# Non-trivial bounds on 2, 3, 4, and 5-torsion in class groups of number
fields, conditional on standard $L$-function conjectures

Arul Shankar and Jacob Tsimerman

###### Abstract

We prove new conditional bounds on the $m$-torsion subgroups of class groups
of number fields of any fixed degree, for $m=2$, $3$, $4$, and $5$. Our
methods first recast the problem in the language of class groups of Galois
modules, which allows us to relate these torsion subgroups to Selmer groups of
elliptic curves. We then obtain a global estimate using the refined BSD
conjecture, in a similar way to how one normally uses the Brauer-Siegel bound.

Our methods are potentially very general, but rely on the existence of motives
with very special ${\mathbb{Z}}/m{\mathbb{Z}}$-cohomology. In particular, the
restriction to $m\leq 5$ stems from needing an elliptic curve over
${\mathbb{Q}}$ with $m$-torsion subgroup isomorphic to
${\mathbb{Z}}/m{\mathbb{Z}}\oplus\mu_{m}$.

##  1 Introduction

### History and main results

Let $n,m\geq 2$ be fixed integers. For any number field $K$ of degree $n$, it
is conjectured that the size $h_{m}(K)$ of the $m$-torsion subgroup of the
class group of $K$ satisfies the bound $h_{m}(K)=O_{m,n}(|{\rm
Disc}(K)|^{\epsilon})$ for every $\epsilon>0$. This conjecture is currently
known only for $n=m=2$ (using Gauss’s genus theory). For other degrees and
primes, the best that is known in general is the bound

$h_{m}(K)=O_{n,\epsilon}(|{\rm Disc}(K)|^{1/2+\epsilon}),$ (1)

which follows from the Brauer–Siegel theorem bounding the size $h(K)$ of the
entire class group, and applying the trivial inequality $h_{m}(K)\leq h(K)$.
It has become a problem of much interest to obtain any improvement on the
trivial bound (1) for any pair $(n,m)\neq(2,2)$.

The first such result was obtained independently by Pierce [17] and
Helfgott–Venkatesh [6], who consider the case $n=2$ and $m=3$, and prove the
bounds $O_{\epsilon}(|{\rm Disc}(K)|^{27/56+\epsilon})$ and
$O_{\epsilon}(|{\rm Disc}(K)|^{0.44178...})$, respectively. Subsequently,
these bounds were improved to $O_{\epsilon}(|{\rm Disc}(K)|^{1/3+\epsilon})$
by work of Ellenberg and Venkatesh [5]. In this work, they also achieve the
bound $O_{\epsilon}(|{\rm Disc}(K)|^{1/3+\epsilon})$ for the cases
$(n,m)=(3,3)$ and $(4,3)$. Recent work of Bhargava, Taniguchi, Thorne, Zhao
and the two authors [2] obtain power saving bounds $O(|{\rm
Disc}(K)|^{1/2-1/(2n)+\epsilon})$ for the case $m=2$ for all degrees $n>2$.
Further unconditional results are also known when the number fields $K$ are
constrained by having a fixed Galois group: in the work of Wang [20], Kluners
and Wang [9], and Wang [21], power saving bounds are respectively obtained for
$p$-torsion in the class groups of $K$, where $K$ has Galois group
$({\mathbb{Z}}/q{\mathbb{Z}})^{r}$ (here $q$ is prime and $r>1$), where the
Galois group of $K$ is a $p$-group, and where the Galois group of $K$ is a
nilpotent group $G$ such that every Sylow subgroup of $G$ is non-cyclic and
non-quaternion. Additionally, conditional on the generalized Riemann
hypothesis, Ellenberg and Venkatesh prove a bound of $O(|{\rm
Disc}(K)|^{\frac{1}{2}-\frac{1}{2m(n-1)}+\epsilon})$ for all pairs $(n,m)$.

The purpose of this paper is to introduce a new method for obtaining such
bounds, albeit a conditional one at the moment. Our main result is the
following.

###### Theorem 1

Let $n>1$ be a fixed integer, and let $m=2$, $3$, $4$, or $5$. Assume the
Hasse–Weil conjecture, the refined Birch and Swinnerton-Dyer conjecture, and
the generalized Riemann hypothesis for $L$-functions of elliptic curves over
degree-$n$ number fields. Then
$h_{m}(K)=O_{n,\epsilon}(D^{\frac{1}{4}+\epsilon})$ for any degree-$n$ number
field $K$.

In the case of quadratic fields, we can get a non-trivial result without
assuming either the Hasse–Weil conjceture or the generalized Riemann
hypothesis:

###### Theorem 2

Let $m=4$ or $5$. Assume the refined Birch and Swinnerton-Dyer conjecture for
elliptic curves over ${\mathbb{Q}}$. Then
$h_{m}({\mathbb{Q}}(\sqrt{D}))=O_{\epsilon}(D^{\frac{1}{2}-\frac{\delta}{2}+\epsilon})$
where $\frac{1}{2}-\delta$ is the best subconvex bound we have for
$L$-functions of elliptic curves over ${\mathbb{Q}}$.111The best such bound
the authors are aware of is $\delta=\frac{25}{256}$ (see [11]).

###### Remark 3

We make the following three observations regarding Theorem 1 and its proof.

  * (a)

To our knowledge, this is the first (even conditional) result which gives
pointwise bounds, where the quality of the exponent does not degrade with the
degree $n$ of the number field.

  * (b)

To obtain the conclusion of Theorem 1, we need to assume something weaker than
the three conjectures for all elliptic curves $E/K$. In fact, we only use base
changes of a single elliptic curve.

  * (c)

If, for a fixed elliptic curve $E/{\mathbb{Q}}$, we had a way of bounding the
rank of $E/K$ by $o(\log|{\rm Disc}(K)|)$ unconditionally, we would not need
to assume the generalized Riemann Hypothesis for the $L$-function of $E/K$.
Assuming the Lindelöf conjecture for this $L$-function would be sufficient.
Moreover, any subconvex estimate on the central value of this $L$-function
would also imply a power saving bound on the torsion in the class group.

### Method of proof

Our perspective is as follows.

  1. 1.

First, for an integer $m$, we express the group ${\rm Cl}(K)[m]$ as an
unramified cohomology group of a finite Galois module $M$. In fact, $M$ is
simply ${\mathbb{Z}}/m{\mathbb{Z}}[{\rm Hom}(K,\bar{{\mathbb{Q}}})]$. We
denote this unramified cohomology group by ${\rm Cl}(M)$.

  2. 2.

Our main idea (philosophically, and somewhat imprecisely) is to find other
‘motives’ $X$ whose mod $m$-reduction is (essentially) isomorphic to a power
of $M$. Then we may hope to reinterpret ${\rm Cl}(M)$ as the $m$-torsion of a
motivic class group $Cl(X)$. There should be a global class number formula for
$Cl(X)$ with a trivial bound that one can use to bound $Cl(M).$ In a sense, we
are obtaining better “trivial bounds” for ${\rm Cl}(M)$ by embedding $M$ into
different motives.

  3. 3.

Concretely, we find an elliptic curve $E$ over ${\mathbb{Q}}$ such that $E[m]$
is isomorphic to $\mu_{m}\oplus({\mathbb{Z}}/m{\mathbb{Z}})$ as Galois modules
over ${\mathbb{Q}}$. It is known that such elliptic curves only exist when $m$
is $2$, $3$, $4$, and $5$, hence our results are limited to these four values
of $m$. Our motive is essentially the Abelian variety
$A=\textrm{Res}_{K/{\mathbb{Q}}}E_{K}$ and its class group corresponds to the
Tate-Shafarevich group. The torsion group scheme $A[n]$ is isomorphic to
$M\oplus M\otimes\mu_{m}$, and we show that ${\rm Cl}(M)$ and ${\rm
Cl}(M\otimes\mu_{m})$ are approximately the same size (in fact they are dual
to each other up to small error).

  4. 4.

The Refined BSD conjecture plays the role of the class number formula which
allows us a method for computing the size of A, assuming we have control on
the rank of $A$ and the size of the central value of the $L$-function. This is
where the generalized Riemann hypothesis enters. We then use GRH to provide an
upper bound for A. This in turn provides a trivial upper bound for
${}_{A}[m]$, which we show is essentially the same size as ${\rm Cl}(A[m])$
(once the size of the rank has been controlled). Combining these observations
gives an upper bound for ${\rm Cl}(M)$.

In fact, we find it easier to work with $E/K$ directly and not form the Weil-
restriction down to ${\mathbb{Q}}$, but this is a minor point done for
technical convenience.

### Acknowledgements

We are happy to thank Ashay Burungale, Jordan Ellenberg, Ananth Shankar, Joe
Silverman, and Tonghai Yang for interesting comments and corrections on a
previous version of this paper.

##  2 Estimating the terms in the refined BSD conjecture

Let $E$ be an elliptic curve over ${\mathbb{Q}}$, and let $K$ be a degree-$n$
number field. Let $L(E/K,s)$ denote the Hasse–Weil $L$-function of $E$ over
$K$, normalized so that the critical line is $\Re(s)=1$. We denote the
completed $L$-function of $E/K$ by $\Lambda(E/K,s)$:

$\Lambda(E/K,s):={\rm Norm}_{K/{\mathbb{Q}}}(N_{E/K})^{s}|{\rm
Disc}(K)|^{2s}\Gamma_{K}(s)L(E/K,s),$

where $N_{E/K}$ is the conductor ideal of $E$ over $K$ and
$\Gamma_{K}(s):=((2\pi)^{-s}\Gamma(s))^{n}$. We begin by recalling three
standard conjectures regarding $E/K$ and its $L$-function. First, the
Hasse–Weil conjecture (HWC) for $E/K$ is the following:

###### Conjecture 2.1

The function $L(E/K,s)$ has an analytic continuation to the entire complex
plane, and satisfies the function equation

$\Lambda(E/K,s)=w(E/K)\Lambda(E/K,2-s),$

where $w(E/K)$ denotes the root number of $E/K$.

Denote the algebraic rank of $E/K$ by $r_{E/K}$. The refined Birch and
Swinnerton-Dyer (BSD) conjecture for $E/K$ is the following:

###### Conjecture 2.2

The algebraic rank $r_{E/K}$ is equal to the order of vanishing of $L(E/K,s)$
at $s=1$. Moreover, the leading term of the Taylor series of $L(E/K,s)$ at
$s=1$ is given by

$\lim_{s\to 1}\frac{L(E/K,s)}{(s-1)^{r_{E/K}}}=|{\rm
Disc}(K)|^{-1/2}\cdot\\#{}_{E/K}\cdot
R_{E/K}\cdot\frac{\Omega_{E/K}C_{E/K}}{\\#E(K)_{{\rm tor}}^{2}}.$ (2)

Above, ${\rm Disc}(K)$ denotes the discriminant of $K$, E/K is the
Tate–Shafarevich group of $E$ over $K$, $R_{E/K}$ is the regulator of $E$ over
$K$, $\Omega_{E,K}:=\Omega_{+}^{r_{1}}\Omega_{-}^{r_{2}}$ is the product of
the two periods of $E$ raised to the number of real and complex embeddings of
$K$, respectively, and $C_{E/K}$ is given by

$C_{E/K}:=\prod_{v}c_{v}(E/K)\Big{|}\frac{\Delta_{E,v,\min}}{\Delta_{E,v}}\Big{|}_{v}^{\frac{1}{12}},$

where the product is over all finite places $v$ of $K$, and where
$\Delta_{E,v,\min}$ denotes the discriminant of a minimal model for $E$ over
$K_{v}$.

Finally, we have the Grand Riemann Hypothesis (GRH) for the $L$-function of
$E/K$.

###### Conjecture 2.3

The nontrivial zeroes of $L(E/K,s)$ lie on the line $\Re(s)=1$.

In the remainder of the section, we obtain bounds on the rank $r_{E/K}$ of
$E/K$ and the size of the Tate–Shafarevich group E/K conditional on the above
three conjectures on $E/K$. Our bounds are in the setting where $E$ and the
degree $n$ of the number field $K$ are fixed, while $K$ is varying.

We begin with the following lemma.

###### Lemma 2.4

We have

$1\ll_{n,E}\frac{\Omega_{E/K}C_{E/K}}{\\#E(K)_{{\rm
tor}}^{2}}\ll_{n,E,\epsilon}1.$

Proof: The quantity $\Omega_{E/K}$ depends only on $E$ and the number of real
and complex embeddings of $K$. By Merel’s generalization [14] of Mazur’s
theorem [13], it follows that we have $\\#E(K)_{{\rm tor}}\ll_{n}1$. We also
have the inequalities

$1\leq\prod_{v}\Big{|}\frac{\Delta_{E,v,\min}}{\Delta_{E}}\Big{|}_{v}\ll_{n,E}1,\quad\quad
1\leq\prod_{v}c_{v}(E/K)\ll_{n,E}1,$

where the first two inequalities are immediate, and the second two follow from
an argument identical to the proof of [16, Lemma 6.2.1]. $\Box$

Next, we we have the following conditional bounds on the rank of $E/K$ from
[8, Proposition 5.21].

###### Lemma 2.5

Assume that $E/K$ satisfies the HWC and the BSD conjecture, and that
$L(E/K,s)$ satisfies GRH. Then we have

$r_{E/K}\ll_{n,E}\frac{\log|{\rm Disc}(K)|}{\log\log|{\rm Disc}(K)|}.$

Next, we obtain a lower bound on the regulator depending on the rank of $E/K$:

###### Lemma 2.6

With notation as above, we have

$R_{E/K}\gg_{n,E}\Bigl{(}\frac{\log|{\rm
Disc}(K)|}{r_{E/K}}\Bigr{)}^{r_{E/K}}.$

Proof: Let $P=(x,y)\in E(K)$ be a rational point. Let $F={\mathbb{Q}}(x,y)$ be
the subfield of $K$ generated by $x$ and $y$. We know that the canonical
height $\hat{h}(P)$ of $P$ is within $O_{E}(1)$ of the Weil height of $P$
which is simply the logarithmic height of $x\in\bar{Q}$. Since $x$ generates
$F$, it follows that we have $\hat{h}(P)\gg_{E}\log|{\rm Disc}{F}|$. Moreover,
by Lemma 2.5, we know that the rank of $E$ over $F$ is $\ll_{n,E}\log|{\rm
Disc}(F)|$. The lemma now follows from Minkowski’s second theorem. $\Box$

###### Remark 2.7

We note that sophisticated lower bounds for the canonical heights of
nontorsion points in elliptic curves $E/K$ are proven by Silverman [19] and
Hindry–Silverman [7]. However, those results are proven in the setting where
$K$ is fixed and $E$ is varying, while we need bounds in the setting where $E$
is fixed and $K$ is varying.

The conductor of $L(E/K,s)$ is ${\rm Disc}(K)^{2}{\rm
Norm}_{K/{\mathbb{Q}}}(A(E/K))\ll_{E}{\rm Disc}(K)^{2}$. This yields the
“convex” bound $|L(E/K,s)|\ll_{E,\epsilon}|{\rm Disc}(K)|^{1/2+|s|+\epsilon}$,
for complex numbers $s$ sufficiently close to $1$. In our next result, we
bound the leading Taylor coefficient of the $L$-function $L(E/K,s)$, assuming
a rank bound on $E/K$ and a subconvex bound on $|L(E/K,s)|$.

###### Lemma 2.8

Assume that we have $|L(E/K,s)|\ll_{E,\epsilon}|{\rm
Disc}(K)|^{1/2+|s|-\eta+\epsilon}$ for some $\eta>0$, and that the rank of $E$
over $K$ is bounded by $r_{E/K}\ll_{E}\log|{\rm Disc}(K)|/\log\log|{\rm
Disc}(K)$. Then we have

$\lim_{s\to 1}\frac{L(E/K,s)}{(s-1)^{r_{E/K}}}\ll_{E,\epsilon}|{\rm
Disc}(K)|^{1/2-\eta+\epsilon}.$

Proof: Denote $r_{E/K}$ by $r$ and note that the result is immediate when
$r=0$. When $r\geq 1$, we write

$\lim_{s\to 1}\frac{L(E/K,s)}{(s-1)^{r}}=L^{(r)}(E/K,1)=r!\cdot{\rm
Res}_{s=0}\frac{L(E/K,1+s)}{s^{r+1}}.$

Therefore, from the residue theorem, we have

$L^{(r)}(E/K,1)=\frac{1}{2\pi i}\int_{|s|=\theta}\frac{L(E/K,1+s)}{s^{r+1}}ds$

for every $\theta>0$. Taking $\theta=(\log\log\log|{\rm Disc}(K)|)^{-1}$ and
using the upper bound on $r$ yields

$L^{(r)}(E/K,1)\ll_{E,\epsilon}|{\rm Disc}(K)|^{\frac{1}{2}-\eta+\epsilon},$
(3)

as necessary. $\Box$

Finally, we combine (2) with Lemmas 2.4, 2.5, 2.6, and 2.8, to obtain the
following conditional upper bound on the Tate–Shafarevich group E/K.

###### Proposition 2.9

Let $E$ be a fixed elliptic curve over ${\mathbb{Q}}$ and let $n\geq 2$ be a
fixed integer. Let $K$ be a number field of degree $n$, and assume Conjectures
2.1 $($HWC$)$, 2.2 $($BSD$)$, and 2.3 $($GRH$)$ for $E/K$ and $L(E/K,s)$. Then
we have

$\\#{}_{E/K}\ll_{n,E,\epsilon}|D|^{1/2+\epsilon}.$

##  3 Comparing torsion in ${\rm Cl}(K)$ to torsion in E/K

###  3.1 Cohomology of finite Galois modules over number fields

Let $A$ be a finite Galois module over a field $F$, i.e., a finite abelian
group $A$, along with an action of the absolute Galois group $G_{F}$ of $F$ on
$A$. We let $H^{i}(F,A)$ denote the $i$’th cohomology group arising from this
action. Now let $A$ be a finite Galois module over a local field $K_{v}$ (with
residue field denoted ${\mathbb{F}}_{v}$). Let $I_{v}$ denote the intertia
subgroup of $G_{K_{v}}$. We say that $A$ is unramified if $I_{v}$ acts
trivially on $A$. When $A$ is unramified, there is a natural action of
$G_{K_{v}}/I_{p}\cong G_{{\mathbb{F}}_{v}}$ on $A$, and we have an injection

$H^{1}({\mathbb{F}}_{v},A)\hookrightarrow H^{1}(K_{v},A).$

Define the subgroup of unramified cohomology classes by

$H^{1}_{\rm ur}(K_{v},A):=\ker\bigl{(}H^{1}(K_{v},A)\to
H^{1}(I_{v},A)\bigr{)}.$

When $A$ is unramified, it follows from the inflation-restriction exact
sequence that the subgroup of unramified cohomology classes is exactly given
by the image of $H^{1}({\mathbb{F}}_{v},A)$ (see [18, Lemma 3.2]).

Next, let $A$ be a finite Galois module over a number field $K$. Let $v$ be a
place of $K$. The absolute Galois group $G_{K_{v}}$ of $K_{v}$ naturally
injects into $G_{K}$, yielding natural maps $H^{1}(K,A)\to H^{1}(K_{v},A)$.
When $v$ is a finite place, we say that $A$ is unramified at $v$ if $A$ is
unramified as a $K_{v}$-Galois module. We define the class group ${\rm Cl}(A)$
of $A$ to be the set of elements $\sigma\in H^{1}(K,A)$ such that for every
$v$ at which $A$ is unramified, the image of $\sigma$ in $H^{1}(K_{v},A)$ is
an unramified cohomology class. That is

${\rm
Cl}(A):=\ker\Bigl{(}H^{1}(K,A)\to\sideset{}{{}^{\prime}}{\prod}_{v}H^{1}(K_{v},A)/H_{\rm
ur}^{1}(K_{v},A)\Bigr{)},$

where the product is over all finite places $v$ at which $A$ is unramified.

###  3.2 Selmer groups of elliptic curves

Let $E$ be an elliptic curve over a field $F$ and let $m$ be a positive
integer. We have the exact sequence

$0\to E[m]\to E\to E\to 0,$

of $G_{F}$-modules, where $E[m]$ is the $m$-torsion subgroup of $E$. This
gives rise to the following exact sequence of Galois cohomology groups:

$0\to E(F)/mE(F)\to H^{1}(F,E[m])\to H^{1}(F,E)\to 0.$ (4)

We now specialize to the arithmetic case. Let $E$ be an elliptic curve over a
number field $K$. We have the exact sequence (4) for the fields $F=K$ and
$F=K_{v}$ for all completions $K_{v}$ of $K$. The $m$-Selmer group
$\operatorname{Sel}_{m}(E)$ is then defined to be

$\operatorname{Sel}_{m}(E):=\ker\Bigl{(}H^{1}(K,E[m])\to\prod_{v}H^{1}(K_{v},E)\Bigr{)}.$
(5)

Above, the product is over all places $v$ of $K$, and the map
$H^{1}(K,E[m])\to H^{1}(K_{v},E)$ is given by composing the map
$H^{1}(K,E[m])\to H^{1}(K_{v},E[m])$ with the map $H^{1}(K_{v},E[m])\to
H^{1}(K_{v},E)$ of (4).

Our next results compare the $m$-Selmer group of $E$ with the class group of
the finite Galois module $E[m]$. We start with the following lemma.

###### Lemma 3.1

Let $E$ be an elliptic curve over a number field $K$, and let $m>1$ be an
integer. Let $v$ be a place not dividing $m$ at which $E[m]$ is unramified and
at which $E$ has good reduction. Then

$\ker\left(H^{1}(K_{v},E[m])\to H^{1}(K_{v},E)\right)=H_{\rm
ur}^{1}(K_{v},E[m]).$

That is, the image of a class $\sigma\in H^{1}(K_{v},E[m])$ in
$H^{1}(K_{v},E)$ is soluble at $p$ if and only if it is unramified.

Proof: This is precisely the content of [4, Lemma 19.3]. $\Box$

###### Proposition 3.2

Let $E$ be an elliptic curve over $K$ of degree $n$, and let $m>1$. Let $s$ be
the number of places of $K$ dividing $m\Delta(E)$. Then we have

$\log\Bigl{(}\frac{\\#\operatorname{Sel}_{m}(E)}{\\#{\rm
Cl}(E[m])}\Bigr{)}=O_{m,n}(s).$

Proof: Both the groups $\operatorname{Sel}_{m}(E)$ and ${\rm Cl}(E[m])$ are
subgroups of the Galois cohomology group $H^{1}(K,E[m])$. The Galois module
$E[m]$ is unramified away from the prime dividing $m\Delta(E)$. It follows
from Lemma 3.1 that the Selmer structures defining $\operatorname{Sel}_{m}(E)$
and ${\rm Cl}(E[m])$ differ at most at $s+n$ ‘bad’ local places (the $n$
coming from the infinite place of $K$). Thus
$\log\frac{\\#\operatorname{Sel}_{m}(E)}{\\#{\rm Cl}(E[m])}$ is bounded by
${\rm ln}\prod_{v\textrm{ bad}}|H^{1}(K_{v},E[m])|$. We are thus finished if
we prove that $|H^{1}(K_{v},E[m])|=O_{m,n}(1)$.

Let $N$ denote the subgroup of $G_{K_{v}}$ which fixes $E[m]$. Then by the
inflation-restriction exact sequence, we have

$\begin{array}[]{rcl}|H^{1}(K_{v},E[m])|&\leq&|H^{1}(G/N,E[m])|\cdot|H^{1}(N,E[m])|\\\\[3.61371pt]
&=&|H^{1}(G/N,E[m])|\cdot|{\rm Hom}(N,E[m]|.\end{array}$

The first term in the second line of the above equation is $O_{m,n}(1)$ by
virtue of having finitely many possibilities for $G/N$ and its action. On the
other hand $|{\rm Hom}(N,E[m])|=|N[m]|$ is bounded by $m$ times the number of
degree $m$ extensions of $L=\overline{K_{v}}^{N}$ and hence by the number of
degree $\leq d$ extensions of ${\mathbb{Q}}_{p}$, where $d=mn\cdot\\#{\rm
Aut}E[m]$. This is well-known to be uniformly bounded. $\Box$

###  3.3 Class groups of number fields

Let $K$ be a number field of degree $n$ over ${\mathbb{Q}}$, and let $m>1$ be
a (fixed) integer. Class field theory provides a bijection between index-$m$
subgroups of ${\rm Cl}(K)$ and degree-$m$ abelian extensions of $K$ which are
unramified at every place. Let $v$ be a place of $K$. The set of degree-$m$
abelian extensions of $K$ (resp. $K_{v}$) is parametrized by
$H^{1}(K,{\mathbb{Z}}/m{\mathbb{Z}})$ (resp.
$H^{1}(K_{v},{\mathbb{Z}}/m{\mathbb{Z}})$). Let $L$ be the extension of $K$
corresponding to $\sigma\in H^{1}(K,\mu_{m})$. Then $L_{v}:=L\otimes_{K}K_{v}$
corresponds to the image of $\sigma$ under the natural map
$H^{1}(K,{\mathbb{Z}}/m{\mathbb{Z}})\rightarrow
H^{1}(K_{v},{\mathbb{Z}}/m{\mathbb{Z}})$. Furthermore, the extension $L_{v}$
of $K_{v}$ is unramified if and only if it corresponds to an unramified class
in $H^{1}(K_{v},{\mathbb{Z}}/m{\mathbb{Z}})$. Therefore, the size of the
$m$-torsion of the class group of $K$ is equal to the size of $H^{1}_{\rm
ur}(K,{\mathbb{Z}}/m{\mathbb{Z}})={\rm Cl}({\mathbb{Z}}/m{\mathbb{Z}})$.

Meanwhile, we have the isomorphism $H^{1}(K,\mu_{m})\cong
K^{\times}/(K^{\times})^{m}$ and, for a finite place $v$ of $K$ not dividing
$m$, a class in $H^{1}(K,\mu_{m})$ is unramified at $v$ if and only if it
corresponds to an element in $K^{\times}/(K^{\times})^{m}$ whose valuation at
$v$ is divisible by $m$. Therefore, defining the group $A_{m,K}$ by

$A_{m,K}\cong\Big{(}\ker
K^{\times}/(K^{\times}_{m})\rightarrow\prod_{v}{\mathbb{Z}}/m{\mathbb{Z}}\Big{)},$

we see that $|{\rm Cl}(\mu_{m})|$ is within a factor of $O_{m,n}(1)$ of
$|A_{m,K}|$.

There is a natural map $\phi:A_{m,K}\rightarrow{\rm Cl}(K)[m]$ given by
$\phi(\alpha)=(\alpha)^{\frac{1}{m}}$ which is surjective with kernel
isomorphic to $U_{K}/U_{K}^{m}$, the $m$-cotorsion of the unit group of $K$.
We thus obtain the following result:

###### Proposition 3.3

Let $K$ be a number field of degree $n$, and consider the $G_{K}$ modules
$\mu_{m},{\mathbb{Z}}/m{\mathbb{Z}}$. The group ${\rm
Cl}({\mathbb{Z}}/m{\mathbb{Z}})$ is isomorphic to the dual of ${\rm
Cl}(K)[m]$, and ${\rm Cl}(\mu_{m})$ has the same size as ${\rm Cl}(K)[m]$ up
to a factor of ${O_{m,n}(1)}$.

##  4 Proofs of Theorem 1 and 2

We begin with the following lemma.

###### Lemma 4.1

Let $m\in\\{2,3,4,5\\}$ be fixed. Then there exists an elliptic curve
$E/{\mathbb{Q}}$ such that we have the isomorphism

$E[m]\cong\mu_{m}\oplus{\mathbb{Z}}/m{\mathbb{Z}}$

as group schemes.

Proof: Since the modular curves $\Gamma(m)$ have genus $0$ for each
$m\in\\{2,3,4,5\\}$, there do exist such curves. Below, we provide specific
examples.

$\begin{array}[]{rl}m=2.&E:y^{2}=x^{3}+x\\\\[2.168pt]
m=3.&E:y^{2}+y=x^{3}\\\\[2.168pt] m=4.&E:y^{2}+xy=x^{3}-4x-1\\\\[2.168pt]
m=5.&E:y^{2}+y=x^{3}-x^{2}-10x-20\end{array}$

These examples have been taken from the database LMFDB [10]. $\Box$

Let $m$ and $E$ be as in the above lemma, and let $K$ be a number field of
degree $n$. Denote the elliptic curve $E$ over $K$ by $E_{K}$. We then have
the isomorphism

$E_{K}[m]\cong\mu_{m}\oplus{\mathbb{Z}}/m{\mathbb{Z}}$

as group schemes. We next prove the following result.

###### Proposition 4.2

The quantities $\\#\operatorname{Sel}_{m}(E_{K})$ and $\\#({\rm
Cl}(K)[m])^{2}$ are within a factor of $O_{\epsilon}(|{\rm
Disc}(K)|^{\epsilon})$ of each other.

Proof: By Proposition 3.2, we see that $|\log\\#{\rm
Cl}(E_{K}[m])-\log\\#\operatorname{Sel}_{m}(E_{K})|$ is bounded by
$O_{n,E}(1)$. By Proposition 3.3, we see that $\\#{\rm
Cl}({\mathbb{Z}}/m{\mathbb{Z}})=\\#{\rm Cl}(K)[m]$ and that $|\log\\#{\rm
Cl}(\mu_{m})-\log\\#{\rm Cl}(K)[m]|$ is bounded by $O_{n,m}(1)$. The result
follows from the structure of $E_{K}[m]$. $\Box$

We are ready to prove the main results.

Proof of Theorem 1: From the exact sequence

$0\rightarrow\frac{E_{K}(K)}{mE_{D}(K)}\rightarrow\operatorname{Sel}_{m}(E_{K})\rightarrow\Sh(E_{K})[m]\rightarrow
0$

it follows that

$\\#\operatorname{Sel}_{m}(E_{K})=\\#\frac{E_{K}(K)}{mE_{K}(K)}\cdot\\#\Sh(E_{K})[m].$

Our conditional rank bound from 2.5 implies that
$\\#\frac{E_{K}(K)}{mE_{K}(\mathbb{K})}=O_{\epsilon}(|{\rm
Disc}(K)|^{\epsilon})$. Theorem 1 now follows immediately by combining
Proposition 4.2 with the conditional bound on $\Sh(E_{K})[m]$ that was proved
in Theorem 2.9. $\Box$

Proof of Theorem 2: let $K={\mathbb{Q}}(\sqrt{D})$ be a quadratic field. Then
the Weil restriction of $E_{K}$ to ${\mathbb{Q}}$ is isomorphic to the abelian
surface $E\oplus E_{D}$, where $E_{D}$ denotes the twist of $E$ by $D$. Thus
the Hasse–Weil conjecture for $E_{K}$ is known by the modularity of $E$ and
$E_{D}$.

Next, denote the Mordell-Weil rank of $E_{D}$ by $r_{D}$. Let
$\operatorname{Sel}_{2}(E_{D})$ denote the $2$-Selmer group of $E_{D}$. From
the exact sequence

$0\to\frac{E_{D}({\mathbb{Q}})}{2E_{D}({\mathbb{Q}})}\to\operatorname{Sel}_{2}(E_{D})\to\Sh(E_{D})[2]\to
0$

we see that $r_{D}\leq\log_{2}\\#\operatorname{Sel}(E_{D})$. Since the field
${\mathbb{Q}}(E_{D}(\bar{{\mathbb{Q}}})[2])$ is the same as
${\mathbb{Q}}(E(\bar{{\mathbb{Q}}})[2])$, it follows from [3, Proposition 7.1]
and [12, Proposition 9.8] that we have
$\log_{2}\\#\operatorname{Sel}(E_{D})\ll\log_{2}\\#\operatorname{Sel}_{2}(E)+O(w_{D})$,
where $w_{D}$ is the number of prime factors of $D$. Therefore, we have the
upper bounds

$r_{E/K}\ll_{E}r_{D}\ll_{E}\left(\frac{\log|D|}{\log\log|D|}\right)$ (6)

on the ranks of $E_{D}/{\mathbb{Q}}$ and $E_{K}$.

Assume that we have the subconvex bound
$|L(E_{D}/{\mathbb{Q}},s)|\ll_{E,\epsilon}|D|^{1/2+|s|-\delta+\epsilon}$ for
some $\delta>0$. This immediately implies the bound
$|L(E/K,s)|\ll_{E,\epsilon}|{\rm Disc}(K)^{1/2}|^{1/2+|s|-\delta+\epsilon}$.
Theorem 2 now follows immediately from Lemma 2.8 by an argument identical to
the proof of Theorem 1. $\Box$

## References

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Influence of relaxation on filtering microflows under harmonic action on the
layer

I.I. Denysiuk1, I.A. Skurativska1111e-mail<EMAIL_ADDRESS>I.V.
Bielinskyi1, O.M. Sizonenko2, I.M. Hubar1

1 S.I.Subbotin Institute of Geophysics of the National Academy of Science of
Ukraine, Kyiv, Ukraine

2 Institute of Impulse Processes and Technologies of the National Academy of
Science of Ukraine, Mykolaiv, Ukraine

> Abstract.
>
> Purpose. Investigation of the velocity fields of non-equilibrium fluid
> filtration in a layer under harmonic action on it and assessment of the
> influence of relaxation effects on the attenuation of the amplitude of the
> initial disturbance within the framework of mathematical modeling of non-
> equilibrium plane-radial filtration.
>
> Methodology. A mathematical model of non-equilibrium plane-radial filtration
> with a generalized dynamic Darcy law in the form of a boundary value problem
> in a half-space with a harmonic excitation law at its boundary is
> considered. Based on the exact solutions of the boundary value problem, the
> attenuation of the amplitude of the initial disturbance from the model’s
> parameters and their influence on the size of the disturbed area is
> investigated.
>
> Findings. A differential equation modeling non-equilibrium filtration
> processes in the massif in the cylindrical reference frame was obtained.
> Using the method of separation of variables, it was constructed the
> solution, bounded at infinity, to the model differential equation subjected
> to harmonic action at the layer boundary. The solution’s asymptotic
> approximation was constructed for large values of the argument. Using the
> asymptotic solution of the boundary value problem, the damping of velocity
> field pulsations during non-equilibrium filtration was analyzed depending on
> the frequency of the harmonic action, the ratio of the piezoconductivity
> coefficients of the layer and the relaxation time. Profiles of the
> dependences of the size of the influence zone on the model parameters were
> plotted and the choice of parameters for optimal influence on the bottom-
> hole zone of the well was analyzed.
>
> Originality. On the basis of the non-equilibrium filtration model, it is
> shown that harmonic disturbances applied to the boundary of a semi-infinite
> layer can penetrate the reservoir over a greater distance under the
> conditions of manifestation of the relaxation mechanism of the fluid-
> skeleton interaction, compared to the equilibrium filtration process. Such
> an effect is observed at a finite interval of disturbance frequencies, while
> at high frequencies relaxation contributes to a more significant damping of
> disturbances. In the parametric space of excitation frequency - relaxation
> time, there is a locus of points that corresponds to the maximum sizes of
> influence zone of disturbances.
>
> Practical value. The obtained results are relevant for research on the
> impact of wave disturbances on the layer with the aim of intensifying
> filtration processes, as well as for the creation of new wave technologies
> to increase the extraction of mineral resources from productive layers.

> Keyword:non-equilibrium filtration, Darcy’s generalized law, porous medium,
> wave action, attenuation, filtration velocity fields.

## Introduction

At the present, the energy stability of Ukraine is connected with the increase
in the production of energy both due to the development of new promising
hydrocarbon deposits and the use of new technologies for the enhancing their
production. In the process of developing oil and gas fields, the filtration
characteristics of the rocks serving as oil and gas collectors deteriorate
significantly, which leads to a decrease in the flow rate of wells and the
degree of mineral resources development.

Most of the highly productive fields of the oil and gas complex of Ukraine
have entered the final stage of development, which is characterized by the
progressive depletion of formation energy, the watering of wells and the
increase in the share of hard-to-extract reserves. The development of fields
with difficult-to-extract oil reserves is carried out at a low rate, and the
final yield of oil in such cases does not exceed 30-40% of the initial balance
of their reserves [1].

In this regard, the most important scientific and technical problem that
arises during the exploitation of deposits is the most complete extraction of
oil while ensuring high rates of development. Therefore, the tasks of applying
new oil production technologies that allow to significantly increase the oil
yield of the layers being developed, from which it is no longer possible to
extract significant residual oil reserves using traditional methods, are
urgent.

To extract these remaining oil reserves, various modern methods of
intensification of oil production are used, in particular, thermal, chemical,
physical, biological and others. Among these methods, physical methods of
increasing the flow rate of production wells play a leading role. They also
include methods of wave action on the near wellbore zone (NWZ) and on the
formation as a whole. Methods of wave action can be divided into several
groups: acoustic (ultrasonic, hydraulic), shock wave and vibroseismic ones
[2].

Knowledge of physical processes and phenomena that are responsible for
restoring the filtration properties of reservoir rocks and fluid mobility is
necessary for a well-founded choice of the method of wave action on the
reservoir and increasing its potential oil extraction capabilities. The rate
of filtration, which is determined using methods of mathematical modeling of
filtration processes, is used to quantify fluid mobility. Therefore, the study
of the mechanisms of the influence of the wave field to restore the filtration
properties of the formation, to solve the problems of increasing oil recovery
is an urgent task. The advantages of the methods of wave action on the
formation are considered to be, firstly, the possibility of adjusting the
parameters of the emitter of acoustic waves (intensity, frequency, duration of
treatment); secondly, ecological purity of the method; thirdly, its high
efficiency.

Literature review. To study the mechanisms of wave action on saturated porous
reservoir media and reservoir fluids, as well as to identify the peculiarities
of the propagation of elastic waves in heterogeneous media, many experimental
studies have been performed. In particular, in works [3, 4], the effect of
ultrasonic waves on the rheological characteristics of various samples of oil
and its processing products was experimentally investigated. Experiments
proved that the viscosity of all liquid samples decreased during the action of
ultrasonic disturbances. It was found [3] that a higher power of ultrasound
leads to a more intense decrease in the viscosity of the liquid.

The results of laboratory studies [4] also showed that the mobility of fluids
increases under ultrasonic action, while the authors of the studies [5], in
addition to a decrease in fluid viscosity, also observed an increase in
reservoir permeability. The effect of increasing the absolute permeability of
saturated porous reservoir media under the influence of high-amplitude
pressure fluctuations in the liquid was experimentally proven in [6].

The publication [7] provides experimental studies on model heterogeneous media
with the aim of confirming the increase in oil mobility under ultrasonic
action in porous media. According to the results of these studies, it was
established that the oil extraction coefficient is proportional to the power
of the wave emitter and depends on the frequency of ultrasonic action.

In order to achieve the maximum efficiency of the wave action on the formation
and NWZ, it is also advisable to use pulse-wave methods with the formation of
shock pulses in a fluid-filled well, starting from the wellhead, using special
devices - pulse generators. In [8], it is proposed to implement pulse-wave
action using a hydraulic generator with a repetition frequency of pressure
pulses 1-50 Hz, which are less intensively absorbed by the reservoir at
distances exceeding 2 m from the well. As a result of the pulse-wave action on
the productive layer, an optimal level of depression is formed, which
contributes to the process of oil extraction and purification of NWZ. To do
this, special pumping equipment is used, which allows one to smoothly change
the pressure on the wellbore over a wide range. As a result, the joint use of
a pump and a hydraulic pulse generator ensures the minimum content of clogging
substances and the optimal mode of extracting reservoir fluids [8]. For the
purification of NWZ from colmatant, work [9] proposes a method of shock-wave
treatment, which is effectively used in the fields of Kazakhstan. The basis of
the method is the use of a special device that forms rarefaction waves. The
use of the shock-wave treatment method together with chemical compounds showed
high efficiency in the process of cleaning the near-breakout zone of the
formation from clogging substances and increasing the acceptability of
injection wells several times (from 20 to 160 m3/day).

In order to study the mechanism of interaction of elastic waves with the
porous medium of the formation, the effect of elastic oscillations on the
change of fluid filtration in the bulk model for the formation is considered
in [10]. The results of experimental studies indicate a significant influence
of the field of elastic oscillations on the filtration of the oil-water
mixture. In addition, during the experiments, other effects were observed: a
decrease in the effective friction between the rock and the fluid, and as a
result, an increase in the mobility of the fluid. During the passage of an
elastic wave, stretching and compression phases are observed, which together
with vibration of the skeleton and liquid causes the effect of a vibration
pump; a change in the shape of the meniscus at the boundary of the separation
of two phases and, as a result, a change in capillary pressure [10].

Understanding the wave process patterns occurring in the well-reservoir system
is incomplete without theoretical research. Such studies are based on the
analysis of the equations of the continuum mechanics for models of the real
medium where a wave process is observed.

The authors of [11] developed a mathematical modeling of the method of
acoustic stimulation of wells. The model takes into account the following
physical processes: reduction of liquid viscosity due to mixing and heating;
excitation of elastic waves on the walls of the well (to reduce the adhesion
forces between formation fluids and rock); excitation of natural frequencies
associated with the vibration of the liquid inside the porous medium. Using
numerical modeling, the optimal radiation frequencies were determined. It is
shown that the well’s productivity can be significantly improved due to the
correct selection of operating frequencies of the acoustic emitter. The
publication [12] deals with the problem of pulse-wave influence on a branched
horizontal well, which is modeled by branched waveguides of a certain radius,
in each of which the fluid movement is described by a wave equation. It is
shown that there are resonant frequencies in branched wells, at which the
pressure value can exceed the amplitude of the applied pressure pulse by
several orders of magnitude. It was established that after pulse-wave
treatment of injection wells in the fields of Oman, their acceptability
increased almost three times.

On the basis of a model differential equation that takes into account the
oscillation damping, the propagation of ultrasonic waves in a viscous liquid
was studied in [13] using the integral Laplace transform. Mathematical
modeling of ultrasonic wave propagation made it possible to highlight the
mechanisms of wave absorption and to study the influence of various parameters
(temperature, relaxation time) on the propagation of waves in viscous liquids.
The results of theoretical studies were verified experimentally by studying
the change in the velocity of ultrasound propagation and the damping parameter
in glycerol depending on temperature and frequency.

The results of research on the pulsating steady motions of a viscous fluid in
the pore channels of the reservoir under the harmonic action of acoustic waves
on them are presented in [14]. To solve the problem, differential equations
describing the laminar motion of a viscous liquid in a cylindrical pore
channel were used. On the basis of the obtained solution, numerical
calculations of dynamic processes in the pore channels of the formation were
carried out. It was found that in the case of acoustic action on the
formation, the speed of fluid movement in pore channels reaches maximum values
in a certain frequency range, depending on the size of the pores and the
kinematic viscosity of the fluid.

The article [15] is devoted to the pulsating movements of a viscous fluid in
the filtration channels of capillary-porous bodies under the action of
harmonic waves. Pulsating movements are accompanied by compression-discharge
waves and sign-changing filtration flows in the filtration channels of
capillary-porous bodies. According to the results of theoretical studies, the
most effective mode of pulse-wave loading was selected depending on the radius
of the pore channel.

It should be noted that the researches on this topic known in the literature
were conducted mainly on the basis of the classical Darcy filtration law
without taking relaxation processes into account. However, to date, a
considerable amount of experimental evidence of deviations from Darcy’s linear
law has been collected [16], especially in relation to non-equilibrium high-
intensity processes, when the strengthening of non-local effects is observed
[17].

To eliminate this gap, the article [18] proposed a mathematical model for the
elastic mode of liquid filtration with a generalized dynamic Darcy filtration
law, which includes a description of nonlocal and nonlinear effects. Within
the framework of this model, the influence of relaxation and the ratio of
permeability coefficients of the reservoir rock on the phase speed of
propagation of small wave disturbances was analyzed. Thus, on the basis of the
analysis of recent publications and research on this problem, it was
established that a significant part of them is devoted to the elucidation of
physical mechanisms and phenomena that affect the restoration of the
filtration characteristics of the porous medium of the reservoir, the fluid
rheological parameters and the increase of fluid mobility under wave action.

At the same time, the survey of scientific information proved that the study
of the attenuation of filtration oscillations, which are formed in the process
of acoustic action and the study of the influence of relaxation effects on the
processes of non-equilibrium filtration, are at the initial stage of their
study and are insufficiently covered. And these factors play a significant
role in the development of methods of wave action on the porous medium of the
formation in order to intensify the filtration processes in them. In this
regard, the purpose of the research is to study on the basis of Darcy’s
generalized dynamic filtration law of pulsating damping filtration microflows
of a fluid in a porous semi-bounded medium of an oil-bearing formation under
harmonic action on it and to determine the critical frequency of the wave
action, which ensures the minimum of the reduced attenuation coefficient
depending on the ratio permeability coefficients and the relaxation parameter.

##  1 The problem statement

It is considered the porous medium with specified initial porosity
$m_{0}$,pores of which are filled with fluid. Let the emitter of acoustic
harmonic waves acting on the formation be located in the well of the radius
$r_{c}$ at the formation level. Then the fluid elements, located at a distance
$r>r_{c}$ from the source of oscillations, perform a one-dimensional
axisymmetric radial non-stationary periodic motion according to the law
determined by the solution of the non-equilibrium filtration equation, which
can be deduced [18, 19] from the following system of equations written in
cylindrical coordinates:

$\begin{split}\tau\left[u_{t}+u\cdot
u_{r}+\frac{k_{f}}{\mu}\left(p_{rt}+up_{rr}\right)\right]+u+\frac{k_{e}}{\mu}p_{r}=0,\\\
(m\rho)_{t}+u\rho_{r}+\rho\left(u_{r}+\frac{u}{r}\right)=0,\\\
\rho=\rho_{0}+\rho_{0}\beta_{0}\left(p-p_{0}\right),\qquad
m=m_{0}+\beta_{s}\left(p-p_{0}\right),\\\ \mu(p,T)=const,\quad
k(p,r)=const,\end{split}$ (1)

where $u$ is the filtration velocity, m/s; $p$, $p_{0}$ are the variable and
initial pressures respectively, Pa; $\rho$, $\rho_{0}$ are the variable and
initial fluid densities, kg/m3; $m$, $m_{0}$ are the variable and initial rock
porosities; $\beta_{0}$, $\beta_{s}$ are coefficients of volume
compressibility of oil and rock skeleton of formation, 1/Pa; $\mu$ is the
coefficient of dynamical viscosity of oil, Pa$\cdot$s; $k_{e}$, $k_{f}$ are
the steady and frozen coefficients of permeability, m2; $\tau$ is the time of
relaxation, s.

##  2 Description of the research methodology.

System of equations (1) is the essentially nonlinear and its analytical
solutions are unknown till now. However, for small deviations from the
equilibrium state that occurs under acoustic wave action on the reservoir,
filtration processes with relaxation can be described with sufficient accuracy
using the proper linearized equations.

Thus, using approximate methods of perturbation theory and methods of
mathematical analysis, the nonlinear system of equations (1) is reduced to the
following form

$\begin{split}\tau\left(u_{t}+\frac{k_{f}}{\mu}p_{rt}\right)+u+\frac{k_{e}}{\mu}p_{r}=0,\\\
\beta p_{t}+u_{r}+\frac{u}{r}=0,\end{split}$ (2)

where $\beta=m_{0}\beta_{0}+\beta_{s}$.

Eliminating the variable from equations (2), we obtain the model equation
describing non-equilibrium filtration processes of a fluid during plane radial
motion

$\begin{split}\tau K_{f}u_{rrt}+K_{e}u_{rr}+\tau
K_{f}\frac{1}{r}u_{rt}+K_{e}\frac{u_{r}}{r}-K_{e}\frac{u}{r^{2}}-\tau
u_{tt}-u_{t}\left(1+\tau K_{f}\frac{1}{r^{2}}\right)=0,\end{split}$ (3)

where $K_{e}=\frac{k_{e}}{\beta\mu},\mathrm{m}/\mathrm{s}^{2}$;
$K_{f}=\frac{k_{f}}{\beta\mu},\mathrm{m}/\mathrm{s}^{2}$ are the coefficients
of piezoconductivity of formation in equilibrium and frozen states
respectively.

Presentation of research results and discussion. To study the effects of wave
harmonic action on filtration in the porous medium of an oil-bearing
reservoir, it is necessary to determine the solution of equation (3) with the
following boundary condition

$u(r,t)=A\sin\omega t\text{ at }r=r_{c}.$ (4)

The constrain providing boundedness of equation’s solution at infinity reads
as follows

$\lim_{r\rightarrow\infty}u(r,t)=0.$

It is convenient to look for the solution of the boundary value problem in the
class of complex-valued functions (with the appropriate modification of the
boundary condition) by the Fourier method (method of separation of variables)
in the form:

$\tilde{u}(r,t)=R(r)\cdot e^{i\omega t},$ (5)

where $\omega$ is the circular frequency, $\mathrm{s}^{-1}$.

Inserting (5) into equation (3), we obtain the Bessel equation with complex
argument

$\frac{d^{2}R}{dr^{2}}+\frac{1}{r}\frac{dR}{dr}-R\left(a^{2}+\frac{1}{r^{2}}\right)=0,$
(6)

where
$a^{2}=\frac{\omega}{\mathrm{K}_{e}}\left(\frac{i-\omega\tau}{1+i\omega\tau\theta}\right)$
and $\theta=K_{f}/K_{e}$.

In general, the parameter $\theta$ can be regarded as a measure of how far
relaxation can take the system from the equilibrium state or how much the
systems can differ in the equilibrium and frozen states. Let us recall that
$\theta<1$ according to thermodynamic constraints [20].

Thus, the solution of equation (6) with modified boundary condition $R(r)=A$
at $r=r_{c}$ and condition at infinity $R(r)\rightarrow 0$ has the form

$R(r)=A\frac{\mathrm{K}_{1}(ra)}{\mathrm{K}_{1}\left(r_{c}a\right)},$ (7)

where $\mathrm{K}_{1}(\cdot)$ is the modified Bessel function of the second
kind of the first order.

The number $a$ in (7) is defined by the well-known relation
$a=\varphi+i\alpha$, where

$\displaystyle\varphi=\sqrt{\frac{\omega}{2\mathrm{~{}K}_{e}}}\sqrt{\sqrt{\frac{1+z^{2}}{1+\theta^{2}z^{2}}}+\frac{z(\theta-1)}{1+\theta^{2}z^{2}}}>0,$
(8)
$\displaystyle\alpha=\sqrt{\frac{\omega}{2\mathrm{~{}K}_{e}}}\sqrt{\frac{1+z^{2}}{1+\theta^{2}z^{2}}-\frac{z(\theta-1)}{1+\theta^{2}z^{2}}},$
(9)

and $z=\omega\tau$ is the dimensionless frequency.

From relations (8) and (9) it follows in particular that in limiting cases
when $z\rightarrow 0$ the quantity
$\varphi\rightarrow\varphi_{e}=\sqrt{\omega/2K_{e}}$,
$\alpha\rightarrow\alpha_{e}=\sqrt{\omega/2K_{e}}$ and when
$z\rightarrow\infty$, the quantity
$\varphi\rightarrow\varphi_{f}=\sqrt{\omega/2K_{f}},\alpha\rightarrow\alpha_{f}=\sqrt{\omega/2K_{f}}$.

Hence, taking into account (7), we arrive to the following expression for
solution (5)

$\tilde{u}(r,t)=Ae^{i\omega
t}\frac{\mathrm{K}_{1}[r(\varphi+i\alpha)]}{\mathrm{K}_{1}\left[r_{c}(\varphi+i\alpha)\right]},$
(10)

which satisfies the boundary conditions and allows one to identify
corresponding solution of the problem (3) – (4) in the real domain

$u(r,t)=\operatorname{Im}\tilde{u}(r,t).$

For further studies, assessing the influence zone of pulsations of filtration
microflows on colmatant region, we need an asymptotics of modified Bessel
function $K_{1}(s)$ for its argument greater than one [21] (p.690, formula
14.127)

$\mathrm{K}_{1}(s)\approx\sqrt{\frac{\pi}{2s}}e^{-s}.$ (11)

Substituting (11) into expression (10), we lead to the following relation

$\tilde{u}(r,t)=A\sqrt{\frac{r_{c}}{r}}e^{i\omega
t}e^{-\left(r-r_{c}\right)(\varphi+i\alpha)}.$ (12)

Then, evaluating the imaginary part of (12), we obtain the real-valued
solution at large argument of the Bessel function for nonequilibrium
filtration

$u(r,t)=A\sqrt{\frac{r_{c}}{r}}\sin\left[\omega
t-\left(r-r_{c}\right)\alpha\right]e^{-\left(r-r_{c}\right)\varphi}.$ (13)

Such solutions describe a pulsating standing damping wave, under the influence
of which oscillating pulsating microflows are formed, which help to increase
the permeability of the formation, washing out the pore channels [5,6].

It is worth to note that similar pulsating fluid oscillations were observed
during experimental studies [10] and in theoretical investigations of the
movement of a viscous fluid in the medium’s pores under the influence of
harmonic wave action [14, 15]. The size of zone of influence of wave fields is
not large due to their significant attenuation, but such a size is sufficient
to affect colmatant area of the NWZ, where the filtration fluid flow is most
significantly suppressed. Therefore, an important objective is to estimate the
attenuation of filtering pulsating oscillations in NWZ in a wide range of
frequencies, depending on the parameter $\theta$ and parameter $\tau$.

Next, based on the theoretical studies outlined above, let us estimate the NWZ
size, which is under the influence of the wave action caused by the operation
of the acoustic emitter mounted directly in a well.

As an estimate of the size of this area, we choose the distance $r_{V}$, at
which the attenuation of the initial disturbance reaches a predetermined
value. In other words, attenuation (or the size of the influence zone) is
conveniently characterized by the following quantity

$\Delta=\frac{\max_{t}u\left(t,r_{V}\right)}{\max_{t}u\left(t,r_{c}\right)}.$

Taking into account solution (12), we get or in relative units

$\Delta=\sqrt{\frac{1}{\bar{r}_{V}}}e^{-\left(\bar{r}_{V}-1\right)r_{c}\varphi}.$
(14)

If we fix the value of $\Delta$, then the derived expression can be regarded
as an algebraic equation with respect to the parameter $\bar{r}_{V}$. It is
convenient to rewrite equation (14) in the following form

$\Phi(y)=\ln y+2yr_{C}\varphi-2\ln M,$ (15)

where $y=\bar{r}_{V}>0$, $M=\Delta^{-1}e^{r_{c}\varphi}$.

Since $\frac{\partial\Phi}{\partial y}=\frac{1}{y}+2r_{c}\varphi>0$, then the
function $\Phi$ is monotonically increasing. Let us show that it can take
values of different signs. To do this, we derive

$\Phi\left(M^{2}\right)=2M^{2}r_{c}\varphi>0.$

Next, let us estimate the value of the function at the point
$\Phi\left(M^{-k}\right)$, where the exponent $k>0$ is chosen from the
condition that $\Phi\left(M^{-k}\right)<0$.

Hence, let $M>1$, then

$\Phi\left(M^{-k}\right)=-k\ln M+2M^{-k}r_{c}\varphi-2\ln M.$

Choose $k$ in such a way that the term $2M^{-k}r_{c}\varphi=\varepsilon$ is
such that $\varepsilon-2\ln M<0$. Then $k=\frac{\ln
2r_{c}\varphi/\varepsilon}{\ln M}>0$, and also $0<\varepsilon<\ln
M^{2}$.Therefore, taking $0<\varepsilon<\min\left\\{2r_{c}\varphi,\ln
M^{2}\right\\}$, we evaluate $k$ and, moreover, it is valid
$\Phi\left(M^{-k}\right)<0$.

Thus, the function $\Phi$ is the continuous at the interval $y\in(0;\infty)$
and admits the values of different signs on the interval $\left\lfloor
M^{-k};M^{2}\right]$. Then by the Intermediate Value Theorem, this function
possesses a root on the specified interval, moreover, due to monotonic
increasing this root is unique.

This root can be evaluated by the simple iteration method. To apply this
method, equation (15) can be rearrange, for instance, in the form

$y=\frac{2\ln M-\ln y}{2r_{c}\varphi}\equiv\psi(y),$ (16)

for which the iteration scheme is as follows $y_{n+1}=\psi\left(y_{n}\right)$.
Let us note that for a simple iteration, a sufficient condition for
convergence is the constraint $\left|\frac{d\psi}{dy}\right|<1$. Since
$\left|\frac{d\psi}{dy}\right|=\frac{1}{2r_{c}\varphi}$, then on the interval
$y\in\left[M^{-k};M^{2}\right]$ the derivative
$\left|\frac{d\psi}{dy}\right|<\frac{M^{k}}{2r_{c}\varphi}=\frac{1}{\varepsilon}<1$.

It is worth to note that at low frequencies it is possible to occur the case
when $\min\left\\{2r_{c}\varphi,\ln M^{2}\right\\}<1$ and then application of
algorithm is impossible.

But the condition $\varepsilon>1$ can be eliminated if we choose another form
of representation (16). In particular, let us multiply equation (15) by a
number $(-\lambda)$ (here $\lambda$ is positive), add $y$ to both parts and
finally obtain

$y-\lambda\left(\ln y+2yr_{c}\varphi-\ln M\right)=y$

The quantity $\lambda$ is chosen from the condition that

$\left|\frac{d\left(y-\lambda\left(\ln y+2yr_{c}\varphi-\ln
M\right)\right)}{dy}\right|=\left|1-\lambda\left(\frac{1}{y}+2r_{c}\varphi\right)\right|<1.$

Since $0<M^{-k}<y$, then $1-\lambda\left(\frac{1}{y}+2r_{c}\varphi\right)<1$
is valid for all positive $\lambda$. Another part of inequality

$-1<1-\lambda\left(\frac{1}{y}+2r_{c}\varphi\right)\Rightarrow\lambda\left(\frac{1}{y}+2r_{c}\varphi\right)<2$

is valid if we take $\lambda=\frac{2}{M^{k}+2r_{C}\varphi}$. Indeed,

$\lambda\left(\frac{1}{y}+2r_{c}\varphi\right)<\frac{2}{M^{k}+2r_{c}\varphi}\left(M^{k}+2r_{c}\varphi\right)=2.$

Thus, the iteration process

$y_{n+1}=y_{n}-\frac{2}{M^{k}+2r_{c}\varphi}\left(\ln
y_{n}+2y_{n}r_{c}\varphi-\ln M\right).$ (17)

is convergent for arbitrary initial data without auxiliary constraints for
$\varepsilon$. Similar studies can be performed also for the case $M<1$.

To show the algorithm’s work, let us fix the parameter values
$\omega=100\mathrm{~{}s}^{-1}$, $\tau=0.03\mathrm{~{}s}$, $\theta=0.05$,
$K_{e}=2\mathrm{~{}m}/\mathrm{s}^{2}$, $r_{C}=0.1\mathrm{~{}m}$, $\Delta=0.1$.
Then the quantity $M=13.385>1$. Since $\min\left\\{2r_{c}\varphi,\ln
M^{2}\right\\}=\min\\{0.583,5.188\\}=0.583<1$, then we can choose
$\varepsilon$ for instance as $\varepsilon=0.5$ and apply algorithm (17). The
process of algorithm convergence is shown in Fig. 1 for different initial
data. Analysis of Fig. 1 indicates that the algorithm is convergent for a wide
range of initial data (as it should be in accordance with the fulfillment of
the sufficient convergence condition) with quite good rate of convergence (in
fact, the third iteration provides the root).

Figure 1: Iteration process for finding the solution of equation (14) by means
of algorithm (17) at fixed parameter values $\omega=100$ s-1, $\tau=0.03$s,
$\theta=0.05$, $K_{e}=2$ m/$s^{2}$, $r_{c}=0.1$ m. Dashed line corresponds to
the root value evaluated numerically by the tools in-built in Mathematica
package.

Using the results obtained above, the relative influence radii $\bar{r}_{V}$
are calculated for different values of perturbation frequency $\omega$, time
of relaxations $\tau$ and parameter $\theta$ when other parameters are fixed:
$r_{c}=0.1\mathrm{~{}m},\mathrm{~{}K}_{e}=2\mathrm{~{}m}^{2}/\mathrm{s}$,
parameter $\Delta=0.1$.

Figure 2: The dependence of $\bar{r}_{V}$ on the frequency $\omega$ at fixed
$\tau$ and $\theta$. The other parameters are
$K_{e}=2\mathrm{~{}m}/\mathrm{s}^{2},r_{c}=0.1\mathrm{~{}m}$.

Thus, Fig. 2 represents the solutions of equation (14) as a function
$\bar{r}_{V}(\omega;\tau,\theta)$ of frequency $\omega$, when $\tau$ and
$\theta$ take discrete values, e.g. $\tau=0$ (absence of relaxing effects),
$\tau=0.01,\quad\tau=0.03$, while $\theta=0.05$ and $\theta=0.1$ (dashed
curves).

It is quite interesting the behavior of graphs at the edge points of domain of
the function. When $\omega\rightarrow 0$ ( $\tau$ is fixed), the curves
$\bar{r}_{V}$ approach the point $\bar{r}_{V}(0)=\Delta^{-2}$. When
$\omega\rightarrow\infty$, then equation (14) reduces to the form

$\sqrt{\bar{r}_{V}}=\Delta^{-1}\exp\left(\left(1-\bar{r}_{V}\right)r_{c}\varphi\right),$

in which $\varphi\rightarrow\varphi_{f}\rightarrow\infty$ at
$\omega\rightarrow\infty$. In this case the equation possesses the bounded
solution if $\bar{r}_{V}\rightarrow 1$. Thus, we encounter the $0\cdot\infty$
uncertainty, which can lead us to bounded result. Therefore, if solution (14)
exists at $\omega\rightarrow\infty$, then this can be realized at
$\bar{r}_{V}=1$ only, that is confirmed by the asymptotics of the graphs in
Fig. 2. Hence, we can state that all curves $\bar{r}_{V}(\omega)$ possess two
common points: $\omega=0$ and $\omega=\infty$.

No less interesting is the behavior of the graphs at varying $\tau$. Near
$\omega=0$ for each curve $\bar{r}_{V}(\omega)$ there exists the interval (its
size decreases when $\tau$ grows), on which the curve is close to equilibrium
curve $\bar{r}_{V}(\omega;\tau=0)$. This indicates the weak influence of
relaxation on the system’s dynamics at low frequencies. Moreover, the
amplitude of initial perturbation attenuates slowly such that its 10 -fold
decrease is observed at distances of 6-8 well radii $r_{c}$. However, for
higher $\omega$ the values $\bar{r}_{V}(\omega)$ attenuates quickly and for
$\omega>200\mathrm{~{}s}^{-1}$ the radius of influence zone is about $3r_{c}$,
i.e., high-frequency disturbances do not penetrate far into the formation,
which is consistent with classical results.

As $\omega$ increases, relaxation plays a more prominent role, i.e. the more
$\tau$, the more $\bar{r}_{V}$. However, due to the faster decline of the
function $\bar{r}_{V}$, the intersection points of the curves corresponding to
different $\tau$ appear. This means that only in a limited range of
frequencies relaxation effects contribute to the growth of the influence zone,
that is, the size of $\bar{r}_{V}$. For frequencies beyond this interval,
relaxation behaves as an additional mechanism for dissipating the energy of
oscillatory motion. As can be seen for the curve
$\bar{r}_{V}(\omega;\tau=0.03)$, at high frequencies the amplitude of the
initial disturbance decreases by a factor of 10 already at distances of the
order of $2r_{c}$, which is less than for relaxation processes with shorter
$\tau$ and under equilibrium conditions.

Figure 3 exhibits the graphs of dependence $\bar{r}_{V}(\tau;\omega,\theta)$
on the time of relaxation $\tau$ at fixed $\omega=50\mathrm{~{}s}^{-1}$ and
$\omega=150\mathrm{~{}s}^{-1}$ (solid curves) when $\theta=0.05$.

In contrast to the graphs in Fig. 2, in intervals of small $\tau$, the curves
$\bar{r}_{V}(\tau)$ in Fig. 3 have local maxima on their profiles. To
calculate their coordinates, we use the necessary extremum condition, applying
it to equation (14) as an implicitly defined function. If we assume that
$\bar{r}_{V}=y=y(\tau)$ and also according to (8) $\varphi=\varphi(\tau)$,
then the derivative is as follows

$\frac{dy}{d\tau}=2r_{c}\frac{\Delta^{-1}\sqrt{y}e^{(1-y)r_{c}\varphi}-y^{2}}{1+2yr_{c}\varphi}\cdot\frac{d\varphi}{d\tau}$

It is obvious that at the critical point $\tau=\tau_{cr}$ we get
$\frac{dy}{d\tau}=0$, when $\frac{d\varphi}{d\tau}=0$. From this it follows
the condition of existence of the local extreme value

$\omega\tau_{cr}=\sqrt{\frac{3(1+\theta)-\sqrt{9+14\theta+9\theta^{2}}}{2\theta^{2}}}.$
(18)

Using expression (18), it is easy to show that at increasing $\theta$ the
critical point shifts to the left.

Figure 3: The dependence $\bar{r}_{V}$ on time of relaxation $\tau$ at fixed
$\omega$ and $\theta$.The parameters are as follows
$K_{e}=2\mathrm{~{}m}/\mathrm{s}^{2},r_{c}=0.1\mathrm{~{}m}$.

If $\theta$ increases to 0.1, we get graphs $\bar{r}_{V}$ marked with dashed
lines (Fig.3). Note that the curves corresponding to different $\theta$,
intersect (compare the solid and dashed curves).

Figure 4: Contour plot for the function $\bar{r}_{V}$ of $\omega$ and $\tau$.
The graph of function (18) is depicted by red solid curve. Here $\theta=0.05$,
$K_{e}=2\mathrm{~{}m}/\mathrm{s}^{2}$, $r_{c}=0.1\mathrm{~{}m}$.

This means that increasing $\theta$ affects differently $\bar{r}_{V}$ on whole
interval $\tau$. In the most interesting interval $\tau$, growth $\theta$
causes suppression of the maximum $\bar{r}_{V}$ and its shift to small values
of $\tau$.

The mutual influence of frequency and relaxation time is clearly visible on
the contour plot of Fig. 4. The red solid line marks the locus of local
maxima, which are determined by condition (18) at $\theta=0.05$.

Thus, the selection of frequency $\omega$ for NWZ processing should be carried
out in such a way as to stay close to the local maxima of the graph
$\bar{r}_{V}(\tau;\omega,\theta)$. This, in turn, requires more detailed
information about the reservoir and its saturating fluid.

##  3 Concluding remark

In the conducted research, using the generalized dynamic Darcy law with one
relaxation parameter, a mathematical model of the elastic regime of non-
stationary non-equilibrium fluid filtration in a porous semi-bounded medium of
a circular formation is considered. The research deals with the boundary value
problem of non-equilibrium filtering with harmonic perturbation at the
boundary of a semi-confined reservoir and the additional condition that the
solution is bounded at infinity. By the method of separation of variables, a
non-stationary complex-valued solution is obtained in the form of a product of
a harmonic function of time and a modified Bessel function of the second kind
of the first order with respect to the spatial coordinate. Based on this
solution, an asymptotic solution of the problem for large values of the
argument of the Bessel function is constructed. This solution determines the
pulsating fluid motion in a porous formation. Using the derived solution, the
damping of pulsations during non-equilibrium filtering is analyzed depending
on the frequency of the wave action, the ratio of the piezoconductivity
coefficients $K_{f}/K_{e}$, and relaxation parameter.

Graphs of the dependences of the size of the influence zone on the model
parameters are plotted and the choice of parameters for optimal influence on
the NWZ is analyzed. It is established that the sizes of the zones affected by
vibration during non-equilibrium filtration exceed the sizes of these zones
during equilibrium filtration processes. A mathematical model of non-
equilibrium fluid filtration is proposed and the results of theoretical
studies are obtained, which are relevant for the development of wave
technologies for the intensification of mineral resource extraction [20]. In
the future, studies of non-equilibrium filtration in the case of a larger
number of relaxation processes, as well as in the case of the dependence of
formation permeability coefficients on the spatial coordinate, are of
scientific and practical interest.

## Acknowledgments

The authors express their gratitude to D. Sci. Mykulyak Sergiy and D. Sci.
Skurativskyi Sergii for useful discussions of the research findings.

The research is partly supported by the National Academy of Science of
Ukraine, Project 0123U100181 and Project 0123U100183.

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