Probabilistic Modeling

The primary aim of the model is to estimate the compositional volume fractions of a reservoir on the basis of borehole measurements. Due to incomplete knowledge, limited amount of measurements, and noise in the measurements, there will be uncertainty in the volume fractions. We will use Bayesian inference to deal with this uncertainty.

The starting point is a model for the probability distribution P(v, m) of the compositional volume fractions v and borehole measurements m. A causal argument “The composition is given by the (unknown) volume fractions, and the volume fractions determine the distribution measurement outcomes of each of the tools” leads us to a Bayesian network formulation of the probabilistic model,

In this model, P(v) is the so-called prior, the prior probability distribution of volume fractions before having seen any data. In principle, the prior encodes the generic geological and petrophysical knowledge and beliefs [30]. The factor nZ=1 P(miv) is the observation model. The observation model relates volume fractions v to measurement outcomes mi of each of the Z tools i. The observation model assumes that given the underlying volume fractions, measurement outcomes of the different tools are independent. Each term in the observation model gives the probability density of observing outcome mi for tool i given that the composition is v. Now given a set of measurement outcomes mo of a subset Obs of tools, the probability distribution of the volume fractions can be updated in a principled way by applying Bayes’ rule,

The updated distribution is called the posterior distribution. The constant in the denominator P(mo) = fv П ieObsP(m°iv)P(v)dv is called the evidence.

In our model, v is a 13 dimensional vector. Each component represents the volume fraction of one of 13 most common minerals and fluids (water, calcite, quartz, oil, etc.). So each component is bounded between zero and one. The components sum up to one. In other words, the volume fractions are confined to a simplex SK = {v|0 < Vj < 1,"LkVk = 1}. There are some additional physical constraints on the distribution of v, for instance that the total amount of fluids should not exceed 40% of the total formation. The presence of more fluids would cause a collapse of the formation.

Each tool measurement gives a one-dimensional continuous value. The relation between composition and measurement outcome is well understood. Based on the physics of the tools, petrophysicists have expressed these relations in terms of deterministic functions fj (v) that provide the idealized noiseless measurement outcomes of tool j given the composition v [30]. In general, the functions fj are nonlinear. For most tools, the noise process is also reasonably well understood — and can be described by either a Gaussian (additive noise) or a log-Gaussian (multiplicative noise) distribution.

A straightforward approach to model a Bayesian network would be to discretize the variables and create conditional probability tables for priors and conditional distributions. However, due to the dimensionality of the volume fraction vector, any reasonable discretization would result in an infeasible large state space of this variable. We therefore decided to remain in the continuous domain.

The remainder of this section describes the prior and observation model, as well as the approximate inference method to obtain the posterior.

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