Auto-characterization of naturally fractured reservoirs drilled by horizontal well using multi-output least squares support vector regression

Abstract

Pressure transient response (PTR) of horizontal well in naturally fractured reservoirs (NFR) has a particular characteristic shape. This PTR is often used to estimate parameters of NFRs and detect their wellbore and boundary regimes. Interporosity flow coefficient (λ) and storativity ratio (ω) are two important parameters of the NFR that often estimated by matching process on the PTR. Since the matching techniques’ results are not often unique, in this study, the multi-output least squares support vector regression (MLS-SVR) is employed for simultaneous estimation of λ and ω. A databank of 500 PTRs for horizontal wells in naturally fractured reservoirs is generated by the finite element method, converted to the pressure derivative (PD) curves, and then used to develop and evaluate this auto-characterization paradigm. The predictive accuracy of the model is checked and validated by both smooth and noisy PTRs. The proposed model predicts ω and λ with overall absolute average relative deviations (AARD) of 0.186% and 3.754%, respectively. The correlation coefficients (R²) of 1 and 0.99992 are obtained for the prediction of ω and λ, respectively. The Leverage outlier detection technique justified that only less than 6% of the predictions are within the suspect region. This MLS-SVR model can be simply integrated with commercial pressure transient analysis (PTA) packages for accurate prediction of ω and λ even from the noisy PTRs.

Appendix 1. Multi-output least squares support vector regression

Support vector regression maps nonlinear patterns into higher dimensional feature space that can approach infinite dimensions (Khandelwal and Kankar 2011; Tikhamarine et al. 2019; Quan et al. 2020). It then applies linear regression to the mapped feature space (Chao et al. 2018; Xu and Chen 2019). Considering a dataset of N records existing in multi-dimensional feature space, its ith data record or element can be expressed as:

$$ \left[\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right),\mathrm{i}=1,2,\dots, \mathrm{N}\right] $$

Here, x_i and y_i represent an actual and predicted value of the ith data record. For such a data set, the support vector regression can be expressed as follows (Xu et al. 2013):

$$ \mathrm{F}\left(\mathrm{x}\right)=<\mathrm{w},\mathrm{x}>+\mathrm{b} $$

(9)

where <,> denotes the dot product of the matrix elements involving all the x data records, w is the weight vector of the SVR regression function, and b is an intercept of the SVR regression function.

Accepting a certain level of error (ε), the objective of SVR is to establish a function F(x) that estimates the values of y from x data for a training dataset that maintains deviations at or less than the value specified for ε. The risk function (R) can then be solved using appropriate optimization techniques (Chen et al. 2017; Deng et al. 2019; Liu et al. 2019a; Cao et al. 2020a, b, c; Qu et al. 2020).

$$ \mathrm{R}\left(\mathrm{f}\right)=\frac{1}{\mathrm{N}}\sum \limits_{\mathrm{i}=1}^{\mathrm{N}}\mathrm{L}\left(\mathrm{f}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{y}}_{\mathrm{i}}\right)+\frac{1}{2}{\left\Vert \mathrm{w}\right\Vert}^2 $$

(10)

where:

$$ \mathrm{L}\left(\mathrm{f}\left(\mathrm{x}\right)-\mathrm{y}\right)=\left\{\begin{array}{c}\left\Vert \mathrm{L}\left(\mathrm{f}\left(\mathrm{x}\right)-\mathrm{y}\right)\right\Vert -\upvarepsilon \kern3.25em \mathrm{if}\ \left.|\mathrm{f}\left(\mathrm{x}\right)-\mathrm{y}|\right\rangle 0\ \\ {}0\kern10.75em \mathrm{otherwise}\ \end{array}\right. $$

(11)

Equation (4) expresses an error insensitive loss function. ε determines the regression’s precision by essentially defining the radius of a cylinder surrounding the regression function, f(x), within which acceptable values may exist. By substitution of Eq. (3) into Eq. (2), it is possible to determine functions that fit the records of the SVR training subset with deviations of no more than ε.

The acceptable error components associated with SVR minimization can be further defined as follows (Smola and Schölkopf 2004; Chen et al. 2019):

(12)

This minimization involves the constraints listed in Eq. (5):

(13)

where C is a positive regularization constant, and are positive slack variables.

C is a metric that establishes a trade-off between a solution’s ability to be generalized across all the elements of a data subset (e.g., the SVR training subset) versus achieving acceptable levels of accuracy (expressed in terms of error tolerance by ε). and quantify the distance from the values to theboundaries values of the error cylinder defined by ε.

SVR can then be expressed as a dual problem in the form of Eq. (6) to be maximized (Burges 1998; Bian et al. 2016):

$$ \mathrm{maxmize}\kern0.75em \frac{1}{2}\sum \limits_{\mathrm{i},\mathrm{j}=1}^{\mathrm{N}}\left({\upalpha}_{\mathrm{i}}-{\upalpha}_{\mathrm{i}}^{\ast}\right)\left({\upalpha}_{\mathrm{j}}-{\upalpha}_{\mathrm{j}}^{\ast}\right)\left({\mathrm{x}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{j}}\right)-\upvarepsilon \sum \limits_{\mathrm{i}=1}^{\mathrm{N}}\left({\upalpha}_{\mathrm{i}}+{\upalpha}_{\mathrm{i}}^{\ast}\right)+\sum \limits_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{y}}_{\mathrm{i}}\left({\upalpha}_{\mathrm{i}}+{\upalpha}_{\mathrm{i}}^{\ast}\right) $$

(14)

$$ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}\sum \limits_{\mathrm{i}=1}^{\mathrm{N}}\left({\upalpha}_{\mathrm{i}}-{\upalpha}_{\mathrm{i}}^{\ast}\right)=0\\ {}{\upalpha}_{\mathrm{i}},{\upalpha}_{\mathrm{i}}^{\ast }>0\\ {}\mathrm{i}=1,2,\dots .,\mathrm{N}\end{array}\right. $$

(15)

Here, α_i and ${\upalpha}_{\mathrm{i}}^{\ast }$ are Lagrange multipliers derived from quadratic equation solutions.

The SVR single space function can then be mathematically expressed by Eq. (8):

$$ \mathrm{f}\left(\mathrm{x}\right)=\left({\upalpha}_{\mathrm{i}}-{\upalpha_{\mathrm{i}}}^{\ast}\right)\left\langle {\mathrm{x}}_{\mathrm{i}},\mathrm{x}\right\rangle +\mathrm{b} $$

(16)

The SVR dual space regression function is expressed by Eq. (9):

$$ \mathrm{f}\left(\mathrm{x}\right)=\left({\upalpha}_{\mathrm{i}}-{\upalpha_{\mathrm{i}}}^{\ast}\right)\mathrm{k}\left({\mathrm{x}}_{\mathrm{i}},\mathrm{x}\right)+\mathrm{b} $$

(17)

where k (x_i, x) indicates the kernel function that satisfies Mercer’s conditions.

Those data records in the dataset determined by Eq. (9) to have non-zero coefficients are the support vectors. A kernel function commonly applied in Eq. (9) is the radial basis function (RBF) expressed as Eq. (10):

$$ \mathrm{k}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{j}}\right)=\exp \left(-\upgamma \left\Vert {\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{j}}\right\Vert \right),\left(\upgamma >0\right) $$

(18)

where γ shows the width of the RBF.

The critical variables in establishing acceptable SVR optimization solutions (regression functions) and the complexity of those solutions are ε (Eq. 3), C (Eq. 4), and γ (Eq. 10). SVR optimization, therefore, focuses upon optimizing the variables ε, C, and γ.

In MLS-SVR, each data record in the dataset has multiple independent-dependent variables (Yan et al. 2020). For the case of our study, the digitized pressure derivative curve with 28 points is the independent variable, while ω and λ constitute the target vector.