-1 CD A* 3 ^ein^c hOOEw 1 Heosu 2. o. 0U3 MOOt U HoDtL T- 0. OOIfl a. o.Sfc'S 0.2,4. Si pM-iU KoQfcL- 4 M o o s u 1 l.feZ, f tf; •V 0 I / i I •jM j. i ^ i J1 i •Y ; -A I 1 iC '! J : 3 v'' ; -- < ;Jj' J i* o. f i 1 a r-f 1 l-1 n 4 u 0 id H 1 % i ^ < ^ J" > 3 J ; f j i-i (. : « j 2 L! h* 0 J \ or <3 p d i tf ' J> yf a !5 * 5 j | * * , a <r j ? J c * 3 T _. _ j h-V JW _ i — 5 U 'I Q I «" » ii3 . ' j 3 ! <« '/ * [ 2 * 3 H « c £ 3 r z i i I n ^ I . t 7 c . j o: j 3; lV 4 > A - 1 I •r ! l _ , 1—^ -1 ! M d ^ ! Ull f ^ *  J s 3tS i« 7 A «- I J c I-1 1/ • I "I r? 3 ' ?! 2i t- ! 5 1! 3 (I •i r. o « ii * ? V i ,J < f> 3 H ' 'i ? S i, i i L L - * - -"7 o -J u-fl!i i ^ ! 2 ! ?! j i i I* iB u l> O j rtJ E I i i r ) j k ! ^ N 0 a * u 7 cJ d o I t J ct <£ 0 o hi r- * c 1 O a- G (> * i" •s q > j i» 3 ^ r A y tf V 3 A 0 J a li/ / sr J i M> I I • n 5 i i j! 1 c •a : i i r I J J o! < il H I *» 3 r | i : S! J ! O i S\i 5 i 3 ! c I I* P Q 0 t IV r 0 1 r i-(t j 7 3 •1 .1 . T i I." I i 2 I - I f I } ; <• > i: * I ^ ! i* i * e: 2 ; J L I I tr w A at J J <£ J •V a. I 3 i A \» / •J I-9 • u 1 I" id 2 •ii .o i x j' •r v t <r Y\ 1 C J-d o i»J f ! i I _ J "I 1—T ! -• t I a i a ai 5 O 1 31 ; c x » 5 1 0 1 t 3 j 9 ji a 2 <r u 3 a i. <r tf 0-•A * J J <L J k-J ; j J T L l -O ftTlOU T M L C o N C . t ? T u <\ C.OK1 C ^ S N i D fa, V N1 ( N U rvo PoUTifM oC POnoT&p 1>J <d»1lC_* ^V*.*) To ^ CUVt VoikiTi i^WHS-TBiL, uttH •WiiPKJT «»*• T+tiC Kit. ftppeuDix x « 2 The problem, therefore, consist to "minimize E[{z-z) ] the variance with A the constrain E[z(x,y}] - E[z(x,y)] = 0. In appendix three the above quantities are evaluated and was found to be: and The minimization of the variance will be obtained using the method of La-grangian multipliers as followed. The Lagrangian function being "it* The conditions to obtain the minimum are 0 ^ . Q {or o\l o y 4 - 0 ^ ^ ' * The unknowns being 6 and we obtain a linear system of a+fc equa-tions. The differentiation of with respect to fe* and ^ gives: First with respect to tf* . Second with respect to • The system then can be written as inc b b n o . / t k si The covariances c(z a,z 0) and c(z a >z) are obtained in appendix three and are given 1n the following relations: Then the linear system of equations becomes f ~ * M ) * \ C = 2 - l f l V 1 1 ? ^ f K ) = { V « where the b's and m's are the unknown quantities. Therefore, solving this system the estimator of the variable is defined by: A estimate is REFERENCES [1] J. R. Benjamin and C. A. Cornell, ProbabM ity, statistics and decisions for civil engineers, HcGraw-Hill Book Company,' hew York 1970. [2] B. Cambou, Applications of First-order Uncertainty Analysis In the Finite Elements Method In Linear Elasticity. [3] L. Estera, Second Moment Probabilistic Analysis of Statically Loaded Nonlinear Structures, Proceedings of Second International Conference In Auches, Germany on Applications of Statistics and Probability In Soli and Structural Engineering, 1975. [4] G. Keumbeln, Analytic Geology, McGraw-Hill Book Company, New York, 1963-[53 G. Matheron, Le Keigeorge Universal, Edlte par l' Ecola Nationale Superleure des Mines de Paris, 1969. [6] J. D. Padilla and E. H. Vanmarcke, Settlement of Structures on Shallow Foundations: A Probabilistic Analysis, M. I. T. Department of Civil Engineering, Research Report R7*»-9, January 1974. 

Stochastic Analysis of the Effect of Random Permeability Distributions on Confine Seepage

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Stochastic Analysis of the Effect of Random Permeability Distributions on Confine Seepage

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