  We study periodic solutions to the following divergence-form stochastic
partial differential equation with Wick-renormalized gradient on the
$d$-dimensional flat torus $\mathbb{T}^d$,
  \[
  -\nabla\cdot\left(e^{\diamond (- \beta X) }\diamond\nabla U\right)=\nabla
\cdot (e^{\diamond (- \beta X)} \diamond \mathbf{F}),
  \] where $X$ is the log-correlated Gaussian field, $\mathbf{F}$ is a random
vector and $\diamond$ denotes the Wick product. The problem is a variant of the
stochastic pressure equation, in which $U$ is modeling the pressure of a
creeping water-flow in crustal rock that occurs in enhanced geothermal heating.
In the original model, the Wick exponential term $e^{\diamond(-\beta X)}$ is
modeling the random permeability of the rock. The porosity field is given by a
log-correlated Gaussian random field $\beta X$, where $\beta<\sqrt{d}$. We use
elliptic regularity theory in order to define a notion of a solution to this (a
priori very ill-posed) problem, via modifying the $S$-transform from Gaussian
white noise analysis, and then establish the existence and uniqueness of
solutions. Moreover, we show that the solution to the problem can be expressed
in terms of the Gaussian multiplicative chaos measure.
