Numerical analysis of thermal effects during carbon dioxide injection with enhanced gas recovery: a theoretical case study for the Altmark gas field

Abstract

Prediction about reservoir temperature change during carbon dioxide injection requires consideration of all, often subtle, thermal effects. In particular, Joule–Thomson cooling (JTC) and the viscous heat dissipation (VHD) effect are factors that cause flowing fluid temperature to differ from the static formation temperature. In this work, warm-back behavior (thermal recovery after injection completed), as well as JTC and VHD effects, at a multi-layered depleted gas reservoir are demonstrated numerically. OpenGeoSys (OGS) is able to solve coupled partial differential equations for pressure, temperature and mole-fraction of each component of the mixture with a combination of monolithic and staggered approaches. The Galerkin finite element approach is adapted for space discretization of governing equations, whereas for temporal discretization, a generalized implicit single-step scheme is used. For numerical modeling of warm-back behavior, we chose a simplified test case of carbon dioxide injection. This test case is numerically solved by using OGS and FeFlow simulators independently. OGS differs from FeFlow in the capability of representing multi-componential effects on warm-back behavior. We verify both code results by showing the close comparison of shut-in temperature profiles along the injection well. As the JTC cooling rate is inversely proportional to the volumetric heat capacity of the solid matrix, the injection layers are cooled faster as compared to the non-injection layers. The shut-in temperature profiles are showing a significant change in reservoir temperature; hence it is important to account for thermal effects in injection monitoring.

Appendix

Elements of mass, advection, Laplace matrices and right hand side vectors of the Eqs. (9)–(11) are as follow:

$$ \begin{aligned} {\bf C}_{{\rm pp}} &= \int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[ n \frac{1}{\rho}\frac{\partial \rho}{\partial p}\right]{\bf N} d{\varvec{\Upomega}}\\ {\bf C}_{{\rm pT}} &= \int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[ n \frac{1}{\rho}\frac{\partial \rho}{\partial T}\right]{\bf N} d{\varvec{\Upomega}}\\ {\bf K}_{{\rm pp}}&=-\int\limits_{\varvec{\Upomega}}\left(\nabla {\bf N}\right)^T\left[\frac{{{\mathsf{k}}} k}{\mu}\right]\nabla {\bf N} d{\varvec{\Upomega}} \\ {\bf K}_{{\rm pT}}&=-\int\limits_{\varvec{\Upomega}}\left(\nabla {\bf N}\right)^T\left[0\right]\nabla {\bf N} d{\varvec{\Upomega}} \\ {\bf A}_{{\rm pp}}&=\int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[n{\bf v} \frac{1}{\rho}\frac{\partial \rho}{\partial p}\right]\nabla {\bf N} d{\varvec{\Upomega}}\\ {\bf A}_{{\rm pT}}&=\int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[n{\bf v} \frac{1}{\rho}\frac{\partial \rho}{\partial T}\right]\nabla {\bf N} d{\varvec{\Upomega}}\\ {\bf f}_{{\rm p}}&=-\int\limits_{\varvec{\Upomega}} \left(\nabla {\bf N}\right)^T \left[\frac{{{\mathsf{k}}} k}{\mu} \rho {\mathbf g}\right] d{\varvec{\Upomega}} - \int\limits_{\partial \varvec{\Upomega}} {\bf N} \left[Q_{\rm p}\right]d{\partial \varvec{\Upomega}}\\ {\bf C}_{{\rm Tp}} &= \int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[ -n T \beta_{{\rm T}}\right]{\bf N} d{\varvec{\Upomega}} \\ {\bf C}_{{\rm TT}} &= \int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[ \left(\rho c_{\rm p}\right)_{{\rm eff}}\right]{\bf N} d{\varvec{\Upomega}}\\ {\bf K}_{{\rm Tp}}&=-\int\limits_{\varvec{\Upomega}}\left(\nabla {\bf N}\right)^T\left[0\right]\nabla {\bf N} d{\varvec{\Upomega}}\\ {\bf K}_{{\rm TT}}&=-\int\limits_{\varvec{\Upomega}}\left(\nabla {\bf N}\right)^T\left[\kappa_{{\rm eff}}\right]\nabla {\bf N} d{\varvec{\Upomega}}\\ {\bf A}_{{\rm Tp}}&=\int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[n{\bf v} \left(1-T \beta_{{\rm T}}\right) \right]\nabla {\bf N} d{\varvec{\Upomega}}\\ {\bf A}_{{\rm TT}}&=\int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[n{\bf v} \rho c_{\rm p}\right]\nabla {\bf N} d{\varvec{\Upomega}}\\ {\bf f}_{{\rm T}}&=-\int\limits_{\varvec{\Upomega}}{\bf N}\left[Q_{\rm T}\right]d{\partial \varvec{\Upomega}}\\ {\bf C}_{{x_k}} &= \int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[ n \rho \right]{\bf N} d{\varvec{\Upomega}} \\ {\bf K}_{{x_k}}&=-\int\limits_{\varvec{\Upomega}}\left(\nabla {\bf N}\right)^T\left[n\rho {{{\mathsf{D}}}}\right]\nabla {\bf N} d{\varvec{\Upomega}} \\ {\bf A}_{{x_k}}&=\int\limits_{\varvec{\Upomega}}{{\bf N}}^T\left[n{\bf v} \rho \right]\nabla {\bf N} d{\varvec{\Upomega}}\\ {\bf f}_{{x_k}}&=-\int\limits_{\varvec{\Upomega}}{\bf N}\left[Q - x_{{k}} \rho Q_{\rho}\right]d{\partial \varvec{\Upomega}} \end{aligned} $$