Principle of stochastic simulation

For geological phenomena or geological processes, mathematical simulation is an effective test method, which uses a mathematical model to imitate and fit the geological phenomena or geological processes. Mathematical simulation is often to grasp the main geological conditions and geological features to construct various mathematical models. There are various mathematical models, which can be roughly divided into four categories: deterministic static model, deterministic dynamic model, stochastic static model and stochastic dynamic model. However, the phenomena encountered in the actual geological work often have both certainty and randomness, so it should also increase this kind of mixed model. Because many geological phenomena, geological processes are with randomness, so the stochastic mathematical model is often used. With this model for simulation, generally always use the computer to generate a certain statistical characteristics (such as obey a certain probability distribution, with a given mathematical expectation and variance) of the pseudo-random number, and then use a variety of mathematical operations to solve various problems, which is also commonly known as the Monte Carlo method (is a statistical test method).

1. The concept of stochastic simulation

Any unknown number z can be regarded as a random variable (RV) Z. The probability distribution of this random variable describes the uncertainty about z. A random variable is a variable that is distributed according to a certain probability. A random variable is a variable that can take on different values according to a certain probability (frequency) distribution. Random variables are generally denoted by an uppercase letter Z, and the corresponding values are denoted by a lowercase letter z. The model of a random variable, or rather the probability distribution of a random variable, usually depends on the spatial location, so we use the notation Z(u), where u is a vector of location coordinates. The random variable also depends on the information that is already available, that is, its probability distribution changes as the information about the unsampled value Z(u) increases.

The cumulative probability distribution function (cdf) of a continuous random variable Z(u) can be expressed as follows:

F(u; z) = Prob｛Z(u) ≤ z｝

When the cdf is related to some pre-existing information, and if there are n neighboring sample data points Z(uα) = z(uα), with α = 1, 2, .... n then the conditional cumulative distribution function (ccdf) should be used for the random variable Z(u):

F(u; z | (n)) = Prob｛Z(u) ≤ z | (n)}}

In the case of the discrete variable Z(u), there are k different values available (k = 1, 2, ..., K), and a similar conditional cumulative distribution function is:

F(u; k | (n)) = Prob｛Z(u) = k | (n)}

The cdf characterizes the prior probability distribution of the uncertainty of the unsampled value of z(u); the ccdf is the posterior probability distribution of the random variable after an informative sample set (n) has been available. Any statistical or simulation algorithm with a predictive function is a process of obtaining the posterior distribution by constantly modifying the prior probability distribution, only some algorithms have this as their goal, while others have this as their goal directly or indirectly. It is worth noting that the conditional cumulative distribution function (ccdf) is a function of spatial location, the number of samples, the location of the data, and the value of the samples.

With a ccdf, one can derive multiple optimal estimation models for the unsampled point z(u) that are consistent with the mean of the ccdf, as well as multiple realizations of the probability intervals. Further, with a single ccdf it is possible to give an arbitrary number of simulated output values z(l) (l = 1, 2, ..., L) using Monte Carlo sampling. The computation of the posterior ccdf and the Monte Carlo implementation of the output values are at the heart of all stochastic simulation algorithms.

Stochastic simulation is the process of modeling the spatial distribution of Z(u) with a stochastic model that is selectable, equiprobable, and highly accurate. Each realization is denoted by the superscript l: ｛zl(u), u ∈ A｝. A simulation is conditional if there is hard data at certain locations and consistent with the simulation realization zl(uα) = z(uα), otherwise it is unconditional.

Stochastic simulation recovers small spatial variations in reservoir structural property parameters by providing multiple optional, equal probability, high-accuracy spatial distributions of these parameters, and comparing the variability (variability) across the images. It has a big difference with interpolation, which is mainly manifested in three aspects: (1) Interpolation is a local optimal estimation, which only considers the accuracy of the local estimation value without considering the spatial correlation of the estimation value; whereas stochastic simulation emphasizes the overall correlation of the results, which provides a measure of the uncertainty of the reservoir attribute space on the whole. (2) Interpolation has a smooth effect and ignores the subtle changes of reservoir attribute parameters, and it is suitable for the estimation and prediction of reservoir parameters with less drastic changes; whereas stochastic simulation reflects the subtle changes of reservoir attribute parameters by systematically adding "stochastic noise" to the interpolation model, which better represents the fluctuation of the real curve. (3) The interpolation method can only produce a deterministic model; whereas in stochastic modeling, multiple realizations (models) can be produced, and the differences in these realizations (models) are exactly the reflection of the spatial uncertainty of the reservoir parameters.

2. Principle of conditional simulation

In geostatistics, the object of study is a regionalized variable, which is also stochastic, so the Monte Carlo method is also used for mathematical simulation. However, it is higher than the traditional statistical simulation conditions, not only requires pseudo-random numbers to obey a certain probability distribution, with a given mathematical expectation and variance, but also to maintain a certain spatial autocorrelation, that is, to maintain the same covariance function or variance function with the actual data. This is because regionalized variables have not only a random aspect, but also a spatial structural aspect. This is what is known in geostatistics as simulation, or unconditional simulation.

Conditional simulation means to add another condition to the above simulation, that is, to require that the simulated value at each observation point is equal to the measured value at that point. This requirement is also called the simulation conditioning (or conditioning the measured data), so the unconditioned simulation is called unconditioned simulation.

Conditional simulation is unique to geostatistics and can be considered a new Monte Carlo method. It has several features compared with the traditional Monte Carlo simulation: (1) it can keep the spatial autocorrelation function of the variables unchanged, and thus it is more suitable to be applied to the simulation of the regionalized variables; (2) it makes the simulated value at the observation point equal to the measured value, and thus the more observation points, the closer the simulation is to the objective reality; (3) it has the possibility of realizing the simulation of the three-dimensional space because it proposes the " Steering Belt" method can easily extend one-dimensional simulation to three-dimensional simulation, saving a lot of memory and machine time.

In short, conditional simulation occupies a very important position in geostatistics, and it can be used together with kriging estimation to solve many practical problems in geology and mining.

Let Z(x) be a regionalized variable that satisfies the second-order smoothness assumption, E[Z(x)] = m, and there exists a covariance function C(h) and a variance function γ(h). To find the conditional simulation of ZSC(x) for Z(x) is to find a reality of the regionalized variable ZSC(x) that is isomorphic to Z(x) and the simulated value is equal to the real value at the real point, i.e.

ZSC(xa)=Z(xa)

The so-called isomorphism means that ZSC(x) has the same mathematical expectation and the same C(h) or γ(h) with Z(x) ) and distribution histogram.

First, we consider the true value Z(x) at any point x, and its kriging value Their difference is an unknown error, because Z(x) itself is unknown. Thus, Z(x) can be expressed as the sum of (x) and this error:

Geology of Oil and Gas Field Development

The kriging method has a special property that the kriging error is orthogonal to (independent of) the kriging estimate or that is:

Geology of Oil and Gas Field Development

We then consider a regionalized variable, ZS(x), which is independent of and isomorphic to Z(x) (the real is an unconditional simulation of Z(x)). (x) the unconditional analog of Z(x)), when given a realization of ZS(x), the value ZS(xa) at the original observation point xa, a = 1, 2, ..., n is known. When we apply the kriging method to this same data configuration ZS(xa), we will get the kriging valuation

Geology of Oil and Gas Field Development

Since Z(x) is isomorphic to ZS(x) and the data configuration is the same, the weighting coefficients λα are also the same, i.e.

Geology of Oil and Gas Field Development

Geology of Oil and Gas Field Development

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We can prove that ZSC(x) is the conditional simulation of Z(x) that we require, which satisfies all the conditions of the conditional simulation and can be practically derived.

The above equation can be written as:

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or

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That is to say, to request the conditional simulation value ZSC (x), first find a non-conditional simulation value ZS (x), and then carry out the kriging estimation of the difference [Z (xα) - ZS (xα)] on the measured point xα, and then sum them together. ZSC (x) can be obtained by adding the two. This reduces the number of operations to solve the kriging system.

3. Introduction to Stochastic Simulation Methods

The models used in stochastic simulation can be categorized into two main groups, namely, discrete models and continuous models. Discrete models are used to describe discrete geologic features and large-scale inhomogeneities of reservoirs. For example, they are used to describe the spatial location and distribution range of sand and mudstone, as well as the distribution of faults and fractures, and to characterize the spatial distribution of various sedimentary phases or rock phases. Continuous simulation methods are used to describe the geological phenomena of continuous changes in the reservoir, such as reservoir physical properties, seismic velocity, and burial depth of geological interfaces. In some cases, it is necessary to combine the discrete and continuous models to form a two-stage model. Among them, the discrete model describes the large-scale inhomogeneity of the reservoir and the continuous model describes the small-scale variability of the reservoir.

Currently, stochastic simulation is widely used in oil and gas reservoir modeling, and there are many methods, mainly Boolean simulation, sequential Gaussian simulation, sequential indication simulation, truncated Gaussian simulation, probability field simulation, annealing simulation, fractal stochastic domain simulation, Markov random field simulation, mosaic process simulation and so on. The following is a brief introduction to the wider application, the development of a more mature several simulation methods:

◎ Boolean simulation (Boolean simulation): the method is the simplest of the stochastic simulation method, belongs to the non-conditional simulation. It is mainly used to establish discrete models, such as sand body lattice plane, profile or three-dimensional spatial distribution model. Therefore, this simulation can be used to model the morphology, size, location and arrangement of sand bodies in space.

Sequential Gaussian simulation: Based on sequential simulation, the ccdf of each simulation node is determined by kriging the mean and variance. The algorithm is robust and widely used in the realization of generating continuous Gaussian distributed variables. Requires data to be normally distributed, otherwise transformed to normal distribution using normal score.

◎Sequential indicatorsimulation method (Sequential indicatorsimulation): proposed by A.G. Journel and F. Alabert in 1978. Sequential variables and type variables can be simulated. It is almost similar to sequential Gaussian simulation except that the ccdf at each point is determined by the indicator kriging rather than by kriging the mean and variance based on the Gaussian distribution assumption.

Truncated Gaussian simulation: This method first generates a Gaussian random field using an instructed simulation method, and then truncates the Gaussian values to obtain a simulation of the type variable. This method is easy to use, fast, flexible, and can be used to simulate discrete features, and is particularly suited to simulating sedimentary reservoirs with simple phase sequences, such as deltas.

Probability field simulation: In order to save machine time, the estimation of local ccdf is separated from the simulation of probability field. A simulated probability field reflecting spatial correlation is used to post-process the local cumulative distribution function (lccdf), which can be obtained by any estimation method. In a probability field simulation, the probability values used to sample from the lccdf are correlated with each other. In probabilistic field simulations, the user himself can control the correlation, usually utilizing the same spatial correlation pattern as the correlation pattern of the parameter being simulated.

Fractal random field simulation (Fractal simulation): a simulation method that estimates plus simulation error (ESE). Based on the assumption that geological phenomena satisfy fractal characteristics, the geometry of the fractal distribution (discontinuity, i.e., inability to fill space) is quantitatively characterized by the fractal dimension. In an unconditional simulation of a fractal simulation, each value is a weighted average of a number of sine and cosine functions. Once the unconditional simulation is generated, sampling is done at nodes with the original data. This sampling provides us with the same data configuration as that used to interpolate the raw data, but the values may be different. The samples are interpolated for the simulation, and the difference between the smoothed graph and the infinitely sampled unconditional simulation is determined by comparing the difference between the two at the original data points.