An analytical plane-strain solution for surface uplift due to pressurized reservoirs

Abstract

In this paper, we present an analytical plane strain solution for surface uplift above pressurized reservoirs. The solution is based on a Fourier representation of the reservoir pressure. The plane strain model is developed in two stages: First, an exact solution is derived for the displacement field for the reservoir alone subjected to a periodic overpressure distribution of one wavelength. This one-layer model forms the basis for the analytical plane strain solution for a two-layer model-a pressurized reservoir with an overburden. We give an example where numerically computed uplift is quite accurately estimated by a simple 1D estimate, except for in the near well area. The plane-strain solution is well suited to study conditions for when the simple 1D approximation of the uplift is accurate. A condition for the accuracy of the simple 1D approximation is first derived for just the reservoir expanded by a periodic overpressure distribution of one wavelength, which corresponds to one term in a Fourier series. The 1D estimate is accurate for wavelengths larger than 2π times the reservoir thickness. Then, a condition is derived for when the 1D estimate is accurate for the two-layer model. We show that the wavelength of the overpressure distribution must be larger than 2π times the maximum of the reservoir thickness and the overburden thickness for the 1D approximation to be accurate. We demonstrate how uplift is computed from a Fourier decomposition of the reservoir overpressure. The resulting uplift is analysed in terms of Fourier coefficients, using the knowledge of how a single wavelength behaves. The analytical results for the displacement field and the uplift are tested by comparison with finite element simulations, and the match is excellent.

A Three-Dimensional Analytical Solution for Reservoir Expansion, Surface Uplift and Caprock Stress Due to Pressurized Reservoirs

“…which becomes the same as the dimensionless uplift reported for the two-layer plainstrain model by Wangen et al (2018) by setting either k 1 = 0 or k 2 = 0. It should be noted that the dimensionless uplift is given by a different expression by Wangen et al (2018), but it is easily rewritten to be as in Eq. (59).…”

“…This is a generalization of the condition for uplift found with the two-dimensional plain-strain model of Wangen et al (2018), which has a reservoir layer and an overburden, but no underburden. The dimensionless uplift (49) gives exactly the same dimensionless uplift as in Wangen et al (2018) under plain strain conditions (k 2 = 0) and with an underburden of zero thickness (h 1 = 0). The surface uplift comes from the expansion of the reservoir.…”

“…In the present article, a different approach is taken, by first solving the poro-elastic equations for a three-dimensional model with three layers when the reservoir pressure is just one term in a Fourier series. The same approach was taken by Wangen et al (2018) with a two-dimensional model of a reservoir with an overburden, but no underburden. The three layers in the present model are an overburden, a reservoir and an underburden, where the underburden is placed on a rigid basement.…”

Versatile Modeling Of Deformation (VMOD) Inversion Framework: Application to 20 Years of Observations at Westdahl Volcano and Fisher Caldera, Alaska, US

“…The framework includes the most common elastic models such as the point source (Mogi, 1958), the spherical cavity (McTigue, 1987), the spheroidal cavity (X. M. Yang et al., 1988), penny‐shaped crack (Fialko et al., 2001), and rectangular dislocation (Okada, 1985). We also include some time‐dependent models such as the open conduit model (Nishimura, 2009) and an implementation for injection wells (Wangen et al., 2018). VMOD's architecture allows for easy integration of new geodetic data types and new geodetic models.…”

“…Furthermore, for sources that are too deep the models cannot give accurate information about dimensions or overpressures (Segall, 2019). Therefore, besides the steady state models implemented in similar frameworks, VMOD distinguishes itself by also implementing models involving time‐dependent crustal rheology such as a spherical source embedded in a viscoelastic half‐space (Bonafede & Ferrari, 2009), a spherical source within a viscoelastic shell surrounded by an elastic half‐space (Segall, 2010), an analytical solution for a pressurized well in a porelastic half‐space (Wangen et al., 2018), and models involving time‐dependent magma properties such as a pressurized open conduit that accounts for the physical properties of the magma (Nishimura, 2009).…”