WELL Model

The WELL model is designed to flexibly couple to various flow and transport modes either sequentially, quasi-implicitly, or fully implicitly. Depending on coupling and well model type, different governing equations can be solved. Please see each individual flow mode for a description of the well model coupling options available. Right now, well model coupling is available in SCO2 Mode, HYDRATE Mode, and WIPP_FLOW mode.

Structure

The WELL model is a 1-dimensional sub-grid of the primary reservoir domain. Where the reservoir grid is defined by grid cells, the well sub-grid is defined by well segments. Well segments are associated with the reservoir cells through which they pass; mass is exchanged between well and reservoir through source/sink terms. Multiple well segments can exist within one reservoir cell, but one well segment cannot span multiple reservoir cells. The well can have a flexible orientation, but multiple wells cannot intersect each other. The well solves its own set of equations governing the distribution of mass along the 1D wellbore. Those equations are chosen by the user and are restricted based on the flow mode and coupling style.

Governing Equations

FULLY_IMPLICIT Coupling

The FULLY_IMPLICIT coupling option is available in SCO2 Mode mode and HYDRATE Mode mode. With this option, the well model is embedded as an extra equation in the flow mode Jacobian and residual.

HYDROSTATIC Well Model

Currently, the only well model type available for fully implicit coupling is the HYDROSTATIC well model type. This well model type solves one conservation equation in the form:

(1)\[\sum_{i} \Big(Q_{w,j}^{i}) = q_{w,j},\]

where \(Q_{w,j}^{i}\) is the flow rate between well and reservoir of component j in segment i of well w, and \(q_{w,j}\) is the user-defined surface injection rate. For a given well segment i, \(Q_{w,j}\) is determined by a well index and a wellbore pressure vis-a-vis the bottomhole pressure, \(P_B\). \(P_B\) is solved as a primary variable in the flow mode. Once solved, \(P_B\) is then used to compute hydrostatic pressures for all well segments. Those pressures are then used to compute well-reservoir flow rates in each well segment.

Well Index

The well index is used to compute flow rate into or out of an individual well segment as a function of the pressure difference between the wellbore and the reservoir at the well segment centroid. For a given well segment, \(Q_{w,j}\) is determined as follows:

(2)\[Q_{w,j}^{i} = -\frac{WI {\rho}_{j}}{\mu_j} (P_w - (P_r + {\rho}_{j} \boldsymbol{g} z_{w-r})),\]

where WI is the well index, \(P_{w}\) is the wellbore pressure at the well segment centroid, \(P_{r}\) is the pressure of the reservoir cell through which the well segment passes, and \(P_{r}\) is adjusted by a hydrostatic correction to the location of the well segment centroid. In 3D, the well index is computed using a generalized Peacemann relationship using well segment projections (White et al., 2013):

(3)\[ \begin{align}\begin{aligned}WI = C \sqrt{WI_{x}^{2} + WI_{y}^{2} + WI_{z}^{2}}\\WI_{x} = \frac{2\pi\sqrt{k_{y}k_{z}}L_{x}}{\ln{\frac{r_{0,x}}{r_{w}}}+s} WI_{y} = \frac{2\pi\sqrt{k_{x}k_{z}}L_{y}}{\ln{\frac{r_{0,y}}{r_{w}}}+s} WI_{z} = \frac{2\pi\sqrt{k_{x}k_{y}}L_{z}}{\ln{\frac{r_{0,z}}{r_{w}}}+s}\\r_{0,x} = 0.28\frac{({{\frac{k_y}{k_z}}^{0.5}\Delta{z}^{2}+{\frac{k_z}{k_y}}^{0.5}\Delta{y}^{2}})^{0.5}}{{\frac{k_y}{k_z}}^{0.25}+{\frac{k_z}{k_y}}^{0.25}} r_{0,y} = 0.28\frac{({{\frac{k_x}{k_z}}^{0.5}\Delta{z}^{2}+{\frac{k_z}{k_x}}^{0.5}\Delta{x}^{2}})^{0.5}}{{\frac{k_x}{k_z}}^{0.25}+{\frac{k_z}{k_x}}^{0.25}} r_{0,z} = 0.28\frac{({{\frac{k_x}{k_y}}^{0.5}\Delta{y}^{2}+{\frac{k_y}{k_x}}^{0.5}\Delta{x}^{2}})^{0.5}}{{\frac{k_x}{k_y}}^{0.25}+{\frac{k_y}{k_x}}^{0.25}}\end{aligned}\end{align} \]

where s is the user-defined skin factor, \(r_{w}\) is the wellbore radius, \(\Delta{x}\), \(\Delta{y}\), and \(\Delta{z}\) are the reservoir cell thicknesses in each principal dimension,and \(L_{x,y,z}\) are the well segment projections on each principal axis. The well index is scaled by the casing factor C, where C = 0 for a fully-cased well and C=1 for a fully uncased well.