Reservoir properties from well logs using neural networks 1

Today, I read the dissertation ‘Reservoir Properties from Well Logs Using Neural Networks’. The reading parts are Summary, Introduction and some of Chapter 2 Neural Networks.

List of Acronyms:

CM: committee machine

MLP: multilayer perceptron network

MLR: multiple linear regression

MNLR: multiple nonlinear regression

OLC: optimal linear combination

MNN: modular neural network

CPI log: computer processed interpretation log

MWD: measurement while drilling

LMS: least mean square

MLFF: multilayer feed forward

Summary and Introduction

The basic unit of a CM is a MLP whose optimum architecture and size of training dataset has been discovered by using synthetic data for each application.

All the programming has been doing using MATLAB programming language and different functions from the neural network toolbox.

ANNs are most likely to be superior to other methods under the following conditions:

(1)  The data is ‘fuzzy’.

(2)  The pattern is hidden.

(3)  The data exhibits non-linearity.

(4)  The data is chaotic.

A single MLP, when repeatedly trained on the same patterns, will reach different minima of the objective function each time and hence give a different set of neuron weights. A common approach therefore is to train many networks, and then select the one that yields the best generalization performance.

CM, where a number of individually trained networks are combined, in one way or another, improves the accuracy and robustness.

CM was demonstrated to improve porosity and permeability predictions from well logs using ensemble combination of neural networks rather than selecting the single best by trial and error.

Chapter 2: Neural Networks

Mathematically the function of the neuron k can be expressed by

y_k=φ(u_k+b_k), where u_k=∑_(j=1)^m▒〖w_kj x_j 〗.

Hardlimit function

φ(ν)={█(1 if υ≥0@ 0 if υ<0 )┤

A symmetrical hard limit function is described as 

φ(ν)={█(+1 if υ≥0@ -1 if υ<0 )┤

It is most commonly used in pattern recognition problems.

Linear function

φ(ν)=ν

This activation function is used in pattern recognition and in function approximation problems.

Sigmoid function

φ(ν)=1/(1+exp⁡(-aν))

This is the most common form of activation function used in the construction of multilayer networks that are trained using BP algorithm.

Tomorrow, I will read more of the dissertation.

Chapter 4 conclusion

Today, I finish reading the Chapter 4 of the thesis.

Summary

k_along=∑_(i=1)^3▒〖α_i k_i 〗

k_across=1/(∑_(i=1)^3▒α_i/k_i )

From the distribution of possible shale permeabilities, a probability plot of k from very shaly intervals shows an LND with log k~N with log mean and log-variance.

From the LND, k is estimated by the following two equations:

(k_1 ) ̅=e^(logx) ̅ ∙ψ_n (s^2/2)

ψ_n (t)=1+(n-1)/n t+〖(n-1)〗^3/(n^2 (n+1))  t^2/2!+〖(n-1)〗^5/(n^3 (n+1)(n+3))  t^3/3!+⋯

After calculating k_1, we can calculate k_(2+3) by the same way.

This equation k_1 α_1+k_(2+3) (1-α_1 )=k can remove the effect of k_1.

SDR: Schlumberger-Doll Research model

TIM: Timur-Coates model

FFI: free fluid volume

BVI: bound fluid volume

LND: log-normally distributed

XRD: X-ray diffraction

Next week, I will read another book called ‘Reservoir Properties from Well Logs Using Neural Networks’.

Chapter 4 of the thesis

Today, I read the thesis ‘Permeability Characterization and Prediction a Tight Oil Reservoir, Edson Feld, Alberta’ Chapter 4 (4.1-4.4) and make a conclusion of this week's work.

After skimming and scanning on Chapter 4 of the thesis, I think that some parts of them is still helpful for my research. So I want to finish reading it. I will not read other parts of the thesis.

Summary

Extremely low and highly variable values and the changing scales of these variations are major challenges to tight rock permeability prediction.

NMR logs have been used to predict permeability in conventional sandstone rocks with good success but their application in tight rocks has been more problematic.

Two empirical models have been widely used in conventional reservoirs:

SDR: Schlumberger-Doll Research model.

TIM: Timur-Coates model.

SDR does not cover all pore information and ignores contributions from small ppores.

TIM depends on T2 cut-off determination. 33 ms is selected for T2 cut-off in sandstone, which is an overestimate in tight formations. Core analysis may resolve the problem as I see in other papers.

The NMR T2 relaxation time depends on fluid in the rock pores. There are mainly three relaxation components: bulk fluid process, surface relaxation, and magnetic field gradient diffusion.

Many researches show that pore size is log normal distribution (LND). Since T2 is directly proportional to pore size, the T2 spectrum is also expected to be LND.

The volume of investigation of the NMR response will likely include several lithofacies, the log (T2) distribution may be expected to be a mixture of several normal distributions.

People used three Gaussian distribution to decompose the log (T2) spectrum.

, where ,

,  was used to assess the match.

For the pore size related facies model, there are three lithofacies with weights of . If  is consistent with the proportion from core analysis, it suggests that interpretation of  is correct (fine pore-size).

The medium and coarse pore-size related facies are not easily identified in the tight Cardium formations.

In general, the decomposition interpretation is consistent with core characteristics. The results are shown as follows:

Tomorrow, I will finish reading Chapter 4.

Permeability Characterization and Prediction a Tight Oil Reservoir, Edson Feld, Alberta & Basic knowledge of NMR

Today, I read the thesis ‘Permeability Characterization and Prediction a Tight Oil Reservoir, Edson Feld, Alberta’ Chapter 1 (1.2 1.3 1.4) and learn about NMR online.

Summary

Core-based Permeability Prediction

These equations are accurate in relatively clean, consolidated sandstone with medium porosity (15%-25%).

In tight Cardium Formation, the unique and clean lithofacies is uncommon and shale plays an important role, which impairs the accuracy of these relationships.

1.3 In tight formations, it is difficult to directly measure the grain size distribution.

1.4 It is impossible to find an accurate general constant for all situations.

1.5 Recommend the c=250 and 79 for oil and gas, respectively.

1.6 empirical formula from laboratory tests

Conventional Log Permeability Prediction

1.7 Applicable only in homogeneous sandstones without significant amount of shale.

1.8 The key and difficult point of this model was the determination of the parameter w.

1.9 Applicable in tight formations.

1.10 An experimental estimator for permeability with Sw and resistivity.

1.11 A multiple regression model.

1.12 The model gave a weak correlation with core results.

These models were mainly suitable for high permeability or specific areas. Only the geological conditions from 1.9 are similar to Cardium tight foromation.

NMR-based Permeability Prediction

Two Empirical Models

A model based on 1.5: transverse relaxation time cutoff (T2 cutoff) selection, usually 33 ms for the T2 cutoff in sandstone, overestimation in tight formation compared to the core measured Swi, underestimate k.

SDR model: porosity from the NMR is usually underestimated, compared to core porosity in tight formations.

Thick-weighted permeability calculation model: There are not clean sand zones. So this method will overestimate the sand weights and k. Another weakness of this method is ignoring the shale k, which is unrealistic in tight formations.

For the NMR well log, the dominant T2 time is directly proportional to the pore size, ignoring fluid effects.

The mercury intrusion porosimetry (MIP) method has found pore size to be lognormal distribution.

Method in the thesis

Lithofacies and core analysis:

Pore throat aperture

Permeability distribution

Permeability heterogeneity

Clay minerals

Conventional logs-permeability calibration:

Factorial design

Check and connect well logs, probe permeability and core data

NMR prediction for permeability:

T2 spectrum can be obtained

T2 is proportional to pore size, log pore size has a Gaussian distribution

In core image analysis, specific facies (shale) volume proportions can be calculated according to color differences

Calibrate the model

Decrease uncertainty by Monte Carlo

Basic knowledge of NMR

NMR logging, a subcategory of electromagnetic logging, measures the induced magnet moment of hydrogen nuclei (protons) contained within the fluid-filled pore space of porous media (reservoir rocks). Unlike conventional logging measurements (e.g., acoustic, density, neutron, and resistivity), which respond to both the rock matrix and fluid properties and are strongly dependent on mineralogy, NMR-logging measurements respond to the presence of hydrogen protons. Because these protons primarily occur in pore fluids, NMR effectively responds to the volume, composition, viscosity, and distribution of these fluids.

NMR logs provide information about the quantities of fluids present, the properties of these fluids, and the sizes of the pores containing these fluids.

The volume (porosity) and distribution (permeability) of the rock pore space

Rock composition

Type and quantity of fluid hydrocarbons

Hydrocarbon producibility

NMR porosity is independent of matrix minerals, and the total response is very sensitive to fluid properties. Differences in relaxation times and/or fluid diffusivity allow NMR data to be used to differentiate clay-bound water, capillary-bound water, movable water, gas, light oil, and viscous oils. NMR-log data also provide information concerning pore size, permeability, hydrocarbon properties, vugs, fractures, and grain size.

Whether used as a standalone service or in combination with other logs and core data, NMR logs can provide an improved understanding of reservoir petrophysics and producibility. However, NMR logs are the most complex logging service introduced to date and require extensive prejob planning to ensure optimal acquisition of the appropriate data needed to achieve the desired objectives.

NMR physics

Atomic nuclei spin, and this angular moment produces a magnetic moment (i.e., a weak magnetic field). The NMR technique measures the magnetic signal emitted by spinning protons (hydrogen nuclei are the protons of interest in NMR logging) as they return to their original state following stimulation by an applied magnetic field and pulsed radio frequency (RF) energy. These signals, which are observed (measured) as parallel or perpendicular to the direction of the applied magnetic field, are expressed as time constants that are related to the decay of magnetization of the total system.

Polarization is not instantaneous—it grows with a time constant, which is called the longitudinal relaxation time, denoted as T1. Once full polarization (magnetic equilibrium) has been achieved, the applied static magnetic field, B0, is turned off.

T1 relaxation (polarization) curves indicate the degree of proton alignment, or magnetization, as a function of the time that a proton population is exposed to an external magnetic field.

T1 is the time at which the magnetization reaches 63% of its final value, and three times T1 is the time at which 95% polarization is achieved.

T1 is directly related to pore size and viscosity.

After application of a 90° pulse, the proton population dephases and an FID signal can be detected.

The FID signal measured in the x-y plane is called T2 —the transverse or spin-spin relaxation.

The primary objectives in NMR logging are measuring T1 signal amplitude (as a function of polarization), T2 signal amplitude and decay, and their distributions. The total signal amplitude is proportional to the total hydrogen content and is calibrated to give formation porosity independent of lithology effects. Both relaxation times can be interpreted for pore-size information and pore-fluid properties, especially viscosity.

In general, pulse NMR offers better methods to measure relaxation times and quantify liquid displacement in rock.

However, because the gradients produced by NMR-logging tools are relatively constant, they can be accounted for in T2 interpretation. In fact, the existence of these field gradients has actually proved beneficial in NMR logging.

The fundamental basis of Nuclear Magnetic Resonance (NMR) measurements on fluid-bearing rocks is that the decay or relaxation time of the NMR signals (T2) is directly related to the pore size. The NMR signal detected from a fluid-bearing rock therefore contains T2 components from every different pore size in the measured volume. Using a mathematical process known as inversion, these components can be extracted from the total NMR signal to form a T2 spectrum or T2 distribution, which is effectively a pore size distribution. From this distribution, various petrophysical parameters such as porosity, permeability, and free and bound fluid ratios can be measured or inferred.

Careful calibration of petrophysical interpretation models with core data is essential for obtaining accurate permeability and bound fluid answers from NMR logs, as compared to using default parameters in the petrophysical model.

The so-called ‘T2 cut-off’ in a T2 distribution is the T2 value that divides the small pores that are unlikely to be producible from the larger pores that are likely to contain free fluid. The integral of the distribution above the T2 cut-off is a measure of the free fluid (mobile fluid) in the rock, and is clearly influenced by the position of the T2 cut-off point, as shown in Figure 1. The portion of the curve below the cut-off is known as bound fluid and is made up of the clay bound fluid and the capillary bound fluid.

An accurate determination of the T2 cut-off point is essential for an accurate determination of recoverable reserves (mobile fluid). T2 cut-off can be easily determined in the laboratory by using two NMR measurements; one on a cleaned and re-saturated plug, the other on the same plug after it has been spun in a centrifuge to irreducible water saturation. T2 distributions are plotted for both data sets, along with the cumulative values of the distributions. The T2 cut-off is taken to be the point at which the cumulative value of the saturated distribution (yellow horizontal arrow in Figure 2) equals the final cumulative value of the irreducible distribution (vertical yellow arrow in Figure 2). The data plotting and calculation for this measurement are carried out automatically by the LithoMetrix software supplied as standard with every GeoSpec NMR core analyser.

Necessary apparatus

Tools needed to perform NMR log calibrations as described in this note include: a GeoSpec NMR Core Analyser with LithoMetrix or LithoMetrix Plus software; a centrifuge capable of removing all mobile (free) fluid from the core plug; and equipment to measure permeability. Although not required to perform log calibrations, pulsed field gradients on the GeoSpec instrument and optional GIT Systems software can allow users to perform additional advanced measurements to further enhance interpretation of NMR logs.

Tomorrow, I will read Chapter 4 of the thesis.

Construction of synthetic capillary pressure curves from the joint use of NMR log data and conventional well logs

Today, I read the paper ‘Construction of synthetic capillary pressure curves from the joint use of NMR log data and conventional well logs’ and find new ways to synthesize NMR bin porosities.

Summary

Synthesize capillary pressure curves in carbonate reservoirs from conventional and NMR logs by using a two-step approach:

1.     Simulate T2 (longitude relaxation time) distribution values

2.     The Pc values are predicted from CBPs through an inversion process

The correlation between capillary pressure and NMR T2 decay versus saturation data provides a methodology to derive synthetic capillary pressure curves.

Due to reasons such as complex pore type system, mineralogy changes, diagenetic overprints and variations in fluid content and saturation, it is hard for linear method for a sandstone reservoir to have an accurate estimate of pore-throat size and pore size distribution.

NMR T2 decay in porous media is described by equation: .

Assumptions:

(1)  Water-wet to mix-wet wettability

(2)  Lack of diffusion coupling

(3)  Compensation for hydrocarbon shift

,

In the fuzzy set theory, values have partial membership. A fuzzy inference system (FIS) is the process of formulating from a given input to an output using fuzzy logic.

Synthetic NMR log data and mercury injection Pc data were divided into 65% training sets to build the intelligent models and 35% testing sets for evaluating the reliability of the developed models.

As with the bin porosities, the TS-FIS and BP-NN models were employed to predict the Pc values.

The FL model with highest performance (small MSE) was chosen as the optimal model.

The comparison between measured and synthesized Pc curves is shown as follows:

This research shows the feasibility of the joint use of intelligent techniques and NMR log parameters (CBPs) to predict Pc-curves from conventional logs.

Intelligent systems could recognize the quantitative relationships between Pc and well logs data when NMR log and SCAL data are not available continuously throughout the reservoir interval.

Another use of the synthetic Pc data achieved in this study is to determine the Gas-Water Contact (GWC) and related information.

Tomorrow, I will read more papers and books on methods for pseudo NMR.

Application of ANN for well logs

Today, I read the paper ‘Application of ANN for well logs’ and know a little more about ANN process.

Summary

Architecture of an ANN includes a large number of neurons organized in different layers, with the neurons of one layer connected to neurons of another layer by means of adjusting weights.

It begins with randomly generated weights and the iterations are continued till the goal, which is to adjust them so that the error is minimal, is achieved.

The activation function of the input layer is tansig, which convolutes the input layer neuron weights with the input data. This is passed to the hidden layer where the product of weights and input from previous layer is integrated with the activating function purelin. Subsequently, the value is passed to the output layer consisting of a single neuron.

Using backpropagation learning algorithm, the network iterates and updates the weights of the input, output and hidden layer neuron. Iteration continues until the target error goal is reached.

Backpropagation learning algorithm:

1.     Levenberg-Marquardt training algorithm

2.     Error calculated uses Mean Squared Algorithm

One common problem during training is data over-fitting. To overcome the problem and prevent the network to memorize the examples, training data set is divided into three subsets: training set, validation set, test set.

This technique is completely data driven and does not require any prior assumptions.

LMA:

Like other numeric minimization algorithms, the Levenberg–Marquardt algorithm is an iterative procedure. To start a minimization, the user has to provide an initial guess for the parameter vector, β. In cases with only one minimum, an uninformed standard guess like  will work fine; in cases with multiple minima, the algorithm converges to the global minimum only if the initial guess is already somewhat close to the final solution.

The results of cases' comparison is shown below:

The improvements of the paper may be done:

1.     Use more wells’ data and give more cases to prove.

2.     The LMA finds only a local minimum, so maybe there are other algorithms for better results.

(The LMA interpolates between the Gauss-Newton algorithm (GNA) and the method of gradient descent. It is more robust than the GNA.)

Tomorrow, I will read more papers and books on methods for pseudo NMR.