Well inflow modelling in different numerical simulation
approaches are compared for a multi-lateral well. Specifically radial wells
will be investigated, which can be created using Radial Jet Drilling (RJD).
In this technique, powerful hydraulic jets are used to create small diameter
laterals (25–50 mm) of limited length (up to 100 m) from a well. The
laterals, also called radials, leave the backbone at a 90
Well configuration for case 1 (
Setup of case 3, which is based on the Groß Schönebeck case and main fault F21n. Also shown are the different rock types, the well including the laterals connecting to fault F21n. For illustration purposes, around the laterals the injected water at the end of an injection period of 5 years is given (in blue).
Geothermal resources can vary widely in their characteristics, depending on
the downhole temperature, water composition, depth and the reservoir rock
(see e.g.
Settings with fault properties for two fault scenarios.
Total flow and relative difference with the results of AEM for all simulators for case 1: vertical well with four laterals.
Results for all four simulators for case 2: 35
Increase due to stimulation with four laterals for two cases. Case 1
is a well with a vertical backbone and case 2 is a well with a deviated
(35
The semi-analytic method to estimate the productivity (or injectivity) of a radial well as applied in this paper does not require a spatial discretization and is as such entirely distinctive from the other discussed numerical methods. To allow for a semi-analytic approach, certain simplifications are needed for the geometry and geology of the reservoir. We assume that the radial well is positioned in a cylindrically shaped homogeneous and anisotropic reservoir with laterally a constant pressure boundary condition and a no flow boundary condition at the top and bottom. Due to the complex geometry of a radial well it is likely not possible, even with the above simplifications, to obtain a closed formula for the well's performance (e.g. the Productivity Index that relates drawdown to production rate). The semi-analytic method is based on the AEM (Analytic Element Method) (see Egberts et al., 2013, and Egberts and Peters, 2015, for a more in-depth explanation of the method).
Injectivity index (m
Comparison of the inflow profile for the downward pointed radial (track 13 in Fig. 1) for case 2.
Comparison of the inflow profile for a horizontal radial (track 12 and 14 in Fig. 1) for case 2.
The semi-analytic method provides a solution of the pressure field
constructed from analytic solutions for the well segments (backbone and
laterals) of the radial well. An important component to arrive at the
semi-analytic solution is to model each well segment by a line sink (or line
source in case of injection) of the (steady state) pressure equation with a
variable influx along the line. The variable influx profile along a well
segment is described by a polynomial of an order
Eclipse® is an industry-standard simulator used extensively in
the petroleum industry and to a lesser extent for geothermal applications
(Schlumberger, 2016). The simulator uses a finite volume approach and employs
a well model to estimate the pressure drop between the well to the grid block
in which it is located based on the approach of Peaceman (1983). This allows
for much larger (> 10 m) grid blocks than well radii
(
The numerical simulator GOLEM is an open source software for solving parallel tightly coupled non-linear THM processes in fractured reservoir (Cacace and Jacquey, 2017). It is based on the MOOSE framework (Gaston et al., 2009) and its internal architecture relies on state of the art libraries for finite element analysis (libMesh, Kirk et al., 2006) and nonlinear iterative algebraic solvers (PETSc, Balay et al., 2016). GOLEM is based on the finite element method using irregular tetrahedral meshes. The major complication of using irregular meshes is to maintain the internal geometric consistency between well, fracture and matrix elements. This requires each well element to be mapped onto an edge (1-D representation) and each fracture/fault element mapped onto a face (2-D representation) of at least a single 3-D matrix element. In the current study, the well is simulated as a 1-D element superimposed in the 3-D domain, which consists of tetrahedrons. The refinement of the 1-D well-path strongly determines the pressure drop near the well and thus the productivity/injectivity of the well.
Complex Systems Modelling Platform (CSMP) is an object-oriented application programme interface (API), for the simulation of complex geological processes and their interactions (formerly CSP, cf. Matthäi et al., 2001). CSMP is based on the finite element method (FEM) and finite element – finite volume (FE-FV) method. CSMP has been utilised to develop coupled thermo-hydro-mechanical-chemical (THMC) model for simulation of subsurface processes (Salimzadeh et al., 2017, 2018). Fractures/faults and wells are modelled using lower dimension elements, i.e. surface elements for fractures/faults and line elements for wells, while the rock layers are modelled using 3-D volume elements. In CSMP, the spatial discretization can be achieved using both structured and unstructured meshes, providing flexibility for different applications. Since the wells are modelled using 1-D elements, like in GOLEM the pressure response is highly mesh dependent. To remove this well-known issue, the size of the tetrahedral connected to the well has been reduced such that the position of the nearest integration point, to the well, where the integrand is being evaluated numerically, resembles the well radius. With this strategy, CSMP results provide a very good fit to the analytical results for the simple geometry wells as discussed later in the results section. A basic well model based on the Peaceman approach (Peaceman, 1983) is available within CSMP, but was not used in this study.
The first case is a vertical well with a single kickoff with four orthogonal
and horizontal laterals in a homogeneous, anisotropic reservoir of 100 m
thick (from 2500 to 2600 m depth) (see Fig. 1). The initial pressure is
25 MPa (at 2500 m depth) with a hydrostatic pressure distribution in the
vertical. Horizontal permeability is 200 mD, vertical permeability is 20 mD
and porosity is 0.2. The lateral boundary condition is a constant pressure
boundary on the edge of the model, which is at 1000 m from the well. Top and
bottom of the reservoir are no flow boundaries. The reservoir fluid is water
with a salinity of 150.000 ppm, which
has, at reference pressure, a viscosity of 0.54 cP and density of
1110.2 kg m
For GOLEM an element size of 0.32 m for both the backbone and the laterals
was selected, because this gave results that most closely matched the
semi-analytical solution. For CSMP an element size of 1.5 m for the backbone
and 0.2 for the radials was selected for the same reasons. Grid size for the
Eclipse model was 10
For case 2, the reservoir is identical to the reservoir in case 1. Only the
well configuration is changed to a deviated well (see Fig. 1). The backbone
has an inclination of 35
For case 3 a low permeability reservoir is selected with a single large
fault. The case is based on the Groß Schönebeck case and details of
the reservoir properties and the reservoir fluids can be found in
Blöcher et al. (2010) and Blöcher et al. (2015). For simplicity
reasons only 1 fault (F21n) and 1 well were considered for the simulation.
The well itself has 8 radials with 2 kickoff points at
In Table 2, the calculated flow is presented for the vertical well with four
horizontal laterals. The relative difference is reported with respect to the
semi-analytical tool AEM. The flow rates obtained by AEM, Eclipse and CSMP
are 3241.8, 3214.4, and 3161.0 sm
For case 3, only Eclipse and GOLEM are used to simulate the flow, because the semi-analytical tool cannot handle fracture flow and the results of CSMP and GOLEM are very similar if the same gridding is used. The differences between these simulators mostly arise from differences in the elements and refinement of the elements near the well and laterals. Overall the injectivity is lower for Eclipse than for GOLEM (Table 5). The difference between GOLEM and Eclipse is largest for the low permeability fault with laterals: 48 %. In general, connecting to a previously unconnected fault is highly beneficial for the injectivity in the well. It should be noted however, that for commercial rates a considerable number of laterals should be achieved since the diameter of the laterals are small and thus the flow through the laterals is limited.
Although all simulators generally are reasonably close in terms of the total well flow (deviations < 4 % for the homogeneous cases), the distribution of the flow over the different parts of the well can vary up to 20 % for some laterals. For the homogeneous cases (1 and 2), the predictions of increase of flow as a result of stimulation by RJD show a range of variation up to 5 % just from differences between numerical solutions even for a simple setup. In realistic implementations with heterogeneous reservoir properties, larger uncertainty from the numerical solution can be expected for all simulators: for Eclipse because of inaccuracies in the calculation of the well index and for the FE approaches because of difficulties in determining the correct mesh size and large number of elements. In case the flow is dominated by fracture flow (case 3), the results deviate more with up to 50 % difference in the predicted flow rate in the case of radials. Even though these uncertainties are considerably smaller than those arising from uncertainty in the properties and uncertainty in the radial path, it is a source of errors that is often ignored.
No data sets were used in this article.
EP wrote the largest part of the paper, defined cases 1 and 2 and did the Eclipse simulations. GB and MC conducted the FEM simulations performed with GOLEM and OGS. Furthermore case 3 was provided by GB and the related simulations were performed and illustrated. SS conducted the simulations performed with CSMP. PE coded and ran the AEM method.
The authors declare that they have no conflict of interest.
This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 654662. The content of this paper reflects only the authors' view. The Innovation and Networks Executive Agency (INEA) is not responsible for any use that may be made of the information it contains. Edited by: Luke Griffiths Reviewed by: Estanislao Pujades and one anonymous referee