Generating a flow model grid based on truncation error threshold values
In some aspects, locations for nodes are computed for a one-dimensional flow model that models well system fluid flow in a subterranean region. Truncation error threshold data indicate a truncation threshold value for each of the nodes. Discretization data indicate a lowest-order term truncated from a discretized governing flow equation for each of the nodes. The locations for the nodes can be computed based on a scalar cost function, such that each of the lowest order terms is less than or equal to the truncation error threshold value for the respective node.
1. A fluid flow modeling method comprising: accessing, at a computer system, truncation error threshold data for a one-dimensional flow model, the one-dimensional flow model including nodes representing locations along a flow path for well system fluid in a subterranean region, the truncation error threshold data indicating a truncation error threshold value for each of the nodes;
accessing, at the computer system, discretization data for the one-dimensional flow model, the discretization data indicating, for each of the nodes, a lowest-order term truncated from a discretized governing flow equation for the node; and
computing, by operation of the computer system, locations for the nodes based on a scalar cost function, such that each of the lowest order terms is less than or equal to the truncation error threshold value for the respective node, thereby improving efficiency of the computer system by eliminating unnecessary use of resources of the computer system wherein the scalar cost function is defined based on a length of the flow path and a sum of distance s between pairs of the nodes and locations of the nodes are based on the scalar cost function according to an iterative root-finding algorithm wherein on each iteration of the root-finding algorithm:
an estimated spacing for a first pair of the nodes is obtained;
spacings for all other adjacent pairs of the nodes based on the estimated spacing for the first pair are computed, such that the lowest order terms are less than or equal to the truncation error threshold value for each respective node; and
a new estimate for the first pair of the nodes is computed based on the scalar cost function.
2. The method of claim 1 , wherein the lowest-order term includes a third-order term from a finite difference approximation of a governing flow equation.
3. The method of claim 1 , comprising adjusting, within the truncation error threshold value for each of the nodes, an actual truncation error associated with discretizing the governing flow equation for the node.
4. The method of claim 1 , wherein the scalar cost function comprises a single scalar cost function for computing locations for all the nodes on the flow path such that each of the lowest order terms is less than or equal to the truncation error threshold value for the respective node.
5. The method of claim 1 , further comprising operating the flow model based on the computed locations for the nodes.
6. The method of claim 1 , further comprising discretizing the governing flow equations and obtaining the lowest-order term for each node.
7. A non-transitory computer-readable medium storing instructions that, when executed by data processing apparatus, perform operations comprising:
receiving truncation error threshold data for a one-dimensional flow model, the one-dimensional flow model including nodes representing locations along a flow path for well system fluid in a subterranean region, the truncation error threshold data indicating a truncation error threshold value for each of the nodes;
receiving discretization data for the one-dimensional flow model, the discretization data indicating, for each of the nodes, a lowest-order term truncated from a discretized governing flow equation for the node; and
computing locations for the nodes based on a scalar cost function, such that each of the lowest order terms is less than or equal to the truncation error threshold value for the respective node, thereby improving efficiency of the data processing apparatus by eliminating unnecessary use of resources of the data processing apparatus wherein locations of the nodes are based on the scalar cost function according to an iterative root-finding algorithm
wherein on each iteration of the root-finding algorithm;
an estimated spacing for a first pair of the nodes is obtained;
spacings for all other pairs of the nodes based on the estimated spacing for the first pair are computed, such that the lowest order terms are less than or equal to the truncation error threshold value for each respective node; and
a new estimate for the first pair of the nodes is computed based on the scalar cost function.
8. The computer-readable medium of claim 7 , wherein the scalar cost function is described by:
F ( h 0 )=Σ h i −L= 0
where h 0 represents a spacing between a first pair of the nodes, h i represents a spacing between the i th pair of nodes, and L represents the length of the flowpath.
9. The computer-readable medium of claim 7 , wherein the improved estimate h n+1 0 for iteration number (n+1) based on the improved estimate h n 0 for iteration number n according to:
h
0
n
+
1
=
h
0
n
-
F
(
h
0
n
)
∂
∑
h
i
∂
h
0
where
∂
∑
h
i
∂
h
0
=
1
+
∑
1
N
-
1
∂
h
i
∂
h
0
,
where h 0 represents a spacing between a first pair of the nodes, h i represents a spacing between the i th pair of nodes, N represents the number of nodes, and F(h n o ) represents the scalar cost function.
10. The computer-readable medium of claim 7 , wherein the lowest-order term includes a third-order term from a finite difference approximation of a governing flow equation.
11. The computer-readable medium of claim 7 , the operations comprising maximizing, within the truncation error threshold value for each of the nodes, an actual truncation error associated with discretizing the governing flow equation for the node.
12. The computer-readable medium of claim 7 , wherein the scalar cost function comprises a single scalar cost function for computing locations for all the nodes on the flow path such that each of the lowest order terms is less than or equal to the truncation error threshold value for the respective node.
13. A fluid flow modeling system comprising one or more computers that include:
flow model data associated with a one-dimensional flow model for well system fluid, the one-dimensional flow model including nodes representing points along a flow path, the one-dimensional flow model including governing flow equations discretized for each of the nodes; and a grid-generation module operable to:
obtain truncation error threshold data indicating a truncation error threshold value for each of the nodes;
obtain discretization data indicating, for each of the nodes, a lowest-order term truncated from a discretized governing flow equation for the node; and use a scalar cost function to compute locations for the nodes, such that each of the lowest order terms is less than or equal to the truncation error threshold value for the respective node, thereby improving efficiency of the one or more computers by eliminating unnecessary use of resources of the one or more computers wherein the scalar cost function is defined based on a length of the flow path and a sum of distances between pairs of the nodes and locations of the nodes are based on the scalar cost function according to an iterative root finding algorithm wherein on each iteration of the root-finding algorithm:
an estimated spacing for a first pair of the nodes is obtained:
spacings for all other pairs of the nodes based on the estimated spacing for the first pair are computed, such that the lowest order terms are less than or equal to the truncation error threshold value for each respective node; and
a new estimate for the first pair of the nodes is computed based on the scalar cost function, wherein the improved estimate is provided as an initial estimate for each subsequent iteration until the root-finding algorithm terminates.
14. The system of claim 13 , the grid-generating module being operable to maximize, within the truncation error threshold value for each of the nodes, an actual truncation error associated with discretizing the governing flow equation for the node.
15. The system of claim 13 , wherein the scalar cost function comprises a single scalar cost function for computing locations for all the nodes on the flow path such that each of the lowest order terms is less than or equal to the truncation error threshold value for the respective node.