# US 2021/0240889 A1 · Discontinuous Interpolation Using Continuous Formulation, C1 or C0 FEM Discretization, and Scalable Solver > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 2021/0240889 A1 - Document ID: US20210240889A1 - Status: Published application - Kind: A1 - Application number: 17/248,470 - Filed: 2021-01-26 - Published: 2021-08-05 - Earliest claimed: 2020-02-04 - Assignees: ExxonMobil Upstream Research Company - Applicants: ExxonMobil Upstream Research Company - Inventors: Volkan Akcelik - Web page: https://patentgrep.com/patent/US20210240889A1 ## CPC - G06F 30/23: using finite element methods [FEM] or finite difference methods [FDM] - G01V 20/00: Geomodelling in general - G06F 17/11: for solving equations {, e.g. nonlinear equations, general mathematical optimization problems (optimization specially adapted for a specific administrative, business or logistic context G06Q10/04)} - G06F 17/17: Function evaluation by approximation methods, e.g. inter- or extrapolation, smoothing, least mean square method ({G06F17/18 takes precedence } ; interpolation for numerical control G05B19/18) - G06F 17/175: {of multidimensional data} - G06F 30/10: Geometric CAD - G01V 2210/641: Continuity of geobodies - G01V 2210/642: Faults - G06F 2111/04: Constraint-based CAD - G06F 2111/10: Numerical modelling - G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N) - G01V: GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS (means for indicating the location of accidentally buried, e.g. snow-buried, persons A63B29/02) ## Abstract A methodology for discontinuous smooth interpolation in order to generate a curve of a discontinuous volume due to one or more faults in a subsurface is disclosed. Faults in a subsurface result in discontinuities in the subsurface. Hydrocarbon management may seek to determine various surfaces in the subsurface, including across the faults in the subsurface. To generate the various surfaces, a continuous formulation of the interpolation method is followed in which discontinuous smooth interpolation is viewed as a variational optimization problem (such as an energy optimization problem) for the surface curvature function. In this way, the methodology does not require that the input data be located at grid points and discretized with a structured regular grid. Rather, because a continuous function is used, an unstructured grid may also be used to discretize the resulting equation. ## Claims ### Claim 1 A computer-implemented method for discontinuous smooth interpolation in order to generate a curve of a discontinuous volume due to one or more faults in a subsurface, the method comprising: accessing an at least partly continuous formulation; discretizing the at least partly continuous formulation with respect to a subsurface grid in order to generate a set of equations; solving the set of equations using the discontinuous smooth interpolation in order to generate the curve of the discontinuous volume by enforcing at least one of value constraints or orientation constraints; and using the curve in order to manage hydrocarbons or to output an image of the curve. ### Claim 2 (depends on 1) The method of claim 1, wherein the at least partly continuous formulation comprises an energy minimization function. ### Claim 3 (depends on 2) The method of claim 2, wherein the at least partly continuous formulation comprises a second derivative of a surface curvature function integrated over a given domain. ### Claim 4 (depends on 3) The method of claim 3, wherein the at least partly continuous formulation further comprises alternative energy minimization terms. ### Claim 5 (depends on 4) The method of claim 4, wherein the alternative energy minimization terms are based on the value constraints and the orientation constraints. ### Claim 6 (depends on 5) The method of claim 5, wherein the alternative energy minimization terms are one or more least squares terms formed as a least squares difference based on the value constraints and the orientation constraints. ### Claim 7 (depends on 5) The method of claim 5, wherein the alternative energy minimization terms are strictly honored as hard constraints. ### Claim 8 (depends on 3) The method of claim 3, wherein the at least partly continuous formulation comprises a 4th order bi-harmonic equation. ### Claim 9 (depends on 8) The method of claim 8, wherein discretizing the at least partly continuous formulation with respect to a subsurface grid in order to generate a set of equations comprises using a C1 finite element method. ### Claim 10 (depends on 9) The method of claim 9, wherein the discretizing is on an irregular grid. ### Claim 11 (depends on 8) The method of claim 8, wherein the at least partly continuous formulation further comprises one or more additional sparse terms indicative of the at least one of value constraints or orientation constraints. ### Claim 12 (depends on 11) The method of claim 11, wherein the one or more additional sparse terms comprise one or more penalty terms; and wherein the one or more penalty terms include sparse measurements indicative of the value constraints and the orientation constraints. ### Claim 13 (depends on 12) The method of claim 12, wherein the sparse measurements comprise a difference between values of the curve and measured values. ### Claim 14 (depends on 1) The method of claim 1, wherein solving the set of equations comprises solving an objective function that includes a surface curvature function integrated over a given domain and one or more alternative energy minimization terms. ### Claim 15 (depends on 14) The method of claim 14, wherein the one or more alternative energy minimization terms comprise one or more equations indicative of sparse measurements and one or more equations indicative of the orientation constraints; and wherein the one or more equations indicative of sparse measurements and the one or more equations indicative of the orientation constraints are weighted separately according to a desired smoothness of the surface curvature function. ### Claim 16 (depends on 1) The method of claim 1, wherein solving the set of equations comprises preconditioning an iterative solver, wherein the preconditioning includes using a combination of a multi-grid method and at least one other preconditioning method. ### Claim 17 A computer-implemented method for discontinuous smooth interpolation in order to generate a curve of a discontinuous volume due to one or more faults in a subsurface, the method comprising: generating a set of equations based on sparse scalar values and sparse orientation values; solving the set of equations in order to generate the curve by preconditioning an iterative solver, wherein the preconditioning includes using a combination of a multi-grid method and at least one other preconditioning method; and using the curve in order to manage hydrocarbons or to output an image of the curve. ### Claim 18 (depends on 17) The method of claim 17, wherein the at least one other preconditioning method comprises a Jacobi method. ### Claim 19 (depends on 18) The method of claim 18, wherein the preconditioning comprises first applying the Jacobi method and thereafter applying the multi-grid method. ### Claim 20 A computer-implemented method for discontinuous smooth interpolation in order to generate a curve of a discontinuous volume due to one or more faults in a subsurface, the method comprising: generating a set of equations using C1 finite element method indicative of energy minimization; solving the set of equations using the discontinuous smooth interpolation in order to generate the curve of the discontinuous volume by enforcing at least one of value constraints or orientation constraints; and using the curve in order to manage hydrocarbons or to output an image of the curve. ### Claim 21 (depends on 20) The method of claim 20, wherein the set of equations are generated by discretizing an at least partly continuous formulation with respect to a subsurface grid.