Thermodynamically admissible derivation of Biot's poroelastic equations and Gassmann's equations from conservation laws

Alkhimenkov, Yury; Podladchikov, Yury Y.

doi:https://doi.org/10.5194/egusphere-2024-3238

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https://doi.org/10.5194/egusphere-2024-3238

Preprints

21 Nov 2024

| 21 Nov 2024

Thermodynamically admissible derivation of Biot's poroelastic equations and Gassmann's equations from conservation laws

Yury Alkhimenkov and Yury Y. Podladchikov

Abstract. Gassmann's equations, formulated several decades ago, remain a cornerstone in geophysics due to their perceived exactness. However, a concise and rigorous derivation rooted in thermodynamic principles and conservation laws has been missing from the literature. Additionally, recent studies have pointed out potential logical inconsistencies in the original formulation. This paper introduces a derivation of Gassmann’s equations, anchored in fundamental conservation laws and constitutive relations, ensuring their thermodynamic consistency. Alongside this, we extend the discussion to include Biot's poroelastic equations, which are widely used to describe the coupled behavior of fluid-saturated porous media under mechanical deformation. By demonstrating that Gassmann's equations are a specific case within the broader framework of Biot’s theory, we further validate their relevance and applicability in geophysical contexts. Given the numerous independent rederivations and numerical verifications of these equations for diverse pore geometries, we affirm their robustness, provided the underlying assumptions are respected. To facilitate reproducibility and further exploration, symbolic Maple routines are provided for the derivations presented in this study.

Received: 17 Oct 2024 – Discussion started: 21 Nov 2024

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Journal article(s) based on this preprint

29 Oct 2025

Revisiting Gassmann-type relationships within Biot poroelastic theory

Yury Alkhimenkov and Yury Y. Podladchikov

Solid Earth, 16, 1227–1247, https://doi.org/10.5194/se-16-1227-2025,https://doi.org/10.5194/se-16-1227-2025, 2025

Short summary

Yury Alkhimenkov and Yury Y. Podladchikov

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-3238', Anonymous Referee #1, 23 Dec 2024
The validity of Gassmann’s equations and the thermodynamic admissibility of Biot’s equations have been widely discussed in the literature since their first publication. The new manuscript by Alkhimenkov and Podladchikov advocates Gassmann’s assumption that equal changes in pore pressure and total pressure leave the porosity unchanged. They present new logical reasoning that this assumption, which is hard to validate in the experiments, follows from the thermodynamic principles. However, the manuscript suffers from insufficient logic, poor structure, and inaccurate statements.
The introduction is misleading. It appears that the authors are building the manuscript around the statement: “However, despite their widespread use, recent studies have highlighted concerns regarding the logical consistency in the derivation of Gassmann’s equations.” (Their lines 15-16). However, there are no references to this statement, nor do they describe the claims they are arguing against. At the end of the manuscript, they explicitly discuss the recent claims of L. Thomsen on the incorrectness of Gassmann’s equations. However, Thomsen’s arguments were about using two solid bulk moduli commonly known as K^’_s and K^’’_s. Thus, the main message of the current manuscript remains unclear after reading.

Authors claim that “This article aims to address these concerns by presenting a novel derivation of Biot’s poroelastic equations and Gassmann’s equations, which strictly adheres to fundamental conservation laws and thermodynamic principles.” (Lines 19-20) and “… a concise and rigorous derivation rooted in thermodynamic principles and conservation laws has been missing from the literature.” (Lines 2-3). These two statements are incorrect. First, the thermodynamic consistency of Gassmann’s equations was presented by Yarushina and Podladchikov (2015). Second, the manuscript does not present a novel derivation but rather repeats the thermodynamic derivation of Yarushina and Podladchikov (2015). Reformulate these statements and need for the article.

The manuscript's structure is somewhat inconsistent and lacks clarity.
For example, a statement on Gassmann’s assumption in the middle of the thermodynamic section (Lines 126 - 132) is out of place. Please discuss this assumption in the introduction since this is the main Gassmann hypothesis that you are aiming to prove. Besides, after reading this para, what exactly was not assumed as highlighted in the bold text is still unclear. Are you trying to say that no porosity change under equal pore and total pressure changes is a thermodynamic admissibility condition?

Numerical verification in section 4.3 is simply missing. You refer to your recently published paper, making unclear statements in this section. Given that you repeat already existing thermodynamic derivation by other authors for the sake of argument in favor of Gassmann, you could as well repeat your own numerical results for clarity. It is unclear what statement you are arguing here against. In statements such as “…solution converges towards the result obtained from the original Gassmann’s equation…” (Line 262), it is unclear which result is meant. Similarly, in the statement from lines 263-265 “…the pore geometry that was used did not contain any special features (among all possible geometries) that were tailored to make it consistent with Gassmann’s equations (Alkhimenkov, 2024)”, it is unclear what special features can be tailored to suite Gassmann. It is very odd to read such statements given that Gassmann’s equation is independent of geometry assumption as follows from his original paper and as was shown again by Yarushina and Podladchikov (2015).

Section 4.4. discusses the recent arguments by Thomsen. Are there other authors that recently questioned the validity of Gassmann’s equations or tried to rederive them? Or is your article built only around Thomsen’s arguments?

The introduction should contain a clear statement you try to prove. Make a statement that you consider rocks with K^’_s = K^’’_s and explain what this assumption means. This makes the whole argument with Thomsen in section 4.4 nonrelevant as he is simply rederiving the generalization of Gassmann’s equations to the case when the two moduli are different. You can mention that such a generalization exists, and you demonstrated numerically that, indeed, this generalization reduces to Gassmann’s equation under condition K^’_s = K^’’_s.

Keep thermodynamic derivation clear of discussions on Gassmann's equation as discussed above.

Include relevant material from your previous paper in section 4.3 on numerical validation.

Statements like “While the general methodology was outlined by Yarushina and Podladchikov (2015), this study specifically focuses on the rigorous derivation of Biot’s poroelastic, Gassmann’s, and effective stress law equations, along with addressing concerns related to their physical validity” (Lines 28-30) should be more accurate, as thermodynamic derivation of Biot’s equations was already presented in the literature (see e.g., Yarushina and Podladchikov (2015) or Coussy et al., 1998, From mixture theory to Biot’s approach for porous media, Int. J. Solids Structures.)

Line 25: “… validate their integrity …”. What is the integrity of equations, and how do you demonstrate it?

Section 3. Make a statement that you just repeat previously published thermodynamic derivation and not making a new one.

Lines 72 – 74. “In the context of classical non-equilibrium thermodynamics (Lebon et al., 2008), each phase within the porous medium is considered to be locally in thermodynamic equilibrium, which means that intensive variables such as temperature and chemical potential are well-defined at each point.” This assumption is from Yarushina and Podladchikov (2015), not from Lebon et al. 2008.

Line 84. You do not consider chemical reactions in the following text, just poroelasticity. Then why do you include chemical potentials in the thermodynamics?

Line 95: “Building upon the concepts from Lebon et al. (2008) and the nonlinear viscoelastoplastic framework developed by Yarushina and Podladchikov (2015), the derivation of Gassmann’s and Biot’s equations must satisfy the constraints of thermodynamic admissibility.” Which concepts from Lebon?

Line 135. This term presents entropy production due to poroviscous, not poroelastic, deformation, and the coefficient, h_j, is not a poroelastic coefficient but effective viscosity.

Line 143. “For detailed derivations and applications of these principles to specific pore geometries and boundary conditions…” What has it to do with geometry and boundary conditions? These principles are free of any assumptions on boundary conditions or pore geometry.

Line 206-207. “Various poroelastic constants can be calculated numerically (Alkhimenkov, 2023) or measured using physical experimentation in a laboratory (Makhnenko and Podladchikov, 2018)” I would add here also that they can be derived from effective media models with relevant references.

Section 3.5. would benefit by combining it with section 4.4. Try to combine results related to Gassmann’s equation into the same section.

Line 250-251. “(v) Assumption that equal changes in pore (fluid) pressure and confining (total) pressure leave the porosity unchanged (Korringa, 1981; Alkhimenkov, 2024).” Was this assumption formulated in those references?!!

Line 255-256. “(v) is not an assumption but a strict requirement for zero entropy production during reversible poroelastic processes.” There is no proof of this statement. You show the model where this assumption holds, and it is thermodynamically admissible. Can you prove that a system not satisfying this assumption will not be thermodynamically admissible?

Section 4.4. This detailed discussion of Thomsen’s articles here distracts from the main message of the presented manuscript. This manuscript is not about the number of independent elastic parameters, which appear in the specific case when there is a multi-mineralogical composition of the rock, but about a single mineral matrix and a specific assumption of Gassmann on porosity changes. If you want to discuss a multi-mineral matrix, I would suggest making a simulation with grains made of two different minerals and comparing that simulation to Thomsen. Statements like “Alkhimenkov (2023) conducted a numerical convergence study showing that K_M is converging to K_s as the resolution increases…” are misleading without providing full details of the assumptions behind Thomsen’s derivations or your numerical setup. The main question related to the whole section that is still unanswered is: under which assumptions Thomsen’s equation coincides with Gassmann’s?
Citation: https://doi.org/10.5194/egusphere-2024-3238-RC1
- AC1: 'Reply on RC1', Yury Alkhimenkov, 30 Jun 2025
  
  Please find attached.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3238-AC1
RC2:
'Comment on egusphere-2024-3238', Anonymous Referee #2, 12 Mar 2025

The manuscript provides a derivation of the well-know and well-studied Gassmann's equations. It is not clear why another derivation is needed and if so, how different is it from their previous work that they reference.
They first make the assumption about linear, elastic, homogeneous material, but then they give derivations with viscoelastoplasticity and elastic versus non-elastic pore deformation. It would be better to re-structure the more general parts first and then make the assumptions about linear elastic materials. Why the reference to chemical potentials when Gassmann's theory is not about chemical reactions?
Check equation 11. Should be 1/K_phi, otherwise not consistent.
References to their own previous numerical work is to cursory and not clear.
If the basic assumption is about a homogeneous mineralogy then why at the end the discussion about heterogeneous minerals?

Citation: https://doi.org/10.5194/egusphere-2024-3238-RC2
- AC2: 'Reply on RC2', Yury Alkhimenkov, 30 Jun 2025
  
  Please find attached.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3238-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-3238', Anonymous Referee #1, 23 Dec 2024
The validity of Gassmann’s equations and the thermodynamic admissibility of Biot’s equations have been widely discussed in the literature since their first publication. The new manuscript by Alkhimenkov and Podladchikov advocates Gassmann’s assumption that equal changes in pore pressure and total pressure leave the porosity unchanged. They present new logical reasoning that this assumption, which is hard to validate in the experiments, follows from the thermodynamic principles. However, the manuscript suffers from insufficient logic, poor structure, and inaccurate statements.
The introduction is misleading. It appears that the authors are building the manuscript around the statement: “However, despite their widespread use, recent studies have highlighted concerns regarding the logical consistency in the derivation of Gassmann’s equations.” (Their lines 15-16). However, there are no references to this statement, nor do they describe the claims they are arguing against. At the end of the manuscript, they explicitly discuss the recent claims of L. Thomsen on the incorrectness of Gassmann’s equations. However, Thomsen’s arguments were about using two solid bulk moduli commonly known as K^’_s and K^’’_s. Thus, the main message of the current manuscript remains unclear after reading.

Authors claim that “This article aims to address these concerns by presenting a novel derivation of Biot’s poroelastic equations and Gassmann’s equations, which strictly adheres to fundamental conservation laws and thermodynamic principles.” (Lines 19-20) and “… a concise and rigorous derivation rooted in thermodynamic principles and conservation laws has been missing from the literature.” (Lines 2-3). These two statements are incorrect. First, the thermodynamic consistency of Gassmann’s equations was presented by Yarushina and Podladchikov (2015). Second, the manuscript does not present a novel derivation but rather repeats the thermodynamic derivation of Yarushina and Podladchikov (2015). Reformulate these statements and need for the article.

The manuscript's structure is somewhat inconsistent and lacks clarity.
For example, a statement on Gassmann’s assumption in the middle of the thermodynamic section (Lines 126 - 132) is out of place. Please discuss this assumption in the introduction since this is the main Gassmann hypothesis that you are aiming to prove. Besides, after reading this para, what exactly was not assumed as highlighted in the bold text is still unclear. Are you trying to say that no porosity change under equal pore and total pressure changes is a thermodynamic admissibility condition?

Numerical verification in section 4.3 is simply missing. You refer to your recently published paper, making unclear statements in this section. Given that you repeat already existing thermodynamic derivation by other authors for the sake of argument in favor of Gassmann, you could as well repeat your own numerical results for clarity. It is unclear what statement you are arguing here against. In statements such as “…solution converges towards the result obtained from the original Gassmann’s equation…” (Line 262), it is unclear which result is meant. Similarly, in the statement from lines 263-265 “…the pore geometry that was used did not contain any special features (among all possible geometries) that were tailored to make it consistent with Gassmann’s equations (Alkhimenkov, 2024)”, it is unclear what special features can be tailored to suite Gassmann. It is very odd to read such statements given that Gassmann’s equation is independent of geometry assumption as follows from his original paper and as was shown again by Yarushina and Podladchikov (2015).

Section 4.4. discusses the recent arguments by Thomsen. Are there other authors that recently questioned the validity of Gassmann’s equations or tried to rederive them? Or is your article built only around Thomsen’s arguments?

The introduction should contain a clear statement you try to prove. Make a statement that you consider rocks with K^’_s = K^’’_s and explain what this assumption means. This makes the whole argument with Thomsen in section 4.4 nonrelevant as he is simply rederiving the generalization of Gassmann’s equations to the case when the two moduli are different. You can mention that such a generalization exists, and you demonstrated numerically that, indeed, this generalization reduces to Gassmann’s equation under condition K^’_s = K^’’_s.

Keep thermodynamic derivation clear of discussions on Gassmann's equation as discussed above.

Include relevant material from your previous paper in section 4.3 on numerical validation.

Statements like “While the general methodology was outlined by Yarushina and Podladchikov (2015), this study specifically focuses on the rigorous derivation of Biot’s poroelastic, Gassmann’s, and effective stress law equations, along with addressing concerns related to their physical validity” (Lines 28-30) should be more accurate, as thermodynamic derivation of Biot’s equations was already presented in the literature (see e.g., Yarushina and Podladchikov (2015) or Coussy et al., 1998, From mixture theory to Biot’s approach for porous media, Int. J. Solids Structures.)

Line 25: “… validate their integrity …”. What is the integrity of equations, and how do you demonstrate it?

Section 3. Make a statement that you just repeat previously published thermodynamic derivation and not making a new one.

Lines 72 – 74. “In the context of classical non-equilibrium thermodynamics (Lebon et al., 2008), each phase within the porous medium is considered to be locally in thermodynamic equilibrium, which means that intensive variables such as temperature and chemical potential are well-defined at each point.” This assumption is from Yarushina and Podladchikov (2015), not from Lebon et al. 2008.

Line 84. You do not consider chemical reactions in the following text, just poroelasticity. Then why do you include chemical potentials in the thermodynamics?

Line 95: “Building upon the concepts from Lebon et al. (2008) and the nonlinear viscoelastoplastic framework developed by Yarushina and Podladchikov (2015), the derivation of Gassmann’s and Biot’s equations must satisfy the constraints of thermodynamic admissibility.” Which concepts from Lebon?

Line 135. This term presents entropy production due to poroviscous, not poroelastic, deformation, and the coefficient, h_j, is not a poroelastic coefficient but effective viscosity.

Line 143. “For detailed derivations and applications of these principles to specific pore geometries and boundary conditions…” What has it to do with geometry and boundary conditions? These principles are free of any assumptions on boundary conditions or pore geometry.

Line 206-207. “Various poroelastic constants can be calculated numerically (Alkhimenkov, 2023) or measured using physical experimentation in a laboratory (Makhnenko and Podladchikov, 2018)” I would add here also that they can be derived from effective media models with relevant references.

Section 3.5. would benefit by combining it with section 4.4. Try to combine results related to Gassmann’s equation into the same section.

Line 250-251. “(v) Assumption that equal changes in pore (fluid) pressure and confining (total) pressure leave the porosity unchanged (Korringa, 1981; Alkhimenkov, 2024).” Was this assumption formulated in those references?!!

Line 255-256. “(v) is not an assumption but a strict requirement for zero entropy production during reversible poroelastic processes.” There is no proof of this statement. You show the model where this assumption holds, and it is thermodynamically admissible. Can you prove that a system not satisfying this assumption will not be thermodynamically admissible?

Section 4.4. This detailed discussion of Thomsen’s articles here distracts from the main message of the presented manuscript. This manuscript is not about the number of independent elastic parameters, which appear in the specific case when there is a multi-mineralogical composition of the rock, but about a single mineral matrix and a specific assumption of Gassmann on porosity changes. If you want to discuss a multi-mineral matrix, I would suggest making a simulation with grains made of two different minerals and comparing that simulation to Thomsen. Statements like “Alkhimenkov (2023) conducted a numerical convergence study showing that K_M is converging to K_s as the resolution increases…” are misleading without providing full details of the assumptions behind Thomsen’s derivations or your numerical setup. The main question related to the whole section that is still unanswered is: under which assumptions Thomsen’s equation coincides with Gassmann’s?
Citation: https://doi.org/10.5194/egusphere-2024-3238-RC1
- AC1: 'Reply on RC1', Yury Alkhimenkov, 30 Jun 2025
  
  Please find attached.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3238-AC1
RC2:
'Comment on egusphere-2024-3238', Anonymous Referee #2, 12 Mar 2025

The manuscript provides a derivation of the well-know and well-studied Gassmann's equations. It is not clear why another derivation is needed and if so, how different is it from their previous work that they reference.
They first make the assumption about linear, elastic, homogeneous material, but then they give derivations with viscoelastoplasticity and elastic versus non-elastic pore deformation. It would be better to re-structure the more general parts first and then make the assumptions about linear elastic materials. Why the reference to chemical potentials when Gassmann's theory is not about chemical reactions?
Check equation 11. Should be 1/K_phi, otherwise not consistent.
References to their own previous numerical work is to cursory and not clear.
If the basic assumption is about a homogeneous mineralogy then why at the end the discussion about heterogeneous minerals?

Citation: https://doi.org/10.5194/egusphere-2024-3238-RC2
- AC2: 'Reply on RC2', Yury Alkhimenkov, 30 Jun 2025
  
  Please find attached.
  
  Citation: https://doi.org/10.5194/egusphere-2024-3238-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Yury Alkhimenkov on behalf of the Authors (30 Jun 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (14 Jul 2025) by Taras Gerya

RR by Anonymous Referee #1 (28 Jul 2025)

ED: Publish as is (28 Jul 2025) by Taras Gerya

ED: Publish as is (06 Aug 2025) by Susanne Buiter (Executive editor)

AR by Yury Alkhimenkov on behalf of the Authors (09 Aug 2025) Author's response Manuscript

Journal article(s) based on this preprint

29 Oct 2025

Revisiting Gassmann-type relationships within Biot poroelastic theory

Yury Alkhimenkov and Yury Y. Podladchikov

Solid Earth, 16, 1227–1247, https://doi.org/10.5194/se-16-1227-2025,https://doi.org/10.5194/se-16-1227-2025, 2025

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Yury Alkhimenkov and Yury Y. Podladchikov

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This paper presents a rigorous derivation of Gassmann's equations, grounded in thermodynamic principles and conservation laws, addressing gaps and potential inconsistencies in the original formulation. It also explores Biot's poroelastic equations, demonstrating that Gassmann's equations are a specific case within Biot’s framework. The study affirms the robustness of Gassmann's equations when assumptions are met, and symbolic Maple routines are provided to ensure reproducibility of the results.


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