WellPINN: Accurate Well Representation for Transient Fluid Pressure Diffusion in Subsurface Reservoirs with Physics-Informed Neural Networks

This paper introduces WellPINN, a novel workflow that utilizes sequentially trained physics-informed neural networks on shrinking subdomains to accurately model fluid pressure diffusion around wells throughout the entire injection period, overcoming previous limitations in capturing early-stage pressure dynamics.

The Gist

The Big Problem: The "Pixelated" Well

Imagine you are trying to draw a map of water pressure in a giant underground reservoir (like a massive sponge). In the middle of this sponge, there is a tiny well where water is being pumped out.

The problem is that the well is tiny (about the width of a pencil), but the reservoir is huge (the size of a football field).

If you try to draw this map using standard computer models (or even standard AI), the computer gets confused. It's like trying to draw a single, sharp pixel on a giant canvas. The AI tries to smooth things out because it prefers smooth lines, but the pressure right next to the well changes very sharply. Standard AI models often "blur" this sharp change, making the pressure look too low or missing the rapid changes that happen right when pumping starts. It's like trying to see a sharp mountain peak through a foggy window.

The Solution: WellPINN (The "Zoom-In" Strategy)

The authors created a new method called WellPINN. Instead of trying to draw the whole map perfectly in one go, they use a "zoom-in" strategy.

Think of it like taking a series of photographs to capture a landscape:

1. Photo 1 (The Wide Shot): You take a picture of the entire reservoir. You can see the general shape of the hills and valleys (the pressure far away from the well), but the tiny well in the center looks like a blurry dot.
2. Photo 2 (The Medium Zoom): You zoom in on the area where the well is. You take a new picture of just that smaller area. Now you can see the well better, but the very center is still a bit fuzzy.
3. Photo 3 (The Close-Up): You zoom in one last time, focusing only on the immediate area around the well. Now you can see the sharp details of the well perfectly.

WellPINN does this mathematically. It trains three separate AI models in a sequence:

• The first model learns the big picture.
• The second model learns the middle ground, using the first model's answer as a starting point.
• The third model learns the tiny area right around the well, using the second model's answer.

Finally, it stitches these three "photos" together into one perfect, high-definition map that is accurate from the edge of the reservoir all the way to the center of the well.

The Secret Ingredients

To make this work, the authors had to tweak two things in their AI recipe:

1. The "Time Lens" (Logarithmic Scaling):
   When water starts pumping, the pressure changes incredibly fast in the first few seconds, then slows down. Standard AI looks at time like a ruler with equal marks (1 second, 2 seconds, 3 seconds). This misses the fast action at the start.
   The authors changed the "ruler" to a logarithmic scale. Imagine a ruler where the first inch is huge (to see the fast changes) and the later inches get smaller and smaller. This lets the AI pay extra attention to the critical early moments of pumping.

2. The "Hard Fence" (Hard Constraints):
   Usually, AI guesses where the boundaries are. The authors built a "hard fence" into the math. This forces the AI to know exactly where the edge of the reservoir is and that the pressure must be zero there. It's like telling the AI, "You cannot draw outside these lines," which keeps the model from getting confused at the edges.

What They Found

The team tested this on a computer simulation of a 100-meter square reservoir with a 10-centimeter well.

• Old Way: The AI missed the pressure changes right next to the well and got the early timing wrong.
• WellPINN: The AI successfully predicted the pressure at the well with high accuracy, capturing both the fast changes at the start and the steady state later on.

They found that for this "zoom-in" method to work best, each zoomed-in area should be about 17% the size of the previous area. If the zoom is too aggressive, the AI gets confused again; if it's too gentle, it doesn't get close enough to the well.

The Bottom Line

This paper introduces a new way to use AI for underground fluid modeling. By breaking the problem into smaller, manageable steps (like zooming in with a camera) and adjusting how time is measured, they solved a long-standing problem: making AI models accurate enough to see the tiny, sharp details of a well inside a massive underground reservoir. This is a big step forward for simulating how reservoirs behave during real-world operations.

Technical Summary

Technical Summary: WellPINN

Problem Statement
Accurate representation of wells is critical for reliable reservoir characterization and the simulation of operational scenarios in subsurface flow models. While Physics-Informed Neural Networks (PINNs) offer a promising hybrid approach by integrating monitoring data with governing physical equations, existing PINN-based studies struggle to capture fluid pressure fields near wells, particularly during the early stages of injection. This limitation arises because analytical solutions treat wells as singularities (point sources), whereas Artificial Neural Networks (ANNs) perform best on smooth, continuous, and similarly scaled variables. Previous attempts to resolve this using "equivalent well functions" (e.g., Gaussian, Lorentz, or piecewise functions) typically require a large equivalent well radius (often $\ge$ 10% of the domain size) to achieve stable convergence. While sufficient for far-field predictions at late injection times, these approaches significantly underestimate well pressure ($p_w$) by up to 30% during late injection and fail entirely to capture early-time diffusion. This deficiency undermines the reliability of PINNs for history matching based on flow rates ($q_w$).

Methodology: The WellPINN Workflow
To address the multiscale challenge of representing a centimeter-scale well within a reservoir-scale domain (spanning three orders of magnitude), the authors propose WellPINN, a sequential training workflow that combines multiple PINN models. The methodology involves the following key components:

1. Domain Decomposition and Sequential Training: The modeling domain is decomposed into nested subdomains. The workflow begins by training a PINN ($D_1$) across the entire domain to capture the far-field pressure. Subsequent PINNs ($D_2, D_3, \dots$) are trained on progressively smaller subdomains centered on the well. The size of each new subdomain is defined by the equivalent well radius ($\bar{r}_{weq}$) of the previously trained model.
2. Iterative Radius Reduction: The equivalent well radius is reduced stepwise ($\bar{r}_{weq, D} = b \cdot \bar{x}_{max, D}$) until it matches the physical well radius. This allows the network to progressively refine the solution near the well without requiring a single network to resolve steep gradients across the entire domain simultaneously.
3. Hard Constraints and Composite Solution: To ensure continuity and strict adherence to boundary conditions, the authors employ a hard constraint approach. A composite pressure function $\hat{p}_c$ is constructed by summing the outputs of the trained PINNs, each weighted by a factor $\epsilon_n$ and multiplied by a piecewise approximate distance function $\tau_n$. This function $\tau$ ensures the solution is zero at domain boundaries and at the initial time.
4. Loss Function and Optimization: The training utilizes a single loss term based on the residual of the dimensionless governing partial differential equation (PDE). To avoid accumulating computational overhead, residuals from previous training stages are precomputed and added as fixed terms to the loss function of the current stage. The training uses the Adam optimizer followed by L-BFGS.
5. Input Scaling: The input time variable is log-scaled to account for early-time nonlinearity in pressure diffusion, a technique noted as novel for well representation in PINNs.

Key Results
The workflow was validated on a synthetic 2D case study involving a single injection well ($r = 10$ cm) in a $100 \text{ m} \times 100 \text{ m}$ domain with homogeneous permeability.

• Accuracy: The sequential training successfully reduced the maximum absolute error ($AE_{max}$) near the well from 0.53 in the initial domain-wide model ($D_1$) to 0.11 in the final model ($D_3$).
• Temporal Resolution: Unlike previous methods, WellPINN accurately captured pressure evolution throughout the entire injection period, including the critical early-time diffusion phase.
• Parameter Optimization: A parameter study identified an optimal ratio $b$ (equivalent well radius to subdomain size) between 0.1 and 0.17. The authors selected $b = 0.17$, which allowed three subdomains to effectively resolve the well.
• Computational Efficiency: By isolating PDE terms for each stage, the method maintained consistent training times and avoided the computational cost of evaluating all networks at every epoch.

Significance and Claims
The paper claims that WellPINN is the first PINN-based workflow capable of accurately inferring fluid pressure from pumping rates throughout the entire injection period, including early times, within a large domain. Its primary significance lies in:

• Bridging the Multiscale Gap: It provides a scalable solution for representing wells where the physical radius is orders of magnitude smaller than the reservoir domain, a challenge that single-network PINNs with standard tanh activation functions cannot solve.
• Inverse Modeling Potential: By enabling accurate inference of well pressure ($p_w$) based on flow rates ($q_w$) without relying on dense observation data, the workflow advances the potential of PINNs for inverse modeling and history matching of pumping well tests.
• Methodological Advancement: The study demonstrates that domain decomposition combined with logarithmic time scaling and hard constraints is necessary for accurate well representation using PINNs with tanh activation functions.

The authors note that while the current study focuses on a single well and homogeneous permeability, the workflow is theoretically extendable to heterogeneous domains and multi-well configurations, though the latter may require further investigation into computational efficiency and activation function scaling.