<p>Reservoir rocks are highly heterogeneous as they are formed by minerals and natural fractures with varying concentration, distribution, and orientation. Precise identification of these inclusions is essential for optimizing reservoir stimulation, energy production from geothermal wells, or storage of greenhouse gases in the subsurface. The estimation of the concentration of fractures (or inclusions) relies on evaluating the elastic properties of a rock through ultrasonic (dynamic) and uniaxial (static) measurements. The measured effective elastic properties are subsequently employed to estimate fracture density using effective medium theories (EMTs). This study identifies the primary factors influencing the accuracy of higher-order EMT models when estimating the elastic properties of fractured or heterogenous reservoirs. Finite element simulation is used to calculate the static and dynamic elastic properties of two-dimensional rocks with natural cracks exhibiting vertical transversely isotropic (VTI) symmetry. The cracks are horizontally, vertically, or randomly oriented and filled with water. The calculated properties are then compared with two EMT models, namely, the self-consistent approximation (SCA) and the differential effective medium theory (DEM) for varying crack densities. The calculations show that the effective elastic properties match those of EMT only for specific cases that we identify by defining three dimensionless factors: the scattering <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> ratio, the distance <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$D$</EquationSource> <EquationSource Format="MATHML"><math> <mi>D</mi> </math></EquationSource> </InlineEquation> ratio, and the homogeneity <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$H$</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation> ratio. The <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> ratio is defined as the wavelength over the crack size, the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$D$</EquationSource> <EquationSource Format="MATHML"><math> <mi>D</mi> </math></EquationSource> </InlineEquation> ratio is defined as the wavelength over the distance between cracks, and the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$H$</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation> ratio is defined as the ratio of the longitudinal to the transverse distance between the cracks multiplied by the inverse of the aspect ratio. Results show that the ultrasonic models match EMTs for higher values of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>. The <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4433_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$D$</EquationSource> <EquationSource Format="MATHML"><math> <mi>D</mi> </math></EquationSource> </InlineEquation> ratio exerts minimal influence on the accuracy of elastic property estimations with respect to EMT. Moreover, lower frequencies of the ultrasonic pulse led to the closest match between measured and estimated elastic properties from EMT. Static calculations align with those of EMT, provided that the cracks are homogenously distributed within a sample. The findings highlight the cases for which it is appropriate to use EMT for rock property inference and the percent deviation expected for heterogeneous reservoirs.</p>

Accuracy of Effective Medium Theories in Measuring the Elastic Properties of Heterogeneous Rocks

期刊论文
发表日期：

2025年3月9日

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Hawraa Kareem,
Mohamad Abdo,
Elsa Maalouf

摘要

Reservoir rocks are highly heterogeneous as they are formed by minerals and natural fractures with varying concentration, distribution, and orientation. Precise identification of these inclusions is essential for optimizing reservoir stimulation, energy production from geothermal wells, or storage of greenhouse gases in the subsurface. The estimation of the concentration of fractures (or inclusions) relies on evaluating the elastic properties of a rock through ultrasonic (dynamic) and uniaxial (static) measurements. The measured effective elastic properties are subsequently employed to estimate fracture density using effective medium theories (EMTs). This study identifies the primary factors influencing the accuracy of higher-order EMT models when estimating the elastic properties of fractured or heterogenous reservoirs. Finite element simulation is used to calculate the static and dynamic elastic properties of two-dimensional rocks with natural cracks exhibiting vertical transversely isotropic (VTI) symmetry. The cracks are horizontally, vertically, or randomly oriented and filled with water. The calculated properties are then compared with two EMT models, namely, the self-consistent approximation (SCA) and the differential effective medium theory (DEM) for varying crack densities. The calculations show that the effective elastic properties match those of EMT only for specific cases that we identify by defining three dimensionless factors: the scattering $\beta$ $β$ ratio, the distance $D$ $D$ ratio, and the homogeneity $H$ $H$ ratio. The $\beta$ $β$ ratio is defined as the wavelength over the crack size, the $D$ $D$ ratio is defined as the wavelength over the distance between cracks, and the $H$ $H$ ratio is defined as the ratio of the longitudinal to the transverse distance between the cracks multiplied by the inverse of the aspect ratio. Results show that the ultrasonic models match EMTs for higher values of $\beta$ $β$ . The $D$ $D$ ratio exerts minimal influence on the accuracy of elastic property estimations with respect to EMT. Moreover, lower frequencies of the ultrasonic pulse led to the closest match between measured and estimated elastic properties from EMT. Static calculations align with those of EMT, provided that the cracks are homogenously distributed within a sample. The findings highlight the cases for which it is appropriate to use EMT for rock property inference and the percent deviation expected for heterogeneous reservoirs.