i=1 P ( m v) P ( v), (2) P ( m)

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1 1 Introduction In the field of reservoir engineering as conducted at Shell Exploration and Production, the petrophysicist has to figure out the composition of underground formations based on tool readings. These tools however do not just measure one type of component, but are usually responsive to several components. Since these tools are real world appliances, their measurement readings also include some error. Due to these two features, it may be possible to come up with several compositions that would explain the measurements within their uncertainties. It is not unlikely that the measurements are misinterpreted by petrophysicist the composition that most closely matches the expectations is chosen and regarded as the solution. A more proper approach would be to report probabilities of solutions. 2 Petrophysical Model Equations All measurement tools are based on measuring physical properties of formation minerals. These properties include for example acoustic properties, electrical resistivity, neutron capture and natural radioactivity. For each of the tools a model equation exists. These equations model the ideal) outcome of the measurements based on the composition. The tool equations are a map f j : S K ε R defined by K f j v) c ji v i 1) where v i is the volume fraction of mineral i and S K ε is the set of all possible vectors of volume fractions: the region of the K dimensional real space such that K v i = 1, the weights c ji are given in table 1. The table is also given electronically at The composition vector v consists of the minerals listed in the table headers of table Construction of the Distribution In the context of reservoir engineering, there is always without measuring) some information available regarding the composition of any reservoir. This information may result from previous experience, general considerations, geographical location etcetera, and is known in advance. The actual measurements do not change these general beliefs, but they add detail in other words we use the information we gain from measuring to update these prior beliefs. The mathematical equivalent of such an update procedure is given by Bayes rule P v m) = P m v) P v) P m) P m v) P v), 2) where the prior information is included in the P v) term, and the addition of information due to the measurement is included in the likelihood term P m v). The actual form of these two components will be discussed in the next sections. 1

2 2.2 Likelihood The likelihood term models our belief in the measurements m given the actual composition v and the tool equations of course). In other words, given a certain composition this defines the probability distribution over the measurement values. There are of course various options for such a probability distribution to choose from. It seems most natural to model this as a Gaussian distribution because this distribution is symmetric around the expected value which corresponds hopefully to the ideal case value superimposed with some Gaussian distributed error. The Gaussian distribution also has the advantage that it is quite calculation friendly. The measured value can thus be expressed as v 1 v 2 v 3 v 4... v K c j1 cj2 cj3 c j4 c jk c M4 c M3 c M2 c M1 c MK m j... m M Figure 1: Graphical representation of the dependency of the observations on minerals m j = f j + ξ, ξ N 0, σ j ). 3) where f j denotes the value obtained from the tool model, and ξ the Gaussian distributed stochastic error with expectation 0. The graphical model is depicted in figure 1. The likelihood i.e. the probability to observe measurment m given v) is given by P m v) M exp j=1 m j f j v)) 2 2σ 2 j where σ j is the uncertainty of tool j, which is 0.05 for all tools. 2.3 Prior Information ), 4) The second component in equation 2 we need to specify is the prior. This codifies generic expert knowledge. In general, this knowledge may be quite substantial. In this exercise we will only consider the simplest case. 2

3 Based on previous experience or generic geological knowledge, there may exist a prior belief in the composition of a certain formation. This belief must be expressed in a probability distribution. On the set S K ε there exists a natural distribution for which the arguments sum to unity the Dirichlet distribution, given by P v α, µ) = Pv 1, v 2,..., v N α, µ) v αµ1 1 1 v αµ v αµn 1 N δ 1 ) N v i 5) where µ a vector of mean values, and α > 0 a shape/sharpness) parameter. One generally assumes αµ j > 1. In this case, the distribution is concave ), and the maximum is somewhere on the simplex. We assume that we don t know anything, and use µ = {1/N, 1/N,..., 1/N}, e.g. in the center of the set S K ε. In addition, we assume α = N. With this choice the prior does not depend on v. 3 Monte Carlo sampling Given the value of the vector m the objective is to compute the distribution over v. The most general approach, and also the most accurate is to use Monte Carlo sampling. Since the simplex is bounded 0 < v i < 1, it is a good idea to transform the simplex to the real space and sample in this real space instead. In general, if our problem is to sample from p v) and we have a mapping mapping v w, the correct propability distribution in the w space is given by p w) = p v) d v d w A further complication is that the mapping from v w is not bijective, because of the constraint j v j = 1, which makes the determinant of the Jacobian zero. Therefore, we must seek a bijective transformation from the K 1 dimensional v space to the K 1 dimensional w space. 3.1 R K 1 S K 1 ε Transformations Various transformations exist for mapping the set S K ε to K-dimensional reals. Here, we will use the centered logratio transformation clr). The clr is a map ψ : S K ε RK 1 defined by ) vj w j = ψ j v) = ln j = 1,..., K 1 6) g v) Note, that w is defined in terms of the first K 1 components of v and g v) 1 K 1 l=1 ) 1 K K 1 v l u=1 v 1 K u 7) is the geometric mean. The reason for using g as normalization is that equation 6 now maps to real space vectors with the property K j=1 w j = 0. The vice 3

4 versa transformation for the K-1 independent parts 1 j K 1) yields v j = ψ 1 j w) = exp K 1 w i exp w j ) ) + K 1 exp w i) 8) and the dependent K th component is given by v K = 1 K 1 v i. The elements of the Jacobian for transformation 8 are given by J ij = v K 1 i = δ ij v i v i v j + v i 1 v u ) 9) w j The determinant of the K 1 transformation 8 e.g. the transformation of the K-1 independent parts) is given by K 1 K 1 K J 1 v i = v i 10) j=1 v j Combining all information provided in the preceding sections, we obtain the following posterior distribution u=1 pw m) = pvw) m, c)pvw) α, µ) J with pv m, c) the measurement model given by eq. 4 f j given by eq. 1 and c ji given in table 1), pv α, µ) the prior given by eq. 5 and J the determininant of the Jacobian given by Eq Exercise Preparation: 1. Implement the petrophysical model as described, including the likelihood and the prior. 2. Write a program that implements a Metropolis sampling method. Use the transformation from v w as described in the text. Sanity checks: 1. To see whether you have correctly implemented 1) the model, 2) the nonlinear transformation from v to w and 3) your sampling method, reproduce fig by taking K = 3 and no measurement information. 2. Take now α µ = 1. Generate a random vector v 0 from the prior p v) and generate a measurement vector m 0 from the conditional distribution p m v 0 ). Use the MC sampler to estimate the posterior probability distribution p v m 0 ). Check whether the posterior is centered around v 0. Questions 4

5 1. Consider the vector v 1 = v quartz = 0.2, v clay = 0.5, v freewater = 0.1, v oil = 0.1) and all other components zero. Generate a measurement vector m 0 from the conditional distribution p m v 0 ). Use the MC sampler to estimate the posterior probability distribution p v m 0 ). Check whether the posterior is centered around v 0. Provide plots of the marginal distributions of the fraction of oil, freewater, and clay + quartz. The results should coincide with fig Optimize the Metropolis sampler by varying the witdh of the proposal distribution, by observing the fraction of accepted samples. Provide figures of the fraction of accepted samples as a function of the width of the proposal distribution. Figure 2: v 1 = v quartz = 0.2, v clay = 0.5, v freewater = 0.1, v oil = 0.1) References [1] D. MacKay. Information Theory, Inference, and Learning Algorithms. Cambridge University Press,

6 Tool Shale Coal Quartz Dolomite Calcite Clay Halite Pyrite Siderite Cbw Bw Fw Oil Gas L L L L L L L L L L L Table 1: Parameter table. Shale = v 1, Coal = v 2 etc. 6