WellPINN: Accurate Near-Well Pressure Inference

• WellPINN is a workflow using physics-informed neural networks that improves near-well pressure resolution by training on shrinking subdomains centered on the well.
• It employs iterative reduction of the equivalent well radius with Gaussian source regularization and hard-constrained boundary conditions to capture steep early-time gradients.
• Validation studies show marked error reductions across training stages, demonstrating the method’s robustness for transient pressure diffusion in subsurface reservoirs.

Searching arXiv for the cited WellPINN-related papers to ground the article. Tool call: arxiv_search({"query":"WellPINN accurate well representation transient fluid pressure diffusion subsurface reservoirs physics-informed neural networks", "max_results": 5, "sort_by": "relevance"}) WellPINN denotes a PINN workflow for transient fluid pressure diffusion in subsurface reservoirs that is designed to improve pressure resolution in the vicinity of wells, particularly during the early stage after injection begins, by combining the outputs of multiple sequentially trained PINN models on shrinking subdomains with a simultaneously reducing equivalent well radius (Walter et al., 12 Jul 2025). A broader interpretive usage is also suggested by the literature on certified PINNs for parameterized PDEs: the 2022 “certified wavelet-based PINN” is not explicitly named “WellPINN,” but it embodies a well-posed, wavelet-enabled, and certified PINN construction for PPDEs (Ernst et al., 2022).

1. Terminology and conceptual scope

In its strict sense, WellPINN is the name of the 2025 workflow introduced for “Accurate Well Representation for Transient Fluid Pressure Diffusion in Subsurface Reservoirs with Physics-Informed Neural Networks.” The method directly addresses the difficulty of capturing fluid pressure near wells in reservoir models, where singular or near-singular behavior, steep early-time gradients, and severe scale separation challenge standard PINN training. The workflow is defined by three coupled ideas: sequential training on shrinking subdomains centered at the well, iterative reduction of the equivalent well radius toward the actual well radius, and superposition of sub-PINNs under hard-constrained boundary and initial conditions (Walter et al., 12 Jul 2025).

The near-well difficulty arises because pumping or injection induces rapid, highly localized transient changes, while practical reservoir domains span orders of magnitude in length. In the formulation summarized for WellPINN, standard MLP PINNs with smooth activations such as tanh tend to resolve the far field more easily than the near-well region, and equivalent-source replacements of the well can stabilize training only at the cost of reduced pressure fidelity if the equivalent well radius is chosen too large. A central misconception addressed by the workflow is therefore that a single global PINN with a stabilized equivalent well source is sufficient for accurate wellbore-pressure reconstruction over the full injection period; the reported results indicate that such global setups can miss early-time diffusion fronts and under-represent near-well pressure.

A second terminological issue concerns the 2022 certified wavelet-based PINN. That paper consistently refers to a “certified wavelet-based PINN,” not to “WellPINN.” The association is interpretive rather than nominal: the proposed method is well-posed through stable variational formulations, wavelet-informed through residual expansion in adaptive biorthogonal wavelets, and certified through computable error bounds. This suggests a broader taxonomy in which “WellPINN” can refer either to the specific near-well reservoir workflow or, more loosely, to PINN constructions whose training objective is aligned with a stable residual norm and an explicit certification mechanism.

2. Physical model and well representation

The 2025 WellPINN paper considers transient pressure diffusion for a single-phase, slightly compressible fluid in a 2D Cartesian domain, with gravity neglected in the plane. The dimensional PDE is written as

$$S_s' \frac{\partial p}{\partial t} - \frac{1}{\mu}\nabla\cdot(k\nabla p) - f(x,y,t)=0,$$

where $p(x,y,t)$ is pressure, $k(x,y)$ is intrinsic permeability, $\mu$ is viscosity, $S_s'$ is the specific storage expressed in pressure units, and $f(x,y,t)$ is a volumetric source term representing the well (Walter et al., 12 Jul 2025).

The well is represented not by an explicit Neumann condition on a physical well surface, but by a Gaussian equivalent source in a 2D domain of unit thickness,

$$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$

with the well located at $(x_0,y_0)$. In the case study, $Q(t)$ is constant and $\sigma$ is linked to an equivalent well radius $p(x,y,t)$0. This source regularization avoids direct singularity handling while preserving a tunable approximation of the well’s spatial support. The paper explicitly notes that the choice of equivalent radius strongly affects accuracy, especially at early times.

The computational setting is a square domain

$p(x,y,t)$1

with a single injection well at the center and time interval $p(x,y,t)$2. The initial and outer boundary conditions are imposed as

$p(x,y,t)$3

Radial symmetry emerges from the physics of the centered well in a homogeneous setting, but the PINN itself is formulated on the Cartesian domain. The Theis solution is discussed only as analytical context for radially symmetric homogeneous cases; WellPINN does not use that solution in training.

3. Sequential superposition on shrinking subdomains

The defining methodological feature of WellPINN is a staged “zoom-in” construction. A sequence of PINNs is trained on nested subdomains $p(x,y,t)$4 centered at the well. At each stage $p(x,y,t)$5, the method uses a smaller subdomain half-extent $p(x,y,t)$6 and a reduced equivalent well radius $p(x,y,t)$7. The schedule is prescribed by

$p(x,y,t)$8

where $p(x,y,t)$9 is the ratio between equivalent well radius and current subdomain half-extent. The minimum number of stages needed to reach the real well radius $k(x,y)$0 is given by

$k(x,y)$1

For the tested setting, the paper finds $k(x,y)$2 as optimal (Walter et al., 12 Jul 2025).

The Gaussian width at stage $k(x,y)$3 is selected so that the Gaussian amplitude at $k(x,y)$4 equals a target $k(x,y)$5:

$k(x,y)$6

and for a centered well this reduces to

$k(x,y)$7

This ties the equivalent source width directly to the staged reduction of the well radius.

The composite pressure surrogate at stage $k(x,y)$8 is a superposition of all previously trained subnetworks,

$k(x,y)$9

where $\mu$0 is the $\mu$1-th PINN output, $\mu$2 is a hard-constraint “distance” function, and $\mu$3 is a scalar stage weight. The functions $\mu$4 are constructed to vanish at $\mu$5 and on $\mu$6, thereby enforcing the initial and Dirichlet boundary conditions multiplicatively rather than through explicit penalty terms. The composite construction is reported to guarantee continuity across subdomain boundaries without explicit interface loss terms.

Training cost is stabilized by precomputing the PDE residuals from earlier stages and keeping only the current stage trainable. The stage loss is

$\mu$7

Before training stage $\mu$8, the current stage scale is updated by

$\mu$9

According to the reported interpretation, this balances the magnitude of the current operator with the remaining source after subtracting prior residuals, and thereby stabilizes staged optimization.

4. PINN formulation, sampling, and optimization

The general PINN loss given for context is

$S_s'$0

with

$S_s'$1

The WellPINN implementation itself uses only the PDE residual, because BCs and IC are imposed through the hard constraints $S_s'$2:

$S_s'$3

This choice removes the need for separate IC/BC penalty balancing and is presented as one of the reasons the workflow remains stable near the well (Walter et al., 12 Jul 2025).

The network architecture is an MLP with depth $S_s'$4 width equal to $S_s'$5, tanh hidden activations, and softplus output activation. Optimization proceeds in two phases: Adam for $S_s'$6 epochs with exponential learning-rate decay from $S_s'$7 to $S_s'$8, followed by L-BFGS for approximately $S_s'$9 steps. Each stage uses $f(x,y,t)$0 collocation points. Spatial inputs $f(x,y,t)$1 are min-max scaled, while time uses logarithmic scaling; the injection start is shifted to $f(x,y,t)$2 to stabilize early-time learning.

Sampling is explicitly biased toward the difficult regime. Collocation points are radially refined toward the well according to a power-law distribution, and temporal sampling is concentrated toward early times through log-time scaling. The paper interprets the staged decomposition and $f(x,y,t)$3 scaling as a curriculum that focuses each stage on the hardest region, namely the region nearer to the well, while limiting interference from already-resolved far-field components.

The paper also describes inverse-modeling extensions, although they are not exercised in the reported experiment. If monitoring data $f(x,y,t)$4 are available, a data term

$f(x,y,t)$5

can be added. Likewise, parameters such as $f(x,y,t)$6, $f(x,y,t)$7, $f(x,y,t)$8, $f(x,y,t)$9, $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$0, $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$1 or $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$2, and skin can be treated as trainable quantities, and uncertain pumping rate $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$3 can be parameterized by a spline or low-order basis and optimized through automatic differentiation via the source term.

5. Validation, error behavior, and parameter dependence

The validation study uses synthetic ground truth from OpenGeoSys (OGS v6.4.3) on the same diffusion PDE. The setup consists of the square domain $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$4, a central injection well, injection rate $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$5, and real well radius $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$6. The material is homogeneous and granite-like, with

$$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$7

fluid compressibility $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$8, solid compressibility $$f(x,y,t)=Q(t)\frac{1}{2\pi\sigma^2}\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\sigma^2}\right),$$9, Biot coefficient $(x_0,y_0)$0, and

$(x_0,y_0)$1

Initial and outer boundary conditions are $(x_0,y_0)$2 (Walter et al., 12 Jul 2025).

The reported WellPINN configuration uses three subdomains, with the ratio $(x_0,y_0)$3 at each stage, $(x_0,y_0)$4 collocation points per stage, and $(x_0,y_0)$5 training epochs per stage ($(x_0,y_0)$6k Adam and $(x_0,y_0)$7k L-BFGS). On an NVIDIA RTX A4500 GPU, wall-times are reported as $(x_0,y_0)$8 for D1, $(x_0,y_0)$9 for D2, and $Q(t)$0 for D3.

Accuracy improves markedly over the stages. Along the line $Q(t)$1, the maximum absolute error $Q(t)$2 decreases from $Q(t)$3 in D1 to $Q(t)$4 in D2 and to $Q(t)$5 in D3. The mean absolute error in D3 is approximately $Q(t)$6, and the maximum absolute PDE residual decreases from $Q(t)$7 in D1 to $Q(t)$8 in D3. Temporal profiles at $Q(t)$9, $\sigma$0, $\sigma$1, and near the boundary at $\sigma$2 are reported to track the OGS solution well over the entire injection period, with boundary values matched nearly perfectly due to the hard constraints. At early time $\sigma$3, the advancing pressure front is captured, with absolute error typically between $\sigma$4 and $\sigma$5 outside the front; at late time $\sigma$6, the reported $\sigma$7 is approximately $\sigma$8, with the largest discrepancies at corners where the square geometry breaks radial symmetry.

A parameter study varies $\sigma$9 over the range $p(x,y,t)$00, with five realizations per value and metrics given by MAE on the well ($p(x,y,t)$01), domain MAE ($p(x,y,t)$02), and domain mean squared residual ($p(x,y,t)$03). The best overall accuracy is reported for $p(x,y,t)$04. For $p(x,y,t)$05, the shrinking equivalent well area and steeper source gradients degrade accuracy, and adding more collocation points near the source helps little with standard tanh activations. For $p(x,y,t)$06, $p(x,y,t)$07 increases because $p(x,y,t)$08 no longer properly collapses to $p(x,y,t)$09 within three stages, although the paper notes that a fourth stage could compensate. By contrast, $p(x,y,t)$10 tends to decrease as $p(x,y,t)$11 grows, which makes optimization easier but harms near-well fidelity.

6. Relation to certified wavelet-based PINNs, limitations, and implications

The 2022 paper “A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations” treats parameterized elliptic PDEs and constructs PINNs together with a computable upper bound of the error. It considers both a standard variational formulation and an optimally stable ultra-weak formulation. In the standard case, if $p(x,y,t)$12 is an approximation in a Hilbert space $p(x,y,t)$13, the residual functional is

$p(x,y,t)$14

and under coercivity or a uniform inf-sup condition one has the certified bound

$p(x,y,t)$15

In the ultra-weak formulation, with trial space $p(x,y,t)$16 and test space $p(x,y,t)$17 equipped with

$p(x,y,t)$18

the method achieves optimal stability $p(x,y,t)$19 under assumptions (B*1) injectivity and (B*2) density of range, and satisfies the error-residual identity

$p(x,y,t)$20

Residuals are expanded in a dual wavelet basis and the training loss is taken as a weighted sum of residual coefficients, with weights $p(x,y,t)$21; via wavelet norm equivalences this yields computable surrogates for dual norms such as $p(x,y,t)$22 and $p(x,y,t)$23, and the paper reports very good quantitative effectivity of the wavelet-based error bound (Ernst et al., 2022).

This earlier framework is not a reservoir-specific well model, and it does not introduce the name WellPINN. Nevertheless, it is relevant to the concept for two reasons. First, it provides a rigorous template for “well-posed” PINN design: stability is encoded at the formulation level rather than delegated to heuristic loss balancing. Second, it shows how a residual-based loss can be aligned with a computable certification functional. A plausible implication is that the near-well reservoir workflow and the certified wavelet framework occupy complementary positions within PINN methodology: the former addresses singular or steep-gradient resolution near wells through staged superposition and geometric refinement, while the latter addresses reliability through stable weak formulations, wavelet compression, and explicit error control.

The limitations stated for the 2025 WellPINN workflow are correspondingly specific. The reported case study is restricted to single-phase, slightly compressible flow, constant properties within each case, a 2D Cartesian plane with gravity neglected in-plane, unit thickness $p(x,y,t)$24, smooth activations (tanh), and a Gaussian equivalent source. Multi-well interference, strong heterogeneity or anisotropy, multiphase or thermal coupling, and explicit wellbore skin or completion details are not treated. The paper identifies domain decomposition with hard constraints, log-time scaling, and $p(x,y,t)$25 scaling as critical robustness enhancers; it also notes that alternative activations such as Gabor or physical activations and Fourier features may further improve early-time and near-well accuracy. This suggests that “WellPINN,” in the narrow sense, is best understood as a specialized workflow for accurate near-well pressure inference from pumping rates over the full injection period, whereas in the broader interpretive sense it also points toward PINN constructions that combine stable formulations with explicit control of residual approximation and error certification.