Data-driven discovery of reservoir physics for reservoir surveillance

A method of determining bottom-hole pressure in a well for a reservoir in a subterranean formation, the method comprising:

• measuring, via one or more gauges, a plurality of flowrates of fluids being produced from the well;
• receiving the plurality of flowrates for the well;
• training a machine learning model to determine the bottom-hole pressure for the well based on the plurality of flowrates for the well;
• performing a virtual shut-in simulation via the trained machine learning model to determine the bottom-hole pressure for the well to avoid an actual shut-in of the well, wherein the virtual shut-in simulation is performed by an analysis on the machine learning model, the analysis comprising assigning a zero flow rate to the machine learning model to generate virtual pressure buildup data and performing a dynamic mode decomposition with control (DMDc) to formulate evolution of the bottom-hole pressure as a problem characterized by equation X′=AX+BY, wherein:
• X is a matrix representing time-delay embeddings of bottom-hole pressure values;
• X′ is a matrix of time-delay embeddings comprising predicted values for the bottom-hole pressure;
• A is a best-fit linear operator that maps matrix X to matrix X′;
• Y is a matrix of time-delay embeddings for flowrate data;
• and
• B is a linear operator that represents an impact of flowrate embeddings matrix Y onto matrix X′;
• and
• adjusting production of fluids from the reservoir based, at least in part, on the determined bottom-hole pressure from the virtual shut-in simulation that avoided the actual shut-in of the well.

A method of determining a flowrate for a well for a reservoir in a subterranean formation, the method comprising:

• measuring, via one or more gauges, a plurality of bottom-hole pressures for the well while fluids are being produced from the well;
• training a machine learning model to determine the flowrate for the well based, at least in part, on the plurality of the measured bottom-hole pressures for the well, wherein the machine learning model is represented by an evolution equation X′=AX+BY, wherein:
• X is a matrix representing time-delay embeddings of the received plurality of flowrates;
• X′ is a matrix of time-delay embeddings comprising the received plurality of flowrates;
• A is a best-fit linear operator that maps matrix X to matrix X′;
• Y is a matrix of time-delay embeddings for the plurality of the measured bottom-hole pressures;
• and
• B is a linear operator that quantifies an impact of matrix Y onto matrix X′, wherein both the A and B linear operators are determined using dynamic mode decomposition with control (DMDc) based on matrices X′, X comprising the received plurality flowrates, and the matrix Y comprising the plurality of the measured bottom-hole pressures;
• performing a virtual metering simulation via the trained machine learning model to determine flowrates for multiphase production to avoid an actual shut-in of the well, taking as an input a matrix representing time-delay embeddings of known flowrates and a matrix of time-delay embeddings for the plurality of known bottom-hole pressures corresponding to the known flowrates;
• determining a productivity index (PI) of the well based on an average reservoir pressure and at least one of the measured bottom-hole pressures, wherein determining the PI of the well comprises determining an inflow performance relationship for gas systems;
• and
• adjusting production of fluids from the reservoir based, at least in part, on the determined flowrates from the virtual metering simulation that avoided the actual shut-in of the well.