Fast and scalable metamodelling of reservoir flows via machine learning techniques

Presenter:

Pavel Temirchev

Our team:

Dmirty Koroteev, Maxim Simonov, Denis Orlov,
Pavel Temirchev, Egor Illarionov, Dmitry Voloskov,
Ruslan Kostoev, Anna Gubanova, Mohammad Ebadi

Objectives and Tasks

Objectives:

Create a fast and scalable reservoir model based on machine learning algorithms:
- 3D, 3-phase flow
- Arbitrary reservoir and wells' geometry
- Generalizable between reservoirs
Develop history-matching and schedule optimization modules suitable for the model.

Tasks:

Formulate, generate and calculate training data
Develop load / store / dump framework for hydrodynamical models
Develop the architecture of the ML reservoir model, implement several algorithms
Train and test different models
Develop History-Matching procedure as a gradient descent process
Develop schedule optimization in the Reinforcement-Learning-like wayr the model.

Hydrodynamical Reservoir Simulation

Finite-Differences modelling

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

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

\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

u_0

s_0

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

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

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

u_1

u_2

u_3

Prior art

ROMs:

Dynamic Mode Decomposition
P.J. Schmid (2010). “Dynamic mode decomposition of numerical and experimental data”
J.L. Proctor, S.L. Brunton, J.N. Kutz (2014). “Dynamic mode decomposition with control”
Galerkin-POD projection
T. Lassila, A. Manzoni, A. Quarteroni, G. Rozza (2014). “Model order reduction in fluid dynamics: challenges and perspectives”
S. Chaturantabut, D.C. Sorensen (2010). “Nonlinear model reduction via discrete empirical interpolation”
Deep Residual Recurrent Neural Networks
J.N. Kani, A.H. Elsheikh (2018). “Reduced order modeling of subsur-face multiphase flow models using deep residual recurrent neural networks”
Embed to Control
M. Watter, J. Springenberg, J. Boedecker, M. Riedmiller (2015). “Embed to control: A locally linear latent dynamics model for control from raw images”
Z.L Jin, Y. Liu, L.J. Durlofsky (2020). “Deep-learning-based surrogate model for reservoir simulation with time-varying well controls”
LSTM + Variational Autoencoder model
P. Temirchev, M. Simonov, R. Kostoev, E. Burnaev, I. Oseledets, A. Akhmetov, A. Margarit, A. Sit- nikov, D. Koroteev (2020). “Deep neural networks predicting oil movement in a development unit”

Supervised Machine Learning

"Cat"

"Dog"

"Giraffe"

Object

Target variable

The training set of reservoirs

Object - a reservoir

Target variable

forecast of pore pressure and saturations (States)
forecast of the production rates (Rates)

Problem: how to find the target variable for an object?

Solution: let us compute it on the commercial simulator (tNavigator).

initial pore pressure and saturations (initial State)
porosity and permeability (Rock)
PVT, RPP (Tables)
reservoir geometry (Grid)
wells and their working schedule (Wells)

tNavigator

Expanding the training set using reservoir randomization

porosity

wells' control

Applying standard transformations during sampling

NDE-b-ROM metamodel:

Neural Differential Equations based Reduced Order Model

Time

\theta

u_{0:T}

s_0

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

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

z_1

z_2

z_3

ENCODER

\tilde\theta

\tilde{u}_{0:T}

z_0

\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})

(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)

(z_t)

VGG-like autoencoder, LINK

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

ReLU

Conv3d_3x3, 32ch

ReLU

Conv3d_3x3, 4ch

(\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

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

\hat s_t = D(z_t)

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

\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})

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

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)

\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)

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)}

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

where

Tightly approximates
commercial solutions

oil field Х

Tightly approximates
commercial solutions

oil field Х: production rates

view from above

Average on the Z axis:

x_{ij} = \frac{1}{n_z}\sum_k x_{ijk}

wells

Experimental results (on toy example)

tNavigator

NDE-b-ROM

Pressure

Oil saturation

Enhanced History Matching

Technical details

Well-known API

formatted files created either by hand or in model designer

RUNSPEC

GRID

PROPS

SOLUTION

SCHEDULE

.DATA

Can be used as:

```
python library
```
```
separate software with GUI
```

Supports CUDA and Pytorch computations

Provides output within diverse formats:

```
ECLIPSE binary
```
```
Pytorch \ numpy
```
```
other...
```

Computatuional time results

Compared with commercial simulator tNavigator

Time, sec

Model

NDE-b-ROM

tNavigator

1 GPU
20 sec

40 CPU
2400 sec

Tested on:

large real oil reservoir
with more than 3.000.000 of active cells
non-linear well trajectories, fish-bones
40 years of simulation

Publications

Will participate SPE RPT Conference this year as a co-author

Conclusions

We constructed the scalable reservoir model NDE-b-ROM based on Deep Learning techniques
The speed up is around 100x times
The framework for working with datasets was developed
The API similar to conventional hydrodynamical models was developed
History-matching and optimization procedures are under research

Serbia_meeting_8_12_20

By cydoroga

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)

Enhanced History Matching

Enhanced History Matching

Enhanced History Matching

Technical details

Well-known API

Computatuional time results

Compared with commercial simulator tNavigator

Publications

Conclusions

Serbia_meeting_8_12_20

More from cydoroga

Tightly approximates
commercial solutions

Tightly approximates
commercial solutions