Annual Review

Thesis Final Review 2021:

Fast and scalable metamodelling of reservoir flows via machine learning techniques

Student:

Pavel Temirchev

Ph.D. student, 4th year

Scientific advisor:

Dmitry Koroteev

Individual Doctoral Committee:

Dmitry Koroteev

Evgeny Burnaev

Ivan Oseledets

Finite-Differences modelling

The standard approach (ECLIPSE, TNAVIGATOR, OPM-FLOW)

\sim O(n^\alpha), \;\;\;\; \alpha=2\div3

\sim O(n^\alpha), \;\;\;\; \alpha=2\div3

\text{PDE\_solver}(s_0, \theta, u)

\text{PDE\_solver}(s_0, \theta, u)

Time

initial reservoir state:
pore pressure, saturation fields

porosity, permeability, relative permeability and PVT tables

control applied on wells:

BHP, injection rates

The computational complexity depends on the number of computational cells (the complexity of matrix inversion)

\theta

\theta

u_0

u_0

s_0

s_0

\frac{\partial s_0}{\partial t}

\frac{\partial s_0}{\partial t}

\frac{\partial s_1}{\partial t}

\frac{\partial s_1}{\partial t}

\frac{\partial s_2}{\partial t}

\frac{\partial s_2}{\partial t}

u_1

u_1

u_2

u_2

u_3

u_3

NDE-b-ROM metamodel:

Neural Differential Equations based Reduced Order Model

Time

\theta

\theta

u_{0:T}

u_{0:T}

s_0

s_0

\frac{\text{d}z_1}{\text{d}t}

\frac{\text{d}z_1}{\text{d}t}

\frac{\text{d}z_2}{\text{d}t}

\frac{\text{d}z_2}{\text{d}t}

z_1

z_1

z_2

z_2

z_3

z_3

ENCODER

\tilde\theta

\tilde\theta

\tilde{u}_{0:T}

\tilde{u}_{0:T}

z_0

z_0

\frac{\text{d}z_0}{\text{d}t}

\frac{\text{d}z_0}{\text{d}t}

DECODER

DECODING

ENCODING

DERIVATIVES

Conv3d_3x3, 8ch

Conv3d_3x3, 16ch, str=2

Conv3d_3x3, 32ch

Conv3d_3x3, 32ch,

str=2

Conv3d_3x3, 64ch

(s_0, \theta, u_{0:T})

(s_0, \theta, u_{0:T})

(z_0,\tilde \theta, \tilde u_{0:T})

(z_0,\tilde \theta, \tilde u_{0:T})

Transp3d_3x3, 16ch

Transp3d_3x3, 8ch

Transp3d_3x3, 32ch, str=2

Transp3d_3x3, 32ch

Transp3d_3x3 64ch, str=2

Transp3d_3x3 64ch

(s_t)

(s_t)

(z_t)

(z_t)

VGG-like autoencoder, LINK

(z_t,\tilde \theta, \tilde u_t)

(z_t,\tilde \theta, \tilde u_t)

ReLU

Conv3d_3x3, 32ch

ReLU

Conv3d_3x3, 4ch

(\frac{dz_t}{dt})

(\frac{dz_t}{dt})

Neural Ordinary Differential Equations, LINK

Neural Networks training

Minimisation problem

\mathbb{E}_{s_0, \theta, u} \sum_t ||s_t - \hat s_t||^2_2 \rightarrow \min_\phi

\mathbb{E}_{s_0, \theta, u} \sum_t ||s_t - \hat s_t||^2_2 \rightarrow \min_\phi

z_0 = E_s(s_0)

z_0 = E_s(s_0)

z_{t} = z_{t-1} + \frac{dz}{dt}(z_{t-1}, \tilde\theta, \tilde u_{t-1})\Delta t

z_{t} = z_{t-1} + \frac{dz}{dt}(z_{t-1}, \tilde\theta, \tilde u_{t-1})\Delta t

\hat s_t = D(z_t)

\hat s_t = D(z_t)

\tilde\theta = E_\theta(\theta)

\tilde\theta = E_\theta(\theta)

\tilde u = E_u(u)

\tilde u = E_u(u)

$\phi$ - a vector of neural network parameters

$\hat{s}_{0:T}$ - the solution obtained as follows:

1. encoding:

2. latent space forecast:

3. decoding:

\frac{dz}{dt}(z_{t-1}, \tilde\theta, \tilde u_{t-1})

\frac{dz}{dt}(z_{t-1}, \tilde\theta, \tilde u_{t-1})

z_{t} = z_{t-1} + \frac{dz}{dt}(z_{t-1}, \tilde\theta, \tilde u_{t-1})\Delta t

z_{t} = z_{t-1} + \frac{dz}{dt}(z_{t-1}, \tilde\theta, \tilde u_{t-1})\Delta t

stochastic optimization

backpropagation + ADAM

a neural network

0

1

2

3

4

Almost as fast as the standard convolutional layer
Only active cells are stored in the memory
Non-active cells participate the calculations only partially
We can take the distances between cells into account by adding them into layer's weights

Convolutions on regular graphs

		A	A
	A	A	A	A
	A	A	A	A	A	A
A	A	A	A	A	A	A	A
A	A	A	A	A	A	A	A	A
A	A							A	A
									A

x_0^{l+1} = \sigma \Big([ x_0^l, x_1^l, x_2^l, x_3^l, \text{na}^l] \cdot w^l +bias^l\Big)

x_0^{l+1} = \sigma \Big([ x_0^l, x_1^l, x_2^l, x_3^l, \text{na}^l] \cdot w^l +bias^l\Big)

\text{na}^l - \text{filler for non-active cell 4}

\text{na}^l - \text{filler for non-active cell 4}

Computation of production rates

Q_p(p, S_w, S_g, t) = T(t) M_p(p, S_w, S_g) (p - p_w)

Q_p(p, S_w, S_g, t) = T(t) M_p(p, S_w, S_g) (p - p_w)

T(t) = \frac{2 \pi K_{mult}(t) Kh}{\ln(r_o/r_w) + s}

T(t) = \frac{2 \pi K_{mult}(t) Kh}{\ln(r_o/r_w) + s}

M_p = \frac{k_{rp}(S_w, S_g)}{\mu_p(p)}

M_p = \frac{k_{rp}(S_w, S_g)}{\mu_p(p)}

We use a fully physics-based approach for production rates calculation.

where

Thesis Final Review 2021: Fa st and scalable metamodelling of reservoir flows via machine learning techniques Student: Pavel Temirchev Ph.D. student, 4th year Scientific advisor: Dmitry Koroteev Individual Doctoral Committee: Dmitry Koroteev Evgeny Burnaev Ivan Oseledets

Fast and scalable metamodelling of reservoir flows via machine learning techniques

Objectives and Tasks

Hydrodynamical Reservoir Simulation

Finite-Differences modelling

Prior art

ROMs:

Supervised Machine Learning

The training set of reservoirs

Expanding the training set using reservoir randomization

Applying standard transformations during sampling

NDE-b-ROM metamodel:

ENCODER

DECODER

DECODING

ENCODING

DERIVATIVES

Neural Networks training

Convolutions on regular graphs

Computation of production rates

Tightly approximates
commercial solutions

oil field Х

Tightly approximates
commercial solutions

oil field Х: production rates

Experimental results (on toy example)

Technical details

Well-known API

Computatuional time results

Compared with commercial simulator tNavigator

Publications

Publications

Conclusions

Conclusions