ResInsight/ApplicationCode/ReservoirDataModel/RigGeoMechWellLogExtractor.cpp

581 lines
25 KiB
C++

/////////////////////////////////////////////////////////////////////////////////
//
// Copyright (C) Statoil ASA
// Copyright (C) Ceetron Solutions AS
//
// ResInsight is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// ResInsight is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE.
//
// See the GNU General Public License at <http://www.gnu.org/licenses/gpl.html>
// for more details.
//
/////////////////////////////////////////////////////////////////////////////////
//==================================================================================================
///
//==================================================================================================
#include "RigGeoMechWellLogExtractor.h"
#include "RiaDefines.h"
#include "RigFemTypes.h"
#include "RigGeoMechBoreHoleStressCalculator.h"
#include "RigFemPart.h"
#include "RigFemPartCollection.h"
#include "RigGeoMechCaseData.h"
#include "RigFemPartResultsCollection.h"
#include "RigWellLogExtractionTools.h"
#include "RigWellPath.h"
#include "RigWellPathIntersectionTools.h"
#include "cafTensor3.h"
#include "cvfGeometryTools.h"
#include "cvfMath.h"
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
RigGeoMechWellLogExtractor::RigGeoMechWellLogExtractor(RigGeoMechCaseData* aCase,
const RigWellPath* wellpath,
const std::string& wellCaseErrorMsgName)
: RigWellLogExtractor(wellpath, wellCaseErrorMsgName)
, m_caseData(aCase)
{
calculateIntersection();
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
void RigGeoMechWellLogExtractor::curveData(const RigFemResultAddress& resAddr, int frameIndex, std::vector<double>* values)
{
CVF_TIGHT_ASSERT(values);
if (resAddr.resultPosType == RIG_WELLPATH_DERIVED)
{
if (resAddr.fieldName == RiaDefines::wellPathFGResultName().toStdString() || resAddr.fieldName == RiaDefines::wellPathSFGResultName().toStdString())
{
wellBoreWallCurveData(resAddr, frameIndex, values);
return;
}
else if (resAddr.fieldName == "PP" || resAddr.fieldName == "OBG" || resAddr.fieldName == "SH")
{
wellPathScaledCurveData(resAddr, frameIndex, values);
return;
}
else if (resAddr.fieldName == "Azimuth" || resAddr.fieldName == "Inclination")
{
wellPathAngles(resAddr, values);
return;
}
}
if (!resAddr.isValid()) return;
RigFemResultAddress convResAddr = resAddr;
// When showing POR results, always use the element nodal result,
// to get correct handling of elements without POR results
if (convResAddr.fieldName == "POR-Bar") convResAddr.resultPosType = RIG_ELEMENT_NODAL;
CVF_ASSERT(resAddr.resultPosType != RIG_WELLPATH_DERIVED);
const RigFemPart* femPart = m_caseData->femParts()->part(0);
const std::vector<float>& resultValues = m_caseData->femPartResults()->resultValues(convResAddr, 0, frameIndex);
if (!resultValues.size()) return;
values->resize(m_intersections.size());
for (size_t intersectionIdx = 0; intersectionIdx < m_intersections.size(); ++intersectionIdx)
{
(*values)[intersectionIdx] = static_cast<double>(interpolateGridResultValue<float>(convResAddr.resultPosType, resultValues, intersectionIdx, false));
}
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
void RigGeoMechWellLogExtractor::wellPathAngles(const RigFemResultAddress& resAddr, std::vector<double>* values)
{
CVF_ASSERT(values);
CVF_ASSERT(resAddr.fieldName == "Azimuth" || resAddr.fieldName == "Inclination");
values->resize(m_intersections.size(), 0.0f);
const double epsilon = 1.0e-6 * 360;
const cvf::Vec3d trueNorth(0.0, 1.0, 0.0);
const cvf::Vec3d up(0.0, 0.0, 1.0);
for (int64_t intersectionIdx = 0; intersectionIdx < (int64_t)m_intersections.size(); ++intersectionIdx)
{
size_t elmIdx = m_intersectedCellsGlobIdx[intersectionIdx];
cvf::Vec3d wellPathTangent = calculateWellPathTangent(intersectionIdx);
// Deviation from vertical. Since well path is tending downwards we compare with negative z.
double inclination = cvf::Math::toDegrees(std::acos(cvf::Vec3d(0.0, 0.0, -1.0) * wellPathTangent.getNormalized()));
if (resAddr.fieldName == "Azimuth")
{
double azimuth = HUGE_VAL;
// Azimuth is not defined when well path is vertical. We define it as infinite to avoid it showing up in the plot.
if (cvf::Math::valueInRange(inclination, epsilon, 180.0 - epsilon))
{
cvf::Vec3d projectedTangentXY = wellPathTangent;
projectedTangentXY.z() = 0.0;
// Do tangentXY to true north for clockwise angles.
double dotProduct = projectedTangentXY * trueNorth;
double crossProduct = (projectedTangentXY ^ trueNorth) * up;
// http://www.glossary.oilfield.slb.com/Terms/a/azimuth.aspx
azimuth = cvf::Math::toDegrees(std::atan2(crossProduct, dotProduct));
if (azimuth < 0.0)
{
// Straight atan2 gives angle from -PI to PI yielding angles from -180 to 180
// where the negative angles are counter clockwise.
// To get all positive clockwise angles, we add 360 degrees to negative angles.
azimuth = azimuth + 360.0;
}
}
(*values)[intersectionIdx] = azimuth;
}
else
{
(*values)[intersectionIdx] = inclination;
}
}
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
void RigGeoMechWellLogExtractor::wellPathScaledCurveData(const RigFemResultAddress& resAddr, int frameIndex, std::vector<double>* values)
{
CVF_ASSERT(values);
const RigFemPart* femPart = m_caseData->femParts()->part(0);
const RigFemPartGrid* femPartGrid = femPart->structGrid();
const std::vector<cvf::Vec3f>& nodeCoords = femPart->nodes().coordinates;
RigFemPartResultsCollection* resultCollection = m_caseData->femPartResults();
std::string nativeFieldName;
std::string nativeCompName;
double scalingFactor = 1000 * 9.81 / 1.0e5;
if (resAddr.fieldName == "PP")
{
nativeFieldName = "POR-Bar"; // More likely to be in memory than POR
}
else if (resAddr.fieldName == "OBG")
{
nativeFieldName = "ST";
nativeCompName = "S33";
}
else if (resAddr.fieldName == "SH")
{
nativeFieldName = "ST";
nativeCompName = "S3";
}
RigFemResultAddress nativeAddr(RIG_ELEMENT_NODAL, nativeFieldName, nativeCompName);
std::vector<float> unscaledResult = resultCollection->resultValues(nativeAddr, 0, frameIndex);
values->resize(m_intersections.size(), 0.0f);
#pragma omp parallel for
for (int64_t intersectionIdx = 0; intersectionIdx < (int64_t)m_intersections.size(); ++intersectionIdx)
{
size_t elmIdx = m_intersectedCellsGlobIdx[intersectionIdx];
RigElementType elmType = femPart->elementType(elmIdx);
if (!(elmType == HEX8 || elmType == HEX8P)) continue;
const int* elmNodeIndices = femPart->connectivities(elmIdx);
cvf::Vec3f centroid = cellCentroid(elmNodeIndices, nodeCoords);
double trueVerticalDepth = -centroid.z();
double effectiveDepth = trueVerticalDepth + m_rkbDiff;
double hydroStaticPorePressure = effectiveDepth * 9.81 / 100.0;
double unscaledValue = static_cast<double>(interpolateGridResultValue<float>(nativeAddr.resultPosType, unscaledResult, intersectionIdx, false));
if (resAddr.fieldName == "PP" && (unscaledValue == std::numeric_limits<float>::infinity() ||
unscaledValue == -std::numeric_limits<float>::infinity()))
{
unscaledValue = hydroStaticPorePressure;
}
double scaledValue = unscaledValue / (scalingFactor * effectiveDepth);
(*values)[intersectionIdx] = scaledValue;
}
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
void RigGeoMechWellLogExtractor::wellBoreWallCurveData(const RigFemResultAddress& resAddr, int frameIndex, std::vector<double>* values)
{
// TODO: Read in these values:
const double poissonRatio = 0.25; // TODO: Read this in.
// Typical UCS: http://ceae.colorado.edu/~amadei/CVEN5768/PDF/NOTES8.pdf
// Typical UCS for Shale is 5 - 100 MPa -> 50 - 1000 bar.
const double uniaxialStrengthInBars = 100.0;
CVF_ASSERT(values);
CVF_ASSERT(resAddr.fieldName == RiaDefines::wellPathFGResultName().toStdString() || resAddr.fieldName == RiaDefines::wellPathSFGResultName().toStdString());
const RigFemPart* femPart = m_caseData->femParts()->part(0);
const std::vector<cvf::Vec3f>& nodeCoords = femPart->nodes().coordinates;
RigFemPartResultsCollection* resultCollection = m_caseData->femPartResults();
RigFemResultAddress stressResAddr(RIG_ELEMENT_NODAL, std::string("ST"), "");
stressResAddr.fieldName = std::string("ST");
RigFemResultAddress porBarResAddr(RIG_ELEMENT_NODAL, std::string("POR-Bar"), "");
std::vector<caf::Ten3f> vertexStressesFloat = resultCollection->tensors(stressResAddr, 0, frameIndex);
if (!vertexStressesFloat.size()) return;
std::vector<caf::Ten3d> vertexStresses; vertexStresses.reserve(vertexStressesFloat.size());
for (const caf::Ten3f& floatTensor : vertexStressesFloat)
{
vertexStresses.push_back(caf::Ten3d(floatTensor));
}
values->resize(m_intersections.size(), 0.0f);
std::vector<float> porePressures = resultCollection->resultValues(porBarResAddr, 0, frameIndex);
#pragma omp parallel for
for (int64_t intersectionIdx = 0; intersectionIdx < (int64_t) m_intersections.size(); ++intersectionIdx)
{
size_t elmIdx = m_intersectedCellsGlobIdx[intersectionIdx];
RigElementType elmType = femPart->elementType(elmIdx);
if (!(elmType == HEX8 || elmType == HEX8P)) continue;
const int* elmNodeIndices = femPart->connectivities(elmIdx);
cvf::Vec3f centroid = cellCentroid(elmNodeIndices, nodeCoords);
double trueVerticalDepth = -centroid.z();
double porePressure = trueVerticalDepth * 9.81 / 100.0;
if (!porePressures.empty())
{
float interpolatedPorePressure = interpolateGridResultValue(porBarResAddr.resultPosType, porePressures, intersectionIdx, false);
if (interpolatedPorePressure != std::numeric_limits<float>::infinity() &&
interpolatedPorePressure != -std::numeric_limits<float>::infinity())
{
porePressure = static_cast<double>(interpolatedPorePressure);
}
}
caf::Ten3d interpolatedStress = interpolateGridResultValue(stressResAddr.resultPosType, vertexStresses, intersectionIdx, false);
cvf::Vec3d wellPathTangent = calculateWellPathTangent(intersectionIdx);
caf::Ten3d wellPathStressFloat = transformTensorToWellPathOrientation(wellPathTangent, interpolatedStress);
caf::Ten3d wellPathStressDouble(wellPathStressFloat);
RigGeoMechBoreHoleStressCalculator sigmaCalculator(wellPathStressDouble, porePressure, poissonRatio, uniaxialStrengthInBars, 32);
double resultValue = 0.0;
if (resAddr.fieldName == RiaDefines::wellPathFGResultName().toStdString())
{
resultValue = sigmaCalculator.solveFractureGradient();
}
else
{
CVF_ASSERT(resAddr.fieldName == RiaDefines::wellPathSFGResultName().toStdString());
resultValue = sigmaCalculator.solveStassiDalia();
}
double effectiveDepth = trueVerticalDepth + m_rkbDiff;
if (effectiveDepth > 1.0e-8)
{
resultValue *= 100.0 / (effectiveDepth * 9.81);
}
else
{
resultValue = std::numeric_limits<double>::infinity();
}
(*values)[intersectionIdx] = resultValue;
}
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
const RigGeoMechCaseData* RigGeoMechWellLogExtractor::caseData()
{
return m_caseData.p();
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
void RigGeoMechWellLogExtractor::setRkbDiff(double rkbDiff)
{
m_rkbDiff = rkbDiff;
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
template<typename T>
T RigGeoMechWellLogExtractor::interpolateGridResultValue(RigFemResultPosEnum resultPosType,
const std::vector<T>& gridResultValues,
int64_t intersectionIdx,
bool averageNodeElementResults) const
{
const RigFemPart* femPart = m_caseData->femParts()->part(0);
const std::vector<cvf::Vec3f>& nodeCoords = femPart->nodes().coordinates;
size_t elmIdx = m_intersectedCellsGlobIdx[intersectionIdx];
RigElementType elmType = femPart->elementType(elmIdx);
if (!(elmType == HEX8 || elmType == HEX8P)) return T();
if (resultPosType == RIG_ELEMENT)
{
return gridResultValues[elmIdx];
}
cvf::StructGridInterface::FaceType cellFace = m_intersectedCellFaces[intersectionIdx];
if (cellFace == cvf::StructGridInterface::NO_FACE)
{
if (resultPosType == RIG_ELEMENT_NODAL_FACE)
{
return std::numeric_limits<T>::infinity(); // undefined value. ELEMENT_NODAL_FACE values are only defined on a face.
}
// TODO: Should interpolate within the whole hexahedron. This requires converting to locals coordinates.
// For now just pick the average value for the cell.
size_t gridResultValueIdx = femPart->resultValueIdxFromResultPosType(resultPosType, static_cast<int>(elmIdx), 0);
T sumOfVertexValues = gridResultValues[gridResultValueIdx];
for (int i = 1; i < 8; ++i)
{
gridResultValueIdx = femPart->resultValueIdxFromResultPosType(resultPosType, static_cast<int>(elmIdx), i);
sumOfVertexValues = sumOfVertexValues + gridResultValues[gridResultValueIdx];
}
return sumOfVertexValues * (1.0 / 8.0);
}
int faceNodeCount = 0;
const int* elementLocalIndicesForFace = RigFemTypes::localElmNodeIndicesForFace(elmType, cellFace, &faceNodeCount);
const int* elmNodeIndices = femPart->connectivities(elmIdx);
cvf::Vec3d v0(nodeCoords[elmNodeIndices[elementLocalIndicesForFace[0]]]);
cvf::Vec3d v1(nodeCoords[elmNodeIndices[elementLocalIndicesForFace[1]]]);
cvf::Vec3d v2(nodeCoords[elmNodeIndices[elementLocalIndicesForFace[2]]]);
cvf::Vec3d v3(nodeCoords[elmNodeIndices[elementLocalIndicesForFace[3]]]);
std::vector<size_t> nodeResIdx(4, cvf::UNDEFINED_SIZE_T);
for (size_t i = 0; i < nodeResIdx.size(); ++i)
{
if (resultPosType == RIG_ELEMENT_NODAL_FACE)
{
nodeResIdx[i] = gridResultIndexFace(elmIdx, cellFace, static_cast<int>(i));
}
else
{
nodeResIdx[i] = femPart->resultValueIdxFromResultPosType(resultPosType, static_cast<int>(elmIdx), elementLocalIndicesForFace[i]);
}
}
std::vector<T> nodeResultValues;
nodeResultValues.reserve(4);
if (resultPosType == RIG_ELEMENT_NODAL && averageNodeElementResults)
{
// Estimate nodal values as the average of the node values from each connected element.
for (size_t i = 0; i < nodeResIdx.size(); ++i)
{
int nodeIndex = femPart->nodeIdxFromElementNodeResultIdx(nodeResIdx[i]);
const std::vector<int>& elements = femPart->elementsUsingNode(nodeIndex);
const std::vector<unsigned char>& localIndices = femPart->elementLocalIndicesForNode(nodeIndex);
size_t otherGridResultValueIdx = femPart->resultValueIdxFromResultPosType(resultPosType, elements[0], static_cast<int>(localIndices[0]));
T nodeResultValue = gridResultValues[otherGridResultValueIdx];
for (size_t j = 1; j < elements.size(); ++j)
{
otherGridResultValueIdx = femPart->resultValueIdxFromResultPosType(resultPosType, elements[j], static_cast<int>(localIndices[j]));
nodeResultValue = nodeResultValue + gridResultValues[otherGridResultValueIdx];
}
nodeResultValue = nodeResultValue * (1.0 / elements.size());
nodeResultValues.push_back(nodeResultValue);
}
}
else {
for (size_t i = 0; i < nodeResIdx.size(); ++i)
{
nodeResultValues.push_back(gridResultValues[nodeResIdx[i]]);
}
}
T interpolatedValue = cvf::GeometryTools::interpolateQuad<T>(
v0, nodeResultValues[0],
v1, nodeResultValues[1],
v2, nodeResultValues[2],
v3, nodeResultValues[3],
m_intersections[intersectionIdx]
);
return interpolatedValue;
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
size_t RigGeoMechWellLogExtractor::gridResultIndexFace(size_t elementIdx, cvf::StructGridInterface::FaceType cellFace, int faceLocalNodeIdx) const
{
CVF_ASSERT(cellFace != cvf::StructGridInterface::NO_FACE && faceLocalNodeIdx < 4);
return elementIdx * 24 + static_cast<int>(cellFace) * 4 + faceLocalNodeIdx;
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
void RigGeoMechWellLogExtractor::calculateIntersection()
{
CVF_ASSERT(m_caseData->femParts()->partCount() == 1);
std::map<RigMDCellIdxEnterLeaveKey, HexIntersectionInfo > uniqueIntersections;
const RigFemPart* femPart = m_caseData->femParts()->part(0);
const std::vector<cvf::Vec3f>& nodeCoords = femPart->nodes().coordinates;
for (size_t wpp = 0; wpp < m_wellPath->m_wellPathPoints.size() - 1; ++wpp)
{
std::vector<HexIntersectionInfo> intersections;
cvf::Vec3d p1 = m_wellPath->m_wellPathPoints[wpp];
cvf::Vec3d p2 = m_wellPath->m_wellPathPoints[wpp+1];
cvf::BoundingBox bb;
bb.add(p1);
bb.add(p2);
std::vector<size_t> closeCells = findCloseCells(bb);
cvf::Vec3d hexCorners[8];
for (size_t ccIdx = 0; ccIdx < closeCells.size(); ++ccIdx)
{
RigElementType elmType = femPart->elementType(closeCells[ccIdx]);
if (!(elmType == HEX8 || elmType == HEX8P)) continue;
const int* cornerIndices = femPart->connectivities(closeCells[ccIdx]);
hexCorners[0] = cvf::Vec3d(nodeCoords[cornerIndices[0]]);
hexCorners[1] = cvf::Vec3d(nodeCoords[cornerIndices[1]]);
hexCorners[2] = cvf::Vec3d(nodeCoords[cornerIndices[2]]);
hexCorners[3] = cvf::Vec3d(nodeCoords[cornerIndices[3]]);
hexCorners[4] = cvf::Vec3d(nodeCoords[cornerIndices[4]]);
hexCorners[5] = cvf::Vec3d(nodeCoords[cornerIndices[5]]);
hexCorners[6] = cvf::Vec3d(nodeCoords[cornerIndices[6]]);
hexCorners[7] = cvf::Vec3d(nodeCoords[cornerIndices[7]]);
//int intersectionCount = RigHexIntersector::lineHexCellIntersection(p1, p2, hexCorners, closeCells[ccIdx], &intersections);
RigHexIntersectionTools::lineHexCellIntersection(p1, p2, hexCorners, closeCells[ccIdx], &intersections);
}
// Now, with all the intersections of this piece of line, we need to
// sort them in order, and set the measured depth and corresponding cell index
// Inserting the intersections in this map will remove identical intersections
// and sort them according to MD, CellIdx, Leave/enter
double md1 = m_wellPath->m_measuredDepths[wpp];
double md2 = m_wellPath->m_measuredDepths[wpp+1];
insertIntersectionsInMap(intersections,
p1, md1, p2, md2,
&uniqueIntersections);
}
this->populateReturnArrays(uniqueIntersections);
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
std::vector<size_t> RigGeoMechWellLogExtractor::findCloseCells(const cvf::BoundingBox& bb)
{
std::vector<size_t> closeCells;
if (m_caseData->femParts()->partCount())
{
m_caseData->femParts()->part(0)->findIntersectingCells(bb, &closeCells);
}
return closeCells;
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
cvf::Vec3d RigGeoMechWellLogExtractor::calculateLengthInCell(size_t cellIndex, const cvf::Vec3d& startPoint, const cvf::Vec3d& endPoint) const
{
std::array<cvf::Vec3d, 8> hexCorners;
const RigFemPart* femPart = m_caseData->femParts()->part(0);
const std::vector<cvf::Vec3f>& nodeCoords = femPart->nodes().coordinates;
const int* cornerIndices = femPart->connectivities(cellIndex);
hexCorners[0] = cvf::Vec3d(nodeCoords[cornerIndices[0]]);
hexCorners[1] = cvf::Vec3d(nodeCoords[cornerIndices[1]]);
hexCorners[2] = cvf::Vec3d(nodeCoords[cornerIndices[2]]);
hexCorners[3] = cvf::Vec3d(nodeCoords[cornerIndices[3]]);
hexCorners[4] = cvf::Vec3d(nodeCoords[cornerIndices[4]]);
hexCorners[5] = cvf::Vec3d(nodeCoords[cornerIndices[5]]);
hexCorners[6] = cvf::Vec3d(nodeCoords[cornerIndices[6]]);
hexCorners[7] = cvf::Vec3d(nodeCoords[cornerIndices[7]]);
return RigWellPathIntersectionTools::calculateLengthInCell(hexCorners, startPoint, endPoint);
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
cvf::Vec3d RigGeoMechWellLogExtractor::calculateWellPathTangent(int64_t intersectionIdx) const
{
cvf::Vec3d segmentStart, segmentEnd;
m_wellPath->twoClosestPoints(m_intersections[intersectionIdx], &segmentStart, &segmentEnd);
return (segmentEnd - segmentStart).getNormalized();
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
caf::Ten3d RigGeoMechWellLogExtractor::transformTensorToWellPathOrientation(const cvf::Vec3d& wellPathTangent,
const caf::Ten3d& tensor)
{
// Create local coordinate system for well path segment
cvf::Vec3d local_z = wellPathTangent;
cvf::Vec3d local_x = local_z.perpendicularVector().getNormalized();
cvf::Vec3d local_y = (local_z ^ local_x).getNormalized();
// Calculate the rotation matrix from global i, j, k to local x, y, z.
cvf::Mat4d rotationMatrix = cvf::Mat4d::fromCoordSystemAxes(&local_x, &local_y, &local_z);
return tensor.rotated(rotationMatrix.toMatrix3());
}
//--------------------------------------------------------------------------------------------------
///
//--------------------------------------------------------------------------------------------------
cvf::Vec3f RigGeoMechWellLogExtractor::cellCentroid(const int* elmNodeIndices, const std::vector<cvf::Vec3f>& nodeCoords)
{
cvf::Vec3f centroid(0.0, 0.0, 0.0);
for (int i = 0; i < 8; ++i)
{
centroid += nodeCoords[elmNodeIndices[i]];
}
return centroid / 8.0;
}