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

- 207 Downloads

## Abstract

An analytical solution is presented for the displacement, strain and stress of a three-dimensional poro-elastic model with three layers, where the three layers are an underburden, a reservoir with a given fluid pressure, and an overburden. The fluid pressure in the reservoir is assumed symmetrical around the *z*-axis and represented by a Fourier cosine series. The poro-elastic solution is expressed as a superposition of the solutions for each term in the Fourier series. It is shown that the bulk strain in the reservoir layer is proportional to the fluid pressure and that the bulk strain in the underburden and overburden is zero. Using these properties of the bulk strain, a solution is derived for the three-layer model where the fluid flow and mechanics are fully coupled. A particular aim of the model is to study the surface uplift from a given reservoir pressure. The expansion of the reservoir and the uplift of the surface are studied in terms of the wavelengths in the Fourier representation of the pressure. It is shown that the surface uplift can be written in a similar form to the 1D vertical expansion of the reservoir layer, but where the fluid pressure is based on the Fourier series. It is shown that the amplitudes with average wavelengths longer than \(2\pi \) times the thickness of the reservoir give expansion of the reservoir, but average wavelengths much shorter than this limit do not. Similarly, amplitudes with average wavelengths much longer than \(2\pi \) times the thickness of the overburden produce surface uplift, but wavelengths much shorter do not. The stress in the overburden, which is generated by the reservoir fluid pressure, is also analysed in terms of the wavelengths. A case is given where the analytical uplift is compared with the results of a numerical simulation and the agreement is excellent.

## Keywords

Geomechanics Poro-elasticity Surface uplift## 1 Introduction

The global annual emissions of \(\text{ CO }_2\) were 32 Gt in 2016 (International Energy Agency 2016), and the concentration of \(\text{ CO }_2\) in the atmosphere has increased from around 280 ppm in 1900 to more than 400 ppm in 2017. \(\text{ CO }_2\) is a greenhouse gas and its increasing concentration in the atmosphere has been found to be the reason for global warming (Bryant 1997). The injection of large volumes of \(\text{ CO }_2\) into aquifers or depleted oil and gas reservoirs is considered a promising way to reduce \(\text{ CO }_2\) emissions and thereby reduce global warming (Bachu 2008; Benson and Cole 2008; Bickle 2009).

\(\text{ CO }_2\) injection leads to a pressure build-up and an expansion of the reservoir, and in turn to surface uplift. An example of subsurface \(\text{ CO }_2\) injection and surface uplift is the In Salah storage project, where 3.8 Mt has been injected since its start in 2004, and where a total uplift of roughly 15 mm has been measured around three injection wells (Rutqvist 2012). The uplift measured at In Salah has attracted a considerable amount of scientific interest, and a number of modelling studies have been published (Vasco et al. 2008; Rutqvist et al. 2010; Rutqvist 2012; Bissell et al. 2011; Verdon et al. 2011; Zhou et al. 2010; Shi et al. 2012; Rinaldi and Rutqvist 2013; Rucci et al. 2013; White et al. 2014; Vilarrasa et al. 2015, 2017; Rinaldi et al. 2017). Similarly, seabed uplift is expected over offshore \(\text{ CO }_2\) sites, although it is more difficult to measure than land uplift. Land-based water pumping from aquifers and land-based oil and gas production are known to produce surface subsidence as the fluid pressure is lowered.

Geertsma (1973) developed an early model to compute land subsidence, which was applied to the Groningen gas field. The model gives the poro-elastic subsidence above a horizontal disk-shaped reservoir with a constant thickness and a constant pressure. The reservoir is placed at a finite depth in the infinite half-space and has a stress-free surface. The model is based on the nucleus-of-strain concept in the half-space, which was introduced by Mindlin and Cheng (1950) and Sen (1950) as a method to solve thermo-elastic problems. A particular feature of the model of Geertsma (1973) is that a constant pressure reduction gives a maximum subsidence that is approximately 1.5 times larger than the one-dimensional (vertical) compaction of the reservoir layer. The model gives a maximum horizontal displacement (contraction) that is also approximately 1.5 times the one-dimensional compaction.

The classical model of Geertsma (1973) has recently been modified by Tempone et al. (2010): the infinite half-space was replaced by a rigid basement underneath the reservoir. Tempone et al. (2010) compare the displacement, stress and strain of their model with the classical Geertsma model (Geertsma 1973). The comparisons show that a rigid basement close to the reservoir gives more uplift than the classical model, because less of the reservoir compaction goes vertically downward into the layer underneath the reservoir.

Another approach is taken by Fokker and Orlic (2006). They model the poro-elastic subsidence by a semi-analytical approach, which allows for an arbitrary number of layers. They use a numerical method to fit a set of analytical solutions to approximate the boundary condition in an optimal manner.

Selvadurai (2009), Kim and Selvadurai (2015), Selvadurai and Kim (2016) and Niu et al. (2017) have developed analytical models for different configurations of a reservoir and a caprock. Selvadurai and Kim (2016) present analytical poro-elastic solutions for a storage aquifer with a caprock; the solutions are given by an integral representation. The model was used to study how the surface displacement is controlled by the radius and the depth of the injection region. Finally, Chang and Segall (2016) studied how fluid pressure variations from injection or production in reservoirs could induce poroelastic stress changes and fault slip in the basement.

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. Each layer in the model is of infinite lateral extent and of constant thickness. A periodic pressure distribution can be represented by a Fourier series, and the displacement, strain and stress becomes the superposition of the effect of each term in the Fourier series. The model has a stress-free surface, except for the shear stress *xz* and *yz* components, which turn out to be negligible for normal configurations of the reservoir and the overburden. The displacement field and the stress, except for the *xx* and *yy* stress components, are continuous across the internal layer boundaries, which are the top and the base of the reservoir. A particular feature of this model is that it allows a decoupling of the equation for fluid flow from the equations of the displacement field. This decoupling simplifies considerably the derivation of analytical solutions of the fully coupled problem of fluid flow and mechanics. This model is well suited for a study of how the displacement, stress and uplift depend on the wavelengths of the reservoir pressure and the layer thicknesses. ‘Short’ and ‘long’ wavelengths in the pressure distribution produce different displacement fields and stresses, where ‘short’ and ‘long’ are with respect to layer thicknesses.

The present paper is organized as follows: The poro-elastic assumption and notation are introduced, and the expansion of the one-dimensional vertical poro-elastic layer is established. The geometry of the three-layer model is explained, and the boundary conditions are discussed. The solution for the displacement field is presented, and then the limit as the wavelength goes to infinity is investigated. How the fluid pressure can be decomposed in a double Fourier series is demonstrated before a Fourier solution for the fully coupled problem of fluid flow and mechanics is derived. The uplift is studied in terms of the wavelengths of the Fourier decomposition, and a special solution for the displacement for a reservoir on top of a rigid basement is presented. The analytical solution for the uplift is compared with a numerical finite element solution. An example of the analytical solution is given, which demonstrates the displacement field and the stress field. The stress in the caprock is investigated with respect to the wavelength of the Fourier components in the reservoir pressure.

## 2 Poro-elasticity

*i*and

*j*indicate the three spatial dimensions \(x=1\), \(y=2\) and \(z=3\). The stress state \(\sigma _{ij}\) fulfills the equilibrium equations

*g*is the acceleration due to gravity, and \(\delta _{iz}\) is the Kronecker delta, that is, \(\delta _{iz}=1\) for \(i=z\) and \(\delta _{iz}=0\) for \(i\ne z\). Einstein’s summation convention is used in the equilibrium Eq. (2), where summation is understood for each pair of the same indices.

*G*is the shear modulus, \(\varepsilon _{ij}=\frac{1}{2}(u_{i,j} + u_{j,i})\) is the strain tensor, and \(u_i\) is the displacement vector (\(i=x,y,z\)). An upper case \(\varLambda \) is used for the Lamé parameter, since the lower case \(\lambda \) will be used for the wavelength. Notice that normal effective stress is positive when the rock is tensile and negative when it is compressive. The elastic moduli are for drained conditions. The equations for the displacement field \(u_i\) are obtained by inserting the stress from Eq. (7) into the equilibrium Eq. (3), which yields

*G*and \(\alpha \). The gradient of the pressure change appears as an internal load on the left-hand side of the equilibrium equations. Therefore, Eq. (8) gives the displacement field when the pressure is known. The displacement field has a feedback on the pressure. The equation for the fluid pressure that includes poro-elastic deformations is

*t*is the time. The notation is simplified by denoting the pressure change by \(p=p^{(0)}\). The notation for the displacement field is also simplified by writing \(u=u_x\), \(v=u_y\) and \(w=u_z\). For instance, the bulk strain can now be written \(\varepsilon =\partial {u_i}/\partial {x_i}=\partial {u}/\partial {x}+\partial {v}/\partial {y}+\partial {w}/\partial {z}\), where it is noted that summation is understood over pairs of equal indices. In the following, the fluid overpressure is assumed known and the displacement equations are solved with the given overpressure.

## 3 One-dimensional Vertical Uplift

*w*(

*z*) is obtained by integrating Eq. (10) twice

*h*is the thickness of the reservoir layer. In case the reservoir layer has an overburden, the overburden is uplifted by

*w*when the base is fixed. The one-dimensional reservoir expansion (12) turns out to be an important reference for the uplift.

## 4 The Three-Layer Model

The model consists of three layers of infinite extent, as shown in Fig. 1. The middle layer is the reservoir and it has an overburden and an underburden. To simplify the analytical solution for the displacement field, the three layers are assigned the same mechanical properties. Only the reservoir layer is considered permeable and has a change in the fluid pressure. The *z*-axis is positive upwards, and \(z=0\) is the base of the model (the underburden). The base of the reservoir layer is at \(z=z_1\), the top of the reservoir layer is at \(z=z_2\) and the top of the model (the surface) is at \(z=z_3\). The thicknesses of the underburden, the reservoir and the overburden are \(h_1\), \(h_2\) and \(h_3\), respectively, and the layer boundaries become \(z_1=h_1\), \(z_2=h_1+h_2\) and \(z_3=h_1+h_2+h_3\).

## 5 Boundary Conditions

## 6 The Solution for the Displacement Field

*z*-axis. The overpressure (14) is independent of the depth inside the reservoir layer. The amplitude of the fluid pressure is \(p_0\), and the wavenumbers in the

*x*- and

*y*-directions are \(k_1\) and \(k_2\), respectively. These wavenumbers correspond to the wavelengths

*z*. Finally, the overburden (\(z_2\le z \le z_3\)) has the solution

The strain \(\varepsilon _{ij}=\frac{1}{2}(u_{i,j} + u_{j,i})\) is straightforward to compute from the displacement field, and the strain tensor for each layer is given in the “Appendix”. The effective stress follows from the strain by Eq. (7) and the effective stress tensor \(\tau ^{(1)}_{ij}\) for each layer is also given in the “Appendix”. The stress \(\sigma ^{(1)}_{ij}\) is equal to the effective stress in the overburden and the underburden where the fluid overpressure is zero. In the reservoir layer, the normal stress is obtained from the effective stress by subtracting the reservoir pressure multiplied by Biot’s coefficient; see Eq. (14). Finally, it is straightforward to verify that the stress \(\sigma ^{(1)}_{ij}\) fulfills the equilibrium Eq. (3).

*u*,

*v*,

*w*) and the stress components \(\sigma ^{(1)}_{zz}\), \(\sigma ^{(1)}_{xz}\), \(\sigma ^{(1)}_{yz}\) and \(\sigma ^{(1)}_{xy}\) are continuous across the internal boundaries at the base and at the top of the reservoir. In the same way, it can be seen that the stress components \(\sigma ^{(1)}_{xx}\) and \(\sigma ^{(1)}_{yy}\) are discontinuous across the horizontal layer interfaces. The stress is \(\Delta \sigma ^{(1)}_{xx}=\Delta \sigma ^{(1)}_{yy}=\varLambda D_0 p_0\cos (k_1 x)\cos (k_2 y)\) larger in the reservoir than right above in the caprock or right below in the underburden. The following two relations are useful for showing the continuity of the stress field across the internal layers

## 7 The Limit as \(k_1\rightarrow 0\) and \(k_2\rightarrow 0\)

*x*- and

*y*-directions, where the pressure (14) becomes laterally constant with an amplitude \(p_0\). Notice that the four functions \(F_1\) to \(F_4\) depend on the arguments \(cz_1\), \(cz_2\) and \(cz_3\), and that

*c*also goes to zero when both \(k_1\) and \(k_2\) do so. The limit \(k_1\rightarrow 0\) and \(k_2\rightarrow 0\) can be studied by using \(k_1=k_2=k\) and \(c=\sqrt{2}k\), and letting the wavenumber \(k\rightarrow 0\). A fixed value of

*x*gives that

*y*and

*z*. This implies that the factor \(k/c^2\) disappears in all expressions for

*u*(

*x*,

*y*,

*z*) and

*v*(

*x*,

*y*,

*z*), which are Eqs. (16), (17), (22), (23), (27) and (28). Next, it is seen that

The vertical displacement field of the reservoir depends on the function \(F_3\) in the limit \(c\rightarrow 0\), and it can be seen that \(F_3\rightarrow (cz - cz_2)\). Therefore, the vertical displacement field of the reservoir (24) has the limit \(w\rightarrow D_0\,p_0\cdot (z- z_1\)) when \(k_1,\,k_2\rightarrow 0\), which is the solution for the one-dimensional vertical reservoir expansion.

In the overburden, the vertical displacement field depends on the function \(F_4\), which has the limit \((cz_2-cz_1)\) when \(c\rightarrow 0\). Therefore, the vertical displacement of the overburden has the limit \(w\rightarrow D_0\,p_0\cdot (z_2- z_1)\), which is, as expected, the maximum vertical expansion of the reservoir layer.

Furthermore, it follows that the strain and the effective stress tensors are zero in the limit \(k_1,\,k_2\rightarrow 0\) for all layers, with exceptions for the strain \(\varepsilon _{zz}=\alpha p_0/(\varLambda +2G)\) and effective stress \(\tau ^{(1)}_{zz}=\alpha p_0\) for the reservoir layer.

## 8 Fourier Representation of the Reservoir Pressure

*z*-axis

*m*and

*n*are

*x*- and

*y*-directions are \(L_1\) and \(L_2\), respectively. A Fourier series gives an exact representation of any periodic and well-behaved function, with periods \(L_1\) and \(L_2\) in the

*x*- and

*y*-directions, respectively, in the limit \(N_1\rightarrow \infty \) and \(N_2\rightarrow \infty \). The domain size is assumed sufficiently large for the pressure to go to zero before the edges of the domain. A finite representation, where \(N_1\) and \(N_2\) are of order 100, normally gives an excellent approximation of the reservoir overpressure. It is convenient to write the series (40) compactly as

*S*(

*x*,

*y*,

*z*) is a property such as the displacement, the stress or the strain, it implies that

*w*in the vertical direction, the strain \(\varepsilon _{zz}\) and the effective stress \(\tau ^{(1)}_{zz}\), as already shown.

## 9 Solution of the Coupled Biot Equations

## 10 Uplift as a Function of Wavelengths and Layer Thicknesses

## 11 Reservoir and Overburden Above a Rigid Basement

## 12 An Example of Surface Uplift from Reservoir Pressure

*Q*is the injection rate, \(h_2\) is, as before, the thickness of the aquifer, \(r=(x^2+y^2)^{1/2}\) is the radius from the injection well and

*t*is the time. The example has a reservoir thickness \(h_2=50\) m, porosity \(\phi =0.1\), fluid compressibility \(1/K_\mathrm{f}={5}\times 10^{-8}\) \(\text{ Pa }^{-1}\), solid matrix compressibility \(1/K_\mathrm{s}=0\) \(\text{ Pa }^{-1}\), aquifer permeability \(\kappa ={1}\times 10^{-12}\) \(\text{ m }^{2}\) and injection rate \(Q=0.278\) \(\text{ m }^3\,\text{ s }^{-1}\). The overburden thickness is \(h_3=1000\) m, and the reservoir layer is placed on a rigid basement, which gives \(h_1=0\) m. Young’s modulus is \(E=15\) GPa, and the Poisson ratio is \(\nu =0.2\). Since \(E\gg K_\mathrm{f}\), the fluid compressibility dominates the compressibility of the rock, and the right-hand side of Eq. (9) can be ignored.

The Fourier coefficients as a function of the average Fourier number, \({n}_{\mathrm {av}}=(m^2+n^2)^{1/2}\), are plotted in Fig. 7a. Figure 7a shows that the first Fourier coefficients become larger as the plume spreads out. The first three Fourier coefficients for the pressure plume at 1000 days are considerably larger than the others.

Figure 7a shows that the largest Fourier coefficients are found for an average Fourier number less than 2. These Fourier numbers correspond to an average wavelength that is longer than 10 km, as seen from Fig. 7b. The pressure plumes at times 10 and 100 days are not wide enough to have noticeable Fourier coefficients for the lowest Fourier numbers.

The Fourier coefficients are multiplied with the \({f}_{\mathrm {surf}}\)-function, which lies in the range 0.8–1 for Fourier numbers up to 2. For Fourier numbers larger than 4, the \({f}_{\mathrm {surf}}\)-function is less than 0.5, as it converges towards 0. The large Fourier coefficients of large wavelengths, which dominate the pressure plume at time 1000 days, are not reduced very much by the \({f}_{\mathrm {surf}}\)-function, and therefore they contribute to the surface uplift.

The actual uplift at the three times is plotted in Fig. 8a–c. These curves show the uplift computed by expression (62) (the red circles) and the results of a finite element computation (the blue curve). The agreement between the analytical expression and the numerical FE computation is excellent. Figure 8a–c also show the one-dimensional reservoir expansion (12), when applied locally, as the green curve. The uplift from the pressure plume approaches the one-dimensional reservoir expansion as it gets wider and therefore has Fourier components with longer wavelengths. The one-dimensional reservoir expansion becomes a reasonable approximation for the surface uplift when the width of the plume exceeds \(2\pi h_3\approx 6000\) m.

## 13 An Example of Displacement, Strain and Stress from Reservoir Pressure

Another version of the preceding example is shown in Fig. 9. The reservoir layer with the reservoir pressure and the overburden is the same, but a 2000 m underburden is added. Figure 9 shows a two-dimensional *xz*-cross section of the model through the *z*-axis. The fluid overpressure is limited to the reservoir layer, as seen from Fig. 9a, and Fig. 9b shows that the displacement field *u* is symmetric around the origin. The displacement field *u* is also restricted to the surroundings of the reservoir layer. Figure 9c shows the vertical displacement field *w* from the expansion of the reservoir layer, which produces negative vertical displacements in the underburden and positive displacements in the overburden. The positive vertical displacements reach the surface, where it produces an uplift.

*x*-direction, which implies that there is less compressive stress in the

*z*-direction than in the

*x*-direction. The shear stress \(\sigma ^{(1)}_{xz}\) is located near the centre of the pressure plume in the underburden, the reservoir and the overburden. The absolute value of the shear stress is also two orders of magnitude less than the overpressure.

## 14 Stress in the Overburden

The stress in the caprock, Eqs. (178)–(183), shows that all the components of the stress tensor are proportional to the coefficient \(F_4\). The components of the stress tensor depend on the *z*-position in the caprock by either the factor \(\cosh (cz_3-cz)\) or \(\sinh (cz_3-cz)\). This implies that the stress increases nearly exponentially, from the surface towards the base of the overburden, when \({ ch}_3\gg 1\), because \(0\le cz_3-cz\le { ch}_3\). In the other regime, where \({ ch}_3\ll 1\), these two factors can be approximated by \(\cosh (cz_3-cz)\approx 1\) and \(\sinh (cz_3-cz)\approx 0\). It can therefore be concluded that the short wavelengths in the fluid pressure, \({ ch}_3\gg 1\) or \(\lambda \ll 2\pi h_3\), produce stress in the caprock that is maximal towards its base. Long wavelengths, \({ ch}_3\ll 1\) or \(\lambda \gg 2\pi h_3\), produce a weak stress in the caprock for normal basin configurations where the overburden is much thicker than the reservoir (\(h_2\ll h_3\)).

## 15 Conclusions

An analytical solution is presented for the displacement, strain and stress of the poro-elastic equations for a three-dimensional model of three horizontal layers with the same rock properties. The three layers are an underburden on a rigid basement, a reservoir with a pore pressure change, and an overburden. The overburden and the underburden are assumed impermeable with no fluid pressure change. The boundary conditions are zero vertical displacement at the base of the underburden and zero horizontal displacement at the top surface. It has been shown that the top surface is stress-free, except for the shear stress components *xz* and *yz*, which are almost zero for common basin configurations where the overburden is much thicker than the reservoir.

The pressure distribution in the reservoir is assumed symmetrical around the *z*-axis, in which case the pressure can be represented by a double Fourier cosine series. The model provides a solution for the displacement, the strain and the stress as a superposition of contributions from each term in the Fourier series. It is shown that the bulk strain in the overburden and the underburden is zero, and that the bulk strain in the reservoir layer is proportional to the fluid overpressure. By means of this property of the bulk strain, a Fourier series solution is derived where the fluid flow and mechanics are fully coupled.

The contribution from each term in the Fourier series is proportional to the amplitude and a factor of proportionality that is a function of the average wavelength and the layer thicknesses. These factors provide insight into how the fluid pressure deforms the layers with respect to wavelengths and layer thicknesses. It is seen that only terms in the Fourier series with average wavelengths larger than \(2\pi \) times the thickness of the reservoir (\(\lambda \gg 2\pi h_2\)) produce an expansion of the reservoir. A second condition must be fulfilled for the expansion of the reservoir to produce a surface uplift, which is that the average wavelength must be larger than \(2\pi \) times the thickness of the overburden (\(\lambda \gg 2\pi h_3\)). When the second condition is not fulfilled, the expansion of the reservoir does not reach the surface. The expansion of the reservoir can also press down the underburden in this case.

The surface uplift can be written in the same form as the one-dimensional vertical reservoir expansion, where the fluid pressure is based on the Fourier series for the reservoir pressure. This expression shows that the surface uplift in this model can never be larger than the expansion of the reservoir. The analytical expressions for the surface uplift are compared with the numerically computed surface uplift for a case where the reservoir is resting on a rigid basement, and the match is excellent.

The model predicts a discontinuity in the *xx* and *yy* components of the stress tensor at the top and the base of the reservoir. Furthermore, ‘short’ wavelengths (\(\lambda \ll 2\pi h_3\)) in the Fourier series for the fluid pressure produce a stress at the base of the overburden with nearly the same strength as the amplitude in the Fourier series, but the stress does not extend far upwards into the overburden. Terms with ‘long’ average wavelengths (\(\lambda \gg 2\pi h_3\)) produce negligible stress in the overburden.

It remains to study how the model eventually can be extended to layers with different mechanical properties.

## Notes

### Acknowledgements

This research was partially supported by the Research Council of Norway through the Project 280567, “Prediction of CO\(_{2}\) leakage from reservoirs during large scale storage”.

## References

- Bachu S (2008) CO\(_2\) storage in geological media: role, means, status and barriers to deployment. Prog Energy Combust Sci 34:254–273CrossRefGoogle Scholar
- Benson S, Cole RD (2008) CO\(_2\) sequestration in deep sedimentary formations. Elements 4:325–331CrossRefGoogle Scholar
- Bickle M (2009) Geological carbon storage. Nat Geosci 2:815–819Google Scholar
- Bissell R, Vasco D, Atbi M, Hamdani M, Okwelegbe M, Goldwater M (2011) A full field simulation of the In Salah gas production and CO\(_2\) storage project using a coupled geo-mechanical and thermal fluid flow simulator. Energy Proc 4:3290–3297CrossRefGoogle Scholar
- Bryant E (1997) Climate process and change. Cambridge University Press, Cambridge, UKCrossRefGoogle Scholar
- Chang KW, Segall P (2016) Seismicity on basement faults induced by simultaneous fluid injection-extraction. Pure Appl Geophys 173:2621–2636CrossRefGoogle Scholar
- Fokker PA, Orlic B (2006) Semi-analytic modelling of subsidence. Math Geol 38(5):565–589, ISSN 1573-8868Google Scholar
- Geertsma J (1973) Land subsidence above compacting oil and gas reservoirs. J Pet Technol 59(6):734–744CrossRefGoogle Scholar
- International Energy Agency (2016) Energy snapshot, Global energy-related carbon dioxide emissions 1980–2016, https://www.iea.org/newsroom/energysnapshots
- Kim J, Selvadurai A (2015) Ground heave due to line injection sources. Geomech Energy Environ 2:1–14CrossRefGoogle Scholar
- Kreyszig E (2011) Advanced engineering mathematics. Wiley, HobokenGoogle Scholar
- Mindlin RD, Cheng DH (1950) Thermoelastic stress in the semi-infinite solid. J Appl Phys 21(9):931–933CrossRefGoogle Scholar
- Niu Z, Li Q, Wei X (2017) Estimation of the surface uplift due to fluid injection into a reservoir with a clayey interbed. Comput Geotech 87(Supplement C):198–211, ISSN 0266-352XGoogle Scholar
- Rinaldi AP, Rutqvist J (2013) Modeling of deep fracture zone opening and transient ground surface uplift at KB-502 CO\(_2\) injection well, In Salah, Algeria. Int J Greenh Gas Control 12(Supplement C):155–167, ISSN 1750-5836Google Scholar
- Rinaldi AP, Rutqvist J, Finsterle S, Liu HH (2017) Inverse modeling of ground surface uplift and pressure with iTOUGH-PEST and TOUGH-FLAC: the case of \({\rm CO}_{2}\) injection at In Salah, Algeria. Comput Geosci 108(Supplement C):98–109, ISSN 0098-3004Google Scholar
- Rucci A, Vasco DW, Novali F (2013) Monitoring the geologic storage of carbon dioxide using multicomponent SAR interferometry. Geophys J Int 193(1):197–208CrossRefGoogle Scholar
- Rutqvist J (2012) The geomechanics of \({\rm CO}_{2}\) storage in deep sedimentary formations. Geotech Geol Eng 30:525–551CrossRefGoogle Scholar
- Rutqvist J, Vasco D, Myer L (2010) Coupled reservoir-geomechanical analysis of \({\rm CO}_{2}\) injection and ground deformations at In Salah, Algeria. Int J Greenh Gas Control 4:225–230CrossRefGoogle Scholar
- Selvadurai A (2009) Heave of a surficial rock layer due to pressures generated by injected fluids. Geophys Res Lett 36(14):L14302CrossRefGoogle Scholar
- Selvadurai A, Kim J (2016) Poromechanical behaviour of a surficial geological barrier during fluid injection into an underlying poroelastic storage formation. In: Proceedings of the Royal Society A, The Royal Society Publishing, pp 1–22Google Scholar
- Sen B (1950) Note on the stress produced nuclei of thermoplastic strain in a semi-infinite elastic solid. Q Appl Math 8:635Google Scholar
- Shi JQ, Sinayuc C, Durucan S, Korre A (2012) Assessment of carbon dioxide plume behaviour within the storage reservoir and the lower caprock around the KB-502 injection well at In Salah. Int J Greenh Gas Control 7(Supplement C):115–126, ISSN 1750-5836Google Scholar
- Tempone P, Fjær E, Landrø M (2010) Improved solution of displacements due to a compacting reservoir over a rigid basement. Appl Math Modell 34(11):3352–3362, ISSN 0307-904XGoogle Scholar
- Theis C (1935) The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. EOS Trans Am Geophys Union 2:519–524CrossRefGoogle Scholar
- Vasco DW, Ferretti A, Novali F (2008) Reservoir monitoring and characterization using satellite geodetic data: Interferometric synthetic aperture radar observations from the Krechba field, Algeria. Geophysics 73(6):WA113–WA122Google Scholar
- Verdon J, Kendall J, White D, Angus D (2011) Linking microseismic event observations with geomechanical models to minimize the risks of storing \({\rm CO}_{2}\) in geological formations. Earth Planet Sci Lett 305:143–152CrossRefGoogle Scholar
- Vilarrasa V, Rinaldi AP, Rutqvist J (2017) Long-term thermal effects on injectivity evolution during CO\(_2\) storage. Int J Greenh Gas Control 64(Supplement C):314–322, ISSN 1750-5836Google Scholar
- Vilarrasa V, Rutqvist J, Rinaldi AP (2015) Thermal and capillary effects on the caprock mechanical stability at In Salah, Algeria. Greenh Gases Sci Technol 5(4):449–461, ISSN 2152-3878Google Scholar
- Wangen M, Halvorsen G, Gasda S, Bjørnarå T (2018) An analytical plane-strain solution for surface uplift due to pressurized reservoirs. Geomech Energy Environ 13:25–34CrossRefGoogle Scholar
- White JA, Chiaramonte L, Ezzedine S, Foxall W, Hao Y, Ramirez A, McNab W (2014) Geomechanical behavior of the reservoir and caprock system at the In Salah \({\rm CO}_{2}\) storage project. Proc Natl Acad Sci 111(24):8747–8752CrossRefGoogle Scholar
- Zhou R, Huang L, Rutledge J (2010) Microseismic event location for monitoring \({\rm CO}_{2}\) injection using double-difference tomography. Lead Edge 29:201–214Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.