Digital twins for subsurface systems based on algebraic models

One of the key challenges addressed by digital twins (DT) is the long-term modelling and monitoring of subsurface system behaviour. Existing DT technologies primarily rely on physics-based models capable of simulating dynamic processes. Long-term forecasting often suffers from uncertainty in data, modelling equations and their parameters, initial conditions and accumulating errors.

DT for natural systems remains an unexplored opportunity at a juvenile stage. Challenges with DT design for natural systems are largely related to their complex and uncertainty multi-physics nature.

We propose algebraic approach for DT design, where system parameters/attributes are represented as constraints rather than as specific values. This approach enables generation of subsurface scenarios and analysis of possible occurrence of critical system states/event.

We model the system as a collection of interacting entities (agents), whose states are defined by sets of attributes. For instance, a geological layer is considered as an agent characterised by its geometry represented by a 3D mesh (X0) , elasticity (E) , porosity (φ), thermal conductivity (T), and other relevant attributes. The initial state S0 of the agent can be presented as a set of constraints.

S0:        E1≤E≤E2 Ʌ  F1≤φ≤F2 Ʌ T1≤T≤T2 Ʌ X0 ,

The geometry X0 can also be represented as a set of constraints that take into account structural/mesh uncertainty. Thus, constraints can be specified for the set of all agents/layers interacting with each other.

We define the semantics of the agent's actions using formalized transitions that changes the constraints on the attributes/agent's state. An example of such a transition is the change in the layer state according to a function that is constructed from a combination of the equilibrium equations F, the constitutive equation Q, which relates the stress σ and the strain ε, and the kinematic equation of the strain D.

S1=G(S0, F(X0,σ), Q(φ,E,T), D(X0, ε)) .

The next state S1 is determined by the change of the agent state with the modelling of this transition and also represents the conjunction of constraints. The resulting new state is checked for compatibility with the critical state Z(σ, σ_max) following the threshold constraint (eg fracture):

σ <= σ_max .

If conjunction  S1 Ʌ Z(σ, σ_max) is satisfied, then there are such layer attribute values for which it is true. Such attributes are represented by the corresponding constraints generated by the solver. Having such constraints, we can obtain scenarios by the method of backward modelling, which will lead to the initial state.

Formalized transitions can be built by considering other parallel processes that affect the change in the state of the agent, in particular thermal, chemical, fluid flow.

This approach increases capability for long-term forecasting because it operates with subsurface states/events constraints/conditions rather than parameter specific simulations.

DT can combine algebraic modelling with neural networks that classify the predictions of a certain event. Algebraic modelling of the agent's behaviour from the classified state will confirm the correctness of the classification and build the corresponding explanatory scenario.